The following table shows all irreducible, admissible representations $\pi$ of $\GL(2,\Q_2)$ with trivial central character and conductor exponent $a(\pi)\leq7$.
orbit | type of representation | $\pi$ | $a(\pi)$ | $\varepsilon(\frac12,\pi)$ |
$[\chi\times\chi^{-1}]$ | unramified principal series | $\chi\times\chi^{-1}$ | 0 | 1 |
$(\chi\times\chi^{-1})\otimes\chi_5$ | 0 | 1 |
$(\chi\times\chi^{-1})\otimes\chi_3$ | 4 | -1 |
$(\chi\times\chi^{-1})\otimes\chi_7$ | 4 | -1 |
$(\chi\times\chi^{-1})\otimes\chi_2$ | 6 | 1 |
$(\chi\times\chi^{-1})\otimes\chi_{2\cdot5}$ | 6 | 1 |
$(\chi\times\chi^{-1})\otimes\chi_{2\cdot3}$ | 6 | -1 |
$(\chi\times\chi^{-1})\otimes\chi_{2\cdot7}$ | 6 | -1 |
$[\St_{\GL(2)}]$ | Steinberg | $\St_{\GL(2)}$ | 1 | -1 |
$\St_{\GL(2)}\otimes\chi_5$ | 1 | 1 |
$\St_{\GL(2)}\otimes\chi_3$ | 4 | -1 |
$\St_{\GL(2)}\otimes\chi_7$ | 4 | -1 |
$\St_{\GL(2)}\otimes\chi_2$ | 6 | 1 |
$\St_{\GL(2)}\otimes\chi_{2\cdot5}$ | 6 | 1 |
$\St_{\GL(2)}\otimes\chi_{2\cdot3}$ | 6 | -1 |
$\St_{\GL(2)}\otimes\chi_{2\cdot7}$ | 6 | -1 |
$[\tau_2]$ | monomial supercuspidal from $\Q_2(\sqrt{5})$ | $\tau_2$ | 2 | -1 |
$\tau_2\otimes\chi_3$ | 4 | -1 |
$\tau_2\otimes\chi_2$ | 6 | 1 |
$\tau_2\otimes\chi_{2\cdot3}$ | 6 | -1 |
$[\tau_3]$ | exceptional supercuspidal | $\tau_3$ | 3 | 1 |
$\tau_3\otimes\chi_5$ | 3 | -1 |
$\tau_3\otimes\chi_3$ | 4 | 1 |
$\tau_3\otimes\chi_7$ | 4 | 1 |
$\tau_3\otimes\chi_2$ | 6 | 1 |
$\tau_3\otimes\chi_{2\cdot5}$ | 6 | 1 |
$\tau_3\otimes\chi_{2\cdot3}$ | 6 | -1 |
$\tau_3\otimes\chi_{2\cdot7}$ | 6 | -1 |
$[\tau_{5a}]$ | monomial supercuspidal from $\Q_2(\sqrt{3})$ | $\tau_{5a}$ | 5 | 1 |
$\tau_{5a}\otimes\chi_5$ | 5 | -1 |
$\tau_{5a}\otimes\chi_2$ | 6 | 1 |
$\tau_{5a}\otimes\chi_{2\cdot5}$ | 6 | 1 |
$[\tau_{5b}]$ | monomial supercuspidal from $\Q_2(\sqrt{7})$ | $\tau_{5b}$ | 5 | 1 |
$\tau_{5b}\otimes\chi_5$ | 5 | -1 |
$\tau_{5b}\otimes\chi_2$ | 6 | -1 |
$\tau_{5b}\otimes\chi_{2\cdot5}$ | 6 | -1 |
$[\tau_7]$ | exceptional supercuspidal | $\tau_7$ | 7 | 1 |
$\tau_7\otimes\chi_5$ | 7 | -1 |
$\tau_7\otimes\chi_3$ | 7 | -1 |
$\tau_7\otimes\chi_7$ | 7 | 1 |
$\tau_7\otimes\chi_2$ | 7 | -1 |
$\tau_7\otimes\chi_{2\cdot5}$ | 7 | 1 |
$\tau_7\otimes\chi_{2\cdot3}$ | 7 | -1 |
$\tau_7\otimes\chi_{2\cdot7}$ | 7 | 1 |
The table below lists the cuspidal automorphic representations of $\GL(2,\A)$ with trivial central character whose conductor divides 32 and whose archimedean component is a discrete series representation of weight $\leq8$. To each $\pi$ and choice of quadratic Hecke character $\chi$ there is associated a finite packet of automorphic representations $\Pi$ of $\GSp(4,\A)$ with trivial central character such that, up to finitely many Euler factors, $L(s,\Pi)=L(s,\pi)L(s-\frac12,\chi)L(s+\frac12,\chi)$. These are the packets of
$\mathrm{\bf(P)}$. The elements of the packet are parametrized by subsets $S$, satisfying a parity condition, of the set of places where $\pi$ is square-integrable. The table below lists the possibilities for which $S$ contains the archimedean place, in which case $S$ is uniquely determined. The archimedean component $\Pi_\infty$ is then a (limit of) holomorphic discrete series representation.
Tables like the one below can be constructed for any weight in a mechanical way from the data given in the LMFDB, because this data allows for the determination of the 2-component of the underlying automorphic representation for any eigenform in $S_k(\Gamma_0(N))$, where $N\mid32$.
$\GL(2)$ |
|
$\GSp(4)$ |
weight | level |
$\pi$ |
size |
$\pi_2$ |
$a(\pi_2)$ |
$\varepsilon(\frac12,\pi_2)$ |
$\varepsilon(\frac12,\pi)$ |
$\chi$ | $S$ | weight | conductor |
$\Pi=\Pi(\pi,\chi,S)$ |
$\Pi_2$ | $a(\Pi_2)$ | $\varepsilon(\frac12,\Pi_2)$ | $\varepsilon(\frac12,\Pi)$ | para |
2 | 32 |
32.2.a.a |
1 |
$\tau_{5b}\otimes\chi_5$ |
5 |
-1 |
1 |
1 | $\{\infty,2\}$ | 2 | 64 |
P.64.2.0.a |
XIa* | 6 | 1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 512 |
P.512.2.0.a |
XIa* | 9 | 1 | 1 | |
4 | 8 |
8.4.a.a |
1 |
$\tau_3$ |
3 |
1 |
1 |
1 | $\{\infty,2\}$ | 3 | 16 |
P.16.3.0.a |
XIa* | 4 | -1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 128 |
P.128.3.0.a |
XIa* | 7 | -1 | 1 | |
4 | 16 |
16.4.a.a |
1 |
$\tau_3\otimes\chi_7$ |
4 |
1 |
1 |
1 | $\{\infty,2\}$ | 3 | 32 |
P.32.3.0.a |
XIa* | 5 | -1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 256 |
P.256.3.0.a |
XIa* | 8 | -1 | 1 | |
4 | 32 |
32.4.a.a |
1 |
$\tau_{5a}\otimes\chi_5$ |
5 |
-1 |
-1 |
1 | $\{\infty\}$ | 3 | 32 |
P.32.3.0.b |
XIb | 5 | -1 | 1 | 32 |
$\xi$ | $\{\infty,2\}$ | 512 |
P.512.3.0.a |
XIa* | 9 | -1 | 1 | |
4 | 32 |
32.4.a.b |
1 |
$\tau_{5b}$ |
5 |
1 |
1 |
1 | $\{\infty,2\}$ | 3 | 64 |
P.64.3.0.a |
XIa* | 6 | -1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 512 |
P.512.3.0.b |
XIa* | 9 | -1 | 1 | |
4 | 32 |
32.4.a.c |
1 |
$\tau_{5a}$ |
5 |
1 |
1 |
1 | $\{\infty,2\}$ | 3 | 64 |
P.64.3.0.b |
XIa* | 6 | -1 | 1 | |
$\xi$ | $\{\infty\}$ | 512 |
P.512.3.0.c |
XIb | 9 | -1 | 1 | |
6 | 4 |
4.6.a.a |
1 |
$\tau_2$ |
2 |
-1 |
1 |
1 | $\{\infty,2\}$ | 4 | 8 |
P.8.4.0.a |
XIa* | 3 | 1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 64 |
P.64.4.0.a |
XIa* | 6 | 1 | 1 | |
6 | 8 |
8.6.a.a |
1 |
$\tau_3\otimes\chi_5$ |
3 |
-1 |
1 |
1 | $\{\infty,2\}$ | 4 | 16 |
P.16.4.0.a |
XIa* | 4 | 1 | 1 | |
$\xi$ | $\{\infty\}$ | 128 |
P.128.4.0.a |
XIb | 7 | 1 | 1 | |
6 | 16 |
16.6.a.a |
1 |
$\tau_3\otimes\chi_3$ |
4 |
1 |
-1 |
1 | $\{\infty\}$ | 4 | 16 |
P.16.4.0.b |
XIb | 4 | 1 | 1 | 16 |
$\xi$ | $\{\infty,2\}$ | 256 |
P.256.4.0.a |
XIa* | 8 | -1 | -1 | |
6 | 16 |
16.6.a.b |
1 |
$\tau_2\otimes\chi_3$ |
4 |
-1 |
1 |
1 | $\{\infty,2\}$ | 4 | 32 |
P.32.4.0.a |
XIa* | 5 | 1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 256 |
P.256.4.0.b |
XIa* | 8 | 1 | 1 | |
6 | 32 |
32.6.a.a |
1 |
$\tau_{5a}$ |
5 |
1 |
-1 |
1 | $\{\infty\}$ | 4 | 32 |
P.32.4.0.b |
XIb | 5 | 1 | 1 | 32 |
$\xi$ | $\{\infty,2\}$ | 512 |
P.512.4.0.a |
XIa* | 9 | -1 | -1 | |
6 | 32 |
32.6.a.b |
1 |
$\tau_{5b}$ |
5 |
1 |
-1 |
1 | $\{\infty\}$ | 4 | 32 |
P.32.4.0.c |
XIb | 5 | 1 | 1 | 32 |
$\xi$ | $\{\infty\}$ | 256 |
P.256.4.0.c |
XIb | 8 | -1 | -1 | |
6 | 32 |
32.6.a.c |
1 |
$\tau_{5a}\otimes\chi_5$ |
5 |
-1 |
1 |
1 | $\{\infty,2\}$ | 4 | 64 |
P.64.4.0.b |
XIa* | 6 | 1 | 1 | |
$\xi$ | $\{\infty\}$ | 512 |
P.512.4.0.b |
XIb | 9 | 1 | 1 | |
6 | 32 |
32.6.a.d |
2 |
$\tau_{5b}\otimes\chi_5$ |
5 |
-1 |
1 |
1 | $\{\infty,2\}$ | 4 | 64 |
P.64.4.0.c |
XIa* | 6 | 1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 512 |
P.512.4.0.c |
XIa* | 9 | 1 | 1 | |
8 | 2 |
2.8.a.a |
1 |
$\St_{\GL(2)}\otimes\chi_5$ |
1 |
1 |
1 |
1 | $\{\infty,2\}$ | 5 | 4 |
P.4.5.0.a |
Va* | 2 | -1 | 1 | |
$\xi$ | $\{\infty\}$ | 32 |
P.32.5.0.a |
Vb | 5 | -1 | 1 | |
8 | 8 |
8.8.a.a |
1 |
$\tau_3\otimes\chi_5$ |
3 |
-1 |
-1 |
1 | $\{\infty\}$ | 5 | 8 |
P.8.5.0.a |
XIb | 3 | -1 | 1 | 8 |
$\xi$ | $\{\infty,2\}$ | 128 |
P.128.5.0.a |
XIa* | 7 | 1 | -1 | |
8 | 8 |
8.8.a.b |
1 |
$\tau_3$ |
3 |
1 |
1 |
1 | $\{\infty,2\}$ | 5 | 16 |
P.16.5.0.a |
XIa* | 4 | -1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 128 |
P.128.5.0.b |
XIa* | 7 | -1 | 1 | |
8 | 16 |
16.8.a.a |
1 |
$\tau_3\otimes\chi_7$ |
4 |
1 |
1 |
1 | $\{\infty,2\}$ | 5 | 32 |
P.32.5.0.b |
XIa* | 5 | -1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 256 |
P.256.5.0.a |
XIa* | 8 | -1 | 1 | |
8 | 16 |
16.8.a.b |
1 |
$\St_{\GL(2)}\otimes\chi_3$ |
4 |
-1 |
-1 |
1 | $\{\infty\}$ | 5 | 16 |
P.16.5.0.b |
Vb | 4 | -1 | 1 | 16 |
$\xi$ | $\{\infty,2\}$ | 256 |
P.256.5.0.b |
Va* | 8 | 1 | -1 | |
8 | 16 |
16.8.a.c |
1 |
$\tau_3\otimes\chi_3$ |
4 |
1 |
1 |
1 | $\{\infty,2\}$ | 5 | 32 |
P.32.5.0.c |
XIa* | 5 | -1 | 1 | |
$\xi$ | $\{\infty\}$ | 256 |
P.256.5.0.c |
XIb | 8 | -1 | 1 | |
8 | 32 |
32.8.a.a |
1 |
$\tau_{5b}\otimes\chi_5$ |
5 |
-1 |
-1 |
1 | $\{\infty\}$ | 5 | 32 |
P.32.5.0.d |
XIb | 5 | -1 | 1 | 32 |
$\xi$ | $\{\infty\}$ | 512 |
P.512.5.0.a |
XIb | 9 | 1 | -1 | |
8 | 32 |
32.8.a.b |
2 |
$\tau_{5a}$ |
5 |
1 |
1 |
1 | $\{\infty,2\}$ | 5 | 64 |
P.64.5.0.b |
XIa* | 6 | -1 | 1 | |
$\xi$ | $\{\infty\}$ | 512 |
P.512.5.0.b |
XIb | 9 | -1 | 1 | |
8 | 32 |
32.8.a.c |
2 |
$\tau_{5b}$ |
5 |
1 |
1 |
1 | $\{\infty,2\}$ | 5 | 64 |
P.64.5.0.c |
XIa* | 6 | -1 | 1 | |
$\xi$ | $\{\infty,2\}$ | 512 |
P.512.5.0.c |
XIa* | 9 | -1 | 1 | |
8 | 32 |
32.8.a.d |
2 |
$\tau_{5a}\otimes\chi_5$ |
5 |
-1 |
-1 |
1 | $\{\infty\}$ | 5 | 32 |
P.32.5.0.e |
XIb | 5 | -1 | 1 | 32 |
$\xi$ | $\{\infty,2\}$ | 512 |
P.512.5.0.d |
XIa* | 9 | 1 | -1 | |