Properties

Label 304.8.a
Level $304$
Weight $8$
Character orbit 304.a
Rep. character $\chi_{304}(1,\cdot)$
Character field $\Q$
Dimension $63$
Newform subspaces $13$
Sturm bound $320$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(320\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(304))\).

Total New Old
Modular forms 286 63 223
Cusp forms 274 63 211
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(19\)FrickeDim
\(+\)\(+\)$+$\(17\)
\(+\)\(-\)$-$\(14\)
\(-\)\(+\)$-$\(16\)
\(-\)\(-\)$+$\(16\)
Plus space\(+\)\(33\)
Minus space\(-\)\(30\)

Trace form

\( 63 q - 278 q^{5} + 686 q^{7} + 45927 q^{9} + O(q^{10}) \) \( 63 q - 278 q^{5} + 686 q^{7} + 45927 q^{9} - 9190 q^{11} + 6554 q^{13} + 37452 q^{15} + 1454 q^{17} - 20577 q^{19} + 2344 q^{21} + 224348 q^{23} + 1012925 q^{25} - 604800 q^{27} - 240134 q^{29} - 256824 q^{33} - 677286 q^{35} + 622506 q^{37} + 1597476 q^{39} - 220642 q^{41} - 1412982 q^{43} - 4310 q^{45} + 677678 q^{47} + 8428279 q^{49} + 893908 q^{51} - 1915422 q^{53} - 2379650 q^{55} + 5250992 q^{59} + 443810 q^{61} - 2418590 q^{63} - 2306988 q^{65} - 970340 q^{67} + 7143672 q^{69} - 18829148 q^{71} + 8271966 q^{73} + 10662348 q^{75} - 11906456 q^{77} - 14621192 q^{79} + 30437639 q^{81} - 7972268 q^{83} + 3490492 q^{85} + 15654444 q^{87} + 9783534 q^{89} + 2466424 q^{91} - 26260456 q^{93} + 6859000 q^{95} - 4601850 q^{97} - 20314414 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(304))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 19
304.8.a.a 304.a 1.a $1$ $94.965$ \(\Q\) None \(0\) \(-77\) \(440\) \(-951\) $-$ $-$ $\mathrm{SU}(2)$ \(q-77q^{3}+440q^{5}-951q^{7}+3742q^{9}+\cdots\)
304.8.a.b 304.a 1.a $2$ $94.965$ \(\Q(\sqrt{17953}) \) None \(0\) \(11\) \(-69\) \(348\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(6-\beta )q^{3}+(-33-3\beta )q^{5}+(167+\cdots)q^{7}+\cdots\)
304.8.a.c 304.a 1.a $2$ $94.965$ \(\Q(\sqrt{2737}) \) None \(0\) \(61\) \(175\) \(2592\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(31-\beta )q^{3}+(80+15\beta )q^{5}+(1299+\cdots)q^{7}+\cdots\)
304.8.a.d 304.a 1.a $2$ $94.965$ \(\Q(\sqrt{633}) \) None \(0\) \(69\) \(155\) \(2238\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(6^{2}-3\beta )q^{3}+(93-31\beta )q^{5}+(1133+\cdots)q^{7}+\cdots\)
304.8.a.e 304.a 1.a $4$ $94.965$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(-279\) \(-2485\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{3}+(-70-2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
304.8.a.f 304.a 1.a $4$ $94.965$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(14\) \(-222\) \(1246\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(4+\beta _{1}-\beta _{2})q^{3}+(-56-4\beta _{1}-3\beta _{2}+\cdots)q^{5}+\cdots\)
304.8.a.g 304.a 1.a $5$ $94.965$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(14\) \(-280\) \(-414\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}+(-56+\beta _{1}-\beta _{2})q^{5}+\cdots\)
304.8.a.h 304.a 1.a $6$ $94.965$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-40\) \(219\) \(-2105\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-7+\beta _{3})q^{3}+(37+2\beta _{1}-\beta _{2})q^{5}+\cdots\)
304.8.a.i 304.a 1.a $6$ $94.965$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-40\) \(279\) \(1565\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-7+\beta _{1})q^{3}+(47-\beta _{1}-\beta _{2})q^{5}+\cdots\)
304.8.a.j 304.a 1.a $6$ $94.965$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-29\) \(-43\) \(908\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-5+\beta _{1})q^{3}+(-7-\beta _{1}-\beta _{2})q^{5}+\cdots\)
304.8.a.k 304.a 1.a $8$ $94.965$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(2\) \(-305\) \(-1925\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-38+\beta _{1}-\beta _{4})q^{5}+(-241+\cdots)q^{7}+\cdots\)
304.8.a.l 304.a 1.a $8$ $94.965$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(83\) \(-430\) \(133\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(10+\beta _{1})q^{3}+(-54-\beta _{3})q^{5}+(15+\cdots)q^{7}+\cdots\)
304.8.a.m 304.a 1.a $9$ $94.965$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-56\) \(82\) \(-464\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-6-\beta _{1})q^{3}+(10-2\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(304))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(304)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 2}\)