Properties

Label 624.4.q
Level $624$
Weight $4$
Character orbit 624.q
Rep. character $\chi_{624}(289,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $84$
Newform subspaces $14$
Sturm bound $448$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(448\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(624, [\chi])\).

Total New Old
Modular forms 696 84 612
Cusp forms 648 84 564
Eisenstein series 48 0 48

Trace form

\( 84 q + 6 q^{3} - 4 q^{5} + 18 q^{7} - 378 q^{9} + O(q^{10}) \) \( 84 q + 6 q^{3} - 4 q^{5} + 18 q^{7} - 378 q^{9} + 46 q^{13} - 26 q^{17} - 180 q^{19} + 2056 q^{25} - 108 q^{27} - 142 q^{29} + 60 q^{31} + 66 q^{37} + 420 q^{39} - 474 q^{41} + 342 q^{43} + 18 q^{45} - 1488 q^{47} - 2134 q^{49} - 1224 q^{51} + 1356 q^{53} - 1276 q^{55} - 336 q^{57} + 748 q^{59} - 278 q^{61} + 162 q^{63} + 1414 q^{65} + 342 q^{67} - 224 q^{71} + 524 q^{73} + 1602 q^{75} - 368 q^{77} + 3364 q^{79} - 3402 q^{81} - 2680 q^{83} - 34 q^{85} - 360 q^{87} - 396 q^{89} - 3682 q^{91} + 2516 q^{95} - 24 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(624, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
624.4.q.a 624.q 13.c $2$ $36.817$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(14\) \(16\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+7q^{5}+2^{4}\zeta_{6}q^{7}+\cdots\)
624.4.q.b 624.q 13.c $2$ $36.817$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-18\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}-9q^{5}+2\zeta_{6}q^{7}-9\zeta_{6}q^{9}+\cdots\)
624.4.q.c 624.q 13.c $2$ $36.817$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(14\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}+7q^{5}-10\zeta_{6}q^{7}-9\zeta_{6}q^{9}+\cdots\)
624.4.q.d 624.q 13.c $4$ $36.817$ \(\Q(\sqrt{-3}, \sqrt{61})\) None \(0\) \(-6\) \(4\) \(18\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\beta _{1})q^{3}+(1+2\beta _{3})q^{5}+(9\beta _{1}+\cdots)q^{7}+\cdots\)
624.4.q.e 624.q 13.c $4$ $36.817$ \(\Q(\sqrt{-3}, \sqrt{14})\) None \(0\) \(-6\) \(12\) \(20\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{2}q^{3}+(3+2\beta _{3})q^{5}+(10+\beta _{1}+\cdots)q^{7}+\cdots\)
624.4.q.f 624.q 13.c $4$ $36.817$ \(\Q(\sqrt{-3}, \sqrt{673})\) None \(0\) \(6\) \(26\) \(9\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\beta _{2})q^{3}+(6+\beta _{3})q^{5}+(\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots\)
624.4.q.g 624.q 13.c $6$ $36.817$ 6.0.31902363.2 None \(0\) \(-9\) \(-24\) \(-29\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{1}q^{3}+(-4+\beta _{5})q^{5}+(-10+10\beta _{1}+\cdots)q^{7}+\cdots\)
624.4.q.h 624.q 13.c $6$ $36.817$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(9\) \(-24\) \(-17\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3+3\beta _{2})q^{3}+(-4-\beta _{3})q^{5}+(6\beta _{2}+\cdots)q^{7}+\cdots\)
624.4.q.i 624.q 13.c $8$ $36.817$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-12\) \(-12\) \(-14\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{2}q^{3}+(-2+\beta _{4})q^{5}+(-3+3\beta _{2}+\cdots)q^{7}+\cdots\)
624.4.q.j 624.q 13.c $8$ $36.817$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-12\) \(-2\) \(13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\beta _{2})q^{3}+(-\beta _{5}+\beta _{7})q^{5}+\cdots\)
624.4.q.k 624.q 13.c $8$ $36.817$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(-12\) \(6\) \(-15\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\beta _{1})q^{3}+(1+\beta _{4}+\beta _{5})q^{5}+\cdots\)
624.4.q.l 624.q 13.c $8$ $36.817$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(12\) \(14\) \(11\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{2}q^{3}+(2+\beta _{5})q^{5}+(3-\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)
624.4.q.m 624.q 13.c $10$ $36.817$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(15\) \(-22\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\beta _{2})q^{3}+(-2-\beta _{1}-\beta _{8})q^{5}+\cdots\)
624.4.q.n 624.q 13.c $12$ $36.817$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(18\) \(8\) \(10\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{4}q^{3}+(1+\beta _{1})q^{5}+(1-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(624, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(624, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(208, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(312, [\chi])\)\(^{\oplus 2}\)