Properties

Label 624.4.bv
Level $624$
Weight $4$
Character orbit 624.bv
Rep. character $\chi_{624}(49,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $84$
Newform subspaces $10$
Sturm bound $448$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.bv (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 10 \)
Sturm bound: \(448\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

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

\( 84 q - 6 q^{3} - 54 q^{7} - 378 q^{9} + O(q^{10}) \) \( 84 q - 6 q^{3} - 54 q^{7} - 378 q^{9} - 46 q^{13} + 26 q^{17} - 2144 q^{25} + 108 q^{27} + 142 q^{29} - 198 q^{37} + 420 q^{39} + 354 q^{41} + 774 q^{43} + 54 q^{45} + 1774 q^{49} - 1224 q^{51} + 1356 q^{53} + 484 q^{55} - 1884 q^{59} - 278 q^{61} + 486 q^{63} + 194 q^{65} + 1242 q^{67} + 498 q^{75} + 368 q^{77} - 6484 q^{79} - 3402 q^{81} - 1434 q^{85} + 360 q^{87} - 4608 q^{89} - 2170 q^{91} - 1276 q^{95} + 276 q^{97} + O(q^{100}) \)

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.bv.a 624.bv 13.e $2$ $36.817$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(0\) \(-18\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-3+3\zeta_{6})q^{3}+(3-6\zeta_{6})q^{5}+(-12+\cdots)q^{7}+\cdots\)
624.4.bv.b 624.bv 13.e $2$ $36.817$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(-33\) $\mathrm{SU}(2)[C_{6}]$ \(q+(3-3\zeta_{6})q^{3}+(-2+4\zeta_{6})q^{5}+(-22+\cdots)q^{7}+\cdots\)
624.4.bv.c 624.bv 13.e $4$ $36.817$ \(\Q(\sqrt{-3}, \sqrt{-17})\) None \(0\) \(-6\) \(0\) \(66\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-3+3\beta _{2})q^{3}+(3-6\beta _{2}+2\beta _{3})q^{5}+\cdots\)
624.4.bv.d 624.bv 13.e $4$ $36.817$ \(\Q(\zeta_{12})\) None \(0\) \(6\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(3-3\zeta_{12}^{2})q^{3}+(-3+6\zeta_{12}^{2}+4\zeta_{12}^{3})q^{5}+\cdots\)
624.4.bv.e 624.bv 13.e $4$ $36.817$ \(\Q(\zeta_{12})\) None \(0\) \(6\) \(0\) \(42\) $\mathrm{SU}(2)[C_{6}]$ \(q+(3-3\zeta_{12}^{2})q^{3}+(-1+2\zeta_{12}^{2}+2\zeta_{12}^{3})q^{5}+\cdots\)
624.4.bv.f 624.bv 13.e $6$ $36.817$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(-9\) \(0\) \(-9\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-3-3\beta _{1})q^{3}+(-1-3\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
624.4.bv.g 624.bv 13.e $8$ $36.817$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-12\) \(0\) \(-24\) $\mathrm{SU}(2)[C_{6}]$ \(q-3\beta _{2}q^{3}+(2-\beta _{1}-5\beta _{2}-\beta _{3}+\beta _{7})q^{5}+\cdots\)
624.4.bv.h 624.bv 13.e $10$ $36.817$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(15\) \(0\) \(-30\) $\mathrm{SU}(2)[C_{6}]$ \(q+(3-3\beta _{2})q^{3}+(1-2\beta _{2}-\beta _{6}-\beta _{7}+\cdots)q^{5}+\cdots\)
624.4.bv.i 624.bv 13.e $20$ $36.817$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(30\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(3+3\beta _{1})q^{3}+(-\beta _{2}-\beta _{5})q^{5}+(-2+\cdots)q^{7}+\cdots\)
624.4.bv.j 624.bv 13.e $24$ $36.817$ None \(0\) \(-36\) \(0\) \(-42\) $\mathrm{SU}(2)[C_{6}]$

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

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