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Properties

Label 504.4.s
Level $504$
Weight $4$
Character orbit 504.s
Rep. character $\chi_{504}(289,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $60$
Newform subspaces $12$
Sturm bound $384$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.s (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(384\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(504, [\chi])\).

Total New Old
Modular forms 608 60 548
Cusp forms 544 60 484
Eisenstein series 64 0 64

Trace form

\( 60 q + 10 q^{5} + 24 q^{7} + O(q^{10}) \) \( 60 q + 10 q^{5} + 24 q^{7} + 14 q^{11} + 16 q^{13} + 70 q^{17} - 10 q^{19} - 94 q^{23} - 588 q^{25} + 224 q^{29} + 78 q^{31} + 366 q^{35} - 318 q^{37} - 624 q^{41} - 536 q^{43} - 66 q^{47} + 172 q^{49} - 226 q^{53} - 348 q^{55} - 678 q^{59} + 6 q^{61} + 148 q^{65} - 1014 q^{67} - 656 q^{71} + 674 q^{73} + 1018 q^{77} + 626 q^{79} - 1528 q^{83} + 3292 q^{85} + 202 q^{89} + 756 q^{91} - 102 q^{95} - 1840 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(504, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
504.4.s.a 504.s 7.c $2$ $29.737$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-11\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-11+11\zeta_{6})q^{5}+(14-21\zeta_{6})q^{7}+\cdots\)
504.4.s.b 504.s 7.c $2$ $29.737$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{5}+(14-21\zeta_{6})q^{7}-63\zeta_{6}q^{11}+\cdots\)
504.4.s.c 504.s 7.c $2$ $29.737$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{5}+(14-21\zeta_{6})q^{7}+63\zeta_{6}q^{11}+\cdots\)
504.4.s.d 504.s 7.c $2$ $29.737$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(35\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{5}+(21-7\zeta_{6})q^{7}-18\zeta_{6}q^{11}+\cdots\)
504.4.s.e 504.s 7.c $2$ $29.737$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(7\) \(-35\) $\mathrm{SU}(2)[C_{3}]$ \(q+(7-7\zeta_{6})q^{5}+(-14-7\zeta_{6})q^{7}+7\zeta_{6}q^{11}+\cdots\)
504.4.s.f 504.s 7.c $4$ $29.737$ \(\Q(\sqrt{-3}, \sqrt{505})\) None \(0\) \(0\) \(9\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+4\beta _{2})q^{5}+(9-17\beta _{2}-\beta _{3})q^{7}+\cdots\)
504.4.s.g 504.s 7.c $6$ $29.737$ 6.0.\(\cdots\).1 None \(0\) \(0\) \(-11\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4\beta _{3}+\beta _{4}+\beta _{5})q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
504.4.s.h 504.s 7.c $6$ $29.737$ 6.0.11163123.4 None \(0\) \(0\) \(-3\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4}-\beta _{5})q^{5}+\cdots\)
504.4.s.i 504.s 7.c $6$ $29.737$ 6.0.11163123.4 None \(0\) \(0\) \(13\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(5\beta _{1}+\beta _{2}-\beta _{3}-2\beta _{4}-\beta _{5})q^{5}+\cdots\)
504.4.s.j 504.s 7.c $8$ $29.737$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(4\) \(18\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}-\beta _{6})q^{5}+(2+\beta _{5})q^{7}+(-1+\cdots)q^{11}+\cdots\)
504.4.s.k 504.s 7.c $10$ $29.737$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(-6\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}-\beta _{3})q^{5}+(2+3\beta _{2}+\beta _{4})q^{7}+\cdots\)
504.4.s.l 504.s 7.c $10$ $29.737$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(6\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{2}+\beta _{3})q^{5}+(2+3\beta _{2}+\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(504, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)