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Properties

Label 784.3.s
Level $784$
Weight $3$
Character orbit 784.s
Rep. character $\chi_{784}(129,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $76$
Newform subspaces $11$
Sturm bound $336$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 784.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 11 \)
Sturm bound: \(336\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(784, [\chi])\).

Total New Old
Modular forms 496 84 412
Cusp forms 400 76 324
Eisenstein series 96 8 88

Trace form

\( 76 q - 3 q^{3} + 3 q^{5} + 103 q^{9} + O(q^{10}) \) \( 76 q - 3 q^{3} + 3 q^{5} + 103 q^{9} + 15 q^{11} - 10 q^{15} + 3 q^{17} - 51 q^{19} + 7 q^{23} + 159 q^{25} + 72 q^{29} + 93 q^{31} - 69 q^{33} + 49 q^{37} - 64 q^{39} + 232 q^{43} + 150 q^{45} + 141 q^{47} - 117 q^{51} + 89 q^{53} - 102 q^{57} - 99 q^{59} - 69 q^{61} - 160 q^{65} - 25 q^{67} - 824 q^{71} - 117 q^{73} - 510 q^{75} + 103 q^{79} - 374 q^{81} - 22 q^{85} + 186 q^{87} + 75 q^{89} - 211 q^{93} + 315 q^{95} + 1688 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(784, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
784.3.s.a 784.s 7.d $2$ $21.362$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q-9\zeta_{6}q^{9}+(-6+6\zeta_{6})q^{11}+18\zeta_{6}q^{23}+\cdots\)
784.3.s.b 784.s 7.d $2$ $21.362$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}-6\zeta_{6}q^{9}+\cdots\)
784.3.s.c 784.s 7.d $4$ $21.362$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-6\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
784.3.s.d 784.s 7.d $4$ $21.362$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{3}+(\beta _{2}-\beta _{3})q^{5}+(15+15\beta _{1}+\cdots)q^{9}+\cdots\)
784.3.s.e 784.s 7.d $8$ $21.362$ 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{3}-\beta _{7})q^{3}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
784.3.s.f 784.s 7.d $8$ $21.362$ 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{5}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(\beta _{3}+\beta _{6})q^{9}+\cdots\)
784.3.s.g 784.s 7.d $8$ $21.362$ 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{3}q^{3}+(\beta _{1}-\beta _{2}+2\beta _{6})q^{5}+(1-\beta _{4}+\cdots)q^{9}+\cdots\)
784.3.s.h 784.s 7.d $8$ $21.362$ 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{3}+(-\beta _{3}-\beta _{7})q^{5}+(-\beta _{4}-\beta _{5}+\cdots)q^{9}+\cdots\)
784.3.s.i 784.s 7.d $8$ $21.362$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{5}q^{3}-\beta _{4}q^{5}+(5\beta _{1}-\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
784.3.s.j 784.s 7.d $8$ $21.362$ 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2\beta _{1}+2\beta _{3})q^{3}+(\beta _{3}-4\beta _{5}+\beta _{7})q^{5}+\cdots\)
784.3.s.k 784.s 7.d $16$ $21.362$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{2}q^{3}+(\beta _{5}+\beta _{9}+\beta _{10})q^{5}+(8+3\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(784, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 2}\)