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Properties

Label 392.3.g
Level $392$
Weight $3$
Character orbit 392.g
Rep. character $\chi_{392}(99,\cdot)$
Character field $\Q$
Dimension $77$
Newform subspaces $15$
Sturm bound $168$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(168\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(392, [\chi])\).

Total New Old
Modular forms 120 87 33
Cusp forms 104 77 27
Eisenstein series 16 10 6

Trace form

\( 77 q + 2 q^{2} + 2 q^{3} + 14 q^{6} - 16 q^{8} + 203 q^{9} + O(q^{10}) \) \( 77 q + 2 q^{2} + 2 q^{3} + 14 q^{6} - 16 q^{8} + 203 q^{9} + 12 q^{10} - 14 q^{11} + 2 q^{12} + 52 q^{16} + 6 q^{17} - 6 q^{18} - 30 q^{19} + 52 q^{20} - 4 q^{22} + 2 q^{24} - 271 q^{25} - 52 q^{26} + 68 q^{27} - 160 q^{30} + 92 q^{32} + 12 q^{33} - 86 q^{34} + 16 q^{36} + 22 q^{38} - 140 q^{40} - 42 q^{41} + 102 q^{43} - 148 q^{44} - 124 q^{46} - 118 q^{48} - 46 q^{50} + 88 q^{51} - 76 q^{52} + 300 q^{54} + 76 q^{58} - 206 q^{59} - 244 q^{60} - 8 q^{62} - 156 q^{64} - 96 q^{65} + 252 q^{66} - 62 q^{67} + 398 q^{68} + 84 q^{72} + 22 q^{73} + 432 q^{74} - 110 q^{75} + 90 q^{76} + 476 q^{78} + 212 q^{80} + 393 q^{81} - 494 q^{82} - 318 q^{83} + 92 q^{86} + 452 q^{88} + 54 q^{89} - 124 q^{90} + 60 q^{92} - 360 q^{94} - 798 q^{96} + 166 q^{97} - 198 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(392, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
392.3.g.a 392.g 8.d $1$ $10.681$ \(\Q\) \(\Q(\sqrt{-2}) \) \(-2\) \(2\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+2q^{3}+4q^{4}-4q^{6}-8q^{8}+\cdots\)
392.3.g.b 392.g 8.d $2$ $10.681$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(-4\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+\beta q^{3}+4q^{4}-2\beta q^{6}-8q^{8}+\cdots\)
392.3.g.c 392.g 8.d $2$ $10.681$ \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) \(-3\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-2+\beta )q^{2}+(2-3\beta )q^{4}+(2+5\beta )q^{8}+\cdots\)
392.3.g.d 392.g 8.d $2$ $10.681$ \(\Q(\sqrt{-3}) \) None \(2\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\zeta_{6})q^{2}-q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
392.3.g.e 392.g 8.d $2$ $10.681$ \(\Q(\sqrt{-3}) \) None \(2\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\zeta_{6})q^{2}+q^{3}+(-2-2\zeta_{6})q^{4}+\cdots\)
392.3.g.f 392.g 8.d $2$ $10.681$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(4\) \(-8\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}+(-4+\beta )q^{3}+4q^{4}+(-8+\cdots)q^{6}+\cdots\)
392.3.g.g 392.g 8.d $2$ $10.681$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(4\) \(8\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}+(4+\beta )q^{3}+4q^{4}+(8+2\beta )q^{6}+\cdots\)
392.3.g.h 392.g 8.d $4$ $10.681$ \(\Q(\sqrt{2}, \sqrt{-7})\) None \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(-2-\beta _{2})q^{3}+(-3+\beta _{3})q^{4}+\cdots\)
392.3.g.i 392.g 8.d $6$ $10.681$ 6.0.15582448.1 None \(-2\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(1+\beta _{1}+\cdots)q^{4}+\cdots\)
392.3.g.j 392.g 8.d $6$ $10.681$ 6.0.15582448.1 None \(-2\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(-\beta _{1}-\beta _{2}-\beta _{3})q^{3}+(1+\cdots)q^{4}+\cdots\)
392.3.g.k 392.g 8.d $6$ $10.681$ 6.0.700560112.1 None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-1+\beta _{1}-\beta _{4})q^{3}+(1+\beta _{3}+\cdots)q^{4}+\cdots\)
392.3.g.l 392.g 8.d $6$ $10.681$ 6.0.700560112.1 None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{2}+(1-\beta _{1}+\beta _{4})q^{3}+(1+\beta _{1}+\cdots)q^{4}+\cdots\)
392.3.g.m 392.g 8.d $8$ $10.681$ 8.0.\(\cdots\).3 None \(1\) \(8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+(1-\beta _{2}-\beta _{3})q^{3}+(1+\beta _{4}+\cdots)q^{4}+\cdots\)
392.3.g.n 392.g 8.d $8$ $10.681$ 8.0.\(\cdots\).5 None \(8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta _{7})q^{2}+\beta _{5}q^{3}+(-1-\beta _{4}-\beta _{6}+\cdots)q^{4}+\cdots\)
392.3.g.o 392.g 8.d $20$ $10.681$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{11}q^{2}-\beta _{6}q^{3}+(-1+\beta _{7})q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(392, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(392, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)