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Properties

Label 392.4.i
Level $392$
Weight $4$
Character orbit 392.i
Rep. character $\chi_{392}(177,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $60$
Newform subspaces $16$
Sturm bound $224$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 16 \)
Sturm bound: \(224\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(392, [\chi])\).

Total New Old
Modular forms 368 60 308
Cusp forms 304 60 244
Eisenstein series 64 0 64

Trace form

\( 60 q - 6 q^{3} + 10 q^{5} - 256 q^{9} + O(q^{10}) \) \( 60 q - 6 q^{3} + 10 q^{5} - 256 q^{9} - 14 q^{11} - 88 q^{13} - 260 q^{15} + 66 q^{17} - 170 q^{19} - 62 q^{23} - 552 q^{25} + 708 q^{27} - 56 q^{29} + 258 q^{31} + 238 q^{33} - 426 q^{37} - 1460 q^{39} - 1512 q^{41} - 232 q^{43} - 292 q^{45} - 66 q^{47} + 726 q^{51} + 214 q^{53} + 3012 q^{55} + 3740 q^{57} - 1446 q^{59} + 1074 q^{61} - 1212 q^{65} - 1146 q^{67} - 7060 q^{69} - 4016 q^{71} + 1258 q^{73} - 2616 q^{75} + 1394 q^{79} + 606 q^{81} + 7504 q^{83} + 4956 q^{85} + 548 q^{87} + 2570 q^{89} - 2374 q^{93} - 5378 q^{95} - 4520 q^{97} - 2976 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(392, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
392.4.i.a 392.i 7.c $2$ $23.129$ \(\Q(\sqrt{-3}) \) None \(0\) \(-6\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-6+6\zeta_{6})q^{3}-8\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
392.4.i.b 392.i 7.c $2$ $23.129$ \(\Q(\sqrt{-3}) \) None \(0\) \(-4\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\zeta_{6})q^{3}-2\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
392.4.i.c 392.i 7.c $2$ $23.129$ \(\Q(\sqrt{-3}) \) None \(0\) \(-4\) \(12\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\zeta_{6})q^{3}+12\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
392.4.i.d 392.i 7.c $2$ $23.129$ \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-2^{4}\zeta_{6}q^{5}+23\zeta_{6}q^{9}+\cdots\)
392.4.i.e 392.i 7.c $2$ $23.129$ \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{3}+2^{4}\zeta_{6}q^{5}+23\zeta_{6}q^{9}+\cdots\)
392.4.i.f 392.i 7.c $2$ $23.129$ \(\Q(\sqrt{-3}) \) None \(0\) \(4\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\zeta_{6})q^{3}-12\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
392.4.i.g 392.i 7.c $2$ $23.129$ \(\Q(\sqrt{-3}) \) None \(0\) \(4\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\zeta_{6})q^{3}+2\zeta_{6}q^{5}+11\zeta_{6}q^{9}+\cdots\)
392.4.i.h 392.i 7.c $2$ $23.129$ \(\Q(\sqrt{-3}) \) None \(0\) \(6\) \(8\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(6-6\zeta_{6})q^{3}+8\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots\)
392.4.i.i 392.i 7.c $4$ $23.129$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(-2\) \(22\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1}+\beta _{3})q^{3}+(-11\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
392.4.i.j 392.i 7.c $4$ $23.129$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+5\beta _{2}q^{9}+\cdots\)
392.4.i.k 392.i 7.c $4$ $23.129$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(10\beta _{1}+10\beta _{3})q^{5}-5^{2}\beta _{2}q^{9}+\cdots\)
392.4.i.l 392.i 7.c $4$ $23.129$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(2\) \(-22\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{3})q^{3}+(11\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
392.4.i.m 392.i 7.c $6$ $23.129$ 6.0.11163123.4 None \(0\) \(-7\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2\beta _{1}+\beta _{5})q^{3}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
392.4.i.n 392.i 7.c $6$ $23.129$ 6.0.11163123.4 None \(0\) \(1\) \(13\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{4}q^{3}+(4-4\beta _{1}+\beta _{5})q^{5}+(-15+\cdots)q^{9}+\cdots\)
392.4.i.o 392.i 7.c $8$ $23.129$ 8.0.\(\cdots\).19 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}-\beta _{4}-\beta _{5})q^{3}+(\beta _{5}+10\beta _{6})q^{5}+\cdots\)
392.4.i.p 392.i 7.c $8$ $23.129$ 8.0.\(\cdots\).19 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{4}-\beta _{6})q^{3}+(-2\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(392, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(392, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)