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Properties

Label 392.6.i
Level $392$
Weight $6$
Character orbit 392.i
Rep. character $\chi_{392}(177,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $100$
Newform subspaces $18$
Sturm bound $336$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 18 \)
Sturm bound: \(336\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(392, [\chi])\).

Total New Old
Modular forms 592 100 492
Cusp forms 528 100 428
Eisenstein series 64 0 64

Trace form

\( 100 q + 18 q^{3} - 50 q^{5} - 4240 q^{9} + O(q^{10}) \) \( 100 q + 18 q^{3} - 50 q^{5} - 4240 q^{9} + 10 q^{11} + 792 q^{13} + 1372 q^{15} - 1698 q^{17} - 242 q^{19} + 7186 q^{23} - 30744 q^{25} - 16716 q^{27} - 9416 q^{29} + 5490 q^{31} - 1886 q^{33} + 11970 q^{37} - 5252 q^{39} + 22536 q^{41} - 3400 q^{43} - 2332 q^{45} - 9426 q^{47} - 77586 q^{51} - 4286 q^{53} + 34724 q^{55} - 82972 q^{57} + 16962 q^{59} - 11194 q^{61} - 46068 q^{65} + 37358 q^{67} + 73700 q^{69} + 323536 q^{71} + 18326 q^{73} + 154248 q^{75} - 89582 q^{79} - 600894 q^{81} - 247088 q^{83} + 103380 q^{85} - 15916 q^{87} - 184906 q^{89} + 197246 q^{93} + 282046 q^{95} + 385608 q^{97} - 572928 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(392, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
392.6.i.a 392.i 7.c $2$ $62.870$ \(\Q(\sqrt{-3}) \) None 56.6.a.b \(0\) \(-30\) \(-32\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-30+30\zeta_{6})q^{3}-2^{5}\zeta_{6}q^{5}-657\zeta_{6}q^{9}+\cdots\)
392.6.i.b 392.i 7.c $2$ $62.870$ \(\Q(\sqrt{-3}) \) None 8.6.a.a \(0\) \(-20\) \(74\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-20+20\zeta_{6})q^{3}+74\zeta_{6}q^{5}-157\zeta_{6}q^{9}+\cdots\)
392.6.i.c 392.i 7.c $2$ $62.870$ \(\Q(\sqrt{-3}) \) None 56.6.a.a \(0\) \(-6\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-6+6\zeta_{6})q^{3}+4\zeta_{6}q^{5}+207\zeta_{6}q^{9}+\cdots\)
392.6.i.d 392.i 7.c $2$ $62.870$ \(\Q(\sqrt{-3}) \) None 56.6.a.a \(0\) \(6\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(6-6\zeta_{6})q^{3}-4\zeta_{6}q^{5}+207\zeta_{6}q^{9}+\cdots\)
392.6.i.e 392.i 7.c $2$ $62.870$ \(\Q(\sqrt{-3}) \) None 8.6.a.a \(0\) \(20\) \(-74\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(20-20\zeta_{6})q^{3}-74\zeta_{6}q^{5}-157\zeta_{6}q^{9}+\cdots\)
392.6.i.f 392.i 7.c $2$ $62.870$ \(\Q(\sqrt{-3}) \) None 56.6.a.b \(0\) \(30\) \(32\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(30-30\zeta_{6})q^{3}+2^{5}\zeta_{6}q^{5}-657\zeta_{6}q^{9}+\cdots\)
392.6.i.g 392.i 7.c $4$ $62.870$ \(\Q(\sqrt{-3}, \sqrt{-59})\) None 56.6.a.c \(0\) \(-26\) \(-62\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-13-13\beta _{1}+\beta _{3})q^{3}+(31\beta _{1}+5\beta _{2}+\cdots)q^{5}+\cdots\)
392.6.i.h 392.i 7.c $4$ $62.870$ \(\Q(\sqrt{-3}, \sqrt{193})\) None 56.6.a.d \(0\) \(-14\) \(42\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-7\beta _{1}-\beta _{2})q^{3}+(21-21\beta _{1}-5\beta _{2}+\cdots)q^{5}+\cdots\)
392.6.i.i 392.i 7.c $4$ $62.870$ \(\Q(\sqrt{-3}, \sqrt{-115})\) None 56.6.a.e \(0\) \(-6\) \(82\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3\beta _{1}+\beta _{2}-\beta _{3})q^{3}+(41+41\beta _{1}+\cdots)q^{5}+\cdots\)
392.6.i.j 392.i 7.c $4$ $62.870$ \(\Q(\sqrt{-3}, \sqrt{-115})\) None 56.6.a.e \(0\) \(6\) \(-82\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3+3\beta _{1}-\beta _{3})q^{3}+(41\beta _{1}-3\beta _{2}+\cdots)q^{5}+\cdots\)
392.6.i.k 392.i 7.c $4$ $62.870$ \(\Q(\sqrt{-3}, \sqrt{193})\) None 56.6.a.d \(0\) \(14\) \(-42\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(7\beta _{1}-\beta _{2})q^{3}+(-21+21\beta _{1}-5\beta _{2}+\cdots)q^{5}+\cdots\)
392.6.i.l 392.i 7.c $4$ $62.870$ \(\Q(\sqrt{-3}, \sqrt{-59})\) None 56.6.a.c \(0\) \(26\) \(62\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-13\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(31+31\beta _{1}+\cdots)q^{5}+\cdots\)
392.6.i.m 392.i 7.c $8$ $62.870$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 392.6.a.h \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{2}+\beta _{6})q^{3}+(-2\beta _{2}+\beta _{3})q^{5}+\cdots\)
392.6.i.n 392.i 7.c $8$ $62.870$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 392.6.a.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{3}-\beta _{5}q^{5}+(23-23\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
392.6.i.o 392.i 7.c $10$ $62.870$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None 56.6.i.b \(0\) \(5\) \(-81\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{1}+\beta _{3})q^{3}+(-\beta _{1}+\beta _{2}+2^{4}\beta _{3}+\cdots)q^{5}+\cdots\)
392.6.i.p 392.i 7.c $10$ $62.870$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 56.6.i.a \(0\) \(13\) \(31\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3+3\beta _{1}-\beta _{4})q^{3}+(-6\beta _{1}-\beta _{6}+\cdots)q^{5}+\cdots\)
392.6.i.q 392.i 7.c $12$ $62.870$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 392.6.a.m \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{6}q^{3}+(\beta _{4}-\beta _{5})q^{5}+(20+20\beta _{1}+\cdots)q^{9}+\cdots\)
392.6.i.r 392.i 7.c $16$ $62.870$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 392.6.a.n \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(\beta _{5}+\beta _{8})q^{5}+(-116-117\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(392, [\chi])\) into lower level spaces

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