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Properties

Label 196.6.e
Level $196$
Weight $6$
Character orbit 196.e
Rep. character $\chi_{196}(165,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $34$
Newform subspaces $12$
Sturm bound $168$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 196.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(168\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(196, [\chi])\).

Total New Old
Modular forms 304 34 270
Cusp forms 256 34 222
Eisenstein series 48 0 48

Trace form

\( 34 q + 9 q^{3} + 61 q^{5} - 1488 q^{9} + O(q^{10}) \) \( 34 q + 9 q^{3} + 61 q^{5} - 1488 q^{9} + 291 q^{11} + 724 q^{13} - 4658 q^{15} + 1049 q^{17} + 1767 q^{19} - 6511 q^{23} - 14122 q^{25} - 6822 q^{27} - 12356 q^{29} + 4837 q^{31} + 24439 q^{33} + 3855 q^{37} - 22258 q^{39} - 3876 q^{41} - 71144 q^{43} + 41470 q^{45} + 34299 q^{47} - 60859 q^{51} - 36369 q^{53} - 61414 q^{55} - 80310 q^{57} + 81001 q^{59} + 111625 q^{61} - 31502 q^{65} - 167489 q^{67} - 49706 q^{69} - 161056 q^{71} + 24533 q^{73} + 142172 q^{75} - 86399 q^{79} - 215809 q^{81} + 106712 q^{83} - 599034 q^{85} + 5378 q^{87} + 114405 q^{89} - 213143 q^{93} + 67463 q^{95} - 284740 q^{97} + 576892 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(196, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
196.6.e.a 196.e 7.c $2$ $31.435$ \(\Q(\sqrt{-3}) \) None \(0\) \(-26\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-26+26\zeta_{6})q^{3}-2^{4}\zeta_{6}q^{5}-433\zeta_{6}q^{9}+\cdots\)
196.6.e.b 196.e 7.c $2$ $31.435$ \(\Q(\sqrt{-3}) \) None \(0\) \(-19\) \(19\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-19+19\zeta_{6})q^{3}+19\zeta_{6}q^{5}-118\zeta_{6}q^{9}+\cdots\)
196.6.e.c 196.e 7.c $2$ $31.435$ \(\Q(\sqrt{-3}) \) None \(0\) \(-16\) \(16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2^{4}+2^{4}\zeta_{6})q^{3}+2^{4}\zeta_{6}q^{5}-13\zeta_{6}q^{9}+\cdots\)
196.6.e.d 196.e 7.c $2$ $31.435$ \(\Q(\sqrt{-3}) \) None \(0\) \(-12\) \(54\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-12+12\zeta_{6})q^{3}+54\zeta_{6}q^{5}+99\zeta_{6}q^{9}+\cdots\)
196.6.e.e 196.e 7.c $2$ $31.435$ \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(-96\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-96\zeta_{6}q^{5}+239\zeta_{6}q^{9}+\cdots\)
196.6.e.f 196.e 7.c $2$ $31.435$ \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(96\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{3}+96\zeta_{6}q^{5}+239\zeta_{6}q^{9}+\cdots\)
196.6.e.g 196.e 7.c $2$ $31.435$ \(\Q(\sqrt{-3}) \) None \(0\) \(12\) \(-54\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(12-12\zeta_{6})q^{3}-54\zeta_{6}q^{5}+99\zeta_{6}q^{9}+\cdots\)
196.6.e.h 196.e 7.c $2$ $31.435$ \(\Q(\sqrt{-3}) \) None \(0\) \(16\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2^{4}-2^{4}\zeta_{6})q^{3}-2^{4}\zeta_{6}q^{5}-13\zeta_{6}q^{9}+\cdots\)
196.6.e.i 196.e 7.c $2$ $31.435$ \(\Q(\sqrt{-3}) \) None \(0\) \(26\) \(16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(26-26\zeta_{6})q^{3}+2^{4}\zeta_{6}q^{5}-433\zeta_{6}q^{9}+\cdots\)
196.6.e.j 196.e 7.c $4$ $31.435$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{1}q^{3}+(-46\beta _{1}-46\beta _{3})q^{5}-15^{2}\beta _{2}q^{9}+\cdots\)
196.6.e.k 196.e 7.c $4$ $31.435$ \(\Q(\sqrt{-3}, \sqrt{109})\) None \(0\) \(28\) \(42\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(14-14\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(21\beta _{1}+\cdots)q^{5}+\cdots\)
196.6.e.l 196.e 7.c $8$ $31.435$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}-\beta _{5}-\beta _{6})q^{3}+(\beta _{2}+7\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(196, [\chi])\) into lower level spaces

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