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Properties

Label 616.2.q
Level $616$
Weight $2$
Character orbit 616.q
Rep. character $\chi_{616}(177,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $40$
Newform subspaces $6$
Sturm bound $192$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(192\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(616, [\chi])\).

Total New Old
Modular forms 208 40 168
Cusp forms 176 40 136
Eisenstein series 32 0 32

Trace form

\( 40 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 16 q^{9} + O(q^{10}) \) \( 40 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 16 q^{9} + 8 q^{13} + 16 q^{15} - 4 q^{19} + 12 q^{21} - 8 q^{23} - 20 q^{25} - 56 q^{27} + 24 q^{29} - 16 q^{31} - 4 q^{33} + 16 q^{35} - 8 q^{41} + 32 q^{43} - 24 q^{45} + 12 q^{47} + 4 q^{51} - 16 q^{53} - 40 q^{57} + 44 q^{59} + 20 q^{61} + 40 q^{63} + 8 q^{65} - 4 q^{67} - 32 q^{69} + 16 q^{71} + 12 q^{73} - 24 q^{75} - 36 q^{79} - 4 q^{81} + 16 q^{83} - 24 q^{87} - 36 q^{89} - 32 q^{91} + 12 q^{93} - 28 q^{95} - 8 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(616, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
616.2.q.a 616.q 7.c $2$ $4.919$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-2\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
616.2.q.b 616.q 7.c $4$ $4.919$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
616.2.q.c 616.q 7.c $6$ $4.919$ 6.0.64827.1 None \(0\) \(5\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{4}+\beta _{5})q^{3}+\cdots\)
616.2.q.d 616.q 7.c $8$ $4.919$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(2\) \(4\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{2}+\beta _{4})q^{3}+(1-\beta _{1}+\beta _{2}-\beta _{6}+\cdots)q^{5}+\cdots\)
616.2.q.e 616.q 7.c $10$ $4.919$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-3\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{4}+\beta _{7})q^{3}+(-1-\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{5}+\cdots\)
616.2.q.f 616.q 7.c $10$ $4.919$ 10.0.\(\cdots\).1 None \(0\) \(1\) \(-4\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{2}-\beta _{8})q^{3}+(-1+\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(616, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(616, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(154, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(308, [\chi])\)\(^{\oplus 2}\)