Properties

Label 351.2.e
Level $351$
Weight $2$
Character orbit 351.e
Rep. character $\chi_{351}(118,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $24$
Newform subspaces $3$
Sturm bound $84$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(84\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(351, [\chi])\).

Total New Old
Modular forms 96 24 72
Cusp forms 72 24 48
Eisenstein series 24 0 24

Trace form

\( 24 q + 2 q^{2} - 12 q^{4} + 2 q^{5} - 12 q^{8} + O(q^{10}) \) \( 24 q + 2 q^{2} - 12 q^{4} + 2 q^{5} - 12 q^{8} + 6 q^{11} - 14 q^{14} - 12 q^{16} + 24 q^{17} - 12 q^{19} - 10 q^{20} + 6 q^{22} - 2 q^{23} - 6 q^{25} - 12 q^{26} - 12 q^{29} + 6 q^{31} + 14 q^{32} - 6 q^{34} - 12 q^{35} - 12 q^{37} + 16 q^{38} - 12 q^{40} + 10 q^{41} + 6 q^{43} + 16 q^{44} + 12 q^{47} + 38 q^{50} + 56 q^{53} + 12 q^{55} - 40 q^{56} - 12 q^{58} - 2 q^{59} + 12 q^{61} - 16 q^{62} + 48 q^{64} + 8 q^{65} + 12 q^{67} - 48 q^{68} - 56 q^{71} - 36 q^{73} + 34 q^{74} + 6 q^{76} - 10 q^{77} - 12 q^{79} - 80 q^{80} - 12 q^{82} - 6 q^{86} + 6 q^{88} - 56 q^{89} - 56 q^{92} - 44 q^{95} + 24 q^{97} + 124 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(351, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
351.2.e.a 351.e 9.c $2$ $2.803$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(4\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+4\zeta_{6}q^{5}+\cdots\)
351.2.e.b 351.e 9.c $10$ $2.803$ 10.0.\(\cdots\).1 None \(2\) \(0\) \(1\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{9}q^{2}+(-\beta _{1}+\beta _{6}-\beta _{9})q^{4}+(\beta _{4}+\cdots)q^{5}+\cdots\)
351.2.e.c 351.e 9.c $12$ $2.803$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-2\) \(0\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{2}+(-1+\beta _{1}-\beta _{5}+\beta _{7}-\beta _{8}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(351, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(351, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(117, [\chi])\)\(^{\oplus 2}\)