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Properties

Label 350.2.q
Level $350$
Weight $2$
Character orbit 350.q
Rep. character $\chi_{350}(11,\cdot)$
Character field $\Q(\zeta_{15})$
Dimension $160$
Newform subspaces $3$
Sturm bound $120$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.q (of order \(15\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 175 \)
Character field: \(\Q(\zeta_{15})\)
Newform subspaces: \( 3 \)
Sturm bound: \(120\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(350, [\chi])\).

Total New Old
Modular forms 512 160 352
Cusp forms 448 160 288
Eisenstein series 64 0 64

Trace form

\( 160 q + 20 q^{4} - 2 q^{5} - 8 q^{6} + 8 q^{7} + 20 q^{9} + O(q^{10}) \) \( 160 q + 20 q^{4} - 2 q^{5} - 8 q^{6} + 8 q^{7} + 20 q^{9} - 2 q^{10} + 6 q^{11} + 48 q^{15} + 20 q^{16} - 24 q^{17} + 8 q^{18} + 4 q^{19} + 4 q^{20} + 12 q^{21} + 32 q^{22} + 2 q^{23} - 16 q^{24} + 4 q^{25} - 48 q^{26} + 96 q^{27} - 14 q^{28} + 24 q^{29} - 18 q^{30} + 6 q^{31} + 26 q^{33} - 16 q^{34} - 16 q^{35} - 40 q^{36} - 4 q^{37} - 8 q^{38} - 32 q^{39} - 2 q^{40} - 68 q^{41} - 14 q^{42} - 40 q^{43} - 4 q^{44} + 36 q^{45} - 12 q^{46} + 20 q^{47} + 24 q^{49} - 12 q^{51} - 56 q^{53} + 16 q^{54} - 16 q^{55} - 136 q^{57} + 24 q^{59} + 6 q^{60} + 8 q^{61} - 72 q^{62} - 136 q^{63} - 40 q^{64} - 2 q^{65} + 56 q^{68} - 116 q^{69} + 34 q^{70} - 68 q^{71} + 8 q^{72} + 12 q^{73} + 16 q^{74} - 60 q^{75} + 32 q^{76} - 72 q^{77} + 16 q^{78} - 8 q^{79} - 2 q^{80} - 24 q^{81} + 64 q^{82} - 128 q^{83} - 18 q^{84} - 4 q^{85} - 12 q^{86} + 24 q^{87} + 14 q^{88} + 54 q^{89} - 160 q^{90} - 44 q^{91} + 16 q^{92} - 52 q^{93} - 36 q^{95} + 4 q^{96} - 84 q^{97} - 64 q^{98} - 24 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.2.q.a 350.q 175.q $8$ $2.795$ \(\Q(\zeta_{15})\) None \(-1\) \(-3\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{15}]$ \(q-\zeta_{15}q^{2}+(1-\zeta_{15}^{2}+\zeta_{15}^{3}-\zeta_{15}^{4}+\cdots)q^{3}+\cdots\)
350.2.q.b 350.q 175.q $72$ $2.795$ None \(-9\) \(1\) \(0\) \(8\) $\mathrm{SU}(2)[C_{15}]$
350.2.q.c 350.q 175.q $80$ $2.795$ None \(10\) \(2\) \(-2\) \(4\) $\mathrm{SU}(2)[C_{15}]$

Decomposition of \(S_{2}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)