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Properties

Label 385.2.b
Level $385$
Weight $2$
Character orbit 385.b
Rep. character $\chi_{385}(309,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $4$
Sturm bound $96$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(385, [\chi])\).

Total New Old
Modular forms 52 32 20
Cusp forms 44 32 12
Eisenstein series 8 0 8

Trace form

\( 32 q - 28 q^{4} + 6 q^{5} - 8 q^{6} - 44 q^{9} + O(q^{10}) \) \( 32 q - 28 q^{4} + 6 q^{5} - 8 q^{6} - 44 q^{9} + 8 q^{10} + 8 q^{11} - 8 q^{14} + 2 q^{15} + 20 q^{16} + 16 q^{19} + 12 q^{20} - 32 q^{24} - 6 q^{25} - 16 q^{26} + 16 q^{30} - 4 q^{31} + 40 q^{34} + 4 q^{35} + 84 q^{36} - 24 q^{40} - 8 q^{41} - 20 q^{44} - 24 q^{45} - 40 q^{46} - 32 q^{49} + 40 q^{50} + 8 q^{51} - 72 q^{54} - 6 q^{55} + 24 q^{56} + 20 q^{59} + 60 q^{60} - 4 q^{64} - 24 q^{65} + 28 q^{69} + 12 q^{70} + 20 q^{71} - 66 q^{75} - 104 q^{76} - 8 q^{79} - 40 q^{80} + 88 q^{81} - 28 q^{85} + 80 q^{86} - 36 q^{89} + 16 q^{90} + 16 q^{91} + 128 q^{94} - 44 q^{95} - 24 q^{96} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(385, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
385.2.b.a 385.b 5.b $2$ $3.074$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-2iq^{3}+q^{4}+(1+2i)q^{5}+\cdots\)
385.2.b.b 385.b 5.b $2$ $3.074$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-2iq^{3}+q^{4}+(1-2i)q^{5}+\cdots\)
385.2.b.c 385.b 5.b $12$ $3.074$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{2}-\beta _{10})q^{3}+(-1+\beta _{4}+\cdots)q^{4}+\cdots\)
385.2.b.d 385.b 5.b $16$ $3.074$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{7}+\beta _{13})q^{3}+(-2+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(385, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(385, [\chi]) \cong \)