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Properties

Label 385.2.n
Level $385$
Weight $2$
Character orbit 385.n
Rep. character $\chi_{385}(36,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $96$
Newform subspaces $6$
Sturm bound $96$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.n (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 6 \)
Sturm bound: \(96\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 208 96 112
Cusp forms 176 96 80
Eisenstein series 32 0 32

Trace form

\( 96 q + 8 q^{2} + 8 q^{3} - 20 q^{4} - 12 q^{6} + 4 q^{7} + 4 q^{8} - 6 q^{9} + O(q^{10}) \) \( 96 q + 8 q^{2} + 8 q^{3} - 20 q^{4} - 12 q^{6} + 4 q^{7} + 4 q^{8} - 6 q^{9} - 6 q^{11} + 24 q^{12} - 12 q^{13} - 6 q^{14} - 6 q^{15} - 16 q^{16} + 32 q^{17} - 30 q^{18} - 36 q^{19} + 4 q^{21} + 24 q^{22} - 16 q^{23} - 4 q^{24} - 24 q^{25} + 40 q^{26} - 4 q^{27} - 22 q^{28} - 16 q^{29} + 40 q^{30} + 8 q^{31} - 64 q^{32} + 80 q^{34} + 8 q^{35} - 120 q^{36} - 48 q^{37} - 44 q^{38} - 40 q^{39} + 64 q^{41} - 16 q^{43} + 14 q^{44} + 54 q^{46} - 16 q^{47} + 120 q^{48} - 24 q^{49} + 8 q^{50} - 8 q^{51} + 36 q^{52} - 16 q^{53} - 96 q^{54} - 32 q^{55} + 12 q^{56} + 60 q^{57} - 42 q^{58} + 20 q^{59} + 16 q^{60} - 32 q^{61} + 40 q^{62} + 20 q^{63} - 4 q^{64} - 60 q^{65} - 152 q^{66} + 72 q^{67} - 68 q^{68} - 24 q^{69} + 12 q^{71} + 174 q^{72} + 40 q^{73} - 20 q^{74} - 12 q^{75} - 104 q^{76} + 8 q^{77} + 120 q^{78} - 40 q^{79} - 48 q^{80} - 22 q^{81} - 124 q^{82} - 40 q^{83} - 24 q^{84} + 20 q^{85} + 130 q^{86} + 56 q^{87} + 64 q^{88} - 16 q^{89} + 32 q^{90} - 18 q^{91} + 30 q^{92} - 12 q^{93} + 32 q^{94} + 32 q^{95} - 24 q^{96} - 144 q^{97} - 12 q^{98} + 40 q^{99} + O(q^{100}) \)

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.n.a 385.n 11.c $4$ $3.074$ \(\Q(\zeta_{10})\) None \(1\) \(-3\) \(-1\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-\zeta_{10}+\cdots)q^{3}+\cdots\)
385.2.n.b 385.n 11.c $4$ $3.074$ \(\Q(\zeta_{10})\) None \(5\) \(3\) \(1\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1+\zeta_{10}-\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(\zeta_{10}+\cdots)q^{3}+\cdots\)
385.2.n.c 385.n 11.c $8$ $3.074$ 8.0.13140625.1 None \(2\) \(4\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-\beta _{3}+\beta _{5}+\beta _{6})q^{2}+(1-\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)
385.2.n.d 385.n 11.c $16$ $3.074$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-4\) \(4\) \(4\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\beta _{4}+\beta _{7}-\beta _{8}-\beta _{9}-\beta _{10}+\cdots)q^{2}+\cdots\)
385.2.n.e 385.n 11.c $28$ $3.074$ None \(3\) \(0\) \(7\) \(-7\) $\mathrm{SU}(2)[C_{5}]$
385.2.n.f 385.n 11.c $36$ $3.074$ None \(1\) \(0\) \(-9\) \(9\) $\mathrm{SU}(2)[C_{5}]$

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

\( S_{2}^{\mathrm{old}}(385, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)