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Properties

Label 310.2.h
Level $310$
Weight $2$
Character orbit 310.h
Rep. character $\chi_{310}(101,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $32$
Newform subspaces $5$
Sturm bound $96$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.h (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 31 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 5 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(310, [\chi])\).

Total New Old
Modular forms 208 32 176
Cusp forms 176 32 144
Eisenstein series 32 0 32

Trace form

\( 32 q + 4 q^{3} - 8 q^{4} + 16 q^{7} + O(q^{10}) \) \( 32 q + 4 q^{3} - 8 q^{4} + 16 q^{7} - 12 q^{11} + 4 q^{12} + 8 q^{13} - 12 q^{14} - 8 q^{16} + 4 q^{17} + 16 q^{18} + 16 q^{19} + 28 q^{21} + 12 q^{22} + 20 q^{23} + 32 q^{25} - 16 q^{26} - 20 q^{27} - 24 q^{28} - 4 q^{29} - 16 q^{30} - 28 q^{31} - 44 q^{33} - 12 q^{34} - 12 q^{35} + 32 q^{37} - 16 q^{38} - 20 q^{39} + 16 q^{41} + 24 q^{42} - 12 q^{43} + 8 q^{44} + 8 q^{45} + 12 q^{46} + 16 q^{47} + 4 q^{48} - 24 q^{51} + 8 q^{52} - 16 q^{53} + 24 q^{54} - 12 q^{55} + 8 q^{56} - 72 q^{57} + 4 q^{58} - 8 q^{59} - 36 q^{62} + 48 q^{63} - 8 q^{64} + 8 q^{65} + 12 q^{66} + 48 q^{67} - 16 q^{68} - 36 q^{69} + 44 q^{71} + 16 q^{72} + 28 q^{73} + 20 q^{74} + 4 q^{75} - 24 q^{76} + 8 q^{77} - 32 q^{78} - 52 q^{79} - 40 q^{81} - 8 q^{82} - 20 q^{83} + 8 q^{84} - 8 q^{85} - 44 q^{86} - 112 q^{87} - 8 q^{88} - 24 q^{89} - 24 q^{90} - 56 q^{91} + 40 q^{92} + 24 q^{93} + 8 q^{95} - 16 q^{97} - 32 q^{98} - 64 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(310, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
310.2.h.a 310.h 31.d $4$ $2.475$ \(\Q(\zeta_{10})\) None \(1\) \(-1\) \(4\) \(7\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{10}q^{2}+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{3}+\cdots\)
310.2.h.b 310.h 31.d $4$ $2.475$ \(\Q(\zeta_{10})\) None \(1\) \(1\) \(4\) \(-3\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{10}q^{2}+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{3}+\cdots\)
310.2.h.c 310.h 31.d $8$ $2.475$ 8.0.484000000.9 None \(-2\) \(0\) \(-8\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\beta _{2}-\beta _{3}+\beta _{5})q^{2}+(\beta _{1}-\beta _{6}+\cdots)q^{3}+\cdots\)
310.2.h.d 310.h 31.d $8$ $2.475$ 8.0.64000000.2 None \(-2\) \(2\) \(8\) \(-2\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\beta _{2}-\beta _{4}-\beta _{6})q^{2}+(-\beta _{2}+\cdots)q^{3}+\cdots\)
310.2.h.e 310.h 31.d $8$ $2.475$ 8.0.64000000.2 None \(2\) \(2\) \(-8\) \(10\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1+\beta _{2}+\beta _{4}+\beta _{6})q^{2}+(-\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(310, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(310, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(31, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(62, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(155, [\chi])\)\(^{\oplus 2}\)