Properties

Label 310.2.a
Level $310$
Weight $2$
Character orbit 310.a
Rep. character $\chi_{310}(1,\cdot)$
Character field $\Q$
Dimension $9$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(310))\).

Total New Old
Modular forms 52 9 43
Cusp forms 45 9 36
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)\(31\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(3\)
Minus space\(-\)\(6\)

Trace form

\( 9 q + q^{2} + 9 q^{4} + q^{5} + 4 q^{6} - 8 q^{7} + q^{8} + 13 q^{9} + O(q^{10}) \) \( 9 q + q^{2} + 9 q^{4} + q^{5} + 4 q^{6} - 8 q^{7} + q^{8} + 13 q^{9} + q^{10} + 4 q^{11} - 14 q^{13} + 4 q^{15} + 9 q^{16} - 6 q^{17} - 3 q^{18} + 4 q^{19} + q^{20} - 16 q^{21} - 16 q^{23} + 4 q^{24} + 9 q^{25} - 10 q^{26} - 8 q^{28} + 14 q^{29} - q^{31} + q^{32} - 8 q^{33} + 10 q^{34} + 13 q^{36} - 14 q^{37} - 4 q^{38} + 16 q^{39} + q^{40} - 14 q^{41} + 8 q^{42} - 24 q^{43} + 4 q^{44} + 5 q^{45} - 8 q^{46} + 32 q^{47} + 17 q^{49} + q^{50} - 24 q^{51} - 14 q^{52} - 30 q^{53} + 16 q^{54} + 4 q^{55} - 8 q^{57} - 14 q^{58} + 28 q^{59} + 4 q^{60} - 18 q^{61} + 7 q^{62} - 24 q^{63} + 9 q^{64} + 6 q^{65} - 24 q^{66} - 20 q^{67} - 6 q^{68} + 32 q^{69} + 8 q^{70} + 8 q^{71} - 3 q^{72} - 14 q^{73} - 2 q^{74} + 4 q^{76} + 8 q^{77} - 32 q^{78} + 32 q^{79} + q^{80} - 39 q^{81} + 10 q^{82} + 24 q^{83} - 16 q^{84} + 6 q^{85} - 12 q^{86} - 8 q^{87} - 6 q^{89} - 3 q^{90} + 8 q^{91} - 16 q^{92} + 8 q^{94} + 12 q^{95} + 4 q^{96} - 6 q^{97} + 9 q^{98} + 20 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(310))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 31
310.2.a.a 310.a 1.a $1$ $2.475$ \(\Q\) None \(1\) \(-2\) \(-1\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}-4q^{7}+\cdots\)
310.2.a.b 310.a 1.a $1$ $2.475$ \(\Q\) None \(1\) \(2\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}-q^{5}+2q^{6}+q^{8}+\cdots\)
310.2.a.c 310.a 1.a $2$ $2.475$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(-2\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots\)
310.2.a.d 310.a 1.a $2$ $2.475$ \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(2\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+q^{5}-\beta q^{6}-2q^{7}+\cdots\)
310.2.a.e 310.a 1.a $3$ $2.475$ 3.3.148.1 None \(3\) \(2\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(310))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(310)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 2}\)