Properties

Label 316.2.a
Level $316$
Weight $2$
Character orbit 316.a
Rep. character $\chi_{316}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $4$
Sturm bound $80$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 316 = 2^{2} \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 316.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(80\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(316))\).

Total New Old
Modular forms 43 6 37
Cusp forms 38 6 32
Eisenstein series 5 0 5

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

\(2\)\(79\)FrickeDim
\(-\)\(+\)$-$\(3\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(3\)
Minus space\(-\)\(3\)

Trace form

\( 6 q + 2 q^{7} + O(q^{10}) \) \( 6 q + 2 q^{7} - 6 q^{11} - 4 q^{13} + 2 q^{15} - 4 q^{17} - 6 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{25} - 12 q^{27} - 8 q^{29} + 10 q^{31} + 22 q^{33} - 4 q^{35} - 2 q^{37} + 6 q^{39} + 4 q^{41} + 6 q^{43} + 22 q^{45} - 12 q^{47} - 4 q^{49} - 4 q^{51} + 2 q^{53} + 6 q^{57} - 2 q^{59} - 18 q^{61} + 6 q^{63} - 6 q^{65} - 10 q^{67} - 6 q^{69} + 6 q^{71} + 6 q^{73} + 18 q^{75} + 18 q^{77} + 6 q^{81} - 8 q^{83} - 14 q^{85} + 4 q^{87} - 8 q^{89} - 14 q^{91} + 6 q^{93} + 42 q^{95} - 12 q^{97} - 22 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(316))\) 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 79
316.2.a.a 316.a 1.a $1$ $2.523$ \(\Q\) None \(0\) \(-3\) \(1\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{5}+q^{7}+6q^{9}-6q^{11}+\cdots\)
316.2.a.b 316.a 1.a $1$ $2.523$ \(\Q\) None \(0\) \(-1\) \(1\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+3q^{7}-2q^{9}+2q^{11}+\cdots\)
316.2.a.c 316.a 1.a $2$ $2.523$ \(\Q(\sqrt{13}) \) None \(0\) \(0\) \(-5\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{5}+(-2+2\beta )q^{7}-3q^{9}+\cdots\)
316.2.a.d 316.a 1.a $2$ $2.523$ \(\Q(\sqrt{13}) \) None \(0\) \(4\) \(3\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+(2-\beta )q^{5}+q^{9}+(1+\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(316))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(316)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(79))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(158))\)\(^{\oplus 2}\)