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Properties

Label 518.2.a
Level $518$
Weight $2$
Character orbit 518.a
Rep. character $\chi_{518}(1,\cdot)$
Character field $\Q$
Dimension $17$
Newform subspaces $6$
Sturm bound $152$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 518 = 2 \cdot 7 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 518.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(152\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(518))\).

Total New Old
Modular forms 80 17 63
Cusp forms 73 17 56
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.

\(2\)\(7\)\(37\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(3\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(5\)
Plus space\(+\)\(5\)
Minus space\(-\)\(12\)

Trace form

\( 17 q + q^{2} + 17 q^{4} - 6 q^{5} - q^{7} + q^{8} + 17 q^{9} + O(q^{10}) \) \( 17 q + q^{2} + 17 q^{4} - 6 q^{5} - q^{7} + q^{8} + 17 q^{9} - 6 q^{10} + 8 q^{11} + 10 q^{13} + 3 q^{14} + 8 q^{15} + 17 q^{16} - 6 q^{17} + 5 q^{18} + 8 q^{19} - 6 q^{20} + 4 q^{21} + 4 q^{22} + 8 q^{23} + 11 q^{25} - 6 q^{26} - 24 q^{27} - q^{28} - 2 q^{29} + 8 q^{30} - 16 q^{31} + q^{32} - 6 q^{34} + 6 q^{35} + 17 q^{36} + q^{37} + 20 q^{38} - 32 q^{39} - 6 q^{40} - 18 q^{41} + 12 q^{43} + 8 q^{44} - 14 q^{45} - 12 q^{46} + 16 q^{47} + 17 q^{49} - 9 q^{50} + 48 q^{51} + 10 q^{52} - 10 q^{53} + 24 q^{54} - 40 q^{55} + 3 q^{56} + 2 q^{58} - 8 q^{59} + 8 q^{60} - 22 q^{61} - 28 q^{62} + 11 q^{63} + 17 q^{64} + 28 q^{65} - 6 q^{68} - 64 q^{69} - 6 q^{70} - 8 q^{71} + 5 q^{72} + 6 q^{73} + 9 q^{74} - 4 q^{75} + 8 q^{76} - 4 q^{77} + 4 q^{78} + 24 q^{79} - 6 q^{80} - 31 q^{81} - 14 q^{82} - 20 q^{83} + 4 q^{84} - 4 q^{85} - 4 q^{86} - 8 q^{87} + 4 q^{88} + 2 q^{89} - 58 q^{90} + 22 q^{91} + 8 q^{92} + 8 q^{93} - 32 q^{94} - 64 q^{95} + 2 q^{97} + q^{98} + 28 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(518))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 7 37
518.2.a.a 518.a 1.a $2$ $4.136$ \(\Q(\sqrt{2}) \) None 518.2.a.a \(-2\) \(0\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+2\beta q^{5}-\beta q^{6}+\cdots\)
518.2.a.b 518.a 1.a $2$ $4.136$ \(\Q(\sqrt{5}) \) None 518.2.a.b \(2\) \(-3\) \(-3\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-2+\beta )q^{5}+\cdots\)
518.2.a.c 518.a 1.a $2$ $4.136$ \(\Q(\sqrt{3}) \) None 518.2.a.c \(2\) \(2\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
518.2.a.d 518.a 1.a $3$ $4.136$ 3.3.229.1 None 518.2.a.d \(-3\) \(-1\) \(-3\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{2}q^{3}+q^{4}+(-1-\beta _{1})q^{5}+\cdots\)
518.2.a.e 518.a 1.a $3$ $4.136$ 3.3.733.1 None 518.2.a.e \(-3\) \(1\) \(3\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(1+\beta _{2})q^{5}-\beta _{1}q^{6}+\cdots\)
518.2.a.f 518.a 1.a $5$ $4.136$ 5.5.2174276.1 None 518.2.a.f \(5\) \(1\) \(-3\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(-1-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(518))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(518)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(259))\)\(^{\oplus 2}\)