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Properties

Label 520.2.a
Level $520$
Weight $2$
Character orbit 520.a
Rep. character $\chi_{520}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $7$
Sturm bound $168$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(168\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(520))\).

Total New Old
Modular forms 92 12 80
Cusp forms 77 12 65
Eisenstein series 15 0 15

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

\(2\)\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(5\)
Minus space\(-\)\(7\)

Trace form

\( 12 q - 4 q^{3} - 2 q^{5} + 20 q^{9} + O(q^{10}) \) \( 12 q - 4 q^{3} - 2 q^{5} + 20 q^{9} - 4 q^{11} - 2 q^{13} + 8 q^{17} - 12 q^{19} + 8 q^{21} + 12 q^{25} - 16 q^{27} + 8 q^{29} + 16 q^{33} - 20 q^{37} - 12 q^{43} + 6 q^{45} - 20 q^{49} - 8 q^{51} + 4 q^{53} + 16 q^{57} - 12 q^{59} - 16 q^{61} - 4 q^{65} - 8 q^{67} - 16 q^{69} + 16 q^{71} + 4 q^{73} - 4 q^{75} - 8 q^{77} - 16 q^{79} + 52 q^{81} + 48 q^{83} - 4 q^{85} + 64 q^{87} - 24 q^{93} - 8 q^{95} - 4 q^{97} + 20 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(520))\) 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 5 13
520.2.a.a 520.a 1.a $1$ $4.152$ \(\Q\) None 520.2.a.a \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{9}-4q^{11}-q^{13}-6q^{17}+\cdots\)
520.2.a.b 520.a 1.a $1$ $4.152$ \(\Q\) None 520.2.a.b \(0\) \(2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+q^{9}+2q^{11}+q^{13}+\cdots\)
520.2.a.c 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{2}) \) None 520.2.a.c \(0\) \(-4\) \(2\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{3}+q^{5}-2q^{7}+(3-4\beta )q^{9}+\cdots\)
520.2.a.d 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{3}) \) None 520.2.a.d \(0\) \(-2\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-q^{5}-2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
520.2.a.e 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{3}) \) None 520.2.a.e \(0\) \(-2\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-q^{5}+2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
520.2.a.f 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{6}) \) None 520.2.a.f \(0\) \(0\) \(2\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{5}+2q^{7}+3q^{9}+(-2+\beta )q^{11}+\cdots\)
520.2.a.g 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{5}) \) None 520.2.a.g \(0\) \(2\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{5}+(3+2\beta )q^{9}+(3-\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(520))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(520)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 2}\)