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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(520))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 13
520.2.a.a \(1\) \(4.152\) \(\Q\) None \(0\) \(0\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(q-q^{5}-3q^{9}-4q^{11}-q^{13}-6q^{17}+\cdots\)
520.2.a.b \(1\) \(4.152\) \(\Q\) None \(0\) \(2\) \(1\) \(0\) \(-\) \(-\) \(-\) \(q+2q^{3}+q^{5}+q^{9}+2q^{11}+q^{13}+\cdots\)
520.2.a.c \(2\) \(4.152\) \(\Q(\sqrt{2}) \) None \(0\) \(-4\) \(2\) \(-4\) \(-\) \(-\) \(+\) \(q+(-2+\beta )q^{3}+q^{5}-2q^{7}+(3-4\beta )q^{9}+\cdots\)
520.2.a.d \(2\) \(4.152\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(-2\) \(0\) \(-\) \(+\) \(-\) \(q+(-1+\beta )q^{3}-q^{5}-2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
520.2.a.e \(2\) \(4.152\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(-2\) \(0\) \(+\) \(+\) \(-\) \(q+(-1+\beta )q^{3}-q^{5}+2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
520.2.a.f \(2\) \(4.152\) \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(2\) \(4\) \(+\) \(-\) \(+\) \(q+\beta q^{3}+q^{5}+2q^{7}+3q^{9}+(-2+\beta )q^{11}+\cdots\)
520.2.a.g \(2\) \(4.152\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-2\) \(0\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{3}-q^{5}+(3+2\beta )q^{9}+(3-\beta )q^{11}+\cdots\)

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

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