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Properties

Label 520.4.a
Level $520$
Weight $4$
Character orbit 520.a
Rep. character $\chi_{520}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $11$
Sturm bound $336$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 520.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(336\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(520))\).

Total New Old
Modular forms 260 36 224
Cusp forms 244 36 208
Eisenstein series 16 0 16

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
\(+\)\(+\)\(+\)$+$\(5\)
\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(-\)\(-\)$+$\(6\)
\(-\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(-\)$+$\(4\)
\(-\)\(-\)\(+\)$+$\(4\)
\(-\)\(-\)\(-\)$-$\(5\)
Plus space\(+\)\(19\)
Minus space\(-\)\(17\)

Trace form

\( 36 q - 4 q^{3} + 10 q^{5} + 236 q^{9} + O(q^{10}) \) \( 36 q - 4 q^{3} + 10 q^{5} + 236 q^{9} - 100 q^{11} + 26 q^{13} + 232 q^{17} + 180 q^{19} - 376 q^{21} + 900 q^{25} - 400 q^{27} - 56 q^{29} + 720 q^{33} + 484 q^{37} - 272 q^{41} - 684 q^{43} + 930 q^{45} + 1540 q^{49} - 392 q^{51} - 676 q^{53} + 2640 q^{57} + 532 q^{59} + 384 q^{61} + 260 q^{65} - 2104 q^{67} - 784 q^{71} - 2324 q^{73} - 100 q^{75} + 536 q^{77} + 944 q^{79} + 1132 q^{81} - 4544 q^{83} + 340 q^{85} - 4704 q^{87} + 3168 q^{89} - 72 q^{93} - 40 q^{95} - 364 q^{97} - 2188 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(520))\) 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 13
520.4.a.a 520.a 1.a $1$ $30.681$ \(\Q\) None \(0\) \(-10\) \(5\) \(-12\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{3}+5q^{5}-12q^{7}+73q^{9}-62q^{11}+\cdots\)
520.4.a.b 520.a 1.a $1$ $30.681$ \(\Q\) None \(0\) \(-2\) \(-5\) \(30\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-5q^{5}+30q^{7}-23q^{9}-12q^{11}+\cdots\)
520.4.a.c 520.a 1.a $2$ $30.681$ \(\Q(\sqrt{14}) \) None \(0\) \(-4\) \(-10\) \(8\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{3}-5q^{5}+4q^{7}+(33-4\beta )q^{9}+\cdots\)
520.4.a.d 520.a 1.a $3$ $30.681$ 3.3.29317.1 None \(0\) \(-6\) \(-15\) \(-26\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-5q^{5}+(-9-\beta _{2})q^{7}-23q^{9}+\cdots\)
520.4.a.e 520.a 1.a $3$ $30.681$ 3.3.7572.1 None \(0\) \(2\) \(-15\) \(-12\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}-5q^{5}-4q^{7}+(14-2\beta _{1}+\cdots)q^{9}+\cdots\)
520.4.a.f 520.a 1.a $4$ $30.681$ 4.4.105984.1 None \(0\) \(-4\) \(-20\) \(8\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{3})q^{3}-5q^{5}+(2-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
520.4.a.g 520.a 1.a $4$ $30.681$ 4.4.4819572.2 None \(0\) \(0\) \(20\) \(-26\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+5q^{5}+(-6-\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
520.4.a.h 520.a 1.a $4$ $30.681$ 4.4.193792.1 None \(0\) \(0\) \(20\) \(-16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{3}+5q^{5}+(-4+2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
520.4.a.i 520.a 1.a $4$ $30.681$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(8\) \(20\) \(16\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{3}+5q^{5}+(4+\beta _{2})q^{7}+(15+\cdots)q^{9}+\cdots\)
520.4.a.j 520.a 1.a $4$ $30.681$ 4.4.2772480.1 None \(0\) \(12\) \(-20\) \(-8\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta _{3})q^{3}-5q^{5}+(-2-\beta _{2})q^{7}+\cdots\)
520.4.a.k 520.a 1.a $6$ $30.681$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(30\) \(38\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+5q^{5}+(6+\beta _{5})q^{7}+(17+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(520))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(520)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 2}\)