Properties

Label 520.2.q
Level $520$
Weight $2$
Character orbit 520.q
Rep. character $\chi_{520}(81,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $28$
Newform subspaces $8$
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.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 8 \)
Sturm bound: \(168\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(520, [\chi])\).

Total New Old
Modular forms 184 28 156
Cusp forms 152 28 124
Eisenstein series 32 0 32

Trace form

\( 28 q + 4 q^{3} - 4 q^{7} - 14 q^{9} + O(q^{10}) \) \( 28 q + 4 q^{3} - 4 q^{7} - 14 q^{9} + 2 q^{11} + 4 q^{13} - 12 q^{17} + 10 q^{19} + 40 q^{21} + 4 q^{23} + 28 q^{25} - 32 q^{27} - 4 q^{29} + 40 q^{31} + 8 q^{33} + 14 q^{35} + 8 q^{37} - 12 q^{39} + 12 q^{41} + 4 q^{43} - 80 q^{47} - 28 q^{49} + 8 q^{51} - 16 q^{53} + 4 q^{55} - 40 q^{57} - 16 q^{59} + 8 q^{61} - 24 q^{63} + 6 q^{65} + 4 q^{67} - 20 q^{69} + 8 q^{71} + 8 q^{73} + 4 q^{75} + 64 q^{77} + 40 q^{79} + 2 q^{81} - 16 q^{83} + 20 q^{87} - 6 q^{89} + 10 q^{91} - 12 q^{93} - 16 q^{97} + 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(520, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
520.2.q.a 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None 520.2.q.a \(0\) \(-2\) \(-2\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-q^{5}-3\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
520.2.q.b 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None 520.2.q.b \(0\) \(1\) \(-2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-q^{5}+3\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots\)
520.2.q.c 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None 520.2.q.c \(0\) \(1\) \(-2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-q^{5}+3\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots\)
520.2.q.d 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None 520.2.q.d \(0\) \(1\) \(2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+q^{5}+3\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots\)
520.2.q.e 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None 520.2.q.e \(0\) \(3\) \(-2\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}-q^{5}-3\zeta_{6}q^{7}-6\zeta_{6}q^{9}+\cdots\)
520.2.q.f 520.q 13.c $4$ $4.152$ \(\Q(\sqrt{-3}, \sqrt{17})\) None 520.2.q.f \(0\) \(1\) \(4\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+q^{5}+(-1+\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots\)
520.2.q.g 520.q 13.c $6$ $4.152$ 6.0.27870912.1 None 520.2.q.g \(0\) \(-1\) \(-6\) \(-9\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{3}-q^{5}+(-3+3\beta _{4})q^{7}+(-1+\cdots)q^{9}+\cdots\)
520.2.q.h 520.q 13.c $8$ $4.152$ 8.0.649638144.4 None 520.2.q.h \(0\) \(0\) \(8\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{4}q^{3}+q^{5}+(1+\beta _{3}+\beta _{4}-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(520, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(520, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)