Properties

Label 520.2.d
Level $520$
Weight $2$
Character orbit 520.d
Rep. character $\chi_{520}(209,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $3$
Sturm bound $168$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(168\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 92 18 74
Cusp forms 76 18 58
Eisenstein series 16 0 16

Trace form

\( 18 q - 22 q^{9} + O(q^{10}) \) \( 18 q - 22 q^{9} + 12 q^{11} - 8 q^{15} - 12 q^{19} - 16 q^{21} + 20 q^{25} + 12 q^{29} - 8 q^{31} + 10 q^{35} - 12 q^{41} + 28 q^{45} - 54 q^{49} + 12 q^{51} + 8 q^{55} - 44 q^{59} + 12 q^{61} - 24 q^{69} - 8 q^{71} + 30 q^{75} + 24 q^{79} + 26 q^{81} + 20 q^{85} + 28 q^{89} + 12 q^{91} + 52 q^{95} - 60 q^{99} + O(q^{100}) \)

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.d.a 520.d 5.b $2$ $4.152$ \(\Q(\sqrt{-1}) \) None 520.2.d.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2+i)q^{5}-4iq^{7}+3q^{9}-2q^{11}+\cdots\)
520.2.d.b 520.d 5.b $6$ $4.152$ 6.0.350464.1 None 520.2.d.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{3}+\beta _{4})q^{3}+(\beta _{1}+\beta _{4})q^{5}+(-\beta _{3}+\cdots)q^{7}+\cdots\)
520.2.d.c 520.d 5.b $10$ $4.152$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 520.2.d.c \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{9}q^{3}-\beta _{1}q^{5}-\beta _{4}q^{7}+(-4-\beta _{5}+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(520, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)