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Properties

Label 500.3.b
Level $500$
Weight $3$
Character orbit 500.b
Rep. character $\chi_{500}(251,\cdot)$
Character field $\Q$
Dimension $96$
Newform subspaces $5$
Sturm bound $225$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 500.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(225\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(500, [\chi])\).

Total New Old
Modular forms 160 96 64
Cusp forms 140 96 44
Eisenstein series 20 0 20

Trace form

\( 96 q - 2 q^{4} - 6 q^{6} - 288 q^{9} + O(q^{10}) \) \( 96 q - 2 q^{4} - 6 q^{6} - 288 q^{9} + 4 q^{14} + 26 q^{16} + 8 q^{21} - 26 q^{24} - 52 q^{26} - 24 q^{29} + 22 q^{34} + 80 q^{36} - 8 q^{41} + 20 q^{44} + 114 q^{46} - 648 q^{49} - 2 q^{54} - 346 q^{56} + 32 q^{61} + 508 q^{64} - 30 q^{66} - 152 q^{69} + 662 q^{74} - 30 q^{76} + 824 q^{81} + 28 q^{84} - 666 q^{86} + 176 q^{89} - 146 q^{94} - 336 q^{96} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(500, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
500.3.b.a 500.b 4.b $8$ $13.624$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-5}) \) 500.3.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{20}q^{2}+(-\zeta_{20}-\zeta_{20}^{3})q^{3}-4q^{4}+\cdots\)
500.3.b.b 500.b 4.b $16$ $13.624$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 500.3.b.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+\beta _{5}q^{3}+(1-\beta _{3})q^{4}+(-2+\cdots)q^{6}+\cdots\)
500.3.b.c 500.b 4.b $24$ $13.624$ None 500.3.b.c \(-5\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
500.3.b.d 500.b 4.b $24$ $13.624$ None 500.3.b.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
500.3.b.e 500.b 4.b $24$ $13.624$ None 500.3.b.c \(5\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(500, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(500, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 2}\)