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Properties

Label 500.2.o
Level $500$
Weight $2$
Character orbit 500.o
Rep. character $\chi_{500}(9,\cdot)$
Character field $\Q(\zeta_{50})$
Dimension $240$
Newform subspaces $1$
Sturm bound $150$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 500.o (of order \(50\) and degree \(20\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 125 \)
Character field: \(\Q(\zeta_{50})\)
Newform subspaces: \( 1 \)
Sturm bound: \(150\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(500, [\chi])\).

Total New Old
Modular forms 1560 240 1320
Cusp forms 1440 240 1200
Eisenstein series 120 0 120

Trace form

\( 240 q - 5 q^{5} + O(q^{10}) \) \( 240 q - 5 q^{5} - 5 q^{11} + 20 q^{15} + 5 q^{17} + 10 q^{19} - 30 q^{23} + 55 q^{25} + 10 q^{29} + 10 q^{31} - 15 q^{33} + 5 q^{35} + 10 q^{37} - 20 q^{39} - 15 q^{41} - 160 q^{45} + 80 q^{47} + 40 q^{49} + 20 q^{51} - 30 q^{53} - 35 q^{55} - 15 q^{59} - 20 q^{61} - 20 q^{63} + 25 q^{65} + 5 q^{67} + 10 q^{69} - 95 q^{71} + 60 q^{73} + 5 q^{75} + 20 q^{77} + 20 q^{79} + 5 q^{81} + 10 q^{83} + 5 q^{85} + 55 q^{87} + 35 q^{89} + 120 q^{91} - 100 q^{93} + 20 q^{95} - 25 q^{97} - 30 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(500, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
500.2.o.a 500.o 125.h $240$ $3.993$ None 500.2.o.a \(0\) \(0\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{50}]$

Decomposition of \(S_{2}^{\mathrm{old}}(500, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(500, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(125, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(250, [\chi])\)\(^{\oplus 2}\)