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Properties

Label 250.2.d
Level $250$
Weight $2$
Character orbit 250.d
Rep. character $\chi_{250}(51,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $28$
Newform subspaces $4$
Sturm bound $75$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 250 = 2 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 250.d (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 4 \)
Sturm bound: \(75\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(250, [\chi])\).

Total New Old
Modular forms 188 28 160
Cusp forms 108 28 80
Eisenstein series 80 0 80

Trace form

\( 28 q + q^{2} + 4 q^{3} - 7 q^{4} + 2 q^{6} + 8 q^{7} + q^{8} - q^{9} + O(q^{10}) \) \( 28 q + q^{2} + 4 q^{3} - 7 q^{4} + 2 q^{6} + 8 q^{7} + q^{8} - q^{9} - 4 q^{11} - 6 q^{12} + 14 q^{13} + 4 q^{14} - 7 q^{16} + 8 q^{17} - 12 q^{18} - 10 q^{19} + 16 q^{21} - 8 q^{22} - 6 q^{23} - 8 q^{24} - 38 q^{26} - 20 q^{27} - 2 q^{28} - 20 q^{29} + 6 q^{31} - 4 q^{32} - 22 q^{33} + 19 q^{34} - q^{36} + 23 q^{37} + 20 q^{38} + 8 q^{39} - 24 q^{41} + 22 q^{42} + 4 q^{43} + 6 q^{44} - 8 q^{46} - 22 q^{47} - 6 q^{48} + 36 q^{49} + 16 q^{51} + 14 q^{52} + 9 q^{53} + 20 q^{54} + 4 q^{56} + 30 q^{58} - 24 q^{61} - 28 q^{62} - 16 q^{63} - 7 q^{64} + 24 q^{66} - 2 q^{67} - 42 q^{68} + 68 q^{69} - 74 q^{71} + 13 q^{72} - 6 q^{73} - 46 q^{74} - 24 q^{77} - 24 q^{78} - 57 q^{81} - 38 q^{82} - 26 q^{83} - 14 q^{84} + 22 q^{86} + 2 q^{88} - 45 q^{89} - 4 q^{91} - 6 q^{92} + 68 q^{93} + 24 q^{94} + 2 q^{96} + 58 q^{97} + 17 q^{98} - 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(250, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
250.2.d.a 250.d 25.d $4$ $1.996$ \(\Q(\zeta_{10})\) None \(-1\) \(1\) \(0\) \(12\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
250.2.d.b 250.d 25.d $8$ $1.996$ \(\Q(\zeta_{20})\) None \(-2\) \(2\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{5}]$ \(q-\zeta_{20}q^{2}+(-\zeta_{20}^{3}+\zeta_{20}^{5})q^{3}+\zeta_{20}^{3}q^{4}+\cdots\)
250.2.d.c 250.d 25.d $8$ $1.996$ \(\Q(\zeta_{20})\) None \(2\) \(-2\) \(0\) \(16\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{20}^{4}q^{2}+(-\zeta_{20}+\zeta_{20}^{2})q^{3}-\zeta_{20}q^{4}+\cdots\)
250.2.d.d 250.d 25.d $8$ $1.996$ 8.0.58140625.2 None \(2\) \(3\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\beta _{2}-\beta _{3}-\beta _{6})q^{2}-\beta _{4}q^{3}-\beta _{2}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(250, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(250, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(125, [\chi])\)\(^{\oplus 2}\)