Properties

Label 750.2.g
Level $750$
Weight $2$
Character orbit 750.g
Rep. character $\chi_{750}(151,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $64$
Newform subspaces $7$
Sturm bound $300$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.g (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 7 \)
Sturm bound: \(300\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(750, [\chi])\).

Total New Old
Modular forms 680 64 616
Cusp forms 520 64 456
Eisenstein series 160 0 160

Trace form

\( 64 q - 2 q^{2} - 16 q^{4} - 2 q^{6} - 12 q^{7} - 2 q^{8} - 16 q^{9} + O(q^{10}) \) \( 64 q - 2 q^{2} - 16 q^{4} - 2 q^{6} - 12 q^{7} - 2 q^{8} - 16 q^{9} + 12 q^{11} - 12 q^{13} - 16 q^{16} + 8 q^{17} + 8 q^{18} + 8 q^{19} - 4 q^{21} + 8 q^{22} + 36 q^{23} + 8 q^{24} + 28 q^{26} - 2 q^{28} + 52 q^{29} + 6 q^{31} + 8 q^{32} + 8 q^{33} - 50 q^{34} - 16 q^{36} - 26 q^{37} - 16 q^{38} + 64 q^{41} + 6 q^{42} + 32 q^{43} - 8 q^{44} + 4 q^{46} - 8 q^{47} + 84 q^{49} + 16 q^{51} - 12 q^{52} + 30 q^{53} - 2 q^{54} - 24 q^{57} - 24 q^{58} + 32 q^{61} + 20 q^{62} + 8 q^{63} - 16 q^{64} - 8 q^{66} - 32 q^{67} - 12 q^{68} - 16 q^{69} - 8 q^{71} - 2 q^{72} + 8 q^{73} + 12 q^{74} + 8 q^{76} - 16 q^{77} - 16 q^{78} + 4 q^{79} - 16 q^{81} + 44 q^{82} + 8 q^{83} - 4 q^{84} - 24 q^{86} - 16 q^{87} - 2 q^{88} - 66 q^{89} - 12 q^{91} - 24 q^{92} + 16 q^{93} - 32 q^{94} - 2 q^{96} - 26 q^{97} + 14 q^{98} - 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(750, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
750.2.g.a 750.g 25.d $4$ $5.989$ \(\Q(\zeta_{10})\) None \(-1\) \(-1\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
750.2.g.b 750.g 25.d $4$ $5.989$ \(\Q(\zeta_{10})\) None \(1\) \(-1\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}-\zeta_{10}^{3}q^{3}+\cdots\)
750.2.g.c 750.g 25.d $8$ $5.989$ \(\Q(\zeta_{20})\) None \(-2\) \(2\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{5}]$ \(q-\zeta_{20}^{4}q^{2}+\zeta_{20}q^{3}-\zeta_{20}q^{4}+(1+\cdots)q^{6}+\cdots\)
750.2.g.d 750.g 25.d $8$ $5.989$ 8.0.1064390625.3 None \(-2\) \(2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{3}q^{2}+(1-\beta _{2}-\beta _{3}-\beta _{4})q^{3}+(-1+\cdots)q^{4}+\cdots\)
750.2.g.e 750.g 25.d $8$ $5.989$ \(\Q(\zeta_{20})\) None \(2\) \(-2\) \(0\) \(8\) $\mathrm{SU}(2)[C_{5}]$ \(q-\zeta_{20}^{3}q^{2}+(-1+\zeta_{20}-\zeta_{20}^{3}+\zeta_{20}^{4}+\cdots)q^{3}+\cdots\)
750.2.g.f 750.g 25.d $16$ $5.989$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-4\) \(-4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{5}q^{2}+\beta _{6}q^{3}+\beta _{6}q^{4}-\beta _{3}q^{6}+\cdots\)
750.2.g.g 750.g 25.d $16$ $5.989$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(4\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{3}q^{2}+\beta _{4}q^{3}-\beta _{4}q^{4}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(750, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(750, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(125, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(250, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(375, [\chi])\)\(^{\oplus 2}\)