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Properties

Label 750.2.h
Level $750$
Weight $2$
Character orbit 750.h
Rep. character $\chi_{750}(49,\cdot)$
Character field $\Q(\zeta_{10})$
Dimension $56$
Newform subspaces $5$
Sturm bound $300$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.h (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 5 \)
Sturm bound: \(300\)
Trace bound: \(11\)
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 56 624
Cusp forms 520 56 464
Eisenstein series 160 0 160

Trace form

\( 56 q + 14 q^{4} + 2 q^{6} + 14 q^{9} + O(q^{10}) \) \( 56 q + 14 q^{4} + 2 q^{6} + 14 q^{9} - 12 q^{11} - 14 q^{16} + 20 q^{17} + 8 q^{19} + 4 q^{21} + 20 q^{22} + 20 q^{23} + 8 q^{24} + 32 q^{26} + 10 q^{28} - 8 q^{29} - 6 q^{31} + 20 q^{33} + 40 q^{34} - 14 q^{36} - 4 q^{41} - 10 q^{42} - 8 q^{44} - 4 q^{46} + 40 q^{47} - 36 q^{49} - 16 q^{51} - 2 q^{54} + 28 q^{61} - 60 q^{62} - 20 q^{63} + 14 q^{64} + 8 q^{66} + 40 q^{67} - 16 q^{69} + 8 q^{71} - 48 q^{74} - 8 q^{76} - 80 q^{77} + 4 q^{79} - 14 q^{81} - 40 q^{83} - 4 q^{84} + 24 q^{86} + 20 q^{87} - 10 q^{88} + 24 q^{89} + 12 q^{91} - 32 q^{94} + 2 q^{96} - 50 q^{97} - 80 q^{98} - 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.h.a 750.h 25.e $8$ $5.989$ \(\Q(\zeta_{20})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+\zeta_{20}q^{2}+\zeta_{20}^{7}q^{3}+\zeta_{20}^{2}q^{4}+(-1+\cdots)q^{6}+\cdots\)
750.2.h.b 750.h 25.e $8$ $5.989$ \(\Q(\zeta_{20})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+\zeta_{20}q^{2}-\zeta_{20}^{7}q^{3}+\zeta_{20}^{2}q^{4}+(1+\cdots)q^{6}+\cdots\)
750.2.h.c 750.h 25.e $8$ $5.989$ \(\Q(\zeta_{20})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+\zeta_{20}q^{2}-\zeta_{20}^{7}q^{3}+\zeta_{20}^{2}q^{4}+(1+\cdots)q^{6}+\cdots\)
750.2.h.d 750.h 25.e $16$ $5.989$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+\beta _{6}q^{2}+\beta _{3}q^{3}+\beta _{8}q^{4}+\beta _{2}q^{6}+\cdots\)
750.2.h.e 750.h 25.e $16$ $5.989$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+\beta _{11}q^{2}+\beta _{7}q^{3}+\beta _{9}q^{4}+(1+\beta _{1}+\cdots)q^{6}+\cdots\)

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

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