Space of modular forms of level 750, weight 2, and Character 750.h

• 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: $11$

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 - 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

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

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
750.2.h.a	$750$	$2$	750.h	25.e	$10$	$8$	$2$	$5.989$	\(\Q(\zeta_{20})\)	None					150.2.g.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\zeta_{20}q^{2}+\zeta_{20}^{7}q^{3}+\zeta_{20}^{2}q^{4}+(-1+\cdots)q^{6}+\cdots\)
750.2.h.b	$750$	$2$	750.h	25.e	$10$	$8$	$2$	$5.989$	\(\Q(\zeta_{20})\)	None					150.2.g.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\zeta_{20}q^{2}-\zeta_{20}^{7}q^{3}+\zeta_{20}^{2}q^{4}+(1+\cdots)q^{6}+\cdots\)
750.2.h.c	$750$	$2$	750.h	25.e	$10$	$8$	$2$	$5.989$	\(\Q(\zeta_{20})\)	None					150.2.h.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\zeta_{20}q^{2}-\zeta_{20}^{7}q^{3}+\zeta_{20}^{2}q^{4}+(1+\cdots)q^{6}+\cdots\)
750.2.h.d	$750$	$2$	750.h	25.e	$10$	$16$	$4$	$5.989$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					150.2.h.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{10}]$	\(q+\beta _{6}q^{2}+\beta _{3}q^{3}+\beta _{8}q^{4}+\beta _{2}q^{6}+\cdots\)
750.2.h.e	$750$	$2$	750.h	25.e	$10$	$16$	$4$	$5.989$	16.0.\(\cdots\).12	None					150.2.g.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{10}]$	\(q+\beta _{11}q^{2}+\beta _{7}q^{3}+\beta _{9}q^{4}+(1+\beta _{1}+\cdots)q^{6}+\cdots\)

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

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