Space of modular forms of level 75, weight 10, and Character 75.b

• Label: 75.10.b
• Level: $75$
• Weight: $10$
• Character orbit: 75.b
• Rep. character: $\chi_{75}(49,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $8$
• Sturm bound: $100$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(100\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(75, [\chi])\).

	Total	New	Old
Modular forms	96	28	68
Cusp forms	84	28	56
Eisenstein series	12	0	12

Trace form

\( 28 q - 8364 q^{4} - 3564 q^{6} - 183708 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.10.b.a	$75$	$10$	75.b	5.b	$2$	$2$	$2$	$38.628$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+6^{2}iq^{2}-3^{4}iq^{3}-28^{2}q^{4}+54^{2}q^{6}+\cdots\)
75.10.b.b	$75$	$10$	75.b	5.b	$2$	$2$	$2$	$38.628$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+22iq^{2}+3^{4}iq^{3}+28q^{4}-1782q^{6}+\cdots\)
75.10.b.c	$75$	$10$	75.b	5.b	$2$	$2$	$2$	$38.628$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+18iq^{2}-3^{4}iq^{3}+188q^{4}+1458q^{6}+\cdots\)
75.10.b.d	$75$	$10$	75.b	5.b	$2$	$2$	$2$	$38.628$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{2}+3^{4}iq^{3}+496q^{4}-18^{2}q^{6}+\cdots\)
75.10.b.e	$75$	$10$	75.b	5.b	$2$	$4$	$4$	$38.628$	\(\Q(i, \sqrt{4729})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-9\beta _{2})q^{2}+3^{4}\beta _{2}q^{3}+(-770+\cdots)q^{4}+\cdots\)
75.10.b.f	$75$	$10$	75.b	5.b	$2$	$4$	$4$	$38.628$	\(\Q(i, \sqrt{241})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2^{4}\beta _{1}-\beta _{2})q^{2}-3^{4}\beta _{1}q^{3}+(-255+\cdots)q^{4}+\cdots\)
75.10.b.g	$75$	$10$	75.b	5.b	$2$	$4$	$4$	$38.628$	\(\Q(i, \sqrt{79})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(18\beta _{1}+\beta _{3})q^{2}+3^{4}\beta _{1}q^{3}+(-2^{7}+\cdots)q^{4}+\cdots\)
75.10.b.h	$75$	$10$	75.b	5.b	$2$	$8$	$8$	$38.628$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}\cdot 5^{4}\cdot 23^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}-3^{4}\beta _{1}q^{3}+(-445-6\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{10}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)