Space of modular forms of level 75, weight 3, and Character 75.d

• Label: 75.3.d
• Level: $75$
• Weight: $3$
• Character orbit: 75.d
• Rep. character: $\chi_{75}(74,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $3$
• Sturm bound: $30$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(30\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(75, [\chi])\).

	Total	New	Old
Modular forms	26	14	12
Cusp forms	14	10	4
Eisenstein series	12	4	8

Trace form

\( 10 q + 24 q^{4} - 2 q^{6} - 28 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(75, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.3.d.a	$75$	$3$	75.d	15.d	$2$	$2$	$2$	$2.044$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3iq^{3}-4q^{4}+11iq^{7}-9q^{9}-12iq^{12}+\cdots\)
75.3.d.b	$75$	$3$	75.d	15.d	$2$	$4$	$4$	$2.044$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(\beta _{1}-\beta _{3})q^{3}+q^{4}+(5-\beta _{2}+\cdots)q^{6}+\cdots\)
75.3.d.c	$75$	$3$	75.d	15.d	$2$	$4$	$4$	$2.044$	\(\Q(i, \sqrt{11})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{2}+\beta _{1}q^{3}+7q^{4}+(-6+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)