Space of modular forms of level 75, weight 12, and trivial character

• Label: 75.12.a
• Level: $75$
• Weight: $12$
• Character orbit: 75.a
• Rep. character: $\chi_{75}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $11$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(75))\).

	Total	New	Old
Modular forms	116	35	81
Cusp forms	104	35	69
Eisenstein series	12	0	12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(9\)
\(+\)	\(-\)	$-$	\(9\)
\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	$+$	\(10\)
Plus space		\(+\)	\(19\)
Minus space		\(-\)	\(16\)

Trace form

\( 35 q + 14 q^{2} - 243 q^{3} + 36944 q^{4} + 18954 q^{6} + 17464 q^{7} - 191220 q^{8} + 2066715 q^{9} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(75))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
75.12.a.a	$75$	$12$	75.a	1.a	$1$	$1$	$1$	$57.626$	\(\Q\)	None	✓	$1$	$1$	\(-78\)	\(243\)	\(0\)	\(27760\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-78q^{2}+3^{5}q^{3}+4036q^{4}-18954q^{6}+\cdots\)
75.12.a.b	$75$	$12$	75.a	1.a	$1$	$1$	$1$	$57.626$	\(\Q\)	None	✓	$1$	$1$	\(56\)	\(243\)	\(0\)	\(-27984\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+56q^{2}+3^{5}q^{3}+1088q^{4}+13608q^{6}+\cdots\)
75.12.a.c	$75$	$12$	75.a	1.a	$1$	$2$	$2$	$57.626$	\(\Q(\sqrt{1801}) \)	None	✓	$1$	$1$	\(13\)	\(486\)	\(0\)	\(-7784\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{2}+3^{5}q^{3}+(-1549-13\beta )q^{4}+\cdots\)
75.12.a.d	$75$	$12$	75.a	1.a	$1$	$2$	$2$	$57.626$	\(\Q(\sqrt{1609}) \)	None	✓	$1$	$0$	\(22\)	\(-486\)	\(0\)	\(10864\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(11-\beta )q^{2}-3^{5}q^{3}+(-318-22\beta )q^{4}+\cdots\)
75.12.a.e	$75$	$12$	75.a	1.a	$1$	$3$	$3$	$57.626$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-729\)	\(0\)	\(53129\)		$+$	$-$	$-$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3^{5}q^{3}+(867+8\beta _{1}+\beta _{2})q^{4}+\cdots\)
75.12.a.f	$75$	$12$	75.a	1.a	$1$	$3$	$3$	$57.626$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(729\)	\(0\)	\(-53129\)		$-$	$+$	$-$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3^{5}q^{3}+(867+8\beta _{1}+\beta _{2})q^{4}+\cdots\)
75.12.a.g	$75$	$12$	75.a	1.a	$1$	$3$	$3$	$57.626$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(-729\)	\(0\)	\(14608\)		$+$	$+$	$+$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3^{5}q^{3}+(1585+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
75.12.a.h	$75$	$12$	75.a	1.a	$1$	$4$	$4$	$57.626$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-46\)	\(-972\)	\(0\)	\(-68372\)		$+$	$+$	$+$	$2^{3}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{1})q^{2}-3^{5}q^{3}+(1145+35\beta _{1}+\cdots)q^{4}+\cdots\)
75.12.a.i	$75$	$12$	75.a	1.a	$1$	$4$	$4$	$57.626$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(46\)	\(972\)	\(0\)	\(68372\)		$-$	$-$	$+$	$2^{3}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(11+\beta _{1})q^{2}+3^{5}q^{3}+(1145+35\beta _{1}+\cdots)q^{4}+\cdots\)
75.12.a.j	$75$	$12$	75.a	1.a	$1$	$6$	$6$	$57.626$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(-1458\)	\(0\)	\(-64368\)		$+$	$-$	$-$	$2^{6}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}-3^{5}q^{3}+(1359+\beta _{1}+\cdots)q^{4}+\cdots\)
75.12.a.k	$75$	$12$	75.a	1.a	$1$	$6$	$6$	$57.626$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(9\)	\(1458\)	\(0\)	\(64368\)		$-$	$-$	$+$	$2^{6}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+3^{5}q^{3}+(1359+\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(75)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)