Space of modular forms of level 75, weight 10, and trivial character

• Label: 75.10.a
• Level: $75$
• Weight: $10$
• Character orbit: 75.a
• Rep. character: $\chi_{75}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $12$
• Sturm bound: $100$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(100\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(75))\).

	Total	New	Old
Modular forms	96	28	68
Cusp forms	84	28	56
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
\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(-\)	$-$	\(8\)
\(-\)	\(+\)	$-$	\(8\)
\(-\)	\(-\)	$+$	\(6\)
Plus space		\(+\)	\(12\)
Minus space		\(-\)	\(16\)

Trace form

\( 28 q - 50 q^{2} + 7424 q^{4} - 4374 q^{6} + 6780 q^{7} - 45540 q^{8} + 183708 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(75))\) 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
75.10.a.a	$75$	$10$	75.a	1.a	$1$	$1$	$1$	$38.628$	\(\Q\)	None	✓	$1$	$0$	\(-22\)	\(81\)	\(0\)	\(5988\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-22q^{2}+3^{4}q^{3}-28q^{4}-1782q^{6}+\cdots\)
75.10.a.b	$75$	$10$	75.a	1.a	$1$	$1$	$1$	$38.628$	\(\Q\)	None	✓	$1$	$1$	\(-18\)	\(-81\)	\(0\)	\(-9128\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-18q^{2}-3^{4}q^{3}-188q^{4}+1458q^{6}+\cdots\)
75.10.a.c	$75$	$10$	75.a	1.a	$1$	$1$	$1$	$38.628$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(-81\)	\(0\)	\(7680\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-3^{4}q^{3}-496q^{4}-18^{2}q^{6}+\cdots\)
75.10.a.d	$75$	$10$	75.a	1.a	$1$	$1$	$1$	$38.628$	\(\Q\)	None	✓	$1$	$0$	\(36\)	\(81\)	\(0\)	\(4480\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6^{2}q^{2}+3^{4}q^{3}+28^{2}q^{4}+54^{2}q^{6}+\cdots\)
75.10.a.e	$75$	$10$	75.a	1.a	$1$	$2$	$2$	$38.628$	\(\Q(\sqrt{79}) \)	None	✓	$1$	$1$	\(-36\)	\(162\)	\(0\)	\(3318\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-18+\beta )q^{2}+3^{4}q^{3}+(2^{7}-6^{2}\beta )q^{4}+\cdots\)
75.10.a.f	$75$	$10$	75.a	1.a	$1$	$2$	$2$	$38.628$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$0$	\(-31\)	\(162\)	\(0\)	\(-14112\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(-2^{4}-\beta )q^{2}+3^{4}q^{3}+(286+31\beta )q^{4}+\cdots\)
75.10.a.g	$75$	$10$	75.a	1.a	$1$	$2$	$2$	$38.628$	\(\Q(\sqrt{4729}) \)	None	✓	$1$	$1$	\(-19\)	\(-162\)	\(0\)	\(11872\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-9-\beta )q^{2}-3^{4}q^{3}+(751+19\beta )q^{4}+\cdots\)
75.10.a.h	$75$	$10$	75.a	1.a	$1$	$2$	$2$	$38.628$	\(\Q(\sqrt{79}) \)	None	✓	$1$	$1$	\(36\)	\(-162\)	\(0\)	\(-3318\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(18+\beta )q^{2}-3^{4}q^{3}+(2^{7}+6^{2}\beta )q^{4}+\cdots\)
75.10.a.i	$75$	$10$	75.a	1.a	$1$	$4$	$4$	$38.628$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-3\)	\(-324\)	\(0\)	\(9834\)		$+$	$-$	$-$	$2^{2}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}-3^{4}q^{3}+(149-2\beta _{1}+\cdots)q^{4}+\cdots\)
75.10.a.j	$75$	$10$	75.a	1.a	$1$	$4$	$4$	$38.628$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(324\)	\(0\)	\(13036\)		$-$	$+$	$-$	$2^{3}\cdot 3\cdot 5^{2}\cdot 23$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3^{4}q^{3}+(445-6\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
75.10.a.k	$75$	$10$	75.a	1.a	$1$	$4$	$4$	$38.628$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(-324\)	\(0\)	\(-13036\)		$+$	$-$	$-$	$2^{3}\cdot 3\cdot 5^{2}\cdot 23$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3^{4}q^{3}+(445-6\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
75.10.a.l	$75$	$10$	75.a	1.a	$1$	$4$	$4$	$38.628$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(3\)	\(324\)	\(0\)	\(-9834\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+3^{4}q^{3}+(149-2\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(75)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)