Space of modular forms of level 45, weight 12, and trivial character

• Label: 45.12.a
• Level: $45$
• Weight: $12$
• Character orbit: 45.a
• Rep. character: $\chi_{45}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $8$
• Sturm bound: $72$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(72\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(45))\).

	Total	New	Old
Modular forms	70	19	51
Cusp forms	62	19	43
Eisenstein series	8	0	8

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
\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(5\)
\(-\)	\(-\)	$+$	\(6\)
Plus space		\(+\)	\(10\)
Minus space		\(-\)	\(9\)

Trace form

\( 19 q + 78 q^{2} + 22608 q^{4} + 3125 q^{5} - 73520 q^{7} + 310044 q^{8} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(45))\) 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
45.12.a.a	$45$	$12$	45.a	1.a	$1$	$1$	$1$	$34.575$	\(\Q\)	None	✓	$1$	$1$	\(-34\)	\(0\)	\(-3125\)	\(-17556\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-34q^{2}-892q^{4}-5^{5}q^{5}-17556q^{7}+\cdots\)
45.12.a.b	$45$	$12$	45.a	1.a	$1$	$1$	$1$	$34.575$	\(\Q\)	None	✓	$1$	$1$	\(56\)	\(0\)	\(-3125\)	\(27984\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+56q^{2}+1088q^{4}-5^{5}q^{5}+27984q^{7}+\cdots\)
45.12.a.c	$45$	$12$	45.a	1.a	$1$	$2$	$2$	$34.575$	\(\Q(\sqrt{1801}) \)	None	✓	$1$	$0$	\(13\)	\(0\)	\(6250\)	\(7784\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{2}+(-1549-13\beta )q^{4}+5^{5}q^{5}+\cdots\)
45.12.a.d	$45$	$12$	45.a	1.a	$1$	$2$	$2$	$34.575$	\(\Q(\sqrt{151}) \)	None	✓	$1$	$0$	\(20\)	\(0\)	\(6250\)	\(57900\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(10+\beta )q^{2}+(3488+20\beta )q^{4}+5^{5}q^{5}+\cdots\)
45.12.a.e	$45$	$12$	45.a	1.a	$1$	$2$	$2$	$34.575$	\(\Q(\sqrt{1609}) \)	None	✓	$1$	$0$	\(22\)	\(0\)	\(6250\)	\(-10864\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(11-\beta )q^{2}+(-318-22\beta )q^{4}+5^{5}q^{5}+\cdots\)
45.12.a.f	$45$	$12$	45.a	1.a	$1$	$3$	$3$	$34.575$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-9375\)	\(-14608\)		$-$	$+$	$-$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1585+3\beta _{1}+\beta _{2})q^{4}-5^{5}q^{5}+\cdots\)
45.12.a.g	$45$	$12$	45.a	1.a	$1$	$4$	$4$	$34.575$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-75\)	\(0\)	\(12500\)	\(-62080\)		$+$	$-$	$-$	$2^{3}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-19+\beta _{1})q^{2}+(1807-15\beta _{1}-\beta _{3})q^{4}+\cdots\)
45.12.a.h	$45$	$12$	45.a	1.a	$1$	$4$	$4$	$34.575$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(75\)	\(0\)	\(-12500\)	\(-62080\)		$+$	$+$	$+$	$2^{3}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(19-\beta _{1})q^{2}+(1807-15\beta _{1}-\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(45)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)