Space of modular forms of level 45, weight 4, and trivial character

• Label: 45.4.a
• Level: $45$
• Weight: $4$
• Character orbit: 45.a
• Rep. character: $\chi_{45}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $24$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(24\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(45))\).

	Total	New	Old
Modular forms	22	5	17
Cusp forms	14	5	9
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
\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(-\)	$-$	\(1\)
\(-\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	$+$	\(2\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(2\)

Trace form

\( 5 q + 36 q^{4} + 5 q^{5} - 58 q^{7} + 36 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(45))\) 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
45.4.a.a	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓	$1$	$1$	\(-5\)	\(0\)	\(5\)	\(-30\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}+5q^{5}-30q^{7}-45q^{8}+\cdots\)
45.4.a.b	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓	$1$	$0$	\(-3\)	\(0\)	\(5\)	\(20\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+5q^{5}+20q^{7}+21q^{8}+\cdots\)
45.4.a.c	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-5\)	\(-24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}-5q^{5}-24q^{7}+15q^{8}+\cdots\)
45.4.a.d	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(5\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+8q^{4}+5q^{5}+6q^{7}+20q^{10}+\cdots\)
45.4.a.e	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓	$1$	$0$	\(5\)	\(0\)	\(-5\)	\(-30\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+17q^{4}-5q^{5}-30q^{7}+45q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

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