Space of modular forms of level 45, weight 6, and trivial character

• Label: 45.6.a
• Level: $45$
• Weight: $6$
• Character orbit: 45.a
• Rep. character: $\chi_{45}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $6$
• Sturm bound: $36$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(36\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(45))\).

	Total	New	Old
Modular forms	34	9	25
Cusp forms	26	9	17
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
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(5\)

Trace form

\( 9 q - 6 q^{2} + 144 q^{4} - 25 q^{5} + 120 q^{7} + 348 q^{8} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(45))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
45.6.a.a	$45$	$6$	45.a	1.a	$1$	$1$	$1$	$7.217$	\(\Q\)	None	✓				15.6.a.b	$1$	$1$	\(-7\)	\(0\)	\(25\)	\(12\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-7q^{2}+17q^{4}+5^{2}q^{5}+12q^{7}+105q^{8}+\cdots\)
45.6.a.b	$45$	$6$	45.a	1.a	$1$	$1$	$1$	$7.217$	\(\Q\)	None	✓				5.6.a.a	$1$	$0$	\(-2\)	\(0\)	\(-25\)	\(192\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-28q^{4}-5^{2}q^{5}+192q^{7}+\cdots\)
45.6.a.c	$45$	$6$	45.a	1.a	$1$	$1$	$1$	$7.217$	\(\Q\)	None	✓				15.6.a.a	$1$	$1$	\(2\)	\(0\)	\(25\)	\(-132\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}+5^{2}q^{5}-132q^{7}+\cdots\)
45.6.a.d	$45$	$6$	45.a	1.a	$1$	$2$	$2$	$7.217$	\(\Q(\sqrt{145}) \)	None	✓	✓	✓	✓	45.6.a.d	$1$	$1$	\(-5\)	\(0\)	\(-50\)	\(80\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+(8+5\beta )q^{4}-5^{2}q^{5}+\cdots\)
45.6.a.e	$45$	$6$	45.a	1.a	$1$	$2$	$2$	$7.217$	\(\Q(\sqrt{409}) \)	None	✓		✓		15.6.a.c	$1$	$0$	\(1\)	\(0\)	\(-50\)	\(-112\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(70+\beta )q^{4}-5^{2}q^{5}+(-2^{6}+\cdots)q^{7}+\cdots\)
45.6.a.f	$45$	$6$	45.a	1.a	$1$	$2$	$2$	$7.217$	\(\Q(\sqrt{145}) \)	None	✓	✓	✓	✓	45.6.a.d	$1$	$0$	\(5\)	\(0\)	\(50\)	\(80\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(13-5\beta )q^{4}+5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

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