Space of modular forms of level 45, weight 18, and trivial character

• Label: 45.18.a
• Level: $45$
• Weight: $18$
• Character orbit: 45.a
• Rep. character: $\chi_{45}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $8$
• Sturm bound: $108$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(108\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(45))\).

	Total	New	Old
Modular forms	106	29	77
Cusp forms	98	29	69
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
\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(9\)
\(-\)	\(-\)	\(+\)	\(8\)
Plus space		\(+\)	\(14\)
Minus space		\(-\)	\(15\)

Trace form

\( 29 q + 286 q^{2} + 2031680 q^{4} - 390625 q^{5} + 20354440 q^{7} + 204595932 q^{8} + O(q^{10}) \)

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(45))\) 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
45.18.a.a	$45$	$18$	45.a	1.a	$1$	$2$	$2$	$82.450$	\(\Q(\sqrt{39}) \)	None	✓				5.18.a.a	$1$	$1$	\(-680\)	\(0\)	\(781250\)	\(-22820700\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-340+\beta )q^{2}+(35072-680\beta )q^{4}+\cdots\)
45.18.a.b	$45$	$18$	45.a	1.a	$1$	$2$	$2$	$82.450$	\(\Q(\sqrt{849}) \)	None	✓				15.18.a.a	$1$	$0$	\(356\)	\(0\)	\(-781250\)	\(-20754552\)		$-$	$+$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(178-\beta )q^{2}+(175688-356\beta )q^{4}+\cdots\)
45.18.a.c	$45$	$18$	45.a	1.a	$1$	$3$	$3$	$82.450$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				5.18.a.b	$1$	$0$	\(-118\)	\(0\)	\(-1171875\)	\(2139308\)		$-$	$+$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-39+\beta _{1})q^{2}+(5574-14^{2}\beta _{1}+\cdots)q^{4}+\cdots\)
45.18.a.d	$45$	$18$	45.a	1.a	$1$	$3$	$3$	$82.450$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				15.18.a.c	$1$	$1$	\(253\)	\(0\)	\(1171875\)	\(-4332484\)		$-$	$-$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(84+\beta _{1})q^{2}+(-2408+137\beta _{1}+\cdots)q^{4}+\cdots\)
45.18.a.e	$45$	$18$	45.a	1.a	$1$	$3$	$3$	$82.450$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				15.18.a.b	$1$	$1$	\(442\)	\(0\)	\(1171875\)	\(4962644\)		$-$	$-$	$+$	$2^{6}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(147-\beta _{1})q^{2}+(99318-14^{2}\beta _{1}+\cdots)q^{4}+\cdots\)
45.18.a.f	$45$	$18$	45.a	1.a	$1$	$4$	$4$	$82.450$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		15.18.a.d	$1$	$0$	\(33\)	\(0\)	\(-1562500\)	\(17583104\)		$-$	$+$	$-$	$2^{6}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(8+\beta _{1})q^{2}+(109856-65\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
45.18.a.g	$45$	$18$	45.a	1.a	$1$	$6$	$6$	$82.450$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	45.18.a.g	$1$	$1$	\(-665\)	\(0\)	\(-2343750\)	\(21788560\)		$+$	$+$	$+$	$2^{11}\cdot 3^{12}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(-111+\beta _{1})q^{2}+(71921-112\beta _{1}+\cdots)q^{4}+\cdots\)
45.18.a.h	$45$	$18$	45.a	1.a	$1$	$6$	$6$	$82.450$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	45.18.a.g	$1$	$0$	\(665\)	\(0\)	\(2343750\)	\(21788560\)		$+$	$-$	$-$	$2^{11}\cdot 3^{12}\cdot 5^{5}$	$\mathrm{SU}(2)$	\(q+(111-\beta _{1})q^{2}+(71921-112\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(45)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)