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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(19\)	\(4\)	\(15\)		\(17\)	\(4\)	\(13\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(17\)	\(4\)	\(13\)		\(15\)	\(4\)	\(11\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(17\)	\(5\)	\(12\)		\(15\)	\(5\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(17\)	\(6\)	\(11\)		\(15\)	\(6\)	\(9\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(36\)	\(10\)	\(26\)		\(32\)	\(10\)	\(22\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(34\)	\(9\)	\(25\)		\(30\)	\(9\)	\(21\)		\(4\)	\(0\)	\(4\)

Trace form

19 * q + 78 * q^2 + 22608 * q^4 + 3125 * q^5 - 73520 * q^7 + 310044 * q^8 - 368750 * q^10 + 724596 * q^11 - 3418614 * q^13 - 3793308 * q^14 + 33047172 * q^16 - 13284486 * q^17 + 45952116 * q^19 - 5387500 * q^20 - 8965768 * q^22 + 27550560 * q^23 + 185546875 * q^25 + 135820128 * q^26 + 207728080 * q^28 + 180282174 * q^29 - 148347680 * q^31 + 83254860 * q^32 + 1148542844 * q^34 + 184375000 * q^35 - 1297504990 * q^37 - 1854515472 * q^38 - 750637500 * q^40 + 503881914 * q^41 - 1490857572 * q^43 - 1169205276 * q^44 + 651611856 * q^46 + 6358265736 * q^47 + 10121546371 * q^49 + 761718750 * q^50 - 27272164864 * q^52 - 2891432610 * q^53 - 4493112500 * q^55 - 4640496420 * q^56 + 37495652764 * q^58 - 19884833052 * q^59 - 2721679494 * q^61 + 36828060600 * q^62 + 48501772012 * q^64 + 8025043750 * q^65 - 6678219884 * q^67 + 17622880704 * q^68 - 47491875000 * q^70 - 27432232632 * q^71 - 2689533330 * q^73 + 14565069192 * q^74 + 156573327944 * q^76 - 12989927760 * q^77 - 6626266208 * q^79 + 16878200000 * q^80 - 141747893900 * q^82 - 45658105284 * q^83 + 68720893750 * q^85 - 114220723344 * q^86 - 231945452064 * q^88 - 28355740398 * q^89 - 217926356944 * q^91 - 46396214160 * q^92 + 100406698712 * q^94 + 11684987500 * q^95 - 374458850794 * q^97 + 142659039006 * q^98

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	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
45.12.a.a	$45$	$12$	45.a	1.a	$1$	$1$	$1$	$34.575$	\(\Q\)	None	✓				5.12.a.a	$1$	$1$	\(-34\)	\(0\)	\(-3125\)	\(-17556\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-34q^{2}-892q^{4}-5^{5}q^{5}-17556q^{7}+\cdots\)
45.12.a.b	$45$	$12$	45.a	1.a	$1$	$1$	$1$	$34.575$	\(\Q\)	None	✓				15.12.a.a	$1$	$1$	\(56\)	\(0\)	\(-3125\)	\(27984\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+56q^{2}+1088q^{4}-5^{5}q^{5}+27984q^{7}+\cdots\)
45.12.a.c	$45$	$12$	45.a	1.a	$1$	$2$	$2$	$34.575$	\(\Q(\sqrt{1801}) \)	None	✓				15.12.a.c	$1$	$0$	\(13\)	\(0\)	\(6250\)	\(7784\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{2}+(-1549-13\beta )q^{4}+5^{5}q^{5}+\cdots\)
45.12.a.d	$45$	$12$	45.a	1.a	$1$	$2$	$2$	$34.575$	\(\Q(\sqrt{151}) \)	None	✓				5.12.a.b	$1$	$0$	\(20\)	\(0\)	\(6250\)	\(57900\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(10+\beta )q^{2}+(3488+20\beta )q^{4}+5^{5}q^{5}+\cdots\)
45.12.a.e	$45$	$12$	45.a	1.a	$1$	$2$	$2$	$34.575$	\(\Q(\sqrt{1609}) \)	None	✓				15.12.a.b	$1$	$0$	\(22\)	\(0\)	\(6250\)	\(-10864\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(11-\beta )q^{2}+(-318-22\beta )q^{4}+5^{5}q^{5}+\cdots\)
45.12.a.f	$45$	$12$	45.a	1.a	$1$	$3$	$3$	$34.575$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		15.12.a.d	$1$	$1$	\(1\)	\(0\)	\(-9375\)	\(-14608\)		$-$	$+$	$-$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1585+3\beta _{1}+\beta _{2})q^{4}-5^{5}q^{5}+\cdots\)
45.12.a.g	$45$	$12$	45.a	1.a	$1$	$4$	$4$	$34.575$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	45.12.a.g	$1$	$1$	\(-75\)	\(0\)	\(12500\)	\(-62080\)		$+$	$-$	$-$	$2^{3}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-19+\beta _{1})q^{2}+(1807-15\beta _{1}-\beta _{3})q^{4}+\cdots\)
45.12.a.h	$45$	$12$	45.a	1.a	$1$	$4$	$4$	$34.575$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	45.12.a.g	$1$	$0$	\(75\)	\(0\)	\(-12500\)	\(-62080\)		$+$	$+$	$+$	$2^{3}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(19-\beta _{1})q^{2}+(1807-15\beta _{1}-\beta _{3})q^{4}+\cdots\)

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

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