Space of modular forms of level 45, weight 8, and trivial character

• Label: 45.8.a
• Level: $45$
• Weight: $8$
• Character orbit: 45.a
• Rep. character: $\chi_{45}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $9$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(45))\).

	Total	New	Old
Modular forms	46	11	35
Cusp forms	38	11	27
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
\(+\)	\(+\)	\(+\)		\(13\)	\(2\)	\(11\)		\(11\)	\(2\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(11\)	\(2\)	\(9\)		\(9\)	\(2\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(11\)	\(3\)	\(8\)		\(9\)	\(3\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(11\)	\(4\)	\(7\)		\(9\)	\(4\)	\(5\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(24\)	\(6\)	\(18\)		\(20\)	\(6\)	\(14\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(22\)	\(5\)	\(17\)		\(18\)	\(5\)	\(13\)		\(4\)	\(0\)	\(4\)

Trace form

11 * q + 22 * q^2 + 368 * q^4 + 125 * q^5 + 40 * q^7 + 156 * q^8 + 2250 * q^10 - 1916 * q^11 - 14006 * q^13 - 9564 * q^14 + 11012 * q^16 + 71626 * q^17 - 52028 * q^19 + 44500 * q^20 + 221688 * q^22 - 58920 * q^23 + 171875 * q^25 - 236624 * q^26 - 238160 * q^28 - 323746 * q^29 + 316720 * q^31 + 963020 * q^32 - 753300 * q^34 + 150000 * q^35 + 395650 * q^37 + 67312 * q^38 + 490500 * q^40 - 1107838 * q^41 - 1428428 * q^43 + 594788 * q^44 - 196752 * q^46 - 708176 * q^47 + 2466411 * q^49 + 343750 * q^50 - 4217216 * q^52 + 2096590 * q^53 - 784500 * q^55 - 5341860 * q^56 - 1187124 * q^58 + 4746148 * q^59 + 3686650 * q^61 - 10413000 * q^62 + 9878828 * q^64 + 770750 * q^65 - 1709156 * q^67 + 15179456 * q^68 - 2595000 * q^70 - 3099784 * q^71 - 11947970 * q^73 + 1932056 * q^74 - 6496184 * q^76 - 9628080 * q^77 + 7145536 * q^79 + 13768000 * q^80 + 11162820 * q^82 + 20973444 * q^83 - 5648250 * q^85 - 31411856 * q^86 + 6073824 * q^88 - 20468118 * q^89 + 22663328 * q^91 - 9162480 * q^92 - 27872712 * q^94 + 16269500 * q^95 + 47226214 * q^97 + 60062534 * q^98

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(45))\) 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
45.8.a.a	$45$	$8$	45.a	1.a	$1$	$1$	$1$	$14.057$	\(\Q\)	None	✓	✓			45.8.a.a	$1$	$0$	\(-10\)	\(0\)	\(-125\)	\(-1170\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}-28q^{4}-5^{3}q^{5}-1170q^{7}+\cdots\)
45.8.a.b	$45$	$8$	45.a	1.a	$1$	$1$	$1$	$14.057$	\(\Q\)	None	✓	✓			45.8.a.b	$1$	$1$	\(-5\)	\(0\)	\(125\)	\(930\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-103q^{4}+5^{3}q^{5}+930q^{7}+\cdots\)
45.8.a.c	$45$	$8$	45.a	1.a	$1$	$1$	$1$	$14.057$	\(\Q\)	None	✓	✓			45.8.a.b	$1$	$0$	\(5\)	\(0\)	\(-125\)	\(930\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}-103q^{4}-5^{3}q^{5}+930q^{7}+\cdots\)
45.8.a.d	$45$	$8$	45.a	1.a	$1$	$1$	$1$	$14.057$	\(\Q\)	None	✓	✓			45.8.a.a	$1$	$1$	\(10\)	\(0\)	\(125\)	\(-1170\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+10q^{2}-28q^{4}+5^{3}q^{5}-1170q^{7}+\cdots\)
45.8.a.e	$45$	$8$	45.a	1.a	$1$	$1$	$1$	$14.057$	\(\Q\)	None	✓				15.8.a.b	$1$	$0$	\(13\)	\(0\)	\(125\)	\(1380\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+13q^{2}+41q^{4}+5^{3}q^{5}+1380q^{7}+\cdots\)
45.8.a.f	$45$	$8$	45.a	1.a	$1$	$1$	$1$	$14.057$	\(\Q\)	None	✓				5.8.a.a	$1$	$1$	\(14\)	\(0\)	\(-125\)	\(-1644\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+14q^{2}+68q^{4}-5^{3}q^{5}-1644q^{7}+\cdots\)
45.8.a.g	$45$	$8$	45.a	1.a	$1$	$1$	$1$	$14.057$	\(\Q\)	None	✓				15.8.a.a	$1$	$0$	\(22\)	\(0\)	\(125\)	\(-420\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+22q^{2}+356q^{4}+5^{3}q^{5}-420q^{7}+\cdots\)
45.8.a.h	$45$	$8$	45.a	1.a	$1$	$2$	$2$	$14.057$	\(\Q(\sqrt{19}) \)	None	✓		✓		5.8.a.b	$1$	$0$	\(-20\)	\(0\)	\(250\)	\(-100\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-10+\beta )q^{2}+(48-20\beta )q^{4}+5^{3}q^{5}+\cdots\)
45.8.a.i	$45$	$8$	45.a	1.a	$1$	$2$	$2$	$14.057$	\(\Q(\sqrt{601}) \)	None	✓		✓		15.8.a.c	$1$	$1$	\(-7\)	\(0\)	\(-250\)	\(1304\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+(31+7\beta )q^{4}-5^{3}q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(45)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)