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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(5\)	\(2\)	\(3\)		\(3\)	\(2\)	\(1\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(12\)	\(3\)	\(9\)		\(8\)	\(3\)	\(5\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(10\)	\(2\)	\(8\)		\(6\)	\(2\)	\(4\)		\(4\)	\(0\)	\(4\)

Trace form

5 * q + 36 * q^4 + 5 * q^5 - 58 * q^7 + 36 * q^8 - 40 * q^10 - 60 * q^11 + 18 * q^13 - 12 * q^14 + 84 * q^16 - 66 * q^17 - 132 * q^19 + 80 * q^20 + 352 * q^22 + 366 * q^23 + 125 * q^25 - 396 * q^26 - 784 * q^28 - 102 * q^29 + 20 * q^31 - 372 * q^32 - 56 * q^34 + 250 * q^35 + 82 * q^37 + 792 * q^38 - 420 * q^40 - 474 * q^41 + 906 * q^43 + 132 * q^44 + 984 * q^46 + 282 * q^47 + 1097 * q^49 - 1064 * q^52 - 114 * q^53 - 280 * q^55 + 60 * q^56 - 304 * q^58 - 624 * q^59 - 1134 * q^61 - 744 * q^62 - 820 * q^64 + 70 * q^65 - 514 * q^67 - 360 * q^68 + 1200 * q^70 + 204 * q^71 - 2166 * q^73 + 1308 * q^74 - 680 * q^76 + 1536 * q^77 - 1192 * q^79 - 880 * q^80 + 5024 * q^82 - 1650 * q^83 - 370 * q^85 + 1680 * q^86 + 4224 * q^88 - 1206 * q^89 + 1924 * q^91 - 432 * q^92 - 560 * q^94 - 20 * q^95 - 2294 * q^97 - 1632 * q^98

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	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
45.4.a.a	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓	✓	✓	✓	45.4.a.a	$1$	$1$	\(-5\)	\(0\)	\(5\)	\(-30\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}+5q^{5}-30q^{7}-45q^{8}+\cdots\)
45.4.a.b	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓				15.4.a.b	$1$	$0$	\(-3\)	\(0\)	\(5\)	\(20\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+5q^{5}+20q^{7}+21q^{8}+\cdots\)
45.4.a.c	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓		✓	✓	15.4.a.a	$1$	$1$	\(-1\)	\(0\)	\(-5\)	\(-24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}-5q^{5}-24q^{7}+15q^{8}+\cdots\)
45.4.a.d	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓				5.4.a.a	$1$	$0$	\(4\)	\(0\)	\(5\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+8q^{4}+5q^{5}+6q^{7}+20q^{10}+\cdots\)
45.4.a.e	$45$	$4$	45.a	1.a	$1$	$1$	$1$	$2.655$	\(\Q\)	None	✓	✓	✓	✓	45.4.a.a	$1$	$0$	\(5\)	\(0\)	\(-5\)	\(-30\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+17q^{4}-5q^{5}-30q^{7}+45q^{8}+\cdots\)

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

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