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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(7\)	\(2\)	\(5\)		\(5\)	\(2\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(9\)	\(3\)	\(6\)		\(7\)	\(3\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(16\)	\(4\)	\(12\)		\(12\)	\(4\)	\(8\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(18\)	\(5\)	\(13\)		\(14\)	\(5\)	\(9\)		\(4\)	\(0\)	\(4\)

Trace form

9 * q - 6 * q^2 + 144 * q^4 - 25 * q^5 + 120 * q^7 + 348 * q^8 + 150 * q^10 - 684 * q^11 - 738 * q^13 + 2484 * q^14 + 2244 * q^16 - 978 * q^17 - 1260 * q^19 - 3100 * q^20 - 6024 * q^22 - 3240 * q^23 + 5625 * q^25 + 2040 * q^26 + 10080 * q^28 + 5442 * q^29 - 9840 * q^31 + 21900 * q^32 + 2724 * q^34 - 5000 * q^35 - 9930 * q^37 - 58224 * q^38 + 3300 * q^40 + 27822 * q^41 + 1596 * q^43 - 16860 * q^44 - 59952 * q^46 + 13248 * q^47 + 26193 * q^49 - 3750 * q^50 - 7392 * q^52 - 43590 * q^53 + 27900 * q^55 + 154140 * q^56 + 141828 * q^58 + 11844 * q^59 - 69666 * q^61 + 24600 * q^62 - 57012 * q^64 - 70550 * q^65 + 41268 * q^67 - 287712 * q^68 + 3000 * q^70 + 178680 * q^71 + 218490 * q^73 + 125184 * q^74 - 147576 * q^76 - 1440 * q^77 - 23136 * q^79 - 121600 * q^80 - 251220 * q^82 - 37668 * q^83 + 90450 * q^85 + 112896 * q^86 + 7392 * q^88 - 81234 * q^89 - 209232 * q^91 + 252960 * q^92 - 184872 * q^94 - 82100 * q^95 - 13902 * q^97 - 276198 * q^98

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

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