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}$