Space of modular forms of level 45, weight 10, and trivial character

• Label: 45.10.a
• Level: $45$
• Weight: $10$
• Character orbit: 45.a
• Rep. character: $\chi_{45}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $8$
• Sturm bound: $60$
• Trace bound: $2$

Defining parameters

Level:	$N$	$=$	$45 = 3^{2} \cdot 5$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	45.a (trivial)
Character field:			$\Q$
Newform subspaces:			$8$
Sturm bound:			$60$
Trace bound:			$2$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{10}(\Gamma_0(45))$.

	Total	New	Old
Modular forms	58	15	43
Cusp forms	50	15	35
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	Dim
$+$	$+$	$+$	$3$
$+$	$-$	$-$	$3$
$-$	$+$	$-$	$5$
$-$	$-$	$+$	$4$
Plus space		$+$	$7$
Minus space		$-$	$8$

Trace form

15 * q - 50 * q^2 + 4352 * q^4 - 625 * q^5 + 2294 * q^7 - 55332 * q^8 + 21250 * q^10 + 109300 * q^11 + 150454 * q^13 + 358836 * q^14 + 1300292 * q^16 - 487142 * q^17 + 128716 * q^19 - 27500 * q^20 + 93704 * q^22 - 1540722 * q^23 + 5859375 * q^25 + 8268832 * q^26 - 1090432 * q^28 + 6467126 * q^29 - 16970860 * q^31 - 13791604 * q^32 - 9891172 * q^34 - 13593750 * q^35 + 32082054 * q^37 - 1435424 * q^38 + 3742500 * q^40 - 8373622 * q^41 + 3264058 * q^43 + 107990564 * q^44 - 31432608 * q^46 - 73981286 * q^47 - 92746061 * q^49 - 19531250 * q^50 + 381376512 * q^52 + 113358778 * q^53 + 29335000 * q^55 + 79453980 * q^56 - 258448052 * q^58 - 40761248 * q^59 + 148618438 * q^61 - 299208168 * q^62 + 339796620 * q^64 + 47701250 * q^65 - 1051294018 * q^67 - 451809280 * q^68 + 299625000 * q^70 + 113955212 * q^71 - 191749778 * q^73 + 134210216 * q^74 - 1156422840 * q^76 + 924082512 * q^77 + 522407096 * q^79 + 196960000 * q^80 - 683209532 * q^82 - 2468939250 * q^83 - 480473750 * q^85 + 913243360 * q^86 + 935247072 * q^88 - 480389682 * q^89 - 1203775868 * q^91 + 3507666816 * q^92 + 4766801480 * q^94 - 808802500 * q^95 - 323550738 * q^97 - 5314785106 * q^98

Decomposition of $S_{10}^{\mathrm{new}}(\Gamma_0(45))$ 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
45.10.a.a	$45$	$10$	45.a	1.a	$1$	$1$	$1$	$23.177$	$\Q$	None	✓				15.10.a.b	$1$	$1$	$-22$	$0$	$625$	$-5988$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-22q^{2}-28q^{4}+5^{4}q^{5}-5988q^{7}+\cdots$
45.10.a.b	$45$	$10$	45.a	1.a	$1$	$1$	$1$	$23.177$	$\Q$	None	✓				15.10.a.a	$1$	$0$	$4$	$0$	$-625$	$-7680$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+4q^{2}-496q^{4}-5^{4}q^{5}-7680q^{7}+\cdots$
45.10.a.c	$45$	$10$	45.a	1.a	$1$	$1$	$1$	$23.177$	$\Q$	None	✓				5.10.a.a	$1$	$1$	$8$	$0$	$625$	$4242$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}-448q^{4}+5^{4}q^{5}+4242q^{7}+\cdots$
45.10.a.d	$45$	$10$	45.a	1.a	$1$	$2$	$2$	$23.177$	$\Q(\sqrt{241})$	None	✓				15.10.a.d	$1$	$0$	$-31$	$0$	$-1250$	$14112$		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	$q+(-2^{4}-\beta )q^{2}+(286+31\beta )q^{4}-5^{4}q^{5}+\cdots$
45.10.a.e	$45$	$10$	45.a	1.a	$1$	$2$	$2$	$23.177$	$\Q(\sqrt{4729})$	None	✓		✓		15.10.a.c	$1$	$1$	$-19$	$0$	$1250$	$-11872$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-9-\beta )q^{2}+(751+19\beta )q^{4}+5^{4}q^{5}+\cdots$
45.10.a.f	$45$	$10$	45.a	1.a	$1$	$2$	$2$	$23.177$	$\Q(\sqrt{1009})$	None	✓				5.10.a.b	$1$	$0$	$10$	$0$	$-1250$	$1700$		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+(5-\beta )q^{2}+(522-10\beta )q^{4}-5^{4}q^{5}+\cdots$
45.10.a.g	$45$	$10$	45.a	1.a	$1$	$3$	$3$	$23.177$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓	✓	✓	45.10.a.g	$1$	$1$	$-25$	$0$	$-1875$	$3890$		$+$	$+$	$+$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	$q+(-8+\beta _{1})q^{2}+(366-10\beta _{1}-\beta _{2})q^{4}+\cdots$
45.10.a.h	$45$	$10$	45.a	1.a	$1$	$3$	$3$	$23.177$	$\mathbb{Q}[x]/(x^{3} - \cdots)$	None	✓	✓	✓	✓	45.10.a.g	$1$	$0$	$25$	$0$	$1875$	$3890$		$+$	$-$	$-$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	$q+(8-\beta _{1})q^{2}+(366-10\beta _{1}-\beta _{2})q^{4}+\cdots$

Decomposition of $S_{10}^{\mathrm{old}}(\Gamma_0(45))$ into lower level spaces

$S_{10}^{\mathrm{old}}(\Gamma_0(45)) \simeq$ $S_{10}^{\mathrm{new}}(\Gamma_0(3))$$^{\oplus 4}$$\oplus$$S_{10}^{\mathrm{new}}(\Gamma_0(5))$$^{\oplus 3}$$\oplus$$S_{10}^{\mathrm{new}}(\Gamma_0(9))$$^{\oplus 2}$$\oplus$$S_{10}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 2}$