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

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