Space of modular forms of level 45, weight 14, and trivial character

• Label: 45.14.a
• Level: $45$
• Weight: $14$
• Character orbit: 45.a
• Rep. character: $\chi_{45}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $8$
• Sturm bound: $84$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(84\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{14}(\Gamma_0(45))\).

	Total	New	Old
Modular forms	82	21	61
Cusp forms	74	21	53
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
\(+\)	\(+\)	\(+\)		\(19\)	\(4\)	\(15\)		\(17\)	\(4\)	\(13\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(21\)	\(4\)	\(17\)		\(19\)	\(4\)	\(15\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(21\)	\(7\)	\(14\)		\(19\)	\(7\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(21\)	\(6\)	\(15\)		\(19\)	\(6\)	\(13\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(40\)	\(10\)	\(30\)		\(36\)	\(10\)	\(26\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(42\)	\(11\)	\(31\)		\(38\)	\(11\)	\(27\)		\(4\)	\(0\)	\(4\)

Trace form

21 * q - 66 * q^2 + 86560 * q^4 - 15625 * q^5 + 262960 * q^7 - 1781412 * q^8 - 1468750 * q^10 - 8105916 * q^11 + 39412382 * q^13 - 140132652 * q^14 + 321998020 * q^16 + 56186262 * q^17 - 277092508 * q^19 - 148187500 * q^20 - 223797464 * q^22 + 182002080 * q^23 + 5126953125 * q^25 + 3239232384 * q^26 - 9155070560 * q^28 - 4760960790 * q^29 + 4476297760 * q^31 - 13821111540 * q^32 + 8959809532 * q^34 + 1928125000 * q^35 - 15017412810 * q^37 + 60406988736 * q^38 + 1697062500 * q^40 - 1280518026 * q^41 - 84115483204 * q^43 + 282194374308 * q^44 - 35940185664 * q^46 - 186031851672 * q^47 + 204917668805 * q^49 - 16113281250 * q^50 + 347760438528 * q^52 - 519500740950 * q^53 + 286083687500 * q^55 - 206467685220 * q^56 - 480238047412 * q^58 - 50057800620 * q^59 + 1319661621422 * q^61 - 905098799400 * q^62 - 594124877748 * q^64 - 418476593750 * q^65 + 608452361428 * q^67 + 5383951613568 * q^68 + 533750625000 * q^70 + 519945988104 * q^71 - 892135964110 * q^73 + 7232962388808 * q^74 - 2921400999864 * q^76 - 6185471727120 * q^77 - 680946529952 * q^79 - 6946366000000 * q^80 + 5621487313220 * q^82 - 855661435428 * q^83 - 1523405968750 * q^85 - 3714377487936 * q^86 + 18374441571552 * q^88 + 20984449605390 * q^89 + 6306621274256 * q^91 - 20630216793120 * q^92 - 52383245759768 * q^94 - 8138331312500 * q^95 + 29907459575898 * q^97 + 11966480015742 * q^98

Decomposition of \(S_{14}^{\mathrm{new}}(\Gamma_0(45))\) 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
45.14.a.a	$45$	$14$	45.a	1.a	$1$	$1$	$1$	$48.254$	\(\Q\)	None	✓				15.14.a.a	$1$	$0$	\(76\)	\(0\)	\(-15625\)	\(-224160\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+76q^{2}-2416q^{4}-5^{6}q^{5}-224160q^{7}+\cdots\)
45.14.a.b	$45$	$14$	45.a	1.a	$1$	$2$	$2$	$48.254$	\(\Q(\sqrt{3121}) \)	None	✓				15.14.a.c	$1$	$1$	\(-131\)	\(0\)	\(31250\)	\(496272\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(-65-\beta )q^{2}+(3055+131\beta )q^{4}+\cdots\)
45.14.a.c	$45$	$14$	45.a	1.a	$1$	$2$	$2$	$48.254$	\(\Q(\sqrt{1609}) \)	None	✓				15.14.a.b	$1$	$1$	\(-14\)	\(0\)	\(31250\)	\(-29832\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{2}+(6338+14\beta )q^{4}+5^{6}q^{5}+\cdots\)
45.14.a.d	$45$	$14$	45.a	1.a	$1$	$2$	$2$	$48.254$	\(\Q(\sqrt{499}) \)	None	✓				5.14.a.a	$1$	$1$	\(80\)	\(0\)	\(31250\)	\(-616300\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(40+\beta )q^{2}+(1392+80\beta )q^{4}+5^{6}q^{5}+\cdots\)
45.14.a.e	$45$	$14$	45.a	1.a	$1$	$3$	$3$	$48.254$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				5.14.a.b	$1$	$0$	\(-142\)	\(0\)	\(-46875\)	\(448292\)		$-$	$+$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-47+\beta _{1})q^{2}+(5932-82\beta _{1}+2\beta _{2})q^{4}+\cdots\)
45.14.a.f	$45$	$14$	45.a	1.a	$1$	$3$	$3$	$48.254$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				15.14.a.d	$1$	$0$	\(65\)	\(0\)	\(-46875\)	\(-497392\)		$-$	$+$	$-$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(22-\beta _{1})q^{2}+(5263-2^{6}\beta _{1}-7\beta _{2})q^{4}+\cdots\)
45.14.a.g	$45$	$14$	45.a	1.a	$1$	$4$	$4$	$48.254$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	45.14.a.g	$1$	$0$	\(-15\)	\(0\)	\(62500\)	\(343040\)		$+$	$-$	$-$	$2^{3}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{2}+(4210-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
45.14.a.h	$45$	$14$	45.a	1.a	$1$	$4$	$4$	$48.254$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	45.14.a.g	$1$	$1$	\(15\)	\(0\)	\(-62500\)	\(343040\)		$+$	$+$	$+$	$2^{3}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(4210-2\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{14}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

\( S_{14}^{\mathrm{old}}(\Gamma_0(45)) \simeq \) \(S_{14}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)