Space of modular forms of level 495, weight 4, and trivial character

• Label: 495.4.a
• Level: $495$
• Weight: $4$
• Character orbit: 495.a
• Rep. character: $\chi_{495}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $16$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(495))\).

	Total	New	Old
Modular forms	224	50	174
Cusp forms	208	50	158
Eisenstein series	16	0	16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(32\)	\(7\)	\(25\)		\(30\)	\(7\)	\(23\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(24\)	\(3\)	\(21\)		\(22\)	\(3\)	\(19\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(28\)	\(3\)	\(25\)		\(26\)	\(3\)	\(23\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(28\)	\(7\)	\(21\)		\(26\)	\(7\)	\(19\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(26\)	\(7\)	\(19\)		\(24\)	\(7\)	\(17\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(30\)	\(9\)	\(21\)		\(28\)	\(9\)	\(19\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(30\)	\(8\)	\(22\)		\(28\)	\(8\)	\(20\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(26\)	\(6\)	\(20\)		\(24\)	\(6\)	\(18\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(120\)	\(31\)	\(89\)		\(112\)	\(31\)	\(81\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(104\)	\(19\)	\(85\)		\(96\)	\(19\)	\(77\)		\(8\)	\(0\)	\(8\)

Trace form

50 * q + 188 * q^4 - 10 * q^5 + 88 * q^7 + 12 * q^8 + 60 * q^10 - 104 * q^13 + 40 * q^14 + 864 * q^16 + 64 * q^17 + 152 * q^19 - 140 * q^20 + 44 * q^22 - 208 * q^23 + 1250 * q^25 + 1108 * q^26 + 1500 * q^28 + 116 * q^29 + 636 * q^31 + 1216 * q^32 + 184 * q^34 + 40 * q^35 + 84 * q^37 - 1088 * q^38 + 600 * q^40 + 484 * q^41 - 712 * q^43 - 308 * q^44 - 128 * q^46 - 1440 * q^47 + 1454 * q^49 + 1036 * q^52 - 1300 * q^53 + 220 * q^55 + 960 * q^56 + 2880 * q^58 + 2584 * q^59 + 2004 * q^61 + 3896 * q^62 + 5116 * q^64 + 160 * q^65 + 2856 * q^67 + 3708 * q^68 - 700 * q^70 + 628 * q^71 - 128 * q^73 + 352 * q^74 + 2224 * q^76 - 792 * q^77 + 256 * q^79 - 400 * q^80 - 3728 * q^82 + 2344 * q^83 - 320 * q^85 - 4084 * q^86 + 528 * q^88 + 312 * q^89 + 952 * q^91 + 3200 * q^92 - 2280 * q^94 + 1560 * q^95 + 3268 * q^97 - 7184 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(495))\) 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	11
495.4.a.a	$495$	$4$	495.a	1.a	$1$	$1$	$1$	$29.206$	\(\Q\)	None	✓				55.4.a.a	$1$	$0$	\(-1\)	\(0\)	\(5\)	\(-9\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}+5q^{5}-9q^{7}+15q^{8}+\cdots\)
495.4.a.b	$495$	$4$	495.a	1.a	$1$	$1$	$1$	$29.206$	\(\Q\)	None	✓				165.4.a.b	$1$	$0$	\(-1\)	\(0\)	\(5\)	\(36\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}+5q^{5}+6^{2}q^{7}+15q^{8}+\cdots\)
495.4.a.c	$495$	$4$	495.a	1.a	$1$	$1$	$1$	$29.206$	\(\Q\)	None	✓				165.4.a.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{4}+5q^{5}+2q^{7}+11q^{11}-22q^{13}+\cdots\)
495.4.a.d	$495$	$4$	495.a	1.a	$1$	$2$	$2$	$29.206$	\(\Q(\sqrt{17}) \)	None	✓				165.4.a.c	$1$	$1$	\(1\)	\(0\)	\(10\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-4+\beta )q^{4}+5q^{5}+(-4+\cdots)q^{7}+\cdots\)
495.4.a.e	$495$	$4$	495.a	1.a	$1$	$2$	$2$	$29.206$	\(\Q(\sqrt{17}) \)	None	✓				55.4.a.b	$1$	$0$	\(7\)	\(0\)	\(-10\)	\(-25\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(4-\beta )q^{2}+(12-7\beta )q^{4}-5q^{5}+(-8+\cdots)q^{7}+\cdots\)
495.4.a.f	$495$	$4$	495.a	1.a	$1$	$3$	$3$	$29.206$	3.3.568.1	None	✓		✓		55.4.a.c	$1$	$1$	\(-5\)	\(0\)	\(15\)	\(-15\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}+\beta _{2})q^{2}+(7+2\beta _{1})q^{4}+\cdots\)
495.4.a.g	$495$	$4$	495.a	1.a	$1$	$3$	$3$	$29.206$	3.3.788.1	None	✓				165.4.a.f	$1$	$0$	\(-1\)	\(0\)	\(-15\)	\(-16\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(-2-\beta _{1})q^{4}-5q^{5}+(-3+\cdots)q^{7}+\cdots\)
495.4.a.h	$495$	$4$	495.a	1.a	$1$	$3$	$3$	$29.206$	3.3.1772.1	None	✓	✓	✓	✓	495.4.a.h	$1$	$1$	\(-1\)	\(0\)	\(-15\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+2\beta _{2})q^{4}-5q^{5}+(2+\cdots)q^{7}+\cdots\)
495.4.a.i	$495$	$4$	495.a	1.a	$1$	$3$	$3$	$29.206$	3.3.1957.1	None	✓				165.4.a.g	$1$	$1$	\(-1\)	\(0\)	\(-15\)	\(6\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{1}+2\beta _{2})q^{4}-5q^{5}+\cdots\)
495.4.a.j	$495$	$4$	495.a	1.a	$1$	$3$	$3$	$29.206$	3.3.1772.1	None	✓	✓	✓	✓	495.4.a.h	$1$	$1$	\(1\)	\(0\)	\(15\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+2\beta _{2})q^{4}+5q^{5}+(2+\cdots)q^{7}+\cdots\)
495.4.a.k	$495$	$4$	495.a	1.a	$1$	$3$	$3$	$29.206$	3.3.47528.1	None	✓				165.4.a.e	$1$	$0$	\(2\)	\(0\)	\(15\)	\(10\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(10+\beta _{2})q^{4}+5q^{5}+\cdots\)
495.4.a.l	$495$	$4$	495.a	1.a	$1$	$3$	$3$	$29.206$	3.3.23612.1	None	✓				165.4.a.d	$1$	$0$	\(4\)	\(0\)	\(15\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)
495.4.a.m	$495$	$4$	495.a	1.a	$1$	$4$	$4$	$29.206$	4.4.1540841.1	None	✓		✓		165.4.a.h	$1$	$0$	\(-4\)	\(0\)	\(-20\)	\(34\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(7-\beta _{1}+\beta _{3})q^{4}+\cdots\)
495.4.a.n	$495$	$4$	495.a	1.a	$1$	$4$	$4$	$29.206$	4.4.1539480.1	None	✓		✓		55.4.a.d	$1$	$1$	\(-1\)	\(0\)	\(-20\)	\(9\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(5+\beta _{1}+\beta _{3})q^{4}-5q^{5}+\cdots\)
495.4.a.o	$495$	$4$	495.a	1.a	$1$	$7$	$7$	$29.206$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	495.4.a.o	$1$	$0$	\(-5\)	\(0\)	\(-35\)	\(30\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(5-\beta _{1}+\beta _{2})q^{4}+\cdots\)
495.4.a.p	$495$	$4$	495.a	1.a	$1$	$7$	$7$	$29.206$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	495.4.a.o	$1$	$0$	\(5\)	\(0\)	\(35\)	\(30\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(5-\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(495))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(495)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 2}\)