Space of modular forms of level 495, weight 6, and trivial character

• Label: 495.6.a
• Level: $495$
• Weight: $6$
• Character orbit: 495.a
• Rep. character: $\chi_{495}(1,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $16$
• Sturm bound: $432$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(432\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(495))\).

	Total	New	Old
Modular forms	368	82	286
Cusp forms	352	82	270
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
\(+\)	\(+\)	\(+\)	\(+\)		\(42\)	\(6\)	\(36\)		\(40\)	\(6\)	\(34\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(50\)	\(10\)	\(40\)		\(48\)	\(10\)	\(38\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(46\)	\(10\)	\(36\)		\(44\)	\(10\)	\(34\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(46\)	\(6\)	\(40\)		\(44\)	\(6\)	\(38\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(48\)	\(13\)	\(35\)		\(46\)	\(13\)	\(33\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(44\)	\(11\)	\(33\)		\(42\)	\(11\)	\(31\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(44\)	\(12\)	\(32\)		\(42\)	\(12\)	\(30\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(48\)	\(14\)	\(34\)		\(46\)	\(14\)	\(32\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(176\)	\(35\)	\(141\)		\(168\)	\(35\)	\(133\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(192\)	\(47\)	\(145\)		\(184\)	\(47\)	\(137\)		\(8\)	\(0\)	\(8\)

Trace form

82 * q + 1268 * q^4 + 50 * q^5 - 44 * q^7 - 852 * q^8 - 100 * q^10 + 1748 * q^13 - 2720 * q^14 + 20160 * q^16 - 956 * q^17 + 2728 * q^19 + 4300 * q^20 - 968 * q^22 + 7556 * q^23 + 51250 * q^25 + 3292 * q^26 - 1028 * q^28 - 10876 * q^29 + 3480 * q^31 - 16064 * q^32 + 13672 * q^34 + 3100 * q^35 + 6020 * q^37 + 61120 * q^38 + 3000 * q^40 + 25852 * q^41 + 47628 * q^43 - 2420 * q^44 + 93512 * q^46 + 9372 * q^47 + 194210 * q^49 + 8972 * q^52 + 95708 * q^53 - 12100 * q^55 - 211872 * q^56 - 140352 * q^58 - 91808 * q^59 + 137852 * q^61 + 60392 * q^62 + 374844 * q^64 + 100 * q^65 - 9564 * q^67 + 231228 * q^68 + 20500 * q^70 - 79232 * q^71 - 93772 * q^73 + 68752 * q^74 + 384528 * q^76 + 76956 * q^77 + 92256 * q^79 + 134000 * q^80 + 392144 * q^82 - 178292 * q^83 - 134300 * q^85 + 522620 * q^86 - 46464 * q^88 - 373284 * q^89 + 90656 * q^91 - 220912 * q^92 + 668768 * q^94 + 19800 * q^95 - 420796 * q^97 + 481168 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(495))\) 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
495.6.a.a	$495$	$6$	495.a	1.a	$1$	$3$	$3$	$79.390$	3.3.307532.1	None	✓				165.6.a.e	$1$	$1$	\(-7\)	\(0\)	\(75\)	\(92\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(24+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
495.6.a.b	$495$	$6$	495.a	1.a	$1$	$3$	$3$	$79.390$	3.3.18257.1	None	✓				165.6.a.c	$1$	$0$	\(-2\)	\(0\)	\(-75\)	\(-68\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-14-\beta _{1}+\beta _{2})q^{4}+\cdots\)
495.6.a.c	$495$	$6$	495.a	1.a	$1$	$3$	$3$	$79.390$	3.3.788.1	None	✓				165.6.a.d	$1$	$1$	\(-2\)	\(0\)	\(-75\)	\(152\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(-8-4\beta _{1})q^{4}-5^{2}q^{5}+\cdots\)
495.6.a.d	$495$	$6$	495.a	1.a	$1$	$3$	$3$	$79.390$	3.3.3368.1	None	✓				165.6.a.b	$1$	$1$	\(2\)	\(0\)	\(75\)	\(-232\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(8+4\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)
495.6.a.e	$495$	$6$	495.a	1.a	$1$	$3$	$3$	$79.390$	3.3.34253.1	None	✓				165.6.a.a	$1$	$1$	\(7\)	\(0\)	\(-75\)	\(-172\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(7+4\beta _{1}+\beta _{2})q^{4}-5^{2}q^{5}+\cdots\)
495.6.a.f	$495$	$6$	495.a	1.a	$1$	$3$	$3$	$79.390$	3.3.21865.1	None	✓				55.6.a.a	$1$	$0$	\(7\)	\(0\)	\(-75\)	\(-102\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{2}+(12+4\beta _{1}+\beta _{2})q^{4}+\cdots\)
495.6.a.g	$495$	$6$	495.a	1.a	$1$	$4$	$4$	$79.390$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				55.6.a.b	$1$	$0$	\(5\)	\(0\)	\(100\)	\(-90\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(15+\beta _{1}-\beta _{2}+\beta _{3})q^{4}+\cdots\)
495.6.a.h	$495$	$6$	495.a	1.a	$1$	$5$	$5$	$79.390$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		55.6.a.c	$1$	$1$	\(-9\)	\(0\)	\(-125\)	\(70\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(24-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
495.6.a.i	$495$	$6$	495.a	1.a	$1$	$5$	$5$	$79.390$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				165.6.a.g	$1$	$0$	\(1\)	\(0\)	\(125\)	\(116\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5^{2}+\beta _{1}+\beta _{3})q^{4}+5^{2}q^{5}+\cdots\)
495.6.a.j	$495$	$6$	495.a	1.a	$1$	$5$	$5$	$79.390$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				165.6.a.f	$1$	$0$	\(2\)	\(0\)	\(125\)	\(184\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2^{4}+\beta _{2})q^{4}+5^{2}q^{5}+(37+\cdots)q^{7}+\cdots\)
495.6.a.k	$495$	$6$	495.a	1.a	$1$	$6$	$6$	$79.390$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	495.6.a.k	$1$	$1$	\(-5\)	\(0\)	\(150\)	\(-80\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(9-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
495.6.a.l	$495$	$6$	495.a	1.a	$1$	$6$	$6$	$79.390$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		55.6.a.d	$1$	$1$	\(-3\)	\(0\)	\(150\)	\(-66\)		$-$	$-$	$+$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(20-\beta _{1}+\beta _{3}+\beta _{5})q^{4}+\cdots\)
495.6.a.m	$495$	$6$	495.a	1.a	$1$	$6$	$6$	$79.390$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	495.6.a.k	$1$	$1$	\(5\)	\(0\)	\(-150\)	\(-80\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(9-2\beta _{1}+\beta _{2})q^{4}-5^{2}q^{5}+\cdots\)
495.6.a.n	$495$	$6$	495.a	1.a	$1$	$7$	$7$	$79.390$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		165.6.a.h	$1$	$0$	\(-1\)	\(0\)	\(-175\)	\(0\)		$-$	$+$	$+$	$-$	$2^{6}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(28+\beta _{2})q^{4}-5^{2}q^{5}+(-1+\cdots)q^{7}+\cdots\)
495.6.a.o	$495$	$6$	495.a	1.a	$1$	$10$	$10$	$79.390$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓	✓	495.6.a.o	$1$	$0$	\(-3\)	\(0\)	\(-250\)	\(116\)		$+$	$+$	$-$	$-$	$2^{4}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(18+\beta _{2})q^{4}-5^{2}q^{5}+(12+\cdots)q^{7}+\cdots\)
495.6.a.p	$495$	$6$	495.a	1.a	$1$	$10$	$10$	$79.390$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓	✓	495.6.a.o	$1$	$0$	\(3\)	\(0\)	\(250\)	\(116\)		$+$	$-$	$+$	$-$	$2^{4}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(18+\beta _{2})q^{4}+5^{2}q^{5}+(12+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(495))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(495)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 2}\)