Space of modular forms of level 585, weight 6, and trivial character

• Label: 585.6.a
• Level: $585$
• Weight: $6$
• Character orbit: 585.a
• Rep. character: $\chi_{585}(1,\cdot)$
• Character field: $\Q$
• Dimension: $100$
• Newform subspaces: $18$
• Sturm bound: $504$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	585.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(504\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(585))\).

	Total	New	Old
Modular forms	428	100	328
Cusp forms	412	100	312
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.

\(3\)	\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(11\)
\(+\)	\(+\)	\(-\)	\(-\)	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)	\(11\)
\(+\)	\(-\)	\(-\)	\(+\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)	\(15\)
\(-\)	\(+\)	\(-\)	\(+\)	\(14\)
\(-\)	\(-\)	\(+\)	\(+\)	\(14\)
\(-\)	\(-\)	\(-\)	\(-\)	\(17\)
Plus space			\(+\)	\(48\)
Minus space			\(-\)	\(52\)

Trace form

100 * q + 12 * q^2 + 1664 * q^4 + 50 * q^5 - 196 * q^7 + 156 * q^8 - 100 * q^10 + 2124 * q^11 - 338 * q^13 - 4432 * q^14 + 28032 * q^16 - 2884 * q^17 - 3864 * q^19 + 2400 * q^20 + 1544 * q^22 + 480 * q^23 + 62500 * q^25 - 2548 * q^28 - 16496 * q^29 - 1028 * q^31 - 12976 * q^32 + 32916 * q^34 + 3100 * q^35 + 11400 * q^37 + 24828 * q^38 + 3000 * q^40 - 40668 * q^41 - 39104 * q^43 + 79124 * q^44 + 36812 * q^46 + 31236 * q^47 + 193516 * q^49 + 7500 * q^50 - 16224 * q^52 + 87352 * q^53 - 1900 * q^55 - 248752 * q^56 - 157132 * q^58 + 37680 * q^59 + 106992 * q^61 - 238084 * q^62 + 607296 * q^64 + 16900 * q^65 - 2484 * q^67 + 146780 * q^68 + 60200 * q^70 - 248728 * q^71 - 281056 * q^73 - 266600 * q^74 + 83800 * q^76 - 67004 * q^77 + 72704 * q^79 + 89600 * q^80 + 141772 * q^82 + 190876 * q^83 - 163200 * q^85 + 165408 * q^86 - 1108 * q^88 + 400648 * q^89 - 37180 * q^91 - 79148 * q^92 - 18160 * q^94 + 75800 * q^95 + 384620 * q^97 + 812052 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(585))\) 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	13
585.6.a.a	$585$	$6$	585.a	1.a	$1$	$1$	$1$	$93.825$	\(\Q\)	None	✓				65.6.a.a	$1$	$1$	\(-5\)	\(0\)	\(25\)	\(-244\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-7q^{4}+5^{2}q^{5}-244q^{7}+195q^{8}+\cdots\)
585.6.a.b	$585$	$6$	585.a	1.a	$1$	$1$	$1$	$93.825$	\(\Q\)	None	✓				195.6.a.a	$1$	$1$	\(-2\)	\(0\)	\(-25\)	\(-168\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-28q^{4}-5^{2}q^{5}-168q^{7}+\cdots\)
585.6.a.c	$585$	$6$	585.a	1.a	$1$	$3$	$3$	$93.825$	3.3.49857.1	None	✓				65.6.a.b	$1$	$1$	\(2\)	\(0\)	\(-75\)	\(-208\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
585.6.a.d	$585$	$6$	585.a	1.a	$1$	$4$	$4$	$93.825$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				195.6.a.e	$1$	$1$	\(-2\)	\(0\)	\(100\)	\(87\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(14-\beta _{1}+\beta _{3})q^{4}+\cdots\)
585.6.a.e	$585$	$6$	585.a	1.a	$1$	$4$	$4$	$93.825$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				195.6.a.d	$1$	$1$	\(3\)	\(0\)	\(-100\)	\(-87\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(3-3\beta _{1}+2\beta _{3})q^{4}+\cdots\)
585.6.a.f	$585$	$6$	585.a	1.a	$1$	$4$	$4$	$93.825$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				195.6.a.c	$1$	$0$	\(7\)	\(0\)	\(100\)	\(73\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(3-2\beta _{1}-\beta _{2}+\beta _{3})q^{4}+\cdots\)
585.6.a.g	$585$	$6$	585.a	1.a	$1$	$4$	$4$	$93.825$	4.4.1878612.1	None	✓				65.6.a.c	$1$	$1$	\(9\)	\(0\)	\(100\)	\(-136\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1}-\beta _{2})q^{2}+(22-4\beta _{2}+\beta _{3})q^{4}+\cdots\)
585.6.a.h	$585$	$6$	585.a	1.a	$1$	$4$	$4$	$93.825$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				195.6.a.b	$1$	$0$	\(12\)	\(0\)	\(-100\)	\(-73\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(13-3\beta _{1}+2\beta _{3})q^{4}+\cdots\)
585.6.a.i	$585$	$6$	585.a	1.a	$1$	$5$	$5$	$93.825$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				195.6.a.g	$1$	$0$	\(-5\)	\(0\)	\(-125\)	\(164\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(8+4\beta _{1}+\beta _{2})q^{4}+\cdots\)
585.6.a.j	$585$	$6$	585.a	1.a	$1$	$5$	$5$	$93.825$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		195.6.a.f	$1$	$1$	\(-1\)	\(0\)	\(125\)	\(128\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(7+4\beta _{1}+\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)
585.6.a.k	$585$	$6$	585.a	1.a	$1$	$6$	$6$	$93.825$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		65.6.a.e	$1$	$0$	\(-2\)	\(0\)	\(-150\)	\(172\)		$-$	$+$	$+$	$-$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(22+\beta _{3})q^{4}-5^{2}q^{5}+(3^{3}+\cdots)q^{7}+\cdots\)
585.6.a.l	$585$	$6$	585.a	1.a	$1$	$6$	$6$	$93.825$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		195.6.a.h	$1$	$1$	\(-2\)	\(0\)	\(-150\)	\(236\)		$-$	$+$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(35+\beta _{3})q^{4}-5^{2}q^{5}+(37+\cdots)q^{7}+\cdots\)
585.6.a.m	$585$	$6$	585.a	1.a	$1$	$6$	$6$	$93.825$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				65.6.a.d	$1$	$0$	\(0\)	\(0\)	\(150\)	\(220\)		$-$	$-$	$-$	$-$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(22+\beta _{2})q^{4}+5^{2}q^{5}+(35+\cdots)q^{7}+\cdots\)
585.6.a.n	$585$	$6$	585.a	1.a	$1$	$7$	$7$	$93.825$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		195.6.a.i	$1$	$0$	\(-2\)	\(0\)	\(175\)	\(32\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(26-\beta _{1}+\beta _{2})q^{4}+5^{2}q^{5}+\cdots\)
585.6.a.o	$585$	$6$	585.a	1.a	$1$	$9$	$9$	$93.825$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	585.6.a.o	$1$	$1$	\(-5\)	\(0\)	\(225\)	\(-153\)		$+$	$-$	$-$	$+$	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2^{4}-\beta _{1}+\beta _{2})q^{4}+\cdots\)
585.6.a.p	$585$	$6$	585.a	1.a	$1$	$9$	$9$	$93.825$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	585.6.a.o	$1$	$0$	\(5\)	\(0\)	\(-225\)	\(-153\)		$+$	$+$	$-$	$-$	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2^{4}-\beta _{1}+\beta _{2})q^{4}-5^{2}q^{5}+\cdots\)
585.6.a.q	$585$	$6$	585.a	1.a	$1$	$11$	$11$	$93.825$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓	✓	585.6.a.q	$1$	$1$	\(-3\)	\(0\)	\(-275\)	\(-43\)		$+$	$+$	$+$	$+$	$2^{13}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(21+\beta _{2})q^{4}-5^{2}q^{5}+(-4+\cdots)q^{7}+\cdots\)
585.6.a.r	$585$	$6$	585.a	1.a	$1$	$11$	$11$	$93.825$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓	✓	585.6.a.q	$1$	$0$	\(3\)	\(0\)	\(275\)	\(-43\)		$+$	$-$	$+$	$-$	$2^{13}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(21+\beta _{2})q^{4}+5^{2}q^{5}+(-4+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(585))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(585)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 2}\)