Space of modular forms of level 704, weight 6, and trivial character

• Label: 704.6.a
• Level: $704$
• Weight: $6$
• Character orbit: 704.a
• Rep. character: $\chi_{704}(1,\cdot)$
• Character field: $\Q$
• Dimension: $100$
• Newform subspaces: $32$
• Sturm bound: $576$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(704))\).

	Total	New	Old
Modular forms	492	100	392
Cusp forms	468	100	368
Eisenstein series	24	0	24

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

\(2\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(120\)	\(24\)	\(96\)		\(114\)	\(24\)	\(90\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(126\)	\(27\)	\(99\)		\(120\)	\(27\)	\(93\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(126\)	\(26\)	\(100\)		\(120\)	\(26\)	\(94\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(120\)	\(23\)	\(97\)		\(114\)	\(23\)	\(91\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(240\)	\(47\)	\(193\)		\(228\)	\(47\)	\(181\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(252\)	\(53\)	\(199\)		\(240\)	\(53\)	\(187\)		\(12\)	\(0\)	\(12\)

Trace form

100 * q + 8100 * q^9 + 464 * q^13 - 808 * q^17 - 8496 * q^21 + 68732 * q^25 + 4304 * q^37 + 4952 * q^41 + 36024 * q^45 + 240100 * q^49 - 38856 * q^53 + 4144 * q^61 - 7680 * q^65 - 97896 * q^69 - 10072 * q^73 + 476740 * q^81 + 264800 * q^85 - 57592 * q^89 - 173544 * q^93 - 160808 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(704))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	11
704.6.a.a	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				22.6.a.c	$1$	$1$	\(0\)	\(-29\)	\(31\)	\(230\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-29q^{3}+31q^{5}+230q^{7}+598q^{9}+\cdots\)
704.6.a.b	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				22.6.a.a	$1$	$1$	\(0\)	\(-21\)	\(-81\)	\(-98\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-21q^{3}-3^{4}q^{5}-98q^{7}+198q^{9}+\cdots\)
704.6.a.c	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				11.6.a.a	$1$	$0$	\(0\)	\(-15\)	\(19\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-15q^{3}+19q^{5}-10q^{7}-18q^{9}+\cdots\)
704.6.a.d	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				44.6.a.a	$1$	$1$	\(0\)	\(-7\)	\(79\)	\(-50\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+79q^{5}-50q^{7}-194q^{9}+\cdots\)
704.6.a.e	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				22.6.a.b	$1$	$0$	\(0\)	\(-1\)	\(51\)	\(-166\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+51q^{5}-166q^{7}-242q^{9}+\cdots\)
704.6.a.f	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				22.6.a.b	$1$	$0$	\(0\)	\(1\)	\(51\)	\(166\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+51q^{5}+166q^{7}-242q^{9}+\cdots\)
704.6.a.g	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				44.6.a.a	$1$	$1$	\(0\)	\(7\)	\(79\)	\(50\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+79q^{5}+50q^{7}-194q^{9}+\cdots\)
704.6.a.h	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				11.6.a.a	$1$	$0$	\(0\)	\(15\)	\(19\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+15q^{3}+19q^{5}+10q^{7}-18q^{9}+\cdots\)
704.6.a.i	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				22.6.a.a	$1$	$1$	\(0\)	\(21\)	\(-81\)	\(98\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+21q^{3}-3^{4}q^{5}+98q^{7}+198q^{9}+\cdots\)
704.6.a.j	$704$	$6$	704.a	1.a	$1$	$1$	$1$	$112.910$	\(\Q\)	None	✓				22.6.a.c	$1$	$1$	\(0\)	\(29\)	\(31\)	\(-230\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+29q^{3}+31q^{5}-230q^{7}+598q^{9}+\cdots\)
704.6.a.k	$704$	$6$	704.a	1.a	$1$	$2$	$2$	$112.910$	\(\Q(\sqrt{793}) \)	None	✓				22.6.a.d	$1$	$0$	\(0\)	\(-29\)	\(13\)	\(-14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-14-\beta )q^{3}+(4+5\beta )q^{5}+(-4+\cdots)q^{7}+\cdots\)
704.6.a.l	$704$	$6$	704.a	1.a	$1$	$2$	$2$	$112.910$	\(\Q(\sqrt{37}) \)	None	✓				88.6.a.a	$1$	$1$	\(0\)	\(-14\)	\(-18\)	\(-48\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}-9q^{5}+(-24-11\beta )q^{7}+\cdots\)
704.6.a.m	$704$	$6$	704.a	1.a	$1$	$2$	$2$	$112.910$	\(\Q(\sqrt{31}) \)	None	✓				44.6.a.b	$1$	$0$	\(0\)	\(-6\)	\(22\)	\(-268\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-3+\beta )q^{3}+(11-2\beta )q^{5}+(-134+\cdots)q^{7}+\cdots\)
704.6.a.n	$704$	$6$	704.a	1.a	$1$	$2$	$2$	$112.910$	\(\Q(\sqrt{31}) \)	None	✓				44.6.a.b	$1$	$0$	\(0\)	\(6\)	\(22\)	\(268\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(11+2\beta )q^{5}+(134+3\beta )q^{7}+\cdots\)
704.6.a.o	$704$	$6$	704.a	1.a	$1$	$2$	$2$	$112.910$	\(\Q(\sqrt{37}) \)	None	✓				88.6.a.a	$1$	$1$	\(0\)	\(14\)	\(-18\)	\(48\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(7+\beta )q^{3}-9q^{5}+(24+11\beta )q^{7}+\cdots\)
704.6.a.p	$704$	$6$	704.a	1.a	$1$	$2$	$2$	$112.910$	\(\Q(\sqrt{793}) \)	None	✓				22.6.a.d	$1$	$0$	\(0\)	\(29\)	\(13\)	\(14\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(15-\beta )q^{3}+(9-5\beta )q^{5}+(10-6\beta )q^{7}+\cdots\)
704.6.a.q	$704$	$6$	704.a	1.a	$1$	$3$	$3$	$112.910$	3.3.54492.1	None	✓				11.6.a.b	$1$	$1$	\(0\)	\(-34\)	\(-24\)	\(84\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{1})q^{3}+(-7-4\beta _{1}-\beta _{2})q^{5}+\cdots\)
704.6.a.r	$704$	$6$	704.a	1.a	$1$	$3$	$3$	$112.910$	3.3.1784453.1	None	✓				88.6.a.b	$1$	$1$	\(0\)	\(-14\)	\(-56\)	\(112\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-5+\beta _{2})q^{3}+(-19-\beta _{1}+\beta _{2})q^{5}+\cdots\)
704.6.a.s	$704$	$6$	704.a	1.a	$1$	$3$	$3$	$112.910$	3.3.1784453.1	None	✓				88.6.a.b	$1$	$1$	\(0\)	\(14\)	\(-56\)	\(-112\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(5-\beta _{2})q^{3}+(-19-\beta _{1}+\beta _{2})q^{5}+\cdots\)
704.6.a.t	$704$	$6$	704.a	1.a	$1$	$3$	$3$	$112.910$	3.3.54492.1	None	✓				11.6.a.b	$1$	$1$	\(0\)	\(34\)	\(-24\)	\(-84\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(11+\beta _{1})q^{3}+(-7-4\beta _{1}-\beta _{2})q^{5}+\cdots\)
704.6.a.u	$704$	$6$	704.a	1.a	$1$	$4$	$4$	$112.910$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				88.6.a.c	$1$	$0$	\(0\)	\(-13\)	\(19\)	\(-58\)		$-$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(5-2\beta _{1}+\beta _{3})q^{5}+\cdots\)
704.6.a.v	$704$	$6$	704.a	1.a	$1$	$4$	$4$	$112.910$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				88.6.a.d	$1$	$0$	\(0\)	\(-5\)	\(-93\)	\(94\)		$-$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-24+2\beta _{1}+\beta _{3})q^{5}+\cdots\)
704.6.a.w	$704$	$6$	704.a	1.a	$1$	$4$	$4$	$112.910$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				88.6.a.d	$1$	$0$	\(0\)	\(5\)	\(-93\)	\(-94\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-24+2\beta _{1}+\beta _{3})q^{5}+\cdots\)
704.6.a.x	$704$	$6$	704.a	1.a	$1$	$4$	$4$	$112.910$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				88.6.a.c	$1$	$0$	\(0\)	\(13\)	\(19\)	\(58\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(5-2\beta _{1}+\beta _{3})q^{5}+(13+\cdots)q^{7}+\cdots\)
704.6.a.y	$704$	$6$	704.a	1.a	$1$	$6$	$6$	$112.910$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				352.6.a.a	$1$	$1$	\(0\)	\(-27\)	\(5\)	\(0\)		$-$	$-$	$+$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{3}+(1+\beta _{5})q^{5}+(-2+\cdots)q^{7}+\cdots\)
704.6.a.z	$704$	$6$	704.a	1.a	$1$	$6$	$6$	$112.910$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				352.6.a.c	$1$	$0$	\(0\)	\(-9\)	\(5\)	\(196\)		$-$	$+$	$-$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(33-\beta _{3}+\cdots)q^{7}+\cdots\)
704.6.a.ba	$704$	$6$	704.a	1.a	$1$	$6$	$6$	$112.910$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				352.6.a.b	$1$	$1$	\(0\)	\(-9\)	\(89\)	\(-196\)		$+$	$+$	$+$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}+(15-\beta _{3})q^{5}+(-33+\cdots)q^{7}+\cdots\)
704.6.a.bb	$704$	$6$	704.a	1.a	$1$	$6$	$6$	$112.910$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				352.6.a.c	$1$	$1$	\(0\)	\(9\)	\(5\)	\(-196\)		$-$	$-$	$+$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(-33+\beta _{3}+\cdots)q^{7}+\cdots\)
704.6.a.bc	$704$	$6$	704.a	1.a	$1$	$6$	$6$	$112.910$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				352.6.a.b	$1$	$0$	\(0\)	\(9\)	\(89\)	\(196\)		$+$	$-$	$-$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{3}+(15-\beta _{3})q^{5}+(33+\beta _{1}+\cdots)q^{7}+\cdots\)
704.6.a.bd	$704$	$6$	704.a	1.a	$1$	$6$	$6$	$112.910$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				352.6.a.a	$1$	$0$	\(0\)	\(27\)	\(5\)	\(0\)		$-$	$+$	$-$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{3}+(1+\beta _{5})q^{5}+(2-3\beta _{1}+\cdots)q^{7}+\cdots\)
704.6.a.be	$704$	$6$	704.a	1.a	$1$	$7$	$7$	$112.910$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		352.6.a.g	$1$	$1$	\(0\)	\(-9\)	\(-61\)	\(0\)		$+$	$+$	$+$	$2^{15}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-9+\beta _{1}+\beta _{2})q^{5}+\cdots\)
704.6.a.bf	$704$	$6$	704.a	1.a	$1$	$7$	$7$	$112.910$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		352.6.a.g	$1$	$0$	\(0\)	\(9\)	\(-61\)	\(0\)		$+$	$-$	$-$	$2^{15}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-9+\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(704))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(704)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 7}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 2}\)