Space of modular forms of level 704, weight 4, and trivial character

• Label: 704.4.a
• Level: $704$
• Weight: $4$
• Character orbit: 704.a
• Rep. character: $\chi_{704}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $28$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(704))\).

	Total	New	Old
Modular forms	300	60	240
Cusp forms	276	60	216
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
\(+\)	\(+\)	\(+\)		\(78\)	\(16\)	\(62\)		\(72\)	\(16\)	\(56\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(72\)	\(13\)	\(59\)		\(66\)	\(13\)	\(53\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(72\)	\(14\)	\(58\)		\(66\)	\(14\)	\(52\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(78\)	\(17\)	\(61\)		\(72\)	\(17\)	\(55\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(156\)	\(33\)	\(123\)		\(144\)	\(33\)	\(111\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(144\)	\(27\)	\(117\)		\(132\)	\(27\)	\(105\)		\(12\)	\(0\)	\(12\)

Trace form

60 * q + 540 * q^9 - 144 * q^13 + 104 * q^17 + 240 * q^21 + 1412 * q^25 + 1008 * q^37 - 472 * q^41 - 984 * q^45 + 2940 * q^49 - 408 * q^53 + 3984 * q^61 - 1536 * q^65 + 1800 * q^69 - 296 * q^73 + 7324 * q^81 - 480 * q^85 + 1784 * q^89 + 10440 * q^93 - 1816 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(704))\) 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
704.4.a.a	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				88.4.a.b	$1$	$1$	\(0\)	\(-7\)	\(-9\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}-9q^{5}+2q^{7}+22q^{9}+11q^{11}+\cdots\)
704.4.a.b	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				22.4.a.a	$1$	$0$	\(0\)	\(-7\)	\(19\)	\(-14\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+19q^{5}-14q^{7}+22q^{9}+11q^{11}+\cdots\)
704.4.a.c	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				44.4.a.a	$1$	$1$	\(0\)	\(-5\)	\(7\)	\(26\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+7q^{5}+26q^{7}-2q^{9}-11q^{11}+\cdots\)
704.4.a.d	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				22.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(-14\)	\(-8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-14q^{5}-8q^{7}-11q^{9}+11q^{11}+\cdots\)
704.4.a.e	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				22.4.a.c	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(-10\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-10q^{7}-26q^{9}-11q^{11}+\cdots\)
704.4.a.f	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				88.4.a.a	$1$	$1$	\(0\)	\(-1\)	\(7\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+7q^{5}+6q^{7}-26q^{9}-11q^{11}+\cdots\)
704.4.a.g	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				22.4.a.c	$1$	$0$	\(0\)	\(1\)	\(3\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}+10q^{7}-26q^{9}+11q^{11}+\cdots\)
704.4.a.h	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				88.4.a.a	$1$	$1$	\(0\)	\(1\)	\(7\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+7q^{5}-6q^{7}-26q^{9}+11q^{11}+\cdots\)
704.4.a.i	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				22.4.a.b	$1$	$1$	\(0\)	\(4\)	\(-14\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-14q^{5}+8q^{7}-11q^{9}-11q^{11}+\cdots\)
704.4.a.j	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				44.4.a.a	$1$	$1$	\(0\)	\(5\)	\(7\)	\(-26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+7q^{5}-26q^{7}-2q^{9}+11q^{11}+\cdots\)
704.4.a.k	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				88.4.a.b	$1$	$1$	\(0\)	\(7\)	\(-9\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-9q^{5}-2q^{7}+22q^{9}-11q^{11}+\cdots\)
704.4.a.l	$704$	$4$	704.a	1.a	$1$	$1$	$1$	$41.537$	\(\Q\)	None	✓				22.4.a.a	$1$	$0$	\(0\)	\(7\)	\(19\)	\(14\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+19q^{5}+14q^{7}+22q^{9}-11q^{11}+\cdots\)
704.4.a.m	$704$	$4$	704.a	1.a	$1$	$2$	$2$	$41.537$	\(\Q(\sqrt{97}) \)	None	✓				44.4.a.b	$1$	$0$	\(0\)	\(-9\)	\(-11\)	\(10\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+(-6+\beta )q^{5}+(8-6\beta )q^{7}+\cdots\)
704.4.a.n	$704$	$4$	704.a	1.a	$1$	$2$	$2$	$41.537$	\(\Q(\sqrt{3}) \)	None	✓				11.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-20\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-1+2\beta )q^{5}+(-10+\cdots)q^{7}+\cdots\)
704.4.a.o	$704$	$4$	704.a	1.a	$1$	$2$	$2$	$41.537$	\(\Q(\sqrt{5}) \)	None	✓				88.4.a.c	$1$	$0$	\(0\)	\(-2\)	\(6\)	\(56\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(3-4\beta )q^{5}+(28+\beta )q^{7}+\cdots\)
704.4.a.p	$704$	$4$	704.a	1.a	$1$	$2$	$2$	$41.537$	\(\Q(\sqrt{3}) \)	None	✓				11.4.a.a	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(20\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1-2\beta )q^{5}+(10-\beta )q^{7}+\cdots\)
704.4.a.q	$704$	$4$	704.a	1.a	$1$	$2$	$2$	$41.537$	\(\Q(\sqrt{5}) \)	None	✓				88.4.a.c	$1$	$0$	\(0\)	\(2\)	\(6\)	\(-56\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(3-4\beta )q^{5}+(-28-\beta )q^{7}+\cdots\)
704.4.a.r	$704$	$4$	704.a	1.a	$1$	$2$	$2$	$41.537$	\(\Q(\sqrt{97}) \)	None	✓				44.4.a.b	$1$	$0$	\(0\)	\(9\)	\(-11\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{3}+(-5-\beta )q^{5}+(-2-6\beta )q^{7}+\cdots\)
704.4.a.s	$704$	$4$	704.a	1.a	$1$	$3$	$3$	$41.537$	3.3.404.1	None	✓				352.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
704.4.a.t	$704$	$4$	704.a	1.a	$1$	$3$	$3$	$41.537$	3.3.11109.1	None	✓				88.4.a.d	$1$	$0$	\(0\)	\(-2\)	\(-8\)	\(24\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-3-\beta _{2})q^{5}+(8+\cdots)q^{7}+\cdots\)
704.4.a.u	$704$	$4$	704.a	1.a	$1$	$3$	$3$	$41.537$	3.3.11109.1	None	✓				88.4.a.d	$1$	$0$	\(0\)	\(2\)	\(-8\)	\(-24\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-3-\beta _{2})q^{5}+(-8+\cdots)q^{7}+\cdots\)
704.4.a.v	$704$	$4$	704.a	1.a	$1$	$3$	$3$	$41.537$	3.3.404.1	None	✓				352.4.a.a	$1$	$0$	\(0\)	\(3\)	\(5\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(2-\beta _{1}+\beta _{2})q^{5}+3\beta _{2}q^{7}+\cdots\)
704.4.a.w	$704$	$4$	704.a	1.a	$1$	$4$	$4$	$41.537$	4.4.2767476.1	None	✓				352.4.a.c	$1$	$1$	\(0\)	\(-9\)	\(11\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(3+\beta _{3})q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
704.4.a.x	$704$	$4$	704.a	1.a	$1$	$4$	$4$	$41.537$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		352.4.a.e	$1$	$1$	\(0\)	\(-3\)	\(-25\)	\(-28\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-7+\beta _{1}-\beta _{2})q^{5}+\cdots\)
704.4.a.y	$704$	$4$	704.a	1.a	$1$	$4$	$4$	$41.537$	4.4.474868.1	None	✓				352.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(11\)	\(28\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}+(3+\beta _{2}-\beta _{3})q^{5}+\cdots\)
704.4.a.z	$704$	$4$	704.a	1.a	$1$	$4$	$4$	$41.537$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		352.4.a.e	$1$	$0$	\(0\)	\(3\)	\(-25\)	\(28\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-7+\beta _{1}-\beta _{2})q^{5}+\cdots\)
704.4.a.ba	$704$	$4$	704.a	1.a	$1$	$4$	$4$	$41.537$	4.4.474868.1	None	✓				352.4.a.d	$1$	$1$	\(0\)	\(3\)	\(11\)	\(-28\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{3}+(3+\beta _{2}-\beta _{3})q^{5}+(-6+\cdots)q^{7}+\cdots\)
704.4.a.bb	$704$	$4$	704.a	1.a	$1$	$4$	$4$	$41.537$	4.4.2767476.1	None	✓				352.4.a.c	$1$	$0$	\(0\)	\(9\)	\(11\)	\(0\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(3+\beta _{3})q^{5}+(\beta _{2}+\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(704))\) into lower level spaces

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