Space of modular forms of level 704, weight 2, and trivial character

• Label: 704.2.a
• Level: $704$
• Weight: $2$
• Character orbit: 704.a
• Rep. character: $\chi_{704}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $16$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(704))\).

	Total	New	Old
Modular forms	108	20	88
Cusp forms	85	20	65
Eisenstein series	23	0	23

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
\(+\)	\(+\)	\(+\)		\(24\)	\(4\)	\(20\)		\(19\)	\(4\)	\(15\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(30\)	\(7\)	\(23\)		\(24\)	\(7\)	\(17\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(30\)	\(6\)	\(24\)		\(24\)	\(6\)	\(18\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(24\)	\(3\)	\(21\)		\(18\)	\(3\)	\(15\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(48\)	\(7\)	\(41\)		\(37\)	\(7\)	\(30\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(60\)	\(13\)	\(47\)		\(48\)	\(13\)	\(35\)		\(12\)	\(0\)	\(12\)

Trace form

20 * q + 20 * q^9 + 16 * q^13 - 8 * q^17 + 16 * q^21 + 12 * q^25 + 16 * q^37 - 8 * q^41 + 24 * q^45 + 20 * q^49 + 24 * q^53 - 16 * q^61 - 8 * q^69 + 8 * q^73 - 12 * q^81 - 32 * q^85 - 24 * q^89 - 8 * q^93 - 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(704))\) 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
704.2.a.a	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				352.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}+6q^{9}+q^{11}+6q^{13}+\cdots\)
704.2.a.b	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				88.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+3q^{5}+2q^{7}+6q^{9}-q^{11}+\cdots\)
704.2.a.c	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				11.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+2q^{7}-2q^{9}+q^{11}-4q^{13}+\cdots\)
704.2.a.d	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				352.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+4q^{7}-2q^{9}-q^{11}+2q^{13}+\cdots\)
704.2.a.e	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				352.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-4q^{7}-2q^{9}-q^{11}+\cdots\)
704.2.a.f	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				44.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}+2q^{7}-2q^{9}+q^{11}+\cdots\)
704.2.a.g	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				352.2.a.c	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-4q^{7}-2q^{9}+q^{11}+2q^{13}+\cdots\)
704.2.a.h	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				11.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}-2q^{9}-q^{11}-4q^{13}+\cdots\)
704.2.a.i	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				44.2.a.a	$1$	$0$	\(0\)	\(1\)	\(3\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}-2q^{7}-2q^{9}-q^{11}+\cdots\)
704.2.a.j	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				352.2.a.b	$1$	$0$	\(0\)	\(1\)	\(3\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}+4q^{7}-2q^{9}+q^{11}+\cdots\)
704.2.a.k	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				352.2.a.a	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}+6q^{9}-q^{11}+6q^{13}+\cdots\)
704.2.a.l	$704$	$2$	704.a	1.a	$1$	$1$	$1$	$5.621$	\(\Q\)	None	✓				88.2.a.a	$1$	$0$	\(0\)	\(3\)	\(3\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+3q^{5}-2q^{7}+6q^{9}+q^{11}+\cdots\)
704.2.a.m	$704$	$2$	704.a	1.a	$1$	$2$	$2$	$5.621$	\(\Q(\sqrt{17}) \)	None	✓				88.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2+\beta )q^{5}-2\beta q^{7}+(1+\beta )q^{9}+\cdots\)
704.2.a.n	$704$	$2$	704.a	1.a	$1$	$2$	$2$	$5.621$	\(\Q(\sqrt{17}) \)	None	✓		✓		352.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2+\beta )q^{5}+(1+\beta )q^{9}-q^{11}+\cdots\)
704.2.a.o	$704$	$2$	704.a	1.a	$1$	$2$	$2$	$5.621$	\(\Q(\sqrt{17}) \)	None	✓				352.2.a.g	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-2+\beta )q^{5}+(1+\beta )q^{9}+q^{11}+\cdots\)
704.2.a.p	$704$	$2$	704.a	1.a	$1$	$2$	$2$	$5.621$	\(\Q(\sqrt{17}) \)	None	✓		✓		88.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-2+\beta )q^{5}+2\beta q^{7}+(1+\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(704))\) into lower level spaces

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