Space of modular forms of level 704, weight 2, and Character 704.s

• Label: 704.2.s
• Level: $704$
• Weight: $2$
• Character orbit: 704.s
• Rep. character: $\chi_{704}(95,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $96$
• Newform subspaces: $5$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	704.s (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 88 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(704, [\chi])\).

	Total	New	Old
Modular forms	432	96	336
Cusp forms	336	96	240
Eisenstein series	96	0	96

Trace form

\( 96 q - 24 q^{9} + 24 q^{25} + 24 q^{33} - 24 q^{49} - 72 q^{81} + 96 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(704, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
704.2.s.a	$704$	$2$	704.s	88.k	$10$	$8$	$2$	$5.621$	\(\Q(\zeta_{20})\)	None		✓			704.2.s.a	$4$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{10}]$	\(q+(2\beta_{6}-\beta_{4}+\beta_{2}-2)q^{3}+(\beta_{5}-\beta_{3})q^{5}+\cdots\)
704.2.s.b	$704$	$2$	704.s	88.k	$10$	$8$	$2$	$5.621$	\(\Q(\zeta_{20})\)	None		✓			704.2.s.b	$4$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(-\zeta_{20}^{5}-\zeta_{20}^{7})q^{3}+(-1-\zeta_{20}^{6}+\cdots)q^{5}+\cdots\)
704.2.s.c	$704$	$2$	704.s	88.k	$10$	$8$	$2$	$5.621$	\(\Q(\zeta_{20})\)	None		✓			704.2.s.b	$4$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(-\zeta_{20}^{5}-\zeta_{20}^{7})q^{3}+(1+\zeta_{20}^{6}+\cdots)q^{5}+\cdots\)
704.2.s.d	$704$	$2$	704.s	88.k	$10$	$8$	$2$	$5.621$	\(\Q(\zeta_{20})\)	None		✓			704.2.s.a	$4$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{10}]$	\(q+(-2\beta_{6}+\beta_{4}-\beta_{2}+2)q^{3}+(-\beta_{5}+\beta_{3})q^{5}+\cdots\)
704.2.s.e	$704$	$2$	704.s	88.k	$10$	$64$	$16$	$5.621$		None		✓	✓		704.2.s.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$

Decomposition of \(S_{2}^{\mathrm{old}}(704, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(704, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(88, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(352, [\chi])\)\(^{\oplus 2}\)