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}$