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

• Label: 704.2.e
• Level: $704$
• Weight: $2$
• Character orbit: 704.e
• Rep. character: $\chi_{704}(703,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $4$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	704.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 44 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	108	26	82
Cusp forms	84	22	62
Eisenstein series	24	4	20

Trace form

\( 22 q + 4 q^{5} - 22 q^{9} + 10 q^{25} - 4 q^{33} - 12 q^{37} + 36 q^{45} + 6 q^{49} - 36 q^{53} + 16 q^{69} + 30 q^{81} - 4 q^{89} + 16 q^{93} - 36 q^{97}+O(q^{100}) \)

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.e.a	$704$	$2$	704.e	44.c	$2$	$2$	$2$	$5.621$	\(\Q(\sqrt{-11}) \)	\(\Q(\sqrt{-11}) \)					176.2.e.a	$4$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{3}-3q^{5}-8q^{9}+\beta q^{11}+3\beta q^{15}+\cdots\)
704.2.e.b	$704$	$2$	704.e	44.c	$2$	$4$	$4$	$5.621$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					44.2.c.a	$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+q^{5}+\beta _{1}q^{7}+(\beta _{1}+\beta _{2})q^{11}+\cdots\)
704.2.e.c	$704$	$2$	704.e	44.c	$2$	$4$	$4$	$5.621$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	\(\Q(\sqrt{-11}) \)					176.2.e.b	$4$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{3}+(2-\beta _{3})q^{5}+(-1+\beta _{3})q^{9}+\cdots\)
704.2.e.d	$704$	$2$	704.e	44.c	$2$	$12$	$12$	$5.621$	12.0.\(\cdots\).1	None			✓		352.2.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{15}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{5}q^{5}+\beta _{8}q^{7}+\beta _{3}q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(704, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(44, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(176, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(352, [\chi])\)\(^{\oplus 2}\)