Space of modular forms of level 680, weight 2, and Character 680.e

• Label: 680.2.e
• Level: $680$
• Weight: $2$
• Character orbit: 680.e
• Rep. character: $\chi_{680}(409,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $3$
• Sturm bound: $216$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(216\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(680, [\chi])\).

	Total	New	Old
Modular forms	116	24	92
Cusp forms	100	24	76
Eisenstein series	16	0	16

Trace form

24 * q - 28 * q^9 + 16 * q^11 - 16 * q^15 - 4 * q^19 - 8 * q^21 + 12 * q^25 + 24 * q^29 - 8 * q^31 - 4 * q^35 + 16 * q^39 - 24 * q^41 + 16 * q^45 - 48 * q^49 + 12 * q^51 - 16 * q^55 + 4 * q^59 - 24 * q^65 - 32 * q^69 - 16 * q^71 - 32 * q^75 + 80 * q^79 + 88 * q^81 + 20 * q^89 - 24 * q^95 - 16 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(680, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
680.2.e.a	$680$	$2$	680.e	5.b	$2$	$2$	$2$	$5.430$	\(\Q(\sqrt{-1}) \)	None		✓			680.2.e.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+(2 i-1)q^{5}-4 i q^{7}+2 q^{9}+\cdots\)
680.2.e.b	$680$	$2$	680.e	5.b	$2$	$10$	$10$	$5.430$	10.0.\(\cdots\).1	None		✓			680.2.e.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}+(-\beta _{3}-\beta _{8})q^{5}+(\beta _{5}+\beta _{7}+\cdots)q^{7}+\cdots\)
680.2.e.c	$680$	$2$	680.e	5.b	$2$	$12$	$12$	$5.430$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		680.2.e.c	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{3})q^{3}+\beta _{7}q^{5}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(680, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(680, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(170, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(340, [\chi])\)\(^{\oplus 2}\)