Space of modular forms of level 680, weight 1, and Character 680.k

• Label: 680.1.k
• Level: $680$
• Weight: $1$
• Character orbit: 680.k
• Rep. character: $\chi_{680}(339,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $5$
• Sturm bound: $108$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	680.k (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 680 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(108\)
Trace bound:			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(680, [\chi])\).

	Total	New	Old
Modular forms	12	12	0
Cusp forms	8	8	0
Eisenstein series	4	4	0

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	8	0	0	0

Trace form

\( 8 q - 4 q^{9} + 8 q^{16} - 4 q^{19} + 4 q^{25} - 4 q^{26} - 4 q^{30} + 4 q^{34} - 4 q^{35} + 4 q^{36} - 4 q^{50} - 4 q^{51} + 4 q^{59} - 4 q^{70} - 4 q^{76} - 4 q^{89} - 4 q^{94}+O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(680, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
680.1.k.a	$680$	$1$	680.k	680.k	$2$	$1$	$1$	$0.339$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-170}) \)	None	✓	✓			680.1.k.a	$2$	$0$	\(-1\)	\(-1\)	\(-1\)	\(0\)	$1$		\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
680.1.k.b	$680$	$1$	680.k	680.k	$2$	$1$	$1$	$0.339$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-170}) \)	None	✓	✓			680.1.k.a	$2$	$0$	\(-1\)	\(1\)	\(1\)	\(0\)	$1$		\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
680.1.k.c	$680$	$1$	680.k	680.k	$2$	$1$	$1$	$0.339$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-170}) \)	None	✓	✓			680.1.k.a	$2$	$0$	\(1\)	\(-1\)	\(1\)	\(0\)	$1$		\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
680.1.k.d	$680$	$1$	680.k	680.k	$2$	$1$	$1$	$0.339$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-170}) \)	None	✓	✓			680.1.k.a	$2$	$0$	\(1\)	\(1\)	\(-1\)	\(0\)	$1$		\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
680.1.k.e	$680$	$1$	680.k	680.k	$2$	$4$	$4$	$0.339$	\(\Q(\zeta_{8})\)	$D_{4}$	\(\Q(\sqrt{-34}) \)	None		✓	✓		680.1.k.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+\zeta_{8}^{2}q^{2}-q^{4}-\zeta_{8}q^{5}+(-\zeta_{8}-\zeta_{8}^{3}+\cdots)q^{7}+\cdots\)