Space of modular forms of level 3400, weight 1, and Character 3400.cs

• Label: 3400.1.cs
• Level: $3400$
• Weight: $1$
• Character orbit: 3400.cs
• Rep. character: $\chi_{3400}(611,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $16$
• Newform subspaces: $3$
• Sturm bound: $540$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3400.cs (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3400 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(540\)
Trace bound:			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(3400, [\chi])\).

	Total	New	Old
Modular forms	32	32	0
Cusp forms	16	16	0
Eisenstein series	16	16	0

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	16	0	0	0

Trace form

\( 16 q - 4 q^{4} - 4 q^{9} - 4 q^{16} - 4 q^{34} - 4 q^{35} - 4 q^{36} - 8 q^{38} + 32 q^{43} + 16 q^{49} - 4 q^{50} + 12 q^{59} - 4 q^{64} - 4 q^{70} - 4 q^{81} - 8 q^{83} - 8 q^{98}+O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(3400, [\chi])\) 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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3400.1.cs.a	$3400$	$1$	3400.cs	3400.bs	$10$	$4$	$1$	$1.697$	\(\Q(\zeta_{10})\)	$D_{5}$	\(\Q(\sqrt{-34}) \)	None		✓			3400.1.cs.a	$4$	$0$	\(-1\)	\(0\)	\(-4\)	\(2\)	$1$		\(q+\zeta_{10}^{2}q^{2}+\zeta_{10}^{4}q^{4}-q^{5}+(-\zeta_{10}^{2}+\cdots)q^{7}+\cdots\)
3400.1.cs.b	$3400$	$1$	3400.cs	3400.bs	$10$	$4$	$1$	$1.697$	\(\Q(\zeta_{10})\)	$D_{5}$	\(\Q(\sqrt{-34}) \)	None		✓			3400.1.cs.a	$4$	$0$	\(-1\)	\(0\)	\(4\)	\(-2\)	$1$		\(q+\zeta_{10}^{2}q^{2}+\zeta_{10}^{4}q^{4}+q^{5}+(\zeta_{10}^{2}+\cdots)q^{7}+\cdots\)
3400.1.cs.c	$3400$	$1$	3400.cs	3400.bs	$10$	$8$	$2$	$1.697$	\(\Q(\zeta_{20})\)	$D_{10}$	\(\Q(\sqrt{-34}) \)	None		✓	✓		3400.1.cs.c	$8$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\zeta_{20}^{4}q^{2}+\zeta_{20}^{8}q^{4}+\zeta_{20}^{5}q^{5}+\cdots\)