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

• Label: 3400.1.bc
• Level: $3400$
• Weight: $1$
• Character orbit: 3400.bc
• Rep. character: $\chi_{3400}(2299,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $540$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3400.bc (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 680 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(540\)
Trace bound:			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	36	20	16
Cusp forms	12	12	0
Eisenstein series	24	8	16

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	12	0	0	0

Trace form

\( 12 q + 12 q^{4} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3400.1.bc.a	$3400$	$1$	3400.bc	680.ac	$4$	$2$	$1$	$1.697$	\(\Q(\sqrt{-1}) \)	$D_{4}$	\(\Q(\sqrt{-2}) \)	None		$4$	$0$	\(-2\)	\(2\)	\(0\)	\(0\)	$1$		\(q-q^{2}+(1-i)q^{3}+q^{4}+(-1+i)q^{6}+\cdots\)
3400.1.bc.b	$3400$	$1$	3400.bc	680.ac	$4$	$2$	$1$	$1.697$	\(\Q(\sqrt{-1}) \)	$D_{4}$	\(\Q(\sqrt{-2}) \)	None		$4$	$0$	\(2\)	\(-2\)	\(0\)	\(0\)	$1$		\(q+q^{2}+(-1+i)q^{3}+q^{4}+(-1+i+\cdots)q^{6}+\cdots\)
3400.1.bc.c	$3400$	$1$	3400.bc	680.ac	$4$	$4$	$2$	$1.697$	\(\Q(\zeta_{12})\)	$D_{12}$	\(\Q(\sqrt{-2}) \)	None		$4$	$0$	\(-4\)	\(-2\)	\(0\)	\(0\)	$1$		\(q-q^{2}+(\zeta_{12}^{4}+\zeta_{12}^{5})q^{3}+q^{4}+(-\zeta_{12}^{4}+\cdots)q^{6}+\cdots\)
3400.1.bc.d	$3400$	$1$	3400.bc	680.ac	$4$	$4$	$2$	$1.697$	\(\Q(\zeta_{12})\)	$D_{12}$	\(\Q(\sqrt{-2}) \)	None		$4$	$0$	\(4\)	\(2\)	\(0\)	\(0\)	$1$		\(q+q^{2}+(-\zeta_{12}+\zeta_{12}^{2})q^{3}+q^{4}+(-\zeta_{12}+\cdots)q^{6}+\cdots\)