Space of modular forms of level 3380, weight 2, and Character 3380.f

• Label: 3380.2.f
• Level: $3380$
• Weight: $2$
• Character orbit: 3380.f
• Rep. character: $\chi_{3380}(3041,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $10$
• Sturm bound: $1092$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(1092\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(3\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3380, [\chi])\).

	Total	New	Old
Modular forms	588	50	538
Cusp forms	504	50	454
Eisenstein series	84	0	84

Trace form

\( 50 q + 50 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3380, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3380.2.f.a	$3380$	$2$	3380.f	13.b	$2$	$2$	$2$	$26.989$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}-iq^{5}+3iq^{7}+6q^{9}+3iq^{11}+\cdots\)
3380.2.f.b	$3380$	$2$	3380.f	13.b	$2$	$2$	$2$	$26.989$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2q^{3}+iq^{5}+2iq^{7}+q^{9}-2iq^{15}+\cdots\)
3380.2.f.c	$3380$	$2$	3380.f	13.b	$2$	$2$	$2$	$26.989$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+iq^{5}+5iq^{7}-2q^{9}-5iq^{11}+\cdots\)
3380.2.f.d	$3380$	$2$	3380.f	13.b	$2$	$2$	$2$	$26.989$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}-iq^{5}+iq^{7}-2q^{9}-3iq^{11}+\cdots\)
3380.2.f.e	$3380$	$2$	3380.f	13.b	$2$	$2$	$2$	$26.989$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}-iq^{5}-iq^{7}-2q^{9}+3iq^{11}+\cdots\)
3380.2.f.f	$3380$	$2$	3380.f	13.b	$2$	$2$	$2$	$26.989$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2q^{3}+iq^{5}+2iq^{7}+q^{9}+4iq^{11}+\cdots\)
3380.2.f.g	$3380$	$2$	3380.f	13.b	$2$	$6$	$6$	$26.989$	6.0.153664.1	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+\beta _{5}q^{5}-\beta _{1}q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots\)
3380.2.f.h	$3380$	$2$	3380.f	13.b	$2$	$6$	$6$	$26.989$	6.0.5089536.1	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{3})q^{3}+\beta _{4}q^{5}+(\beta _{4}+\beta _{5})q^{7}+\cdots\)
3380.2.f.i	$3380$	$2$	3380.f	13.b	$2$	$8$	$8$	$26.989$	8.0.22581504.2	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{2}-\beta _{4})q^{3}-\beta _{5}q^{5}+(-\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots\)
3380.2.f.j	$3380$	$2$	3380.f	13.b	$2$	$18$	$18$	$26.989$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{14}q^{5}+(-\beta _{7}+\beta _{10}-\beta _{16}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3380, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3380, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(338, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(676, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(845, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1690, [\chi])\)\(^{\oplus 2}\)