Space of modular forms of level 416, weight 2, and Character 416.f

• Label: 416.2.f
• Level: $416$
• Weight: $2$
• Character orbit: 416.f
• Rep. character: $\chi_{416}(129,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $5$
• Sturm bound: $112$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(112\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(416, [\chi])\).

	Total	New	Old
Modular forms	64	14	50
Cusp forms	48	14	34
Eisenstein series	16	0	16

Trace form

\( 14 q + 14 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(416, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
416.2.f.a	$416$	$2$	416.f	13.b	$2$	$2$	$2$	$3.322$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(-6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3q^{3}+3iq^{5}-iq^{7}+6q^{9}+4iq^{11}+\cdots\)
416.2.f.b	$416$	$2$	416.f	13.b	$2$	$2$	$2$	$3.322$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+2iq^{5}-3q^{9}+(-3+i)q^{13}-2q^{17}+\cdots\)
416.2.f.c	$416$	$2$	416.f	13.b	$2$	$2$	$2$	$3.322$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3q^{3}+3iq^{5}+iq^{7}+6q^{9}-4iq^{11}+\cdots\)
416.2.f.d	$416$	$2$	416.f	13.b	$2$	$4$	$4$	$3.322$	\(\Q(i, \sqrt{13})\)	\(\Q(\sqrt{-13}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{7}-3q^{9}+(\beta _{1}+\beta _{2})q^{11}-\beta _{3}q^{13}+\cdots\)
416.2.f.e	$416$	$2$	416.f	13.b	$2$	$4$	$4$	$3.322$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}-\beta _{1}q^{5}-\beta _{2}q^{7}+2q^{9}+(2+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(416, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(416, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(208, [\chi])\)\(^{\oplus 2}\)