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

• Label: 416.2.w
• Level: $416$
• Weight: $2$
• Character orbit: 416.w
• Rep. character: $\chi_{416}(225,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $28$
• Newform subspaces: $4$
• Sturm bound: $112$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.w (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(112\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	128	28	100
Cusp forms	96	28	68
Eisenstein series	32	0	32

Trace form

\( 28 q - 14 q^{9} + O(q^{10}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
416.2.w.a	$416$	$2$	416.w	13.e	$6$	$4$	$2$	$3.322$	\(\Q(\zeta_{12})\)	None		✓			416.2.w.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+(-2+4\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
416.2.w.b	$416$	$2$	416.w	13.e	$6$	$4$	$2$	$3.322$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-1}) \)		✓			416.2.w.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{6}]$	\(q+(-1+2\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}+(3-3\zeta_{12}^{2}+\cdots)q^{9}+\cdots\)
416.2.w.c	$416$	$2$	416.w	13.e	$6$	$8$	$4$	$3.322$	8.0.56070144.2	None		✓			416.2.w.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{6}q^{3}+(-\beta _{4}+\beta _{5})q^{5}+\beta _{3}q^{7}+\cdots\)
416.2.w.d	$416$	$2$	416.w	13.e	$6$	$12$	$6$	$3.322$	12.0.\(\cdots\).1	None		✓	✓		416.2.w.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{9}q^{3}+(-2\beta _{4}-\beta _{5}+\beta _{6})q^{5}-\beta _{3}q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(416, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(208, [\chi])\)\(^{\oplus 2}\)