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 + 6 * q^13 - 6 * q^17 - 40 * q^25 + 2 * q^29 + 30 * q^37 + 18 * q^41 - 30 * q^45 + 30 * q^49 - 12 * q^53 - 10 * q^61 + 10 * q^65 + 16 * q^69 + 48 * q^77 + 2 * q^81 - 6 * q^85 - 72 * q^93 - 48 * q^97

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	$4$	$0$	$0$	$0$	$0$	$0$	$2^{2}$	$\mathrm{U}(1)[D_{6}]$	$q+(\beta_{3}+2\beta_{2}-1)q^{5}+(-3\beta_{2}+3)q^{9}+\cdots$
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$$S_{2}^{\mathrm{new}}(52, [\chi])$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(104, [\chi])$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(208, [\chi])$$^{\oplus 2}$