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

• Label: 416.2.i
• Level: $416$
• Weight: $2$
• Character orbit: 416.i
• Rep. character: $\chi_{416}(289,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $28$
• Newform subspaces: $7$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
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 + 4 * q^5 - 14 * q^9 - 6 * q^13 + 6 * q^17 - 16 * q^21 + 16 * q^25 - 2 * q^29 - 26 * q^37 + 22 * q^41 - 10 * q^45 - 14 * q^49 - 12 * q^53 + 64 * q^57 + 6 * q^61 + 30 * q^65 - 16 * q^69 - 28 * q^73 + 16 * q^77 - 30 * q^81 + 18 * q^85 - 28 * q^89 - 8 * q^93 - 12 * 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.i.a	$416$	$2$	416.i	13.c	$3$	$2$	$1$	$3.322$	\(\Q(\sqrt{-3}) \)	None		✓			416.2.i.a	$2$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}+q^{5}-\zeta_{6}q^{9}+(-4+\cdots)q^{11}+\cdots\)
416.2.i.b	$416$	$2$	416.i	13.c	$3$	$2$	$1$	$3.322$	\(\Q(\sqrt{-3}) \)	None		✓			416.2.i.a	$2$	$0$	\(0\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}+q^{5}-\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots\)
416.2.i.c	$416$	$2$	416.i	13.c	$3$	$4$	$2$	$3.322$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			416.2.i.c	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+2\beta _{3}q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
416.2.i.d	$416$	$2$	416.i	13.c	$3$	$4$	$2$	$3.322$	\(\Q(\sqrt{-3}, \sqrt{11})\)	None		✓			416.2.i.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+8\beta _{2}q^{9}+\cdots\)
416.2.i.e	$416$	$2$	416.i	13.c	$3$	$4$	$2$	$3.322$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-1}) \)		✓			416.2.i.e	$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{3}]$	\(q+(\beta_{3}-2\beta_1+1)q^{5}+3\beta_{2} q^{9}+\cdots\)
416.2.i.f	$416$	$2$	416.i	13.c	$3$	$4$	$2$	$3.322$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			416.2.i.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{3}-2\beta _{3}q^{5}+(\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)
416.2.i.g	$416$	$2$	416.i	13.c	$3$	$8$	$4$	$3.322$	8.0.6927565824.3	None		✓	✓		416.2.i.g	$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{2}-\beta _{7})q^{3}+(-1+\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)

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

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