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

• Label: 416.2.b
• Level: $416$
• Weight: $2$
• Character orbit: 416.b
• Rep. character: $\chi_{416}(209,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $3$
• Sturm bound: $112$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(112\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	64	12	52
Cusp forms	48	12	36
Eisenstein series	16	0	16

Trace form

12 * q + 4 * q^7 - 12 * q^9 - 8 * q^15 - 8 * q^23 - 12 * q^25 + 4 * q^31 + 8 * q^33 + 8 * q^39 - 8 * q^41 + 4 * q^47 + 12 * q^49 - 16 * q^55 - 8 * q^57 + 20 * q^63 + 36 * q^71 + 24 * q^79 + 4 * q^81 - 56 * q^87 + 16 * q^89 - 24 * q^95 + 16 * 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.b.a	$416$	$2$	416.b	8.b	$2$	$2$	$2$	$3.322$	\(\Q(\sqrt{-1}) \)	None					104.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+3 i q^{5}-3 q^{7}+2 q^{9}+i q^{13}+\cdots\)
416.2.b.b	$416$	$2$	416.b	8.b	$2$	$4$	$4$	$3.322$	\(\Q(\zeta_{12})\)	None					104.2.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta_1 q^{3}-2\beta_{2} q^{5}+(-\beta_{3}+3)q^{7}+\cdots\)
416.2.b.c	$416$	$2$	416.b	8.b	$2$	$6$	$6$	$3.322$	6.0.399424.1	None			✓		104.2.b.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}-\beta _{2}q^{5}+\beta _{1}q^{7}+(-2+\beta _{1}+\cdots)q^{9}+\cdots\)

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

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