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

• Label: 416.2.e
• Level: $416$
• Weight: $2$
• Character orbit: 416.e
• Rep. character: $\chi_{416}(337,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $3$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 104 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
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	16	48
Cusp forms	48	12	36
Eisenstein series	16	4	12

Trace form

\( 12 q - 12 q^{9} - 8 q^{17} - 8 q^{23} - 4 q^{25} - 16 q^{39} + 12 q^{49} + 56 q^{55} - 16 q^{79} + 4 q^{81} - 8 q^{87} + 16 q^{95}+O(q^{100}) \)

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.e.a	$416$	$2$	416.e	104.e	$2$	$2$	$2$	$3.322$	\(\Q(\sqrt{-1}) \)	None					104.2.e.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-q^{5}-3 i q^{7}+2 q^{9}+2 q^{11}+\cdots\)
416.2.e.b	$416$	$2$	416.e	104.e	$2$	$2$	$2$	$3.322$	\(\Q(\sqrt{-1}) \)	None					104.2.e.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+q^{5}+3 i q^{7}+2 q^{9}-2 q^{11}+\cdots\)
416.2.e.c	$416$	$2$	416.e	104.e	$2$	$8$	$8$	$3.322$	8.0.4521217600.1	None			✓		104.2.e.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{4}q^{5}+\beta _{3}q^{7}+(-3-\beta _{5}+\cdots)q^{9}+\cdots\)

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

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