Space of modular forms of level 396, weight 2, and Character 396.x

• Label: 396.2.x
• Level: $396$
• Weight: $2$
• Character orbit: 396.x
• Rep. character: $\chi_{396}(17,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $16$
• Newform subspaces: $1$
• Sturm bound: $144$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	396.x (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 33 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(144\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(396, [\chi])\).

	Total	New	Old
Modular forms	336	16	320
Cusp forms	240	16	224
Eisenstein series	96	0	96

Trace form

\( 16 q + 48 q^{25} + 8 q^{31} + 36 q^{37} + 24 q^{49} - 88 q^{55} - 80 q^{61} - 64 q^{67} - 20 q^{73} - 40 q^{79} + 60 q^{85} + 12 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(396, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
396.2.x.a	$396$	$2$	396.x	33.f	$10$	$16$	$4$	$3.162$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓	✓	✓	396.2.x.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(-\beta _{4}+2\beta _{5}+\beta _{13}-\beta _{15})q^{5}+(-1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(396, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(396, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(66, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(132, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(198, [\chi])\)\(^{\oplus 2}\)