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

• Label: 396.2.r
• Level: $396$
• Weight: $2$
• Character orbit: 396.r
• Rep. character: $\chi_{396}(19,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $112$
• Newform subspaces: $3$
• Sturm bound: $144$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	396.r (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 44 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(144\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	320	128	192
Cusp forms	256	112	144
Eisenstein series	64	16	48

Trace form

112 * q + 5 * q^2 - q^4 + 6 * q^5 + 5 * q^8 - 10 * q^13 + 6 * q^14 - 9 * q^16 + 10 * q^17 + 6 * q^20 - 19 * q^22 - 34 * q^25 + 42 * q^26 + 10 * q^29 - 30 * q^34 + 18 * q^37 - 20 * q^38 - 40 * q^40 + 30 * q^41 - 6 * q^44 - 30 * q^46 - 2 * q^49 - 55 * q^50 - 50 * q^52 + 26 * q^53 - 48 * q^56 - 34 * q^58 - 10 * q^61 - 70 * q^62 - q^64 - 60 * q^68 - 84 * q^70 + 10 * q^73 - 90 * q^74 + 6 * q^77 - 30 * q^80 + 25 * q^82 + 30 * q^85 + 31 * q^86 - 45 * q^88 - 12 * q^89 - 78 * q^92 + 90 * q^94 + 12 * q^97

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.r.a	$396$	$2$	396.r	44.g	$10$	$16$	$4$	$3.162$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					44.2.g.a	$4$	$0$	\(5\)	\(0\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q-\beta _{2}q^{2}+(-\beta _{2}-\beta _{4}-\beta _{5}+\beta _{6}+\beta _{10}+\cdots)q^{4}+\cdots\)
396.2.r.b	$396$	$2$	396.r	44.g	$10$	$48$	$12$	$3.162$		None					132.2.j.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$
396.2.r.c	$396$	$2$	396.r	44.g	$10$	$48$	$12$	$3.162$		None		✓			396.2.r.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$

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

\( S_{2}^{\mathrm{old}}(396, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(44, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(132, [\chi])\)\(^{\oplus 2}\)