Space of modular forms of level 396, weight 4, and trivial character

• Label: 396.4.a
• Level: $396$
• Weight: $4$
• Character orbit: 396.a
• Rep. character: $\chi_{396}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $10$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	396.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(396))\).

	Total	New	Old
Modular forms	228	13	215
Cusp forms	204	13	191
Eisenstein series	24	0	24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(32\)	\(0\)	\(32\)		\(28\)	\(0\)	\(28\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(26\)	\(0\)	\(26\)		\(22\)	\(0\)	\(22\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(29\)	\(0\)	\(29\)		\(25\)	\(0\)	\(25\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(29\)	\(0\)	\(29\)		\(25\)	\(0\)	\(25\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(28\)	\(3\)	\(25\)		\(26\)	\(3\)	\(23\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(28\)	\(3\)	\(25\)		\(26\)	\(3\)	\(23\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(28\)	\(4\)	\(24\)		\(26\)	\(4\)	\(22\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(28\)	\(3\)	\(25\)		\(26\)	\(3\)	\(23\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(117\)	\(7\)	\(110\)		\(105\)	\(7\)	\(98\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(111\)	\(6\)	\(105\)		\(99\)	\(6\)	\(93\)		\(12\)	\(0\)	\(12\)

Trace form

13 * q - 24 * q^5 - 8 * q^7 - 11 * q^11 - 46 * q^13 - 18 * q^17 + 164 * q^19 + 189 * q^25 - 66 * q^29 - 220 * q^31 + 404 * q^37 + 546 * q^41 - 228 * q^43 + 348 * q^47 + 1101 * q^49 + 354 * q^53 + 198 * q^55 - 588 * q^59 - 1422 * q^61 + 396 * q^65 + 308 * q^67 + 216 * q^71 - 554 * q^73 - 220 * q^77 + 160 * q^79 + 36 * q^83 + 1080 * q^85 - 1656 * q^89 + 336 * q^91 + 168 * q^95 + 2016 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(396))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	11
396.4.a.a	$396$	$4$	396.a	1.a	$1$	$1$	$1$	$23.365$	\(\Q\)	None	✓				132.4.a.c	$1$	$0$	\(0\)	\(0\)	\(-22\)	\(-20\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-22q^{5}-20q^{7}-11q^{11}+22q^{13}+\cdots\)
396.4.a.b	$396$	$4$	396.a	1.a	$1$	$1$	$1$	$23.365$	\(\Q\)	None	✓	✓			396.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-12\)	\(26\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-12q^{5}+26q^{7}-11q^{11}-34q^{13}+\cdots\)
396.4.a.c	$396$	$4$	396.a	1.a	$1$	$1$	$1$	$23.365$	\(\Q\)	None	✓				132.4.a.d	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{5}+8q^{7}+11q^{11}+18q^{13}+\cdots\)
396.4.a.d	$396$	$4$	396.a	1.a	$1$	$1$	$1$	$23.365$	\(\Q\)	None	✓				132.4.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+11q^{11}-88q^{13}+66q^{17}+\cdots\)
396.4.a.e	$396$	$4$	396.a	1.a	$1$	$1$	$1$	$23.365$	\(\Q\)	None	✓				44.4.a.a	$1$	$1$	\(0\)	\(0\)	\(7\)	\(-26\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{5}-26q^{7}+11q^{11}+52q^{13}+\cdots\)
396.4.a.f	$396$	$4$	396.a	1.a	$1$	$1$	$1$	$23.365$	\(\Q\)	None	✓				132.4.a.a	$1$	$0$	\(0\)	\(0\)	\(12\)	\(14\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}+14q^{7}-11q^{11}+56q^{13}+\cdots\)
396.4.a.g	$396$	$4$	396.a	1.a	$1$	$1$	$1$	$23.365$	\(\Q\)	None	✓	✓			396.4.a.b	$1$	$0$	\(0\)	\(0\)	\(12\)	\(26\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{5}+26q^{7}+11q^{11}-34q^{13}+\cdots\)
396.4.a.h	$396$	$4$	396.a	1.a	$1$	$2$	$2$	$23.365$	\(\Q(\sqrt{31}) \)	None	✓	✓	✓		396.4.a.h	$1$	$0$	\(0\)	\(0\)	\(-12\)	\(-24\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-6+\beta )q^{5}+(-12+\beta )q^{7}+11q^{11}+\cdots\)
396.4.a.i	$396$	$4$	396.a	1.a	$1$	$2$	$2$	$23.365$	\(\Q(\sqrt{97}) \)	None	✓		✓		44.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-11\)	\(10\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-5-\beta )q^{5}+(2+6\beta )q^{7}-11q^{11}+\cdots\)
396.4.a.j	$396$	$4$	396.a	1.a	$1$	$2$	$2$	$23.365$	\(\Q(\sqrt{31}) \)	None	✓	✓	✓		396.4.a.h	$1$	$1$	\(0\)	\(0\)	\(12\)	\(-24\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(6+\beta )q^{5}+(-12-\beta )q^{7}-11q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(396))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(396)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)