Space of modular forms of level 352, weight 2, and trivial character

• Label: 352.2.a
• Level: $352$
• Weight: $2$
• Character orbit: 352.a
• Rep. character: $\chi_{352}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $8$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(352))\).

	Total	New	Old
Modular forms	56	10	46
Cusp forms	41	10	31
Eisenstein series	15	0	15

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

\(2\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(12\)	\(2\)	\(10\)		\(9\)	\(2\)	\(7\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(14\)	\(3\)	\(11\)		\(10\)	\(3\)	\(7\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(16\)	\(3\)	\(13\)		\(12\)	\(3\)	\(9\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(14\)	\(2\)	\(12\)		\(10\)	\(2\)	\(8\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(26\)	\(4\)	\(22\)		\(19\)	\(4\)	\(15\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(30\)	\(6\)	\(24\)		\(22\)	\(6\)	\(16\)		\(8\)	\(0\)	\(8\)

Trace form

10 * q + 4 * q^5 + 10 * q^9 - 12 * q^13 - 12 * q^17 - 2 * q^25 + 20 * q^29 + 4 * q^37 - 12 * q^41 + 12 * q^45 - 6 * q^49 - 20 * q^53 + 16 * q^57 + 4 * q^61 + 8 * q^65 - 8 * q^69 - 28 * q^73 - 6 * q^81 + 24 * q^85 - 52 * q^89 - 40 * q^93 - 20 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(352))\) 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	11
352.2.a.a	$352$	$2$	352.a	1.a	$1$	$1$	$1$	$2.811$	\(\Q\)	None	✓	✓			352.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}+6q^{9}+q^{11}-6q^{13}+\cdots\)
352.2.a.b	$352$	$2$	352.a	1.a	$1$	$1$	$1$	$2.811$	\(\Q\)	None	✓	✓			352.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+4q^{7}-2q^{9}-q^{11}+\cdots\)
352.2.a.c	$352$	$2$	352.a	1.a	$1$	$1$	$1$	$2.811$	\(\Q\)	None	✓	✓			352.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-4q^{7}-2q^{9}-q^{11}-2q^{13}+\cdots\)
352.2.a.d	$352$	$2$	352.a	1.a	$1$	$1$	$1$	$2.811$	\(\Q\)	None	✓	✓			352.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}-4q^{7}-2q^{9}+q^{11}+\cdots\)
352.2.a.e	$352$	$2$	352.a	1.a	$1$	$1$	$1$	$2.811$	\(\Q\)	None	✓	✓			352.2.a.c	$1$	$0$	\(0\)	\(1\)	\(1\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+4q^{7}-2q^{9}+q^{11}-2q^{13}+\cdots\)
352.2.a.f	$352$	$2$	352.a	1.a	$1$	$1$	$1$	$2.811$	\(\Q\)	None	✓	✓			352.2.a.a	$1$	$0$	\(0\)	\(3\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}+6q^{9}-q^{11}-6q^{13}+\cdots\)
352.2.a.g	$352$	$2$	352.a	1.a	$1$	$2$	$2$	$2.811$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓		352.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(2-\beta )q^{5}+(1+\beta )q^{9}-q^{11}+\cdots\)
352.2.a.h	$352$	$2$	352.a	1.a	$1$	$2$	$2$	$2.811$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓		352.2.a.g	$1$	$0$	\(0\)	\(1\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-\beta )q^{5}+(1+\beta )q^{9}+q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(352))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(352)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 2}\)