Space of modular forms of level 368, weight 2, and trivial character

• Label: 368.2.a
• Level: $368$
• Weight: $2$
• Character orbit: 368.a
• Rep. character: $\chi_{368}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $9$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	368.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(368))\).

	Total	New	Old
Modular forms	54	11	43
Cusp forms	43	11	32
Eisenstein series	11	0	11

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

\(2\)	\(23\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(9\)	\(3\)	\(6\)		\(7\)	\(3\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(18\)	\(3\)	\(15\)		\(15\)	\(3\)	\(12\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(12\)	\(4\)	\(8\)		\(9\)	\(4\)	\(5\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(15\)	\(1\)	\(14\)		\(12\)	\(1\)	\(11\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(24\)	\(4\)	\(20\)		\(19\)	\(4\)	\(15\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(30\)	\(7\)	\(23\)		\(24\)	\(7\)	\(17\)		\(6\)	\(0\)	\(6\)

Trace form

11 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 7 * q^9 - 2 * q^13 - 2 * q^17 + 4 * q^19 - 3 * q^23 + 5 * q^25 + 2 * q^27 - 2 * q^29 + 10 * q^31 - 8 * q^33 - 10 * q^37 + 14 * q^39 + 6 * q^41 - 10 * q^45 - 22 * q^47 + 11 * q^49 + 4 * q^51 - 10 * q^53 - 24 * q^55 + 8 * q^57 - 4 * q^59 - 18 * q^61 + 32 * q^63 + 12 * q^65 - 8 * q^67 - 6 * q^71 - 2 * q^73 + 6 * q^75 - 8 * q^77 + 28 * q^79 + 3 * q^81 - 20 * q^83 - 12 * q^85 - 46 * q^87 + 6 * q^89 + 4 * q^91 + 12 * q^93 + 8 * q^95 + 6 * q^97 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(368))\) 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	23
368.2.a.a	$368$	$2$	368.a	1.a	$1$	$1$	$1$	$2.938$	\(\Q\)	None	✓				184.2.a.d	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{7}+6q^{9}-5q^{13}-6q^{17}+\cdots\)
368.2.a.b	$368$	$2$	368.a	1.a	$1$	$1$	$1$	$2.938$	\(\Q\)	None	✓		✓	✓	92.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}-2q^{9}-q^{13}-6q^{17}+\cdots\)
368.2.a.c	$368$	$2$	368.a	1.a	$1$	$1$	$1$	$2.938$	\(\Q\)	None	✓				184.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-3q^{9}-6q^{11}-2q^{13}+6q^{17}+\cdots\)
368.2.a.d	$368$	$2$	368.a	1.a	$1$	$1$	$1$	$2.938$	\(\Q\)	None	✓				46.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+4q^{7}-3q^{9}-2q^{11}-2q^{13}+\cdots\)
368.2.a.e	$368$	$2$	368.a	1.a	$1$	$1$	$1$	$2.938$	\(\Q\)	None	✓				184.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-2q^{7}-2q^{9}+4q^{11}+\cdots\)
368.2.a.f	$368$	$2$	368.a	1.a	$1$	$1$	$1$	$2.938$	\(\Q\)	None	✓				184.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+4q^{7}-2q^{9}+2q^{11}+\cdots\)
368.2.a.g	$368$	$2$	368.a	1.a	$1$	$1$	$1$	$2.938$	\(\Q\)	None	✓				92.2.a.a	$1$	$0$	\(0\)	\(3\)	\(-2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{5}+4q^{7}+6q^{9}-2q^{11}+\cdots\)
368.2.a.h	$368$	$2$	368.a	1.a	$1$	$2$	$2$	$2.938$	\(\Q(\sqrt{5}) \)	None	✓		✓		23.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1-\beta )q^{5}+(-1+\beta )q^{7}+\cdots\)
368.2.a.i	$368$	$2$	368.a	1.a	$1$	$2$	$2$	$2.938$	\(\Q(\sqrt{17}) \)	None	✓		✓		184.2.a.e	$1$	$0$	\(0\)	\(1\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2q^{5}+(1+\beta )q^{9}-2\beta q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(368))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(368)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 2}\)