Space of modular forms of level 363, weight 2, and trivial character

• Label: 363.2.a
• Level: $363$
• Weight: $2$
• Character orbit: 363.a
• Rep. character: $\chi_{363}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $10$
• Sturm bound: $88$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(88\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(363))\).

	Total	New	Old
Modular forms	56	19	37
Cusp forms	33	19	14
Eisenstein series	23	0	23

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

\(3\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(-\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(8\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(13\)

Trace form

19 * q - q^2 + q^3 + 21 * q^4 + 2 * q^5 + q^6 - 4 * q^7 + 3 * q^8 + 19 * q^9 + 2 * q^10 - q^12 + 2 * q^13 - 12 * q^14 - 2 * q^15 + 13 * q^16 + 2 * q^17 - q^18 - 10 * q^20 + 4 * q^21 - 12 * q^23 - 3 * q^24 + 21 * q^25 - 10 * q^26 + q^27 + 4 * q^28 + 6 * q^29 - 2 * q^30 + 4 * q^31 - 5 * q^32 - 10 * q^34 + 8 * q^35 + 21 * q^36 - 10 * q^37 - 4 * q^38 - 2 * q^39 - 6 * q^40 + 2 * q^41 + 2 * q^45 - 8 * q^46 - 16 * q^47 - q^48 + 3 * q^49 + q^50 - 2 * q^51 - 2 * q^52 - 10 * q^53 + q^54 - 12 * q^56 - 14 * q^58 - 16 * q^59 - 10 * q^60 - 6 * q^61 + 8 * q^62 - 4 * q^63 - 15 * q^64 - 4 * q^65 - 8 * q^67 - 2 * q^68 + 4 * q^69 - 20 * q^70 - 20 * q^71 + 3 * q^72 + 14 * q^73 - 6 * q^74 - 9 * q^75 - 14 * q^78 + 4 * q^79 - 38 * q^80 + 19 * q^81 - 30 * q^82 - 12 * q^83 - 4 * q^84 - 4 * q^85 - 32 * q^86 - 6 * q^87 - 10 * q^89 + 2 * q^90 + 20 * q^91 - 24 * q^92 + 28 * q^93 - 8 * q^94 + 5 * q^96 + 18 * q^97 - 9 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	11
363.2.a.a	$363$	$2$	363.a	1.a	$1$	$1$	$1$	$2.899$	\(\Q\)	None	✓	✓			363.2.a.a	$1$	$0$	\(-2\)	\(-1\)	\(4\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}+4q^{5}+2q^{6}+\cdots\)
363.2.a.b	$363$	$2$	363.a	1.a	$1$	$1$	$1$	$2.899$	\(\Q\)	None	✓				33.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}-4q^{7}+\cdots\)
363.2.a.c	$363$	$2$	363.a	1.a	$1$	$1$	$1$	$2.899$	\(\Q\)	None	✓	✓			363.2.a.a	$1$	$0$	\(2\)	\(-1\)	\(4\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}+4q^{5}-2q^{6}+\cdots\)
363.2.a.d	$363$	$2$	363.a	1.a	$1$	$2$	$2$	$2.899$	\(\Q(\sqrt{5}) \)	None	✓				33.2.e.b	$1$	$1$	\(-3\)	\(-2\)	\(1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}+(1-\beta )q^{5}+\cdots\)
363.2.a.e	$363$	$2$	363.a	1.a	$1$	$2$	$2$	$2.899$	\(\Q(\sqrt{5}) \)	None	✓		✓	✓	33.2.e.a	$1$	$1$	\(-1\)	\(2\)	\(-3\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
363.2.a.f	$363$	$2$	363.a	1.a	$1$	$2$	$2$	$2.899$	\(\Q(\sqrt{3}) \)	None	✓	✓			363.2.a.f	$2$	$1$	\(0\)	\(-2\)	\(-6\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+q^{4}-3q^{5}-\beta q^{6}-2\beta q^{7}+\cdots\)
363.2.a.g	$363$	$2$	363.a	1.a	$1$	$2$	$2$	$2.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			363.2.a.g	$2$	$0$	\(0\)	\(2\)	\(4\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+3q^{4}+2q^{5}-\beta q^{6}+\cdots\)
363.2.a.h	$363$	$2$	363.a	1.a	$1$	$2$	$2$	$2.899$	\(\Q(\sqrt{5}) \)	None	✓				33.2.e.a	$1$	$0$	\(1\)	\(2\)	\(-3\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
363.2.a.i	$363$	$2$	363.a	1.a	$1$	$2$	$2$	$2.899$	\(\Q(\sqrt{5}) \)	None	✓		✓		33.2.e.b	$1$	$0$	\(3\)	\(-2\)	\(1\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+3\beta q^{4}+(1-\beta )q^{5}+\cdots\)
363.2.a.j	$363$	$2$	363.a	1.a	$1$	$4$	$4$	$2.899$	\(\Q(\sqrt{3}, \sqrt{11})\)	None	✓	✓	✓		363.2.a.j	$2$	$0$	\(0\)	\(4\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{3})q^{4}-\beta _{3}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(363))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(363)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)