Space of modular forms of level 363, weight 4, and trivial character

• Label: 363.4.a
• Level: $363$
• Weight: $4$
• Character orbit: 363.a
• Rep. character: $\chi_{363}(1,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $22$
• Sturm bound: $176$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(176\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(363))\).

	Total	New	Old
Modular forms	144	54	90
Cusp forms	120	54	66
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.

\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(38\)	\(14\)	\(24\)		\(32\)	\(14\)	\(18\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(34\)	\(13\)	\(21\)		\(28\)	\(13\)	\(15\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(34\)	\(10\)	\(24\)		\(28\)	\(10\)	\(18\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(38\)	\(17\)	\(21\)		\(32\)	\(17\)	\(15\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(76\)	\(31\)	\(45\)		\(64\)	\(31\)	\(33\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(68\)	\(23\)	\(45\)		\(56\)	\(23\)	\(33\)		\(12\)	\(0\)	\(12\)

Trace form

54 * q + 4 * q^2 + 196 * q^4 + 16 * q^5 + 12 * q^6 + 32 * q^7 - 36 * q^8 + 486 * q^9 + 88 * q^10 + 24 * q^12 + 116 * q^13 + 132 * q^14 - 60 * q^15 + 804 * q^16 - 152 * q^17 + 36 * q^18 - 8 * q^19 + 556 * q^20 + 84 * q^21 - 168 * q^23 + 324 * q^24 + 1354 * q^25 - 572 * q^26 + 192 * q^28 + 120 * q^29 - 336 * q^30 - 180 * q^31 - 364 * q^32 - 48 * q^34 - 920 * q^35 + 1764 * q^36 + 352 * q^37 + 592 * q^38 + 336 * q^39 + 168 * q^40 + 760 * q^41 - 828 * q^42 - 200 * q^43 + 144 * q^45 + 464 * q^46 + 616 * q^47 + 816 * q^48 + 2266 * q^49 - 148 * q^50 + 624 * q^51 - 264 * q^52 + 808 * q^53 + 108 * q^54 - 300 * q^56 - 84 * q^57 - 668 * q^58 + 688 * q^59 + 600 * q^60 + 1180 * q^61 - 3032 * q^62 + 288 * q^63 + 3688 * q^64 + 880 * q^65 + 1792 * q^67 - 1000 * q^68 - 1320 * q^69 + 1740 * q^70 - 1144 * q^71 - 324 * q^72 - 724 * q^73 - 2256 * q^74 - 528 * q^75 - 2048 * q^76 - 456 * q^78 + 144 * q^79 + 6992 * q^80 + 4374 * q^81 + 3324 * q^82 - 264 * q^83 + 2880 * q^84 + 1840 * q^85 - 796 * q^86 - 1368 * q^87 + 880 * q^89 + 792 * q^90 - 5840 * q^91 - 2496 * q^92 - 1392 * q^93 - 5104 * q^94 + 168 * q^95 + 3084 * q^96 - 348 * q^97 + 8580 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\) 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
363.4.a.a	$363$	$4$	363.a	1.a	$1$	$1$	$1$	$21.418$	\(\Q\)	None	✓	✓			363.4.a.a	$1$	$1$	\(-4\)	\(-3\)	\(-13\)	\(-26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-3q^{3}+8q^{4}-13q^{5}+12q^{6}+\cdots\)
363.4.a.b	$363$	$4$	363.a	1.a	$1$	$1$	$1$	$21.418$	\(\Q\)	None	✓	✓			363.4.a.b	$1$	$1$	\(-3\)	\(3\)	\(-12\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+3q^{3}+q^{4}-12q^{5}-9q^{6}+\cdots\)
363.4.a.c	$363$	$4$	363.a	1.a	$1$	$1$	$1$	$21.418$	\(\Q\)	None	✓	✓			363.4.a.c	$1$	$1$	\(-1\)	\(-3\)	\(7\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}-7q^{4}+7q^{5}+3q^{6}+\cdots\)
363.4.a.d	$363$	$4$	363.a	1.a	$1$	$1$	$1$	$21.418$	\(\Q\)	None	✓				33.4.a.b	$1$	$1$	\(1\)	\(-3\)	\(-4\)	\(26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-7q^{4}-4q^{5}-3q^{6}+\cdots\)
363.4.a.e	$363$	$4$	363.a	1.a	$1$	$1$	$1$	$21.418$	\(\Q\)	None	✓	✓			363.4.a.c	$1$	$1$	\(1\)	\(-3\)	\(7\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-7q^{4}+7q^{5}-3q^{6}+\cdots\)
363.4.a.f	$363$	$4$	363.a	1.a	$1$	$1$	$1$	$21.418$	\(\Q\)	None	✓	✓			363.4.a.b	$1$	$1$	\(3\)	\(3\)	\(-12\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+3q^{3}+q^{4}-12q^{5}+9q^{6}+\cdots\)
363.4.a.g	$363$	$4$	363.a	1.a	$1$	$1$	$1$	$21.418$	\(\Q\)	None	✓	✓			363.4.a.a	$1$	$1$	\(4\)	\(-3\)	\(-13\)	\(26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-3q^{3}+8q^{4}-13q^{5}-12q^{6}+\cdots\)
363.4.a.h	$363$	$4$	363.a	1.a	$1$	$1$	$1$	$21.418$	\(\Q\)	None	✓				33.4.a.a	$1$	$0$	\(5\)	\(3\)	\(-14\)	\(32\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+3q^{3}+17q^{4}-14q^{5}+15q^{6}+\cdots\)
363.4.a.i	$363$	$4$	363.a	1.a	$1$	$2$	$2$	$21.418$	\(\Q(\sqrt{97}) \)	None	✓				33.4.a.c	$1$	$1$	\(-1\)	\(-6\)	\(-14\)	\(-24\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-3q^{3}+(2^{4}+\beta )q^{4}+(-6-2\beta )q^{5}+\cdots\)
363.4.a.j	$363$	$4$	363.a	1.a	$1$	$2$	$2$	$21.418$	\(\Q(\sqrt{33}) \)	None	✓				33.4.a.d	$1$	$0$	\(-1\)	\(6\)	\(16\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{3}+\beta q^{4}+(10-4\beta )q^{5}+\cdots\)
363.4.a.k	$363$	$4$	363.a	1.a	$1$	$2$	$2$	$21.418$	\(\Q(\sqrt{5}) \)	None	✓	✓			363.4.a.k	$2$	$0$	\(0\)	\(-6\)	\(-4\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-3q^{3}-3q^{4}-2q^{5}+3\beta q^{6}+\cdots\)
363.4.a.l	$363$	$4$	363.a	1.a	$1$	$2$	$2$	$21.418$	\(\Q(\sqrt{3}) \)	None	✓	✓			363.4.a.l	$2$	$0$	\(0\)	\(-6\)	\(18\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3\beta q^{2}-3q^{3}+19q^{4}+9q^{5}-9\beta q^{6}+\cdots\)
363.4.a.m	$363$	$4$	363.a	1.a	$1$	$2$	$2$	$21.418$	\(\Q(\sqrt{3}) \)	None	✓	✓			363.4.a.m	$2$	$1$	\(0\)	\(6\)	\(-6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+3q^{3}-5q^{4}-3q^{5}+3\beta q^{6}+\cdots\)
363.4.a.n	$363$	$4$	363.a	1.a	$1$	$2$	$2$	$21.418$	\(\Q(\sqrt{5}) \)	None	✓				33.4.e.a	$1$	$1$	\(0\)	\(6\)	\(-13\)	\(-44\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2-4\beta )q^{2}+3q^{3}+12q^{4}+(-12+\cdots)q^{5}+\cdots\)
363.4.a.o	$363$	$4$	363.a	1.a	$1$	$2$	$2$	$21.418$	\(\Q(\sqrt{5}) \)	None	✓				33.4.e.a	$1$	$0$	\(0\)	\(6\)	\(-13\)	\(44\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2-4\beta )q^{2}+3q^{3}+12q^{4}+(-1+\cdots)q^{5}+\cdots\)
363.4.a.p	$363$	$4$	363.a	1.a	$1$	$4$	$4$	$21.418$	4.4.5225.1	None	✓		✓		33.4.e.b	$1$	$1$	\(-3\)	\(12\)	\(12\)	\(-11\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}-\beta _{2})q^{2}+3q^{3}+(-2+\cdots)q^{4}+\cdots\)
363.4.a.q	$363$	$4$	363.a	1.a	$1$	$4$	$4$	$21.418$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			363.4.a.q	$1$	$0$	\(-1\)	\(12\)	\(14\)	\(20\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(6+\beta _{1}+\beta _{3})q^{4}+\cdots\)
363.4.a.r	$363$	$4$	363.a	1.a	$1$	$4$	$4$	$21.418$	\(\Q(\sqrt{3}, \sqrt{5})\)	None	✓	✓			363.4.a.r	$2$	$0$	\(0\)	\(-12\)	\(16\)	\(0\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}+\beta _{3})q^{2}-3q^{3}+2\beta _{2}q^{4}+\cdots\)
363.4.a.s	$363$	$4$	363.a	1.a	$1$	$4$	$4$	$21.418$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			363.4.a.q	$1$	$0$	\(1\)	\(12\)	\(14\)	\(-20\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{1}+\beta _{3})q^{4}+\cdots\)
363.4.a.t	$363$	$4$	363.a	1.a	$1$	$4$	$4$	$21.418$	4.4.5225.1	None	✓				33.4.e.b	$1$	$0$	\(3\)	\(12\)	\(12\)	\(11\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+3q^{3}+(-2+3\beta _{1}+\cdots)q^{4}+\cdots\)
363.4.a.u	$363$	$4$	363.a	1.a	$1$	$6$	$6$	$21.418$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		33.4.e.c	$1$	$1$	\(-5\)	\(-18\)	\(9\)	\(-1\)		$+$	$-$	$-$	$11$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
363.4.a.v	$363$	$4$	363.a	1.a	$1$	$6$	$6$	$21.418$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		33.4.e.c	$1$	$0$	\(5\)	\(-18\)	\(9\)	\(1\)		$+$	$+$	$+$	$11$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(363))\) into lower level spaces

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