Space of modular forms of level 363, weight 4, and Character 363.d

• Label: 363.4.d
• Level: $363$
• Weight: $4$
• Character orbit: 363.d
• Rep. character: $\chi_{363}(362,\cdot)$
• Character field: $\Q$
• Dimension: $100$
• Newform subspaces: $5$
• Sturm bound: $176$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	363.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 33 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(176\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(363, [\chi])\).

	Total	New	Old
Modular forms	144	116	28
Cusp forms	120	100	20
Eisenstein series	24	16	8

Trace form

100 * q + 4 * q^3 + 372 * q^4 + 36 * q^9 + 102 * q^12 - 32 * q^15 + 1420 * q^16 - 2068 * q^25 - 608 * q^27 - 104 * q^31 - 908 * q^34 + 1382 * q^36 - 140 * q^37 + 220 * q^42 + 1542 * q^45 + 1494 * q^48 - 1044 * q^49 - 2844 * q^58 - 3612 * q^60 + 5136 * q^64 - 2140 * q^67 + 2760 * q^69 - 5356 * q^70 - 3666 * q^75 + 1372 * q^78 + 5732 * q^81 + 5808 * q^82 - 1608 * q^91 - 2336 * q^93 + 3100 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(363, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
363.4.d.a	$363$	$4$	363.d	33.d	$2$	$4$	$4$	$21.418$	\(\Q(\sqrt{-2}, \sqrt{3})\)	\(\Q(\sqrt{-3}) \)		✓			363.4.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3^{2}\cdot 11$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{3}-8q^{4}+(-2\beta _{2}+\beta _{3})q^{7}+\cdots\)
363.4.d.b	$363$	$4$	363.d	33.d	$2$	$4$	$4$	$21.418$	\(\Q(\sqrt{-2}, \sqrt{13})\)	None		✓			363.4.d.b	$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(1-\beta _{1})q^{3}+5q^{4}+4\beta _{1}q^{5}+\cdots\)
363.4.d.c	$363$	$4$	363.d	33.d	$2$	$12$	$12$	$21.418$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			363.4.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{7}q^{2}-\beta _{1}q^{3}+(8-\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots\)
363.4.d.d	$363$	$4$	363.d	33.d	$2$	$40$	$40$	$21.418$		None					33.4.f.a	$4$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
363.4.d.e	$363$	$4$	363.d	33.d	$2$	$40$	$40$	$21.418$		None		✓			363.4.d.e	$4$	$0$	\(0\)	\(8\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{4}^{\mathrm{old}}(363, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(363, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 2}\)