Space of modular forms of level 363, weight 3, and Character 363.b

• Label: 363.3.b
• Level: $363$
• Weight: $3$
• Character orbit: 363.b
• Rep. character: $\chi_{363}(122,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $14$
• Sturm bound: $132$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(132\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(363, [\chi])\).

	Total	New	Old
Modular forms	100	82	18
Cusp forms	76	64	12
Eisenstein series	24	18	6

Trace form

\( 64 q - 108 q^{4} + 2 q^{6} + 12 q^{7} - 16 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(363, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
363.3.b.a	$363$	$3$	363.b	3.b	$2$	$1$	$1$	$9.891$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(-11\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}+4q^{4}-11q^{7}+9q^{9}-12q^{12}+\cdots\)
363.3.b.b	$363$	$3$	363.b	3.b	$2$	$1$	$1$	$9.891$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-3\)	\(0\)	\(11\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}+4q^{4}+11q^{7}+9q^{9}-12q^{12}+\cdots\)
363.3.b.c	$363$	$3$	363.b	3.b	$2$	$2$	$2$	$9.891$	\(\Q(\sqrt{-11}) \)	\(\Q(\sqrt{-11}) \)		$4$	$0$	\(0\)	\(5\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(3-\beta )q^{3}+4q^{4}+(-3+6\beta )q^{5}+\cdots\)
363.3.b.d	$363$	$3$	363.b	3.b	$2$	$2$	$2$	$9.891$	\(\Q(\sqrt{-11}) \)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(16\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{2}+3q^{3}-7q^{4}-2\beta q^{5}-3\beta q^{6}+\cdots\)
363.3.b.e	$363$	$3$	363.b	3.b	$2$	$2$	$2$	$9.891$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-3}) \)	✓	$4$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}+4q^{4}-5\beta q^{7}+9q^{9}+12q^{12}+\cdots\)
363.3.b.f	$363$	$3$	363.b	3.b	$2$	$4$	$4$	$9.891$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(-22\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(-2-2\beta _{1}-\beta _{3})q^{3}+3q^{4}+\cdots\)
363.3.b.g	$363$	$3$	363.b	3.b	$2$	$4$	$4$	$9.891$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(-2-2\beta _{1}-\beta _{3})q^{3}+3q^{4}+\cdots\)
363.3.b.h	$363$	$3$	363.b	3.b	$2$	$4$	$4$	$9.891$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$0$	\(0\)	\(-5\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{2}+\beta _{3})q^{2}+(-1+\beta _{2}+\cdots)q^{3}+\cdots\)
363.3.b.i	$363$	$3$	363.b	3.b	$2$	$4$	$4$	$9.891$	\(\Q(\sqrt{-2}, \sqrt{-7})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-1+\beta _{3})q^{3}-3q^{4}+\beta _{3}q^{5}+\cdots\)
363.3.b.j	$363$	$3$	363.b	3.b	$2$	$6$	$6$	$9.891$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(-3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
363.3.b.k	$363$	$3$	363.b	3.b	$2$	$6$	$6$	$9.891$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(-3+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
363.3.b.l	$363$	$3$	363.b	3.b	$2$	$8$	$8$	$9.891$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(5\)	\(0\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-\beta _{2}+\beta _{3}-\beta _{5}+\beta _{6})q^{3}+\cdots\)
363.3.b.m	$363$	$3$	363.b	3.b	$2$	$8$	$8$	$9.891$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(5\)	\(0\)	\(28\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-\beta _{2}+\beta _{4}+\beta _{5})q^{3}+(-4+\cdots)q^{4}+\cdots\)
363.3.b.n	$363$	$3$	363.b	3.b	$2$	$12$	$12$	$9.891$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(-4+\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(363, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(363, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 2}\)