Space of modular forms of level 363, weight 2, and Character 363.f

• Label: 363.2.f
• Level: $363$
• Weight: $2$
• Character orbit: 363.f
• Rep. character: $\chi_{363}(161,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $112$
• Newform subspaces: $9$
• Sturm bound: $88$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.f (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 33 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(88\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(363, [\chi])\).

	Total	New	Old
Modular forms	224	176	48
Cusp forms	128	112	16
Eisenstein series	96	64	32

Trace form

112 * q + 7 * q^3 - 14 * q^4 + 10 * q^7 - 11 * q^9 - 68 * q^12 + 10 * q^13 + 9 * q^15 - 2 * q^16 - 20 * q^19 + 10 * q^24 - 28 * q^25 + 16 * q^27 + 20 * q^30 + 26 * q^31 - 24 * q^34 + 16 * q^36 + 8 * q^37 - 20 * q^39 - 60 * q^42 - 100 * q^45 - 30 * q^46 - 6 * q^48 - 50 * q^49 - 30 * q^51 - 10 * q^52 + 30 * q^57 - 28 * q^58 + 44 * q^60 + 10 * q^61 + 30 * q^63 + 86 * q^64 - 68 * q^67 + 11 * q^69 + 34 * q^70 + 20 * q^72 + 24 * q^75 - 12 * q^78 - 50 * q^79 + 5 * q^81 + 6 * q^82 - 10 * q^85 - 40 * q^90 + 54 * q^91 - 13 * q^93 + 30 * q^94 - 10 * q^96 - 52 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(363, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
363.2.f.a	$363$	$2$	363.f	33.f	$10$	$8$	$2$	$2.899$	8.0.64000000.1	None		✓			363.2.d.a	$8$	$0$	\(-2\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q-\beta _{6}q^{2}+(-1+\beta _{2}-\beta _{3}-\beta _{4}+\beta _{6}+\cdots)q^{3}+\cdots\)
363.2.f.b	$363$	$2$	363.f	33.f	$10$	$8$	$2$	$2.899$	\(\Q(\zeta_{20})\)	None					33.2.f.a	$4$	$0$	\(0\)	\(-6\)	\(0\)	\(10\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(-\zeta_{20}^{5}-\zeta_{20}^{7})q^{2}+(-1-\zeta_{20}+\cdots)q^{3}+\cdots\)
363.2.f.c	$363$	$2$	363.f	33.f	$10$	$8$	$2$	$2.899$	8.0.228765625.1	\(\Q(\sqrt{-11}) \)					33.2.d.a	$16$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{10}]$	\(q-\beta _{1}q^{3}-2\beta _{7}q^{4}+(-\beta _{4}-2\beta _{5})q^{5}+\cdots\)
363.2.f.d	$363$	$2$	363.f	33.f	$10$	$8$	$2$	$2.899$	\(\Q(\zeta_{20})\)	None					33.2.f.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(10\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(-\zeta_{20}-\zeta_{20}^{7})q^{2}+(1+\zeta_{20}-\zeta_{20}^{2}+\cdots)q^{3}+\cdots\)
363.2.f.e	$363$	$2$	363.f	33.f	$10$	$8$	$2$	$2.899$	\(\Q(\zeta_{20})\)	None					33.2.f.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(-\zeta_{20}-\zeta_{20}^{7})q^{2}+(1-\zeta_{20}-\zeta_{20}^{2}+\cdots)q^{3}+\cdots\)
363.2.f.f	$363$	$2$	363.f	33.f	$10$	$8$	$2$	$2.899$	8.0.64000000.1	None		✓			363.2.d.a	$8$	$0$	\(2\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(1-\beta _{2}+\beta _{4}-\beta _{6})q^{2}+(\beta _{1}-\beta _{3}+\cdots)q^{3}+\cdots\)
363.2.f.g	$363$	$2$	363.f	33.f	$10$	$16$	$4$	$2.899$	16.0.\(\cdots\).7	\(\Q(\sqrt{-3}) \)		✓			363.2.d.d	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$11^{4}$	$\mathrm{U}(1)[D_{10}]$	\(q+(\beta _{1}+\beta _{5}-\beta _{10}-\beta _{13})q^{3}+2\beta _{4}q^{4}+\cdots\)
363.2.f.h	$363$	$2$	363.f	33.f	$10$	$16$	$4$	$2.899$	16.0.\(\cdots\).1	None		✓			363.2.d.c	$16$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(\beta _{2}-2\beta _{6}-\beta _{8}+\beta _{12}+\beta _{13})q^{2}+\cdots\)
363.2.f.i	$363$	$2$	363.f	33.f	$10$	$32$	$8$	$2.899$		None		✓	✓		363.2.d.e	$16$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$

Decomposition of \(S_{2}^{\mathrm{old}}(363, [\chi])\) into lower level spaces

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