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

• Label: 363.3.h
• Level: $363$
• Weight: $3$
• Character orbit: 363.h
• Rep. character: $\chi_{363}(245,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $256$
• Newform subspaces: $18$
• Sturm bound: $132$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.h (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 33 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 18 \)
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	400	320	80
Cusp forms	304	256	48
Eisenstein series	96	64	32

Trace form

256 * q + 5 * q^3 + 118 * q^4 + 23 * q^6 - 2 * q^7 + 31 * q^9 + 56 * q^10 - 130 * q^12 + 10 * q^13 - 55 * q^15 - 62 * q^16 + 3 * q^18 + 22 * q^19 - 76 * q^21 - 111 * q^24 + 64 * q^25 - 4 * q^27 - 342 * q^28 + 132 * q^30 - 136 * q^31 + 36 * q^34 + 163 * q^36 + 88 * q^37 + 238 * q^39 + 242 * q^40 + 132 * q^42 + 420 * q^43 - 76 * q^45 - 2 * q^46 - 200 * q^48 + 78 * q^49 - 171 * q^51 - 362 * q^52 - 1046 * q^54 - 417 * q^57 - 18 * q^58 + 62 * q^60 - 210 * q^61 + 134 * q^63 - 322 * q^64 - 672 * q^67 - 63 * q^69 + 316 * q^70 + 842 * q^72 + 158 * q^73 + 225 * q^75 + 1164 * q^76 + 868 * q^78 + 774 * q^79 + 163 * q^81 - 48 * q^82 - 294 * q^84 - 282 * q^85 - 792 * q^87 - 160 * q^90 - 414 * q^91 + 159 * q^93 - 1042 * q^94 + 662 * q^96 - 446 * q^97

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
363.3.h.a	$363$	$3$	363.h	33.h	$10$	$4$	$1$	$9.891$	\(\Q(\zeta_{10})\)	\(\Q(\sqrt{-3}) \)		✓			363.3.b.a	$8$	$0$	\(0\)	\(3\)	\(0\)	\(-11\)	$1$	$\mathrm{U}(1)[D_{10}]$	\(q+3\zeta_{10}^{3}q^{3}+4\zeta_{10}^{2}q^{4}+11\zeta_{10}^{2}q^{7}+\cdots\)
363.3.h.b	$363$	$3$	363.h	33.h	$10$	$4$	$1$	$9.891$	\(\Q(\zeta_{10})\)	\(\Q(\sqrt{-3}) \)		✓			363.3.b.a	$8$	$0$	\(0\)	\(3\)	\(0\)	\(11\)	$1$	$\mathrm{U}(1)[D_{10}]$	\(q+3\zeta_{10}^{3}q^{3}+4\zeta_{10}^{2}q^{4}-11\zeta_{10}^{2}q^{7}+\cdots\)
363.3.h.c	$363$	$3$	363.h	33.h	$10$	$8$	$2$	$9.891$	8.0.324000000.3	\(\Q(\sqrt{-3}) \)		✓			363.3.b.e	$16$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{10}]$	\(q+(-3-3\beta _{2}-3\beta _{4}-3\beta _{6})q^{3}+4\beta _{2}q^{4}+\cdots\)
363.3.h.d	$363$	$3$	363.h	33.h	$10$	$8$	$2$	$9.891$	8.0.228765625.1	None					33.3.b.a	$8$	$0$	\(0\)	\(-6\)	\(0\)	\(-16\)	$2^{4}$	$\mathrm{SU}(2)[C_{10}]$	\(q+\beta _{7}q^{2}+3\beta _{4}q^{3}-7\beta _{2}q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots\)
363.3.h.e	$363$	$3$	363.h	33.h	$10$	$8$	$2$	$9.891$	8.0.228765625.1	None					33.3.b.a	$8$	$0$	\(0\)	\(-6\)	\(0\)	\(16\)	$2^{4}$	$\mathrm{SU}(2)[C_{10}]$	\(q+\beta _{7}q^{2}+3\beta _{4}q^{3}-7\beta _{2}q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)
363.3.h.f	$363$	$3$	363.h	33.h	$10$	$8$	$2$	$9.891$	8.0.228765625.1	\(\Q(\sqrt{-11}) \)		✓			363.3.b.c	$16$	$0$	\(0\)	\(-5\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{10}]$	\(q+(-\beta _{1}-2\beta _{3})q^{3}+4\beta _{7}q^{4}+(-3\beta _{4}+\cdots)q^{5}+\cdots\)
363.3.h.g	$363$	$3$	363.h	33.h	$10$	$8$	$2$	$9.891$	\(\Q(\zeta_{20})\)	None					33.3.h.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\zeta_{20}q^{2}+(2-2\zeta_{20}-2\zeta_{20}^{2}+\zeta_{20}^{3}+\cdots)q^{3}+\cdots\)
363.3.h.h	$363$	$3$	363.h	33.h	$10$	$8$	$2$	$9.891$	\(\Q(\zeta_{20})\)	None					33.3.h.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(26\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\zeta_{20}q^{2}+(2-2\zeta_{20}-2\zeta_{20}^{2}+\zeta_{20}^{3}+\cdots)q^{3}+\cdots\)
363.3.h.i	$363$	$3$	363.h	33.h	$10$	$8$	$2$	$9.891$	\(\Q(\zeta_{20})\)	None					33.3.h.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(-26\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\zeta_{20}q^{2}+(2+2\zeta_{20}-2\zeta_{20}^{2}-\zeta_{20}^{3}+\cdots)q^{3}+\cdots\)
363.3.h.j	$363$	$3$	363.h	33.h	$10$	$16$	$4$	$9.891$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					33.3.h.b	$4$	$0$	\(0\)	\(-10\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{6}+\beta _{10})q^{3}+(-1+\cdots)q^{4}+\cdots\)
363.3.h.k	$363$	$3$	363.h	33.h	$10$	$16$	$4$	$9.891$	16.0.\(\cdots\).1	None		✓			363.3.b.i	$16$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{16}$	$\mathrm{SU}(2)[C_{10}]$	\(q+\beta _{11}q^{2}+(-\beta _{6}-\beta _{13})q^{3}+3\beta _{9}q^{4}+\cdots\)
363.3.h.l	$363$	$3$	363.h	33.h	$10$	$16$	$4$	$9.891$	16.0.\(\cdots\).1	None					33.3.b.b	$8$	$0$	\(0\)	\(5\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{10}]$	\(q+(\beta _{1}+\beta _{4}+\beta _{5}-\beta _{8}+\beta _{11}-\beta _{13}+\cdots)q^{2}+\cdots\)
363.3.h.m	$363$	$3$	363.h	33.h	$10$	$16$	$4$	$9.891$	16.0.\(\cdots\).1	None					33.3.b.b	$8$	$0$	\(0\)	\(5\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{10}]$	\(q+(\beta _{1}+\beta _{4}+\beta _{5}-\beta _{8}+\beta _{11}-\beta _{13}+\cdots)q^{2}+\cdots\)
363.3.h.n	$363$	$3$	363.h	33.h	$10$	$16$	$4$	$9.891$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					33.3.h.b	$4$	$0$	\(0\)	\(5\)	\(0\)	\(34\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\beta _{4}q^{2}+(-\beta _{2}+\beta _{6}+\beta _{13})q^{3}+(1+\cdots)q^{4}+\cdots\)
363.3.h.o	$363$	$3$	363.h	33.h	$10$	$16$	$4$	$9.891$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					33.3.h.b	$4$	$0$	\(0\)	\(5\)	\(0\)	\(-34\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+\beta _{4}q^{2}+(-\beta _{2}+\beta _{6}-\beta _{12})q^{3}+(1+\cdots)q^{4}+\cdots\)
363.3.h.p	$363$	$3$	363.h	33.h	$10$	$24$	$6$	$9.891$		None		✓			363.3.b.j	$8$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$
363.3.h.q	$363$	$3$	363.h	33.h	$10$	$24$	$6$	$9.891$		None		✓			363.3.b.j	$8$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$
363.3.h.r	$363$	$3$	363.h	33.h	$10$	$48$	$12$	$9.891$		None		✓	✓		363.3.b.n	$16$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$

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

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