Space of modular forms of level 336, weight 4, and Character 336.q

• Label: 336.4.q
• Level: $336$
• Weight: $4$
• Character orbit: 336.q
• Rep. character: $\chi_{336}(193,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $48$
• Newform subspaces: $13$
• Sturm bound: $256$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.q (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$7$
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$13$
Sturm bound:			$256$
Trace bound:			$5$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(336, [\chi])$.

	Total	New	Old
Modular forms	408	48	360
Cusp forms	360	48	312
Eisenstein series	48	0	48

Trace form

48 * q + 6 * q^3 - 18 * q^7 - 216 * q^9 + 20 * q^11 + 318 * q^19 + 80 * q^23 - 516 * q^25 - 108 * q^27 - 400 * q^29 - 402 * q^31 - 12 * q^33 + 468 * q^35 + 8 * q^37 + 210 * q^39 - 592 * q^41 - 892 * q^43 - 204 * q^47 + 592 * q^49 + 16 * q^53 - 376 * q^55 + 168 * q^57 + 688 * q^59 + 200 * q^61 - 162 * q^63 - 280 * q^65 - 750 * q^67 - 1704 * q^71 + 1092 * q^73 + 1050 * q^75 + 184 * q^77 - 942 * q^79 - 1944 * q^81 - 4152 * q^83 + 1296 * q^85 - 1044 * q^87 + 1712 * q^89 - 1138 * q^91 - 456 * q^93 - 36 * q^95 + 2328 * q^97 - 360 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(336, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
336.4.q.a	$336$	$4$	336.q	7.c	$3$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None					168.4.q.a	$2$	$0$	$0$	$-3$	$-7$	$35$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-3+3\zeta_{6})q^{3}-7\zeta_{6}q^{5}+(21-7\zeta_{6})q^{7}+\cdots$
336.4.q.b	$336$	$4$	336.q	7.c	$3$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None					168.4.q.b	$2$	$0$	$0$	$-3$	$-2$	$-35$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-3+3\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-14+\cdots)q^{7}+\cdots$
336.4.q.c	$336$	$4$	336.q	7.c	$3$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None					168.4.q.c	$2$	$0$	$0$	$-3$	$11$	$-7$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-3+3\zeta_{6})q^{3}+11\zeta_{6}q^{5}+(7-21\zeta_{6})q^{7}+\cdots$
336.4.q.d	$336$	$4$	336.q	7.c	$3$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None					42.4.e.b	$2$	$0$	$0$	$-3$	$15$	$-35$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-3+3\zeta_{6})q^{3}+15\zeta_{6}q^{5}+(-21+\cdots)q^{7}+\cdots$
336.4.q.e	$336$	$4$	336.q	7.c	$3$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None					21.4.e.a	$2$	$0$	$0$	$3$	$3$	$7$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(3-3\zeta_{6})q^{3}+3\zeta_{6}q^{5}+(-7+21\zeta_{6})q^{7}+\cdots$
336.4.q.f	$336$	$4$	336.q	7.c	$3$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None					42.4.e.a	$2$	$0$	$0$	$3$	$6$	$7$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(3-3\zeta_{6})q^{3}+6\zeta_{6}q^{5}+(14-21\zeta_{6})q^{7}+\cdots$
336.4.q.g	$336$	$4$	336.q	7.c	$3$	$4$	$2$	$19.825$	$\Q(\sqrt{-3}, \sqrt{505})$	None					168.4.q.d	$2$	$0$	$0$	$-6$	$-9$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-3+3\beta _{2})q^{3}+(-\beta _{1}-4\beta _{2})q^{5}+\cdots$
336.4.q.h	$336$	$4$	336.q	7.c	$3$	$4$	$2$	$19.825$	$\Q(\sqrt{-3}, \sqrt{-19})$	None					84.4.i.b	$2$	$0$	$0$	$-6$	$3$	$20$	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+3\beta _{2}q^{3}+(1+2\beta _{1}+2\beta _{2}-\beta _{3})q^{5}+\cdots$
336.4.q.i	$336$	$4$	336.q	7.c	$3$	$4$	$2$	$19.825$	$\Q(\sqrt{-3}, \sqrt{193})$	None					84.4.i.a	$2$	$0$	$0$	$6$	$-11$	$-6$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(3-3\beta _{2})q^{3}+(\beta _{1}-6\beta _{2})q^{5}+(1-2\beta _{1}+\cdots)q^{7}+\cdots$
336.4.q.j	$336$	$4$	336.q	7.c	$3$	$4$	$2$	$19.825$	$\Q(\sqrt{-3}, \sqrt{1345})$	None					42.4.e.c	$2$	$0$	$0$	$6$	$-5$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(3-3\beta _{2})q^{3}+(-\beta _{1}-2\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots$
336.4.q.k	$336$	$4$	336.q	7.c	$3$	$6$	$3$	$19.825$	6.0.9924270768.1	None					21.4.e.b	$2$	$0$	$0$	$-9$	$-11$	$13$	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	$q+3\beta _{3}q^{3}+(-4+\beta _{2}-4\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots$
336.4.q.l	$336$	$4$	336.q	7.c	$3$	$6$	$3$	$19.825$	6.0.$\cdots$.1	None					168.4.q.e	$2$	$0$	$0$	$9$	$11$	$1$	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	$q+(3-3\beta _{3})q^{3}+(4\beta _{3}-\beta _{4}-\beta _{5})q^{5}+\cdots$
336.4.q.m	$336$	$4$	336.q	7.c	$3$	$8$	$4$	$19.825$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None			✓		168.4.q.f	$2$	$0$	$0$	$12$	$-4$	$-18$	$2^{6}\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	$q-3\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{6})q^{5}+(-2+\cdots)q^{7}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(336, [\chi])$ into lower level spaces

$S_{4}^{\mathrm{old}}(336, [\chi]) \simeq$ $S_{4}^{\mathrm{new}}(7, [\chi])$$^{\oplus 10}$$\oplus$$S_{4}^{\mathrm{new}}(14, [\chi])$$^{\oplus 8}$$\oplus$$S_{4}^{\mathrm{new}}(21, [\chi])$$^{\oplus 5}$$\oplus$$S_{4}^{\mathrm{new}}(28, [\chi])$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(42, [\chi])$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(56, [\chi])$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(84, [\chi])$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(112, [\chi])$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(168, [\chi])$$^{\oplus 2}$