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

• Label: 336.4.bc
• Level: $336$
• Weight: $4$
• Character orbit: 336.bc
• Rep. character: $\chi_{336}(17,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $92$
• Newform subspaces: $6$
• Sturm bound: $256$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.bc (of order $6$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$21$
Character field:			$\Q(\zeta_{6})$
Newform subspaces:			$6$
Sturm bound:			$256$
Trace bound:			$3$
Distinguishing $T_p$:			$5$, $13$

Dimensions

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

	Total	New	Old
Modular forms	408	100	308
Cusp forms	360	92	268
Eisenstein series	48	8	40

Trace form

92 * q + 3 * q^3 - 14 * q^7 - q^9 - 50 * q^15 + 276 * q^19 - 97 * q^21 - 868 * q^25 - 624 * q^31 - 3 * q^33 - 2 * q^37 - 302 * q^39 - 580 * q^43 - 663 * q^45 - 388 * q^49 + 141 * q^51 + 790 * q^57 - 6 * q^61 - 1207 * q^63 - 1192 * q^67 - 330 * q^73 + 222 * q^75 + 1124 * q^79 + 459 * q^81 - 804 * q^85 + 3756 * q^87 - 3822 * q^91 + q^93 - 3566 * 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.bc.a	$336$	$4$	336.bc	21.g	$6$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	$\Q(\sqrt{-3})$					84.4.k.b	$4$	$0$	$0$	$-9$	$0$	$-37$	$1$	$\mathrm{U}(1)[D_{6}]$	$q+(-3-3\zeta_{6})q^{3}+(-18-\zeta_{6})q^{7}+\cdots$
336.4.bc.b	$336$	$4$	336.bc	21.g	$6$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	$\Q(\sqrt{-3})$					84.4.k.a	$4$	$0$	$0$	$9$	$0$	$17$	$1$	$\mathrm{U}(1)[D_{6}]$	$q+(3+3\zeta_{6})q^{3}+(18-19\zeta_{6})q^{7}+3^{3}\zeta_{6}q^{9}+\cdots$
336.4.bc.c	$336$	$4$	336.bc	21.g	$6$	$12$	$6$	$19.825$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None					84.4.k.c	$4$	$0$	$0$	$0$	$0$	$42$	$2^{4}\cdot 3^{7}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\beta _{3}+\beta _{4})q^{3}-\beta _{11}q^{5}+(2-2\beta _{1}+\cdots)q^{7}+\cdots$
336.4.bc.d	$336$	$4$	336.bc	21.g	$6$	$12$	$6$	$19.825$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None					21.4.g.a	$4$	$0$	$0$	$3$	$0$	$56$	$2^{4}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{1}q^{3}+\beta _{4}q^{5}+(4+2\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots$
336.4.bc.e	$336$	$4$	336.bc	21.g	$6$	$16$	$8$	$19.825$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None					42.4.f.a	$4$	$0$	$0$	$0$	$0$	$-80$	$2^{8}\cdot 3^{11}$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{4}q^{3}-\beta _{9}q^{5}+(-6-\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots$
336.4.bc.f	$336$	$4$	336.bc	21.g	$6$	$48$	$24$	$19.825$		None			✓		168.4.u.a	$4$	$0$	$0$	$0$	$0$	$-12$		$\mathrm{SU}(2)[C_{6}]$

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

$S_{4}^{\mathrm{old}}(336, [\chi]) \simeq$ $S_{4}^{\mathrm{new}}(21, [\chi])$$^{\oplus 5}$$\oplus$$S_{4}^{\mathrm{new}}(42, [\chi])$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(84, [\chi])$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(168, [\chi])$$^{\oplus 2}$