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

• Label: 336.4.k
• Level: $336$
• Weight: $4$
• Character orbit: 336.k
• Rep. character: $\chi_{336}(209,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $5$
• Sturm bound: $256$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.k (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$21$
Character field:			$\Q$
Newform subspaces:			$5$
Sturm bound:			$256$
Trace bound:			$7$
Distinguishing $T_p$:			$5$

Dimensions

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

	Total	New	Old
Modular forms	204	50	154
Cusp forms	180	46	134
Eisenstein series	24	4	20

Trace form

46 * q + 20 * q^7 - 2 * q^9 + 56 * q^15 + 94 * q^21 + 1114 * q^25 - 4 * q^37 + 224 * q^39 + 592 * q^43 + 382 * q^49 + 912 * q^51 + 476 * q^57 + 1180 * q^63 + 64 * q^67 + 1528 * q^79 + 462 * q^81 - 1152 * q^85 + 2904 * q^91 + 164 * q^93 - 2080 * 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.k.a	$336$	$4$	336.k	21.c	$2$	$2$	$2$	$19.825$	$\Q(\sqrt{-3})$	$\Q(\sqrt{-3})$					21.4.c.a	$4$	$0$	$0$	$0$	$0$	$20$	$2\cdot 3$	$\mathrm{U}(1)[D_{2}]$	$q+\beta q^{3}+(-3\beta+10)q^{7}-27 q^{9}+\cdots$
336.4.k.b	$336$	$4$	336.k	21.c	$2$	$4$	$4$	$19.825$	$\Q(\sqrt{-6}, \sqrt{-17})$	None					21.4.c.b	$4$	$0$	$0$	$0$	$0$	$-28$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(-\beta _{1}-\beta _{3})q^{3}+(-\beta _{1}-2\beta _{3})q^{5}+\cdots$
336.4.k.c	$336$	$4$	336.k	21.c	$2$	$8$	$8$	$19.825$	8.0.$\cdots$.13	None					42.4.d.a	$4$	$0$	$0$	$0$	$0$	$-4$	$2^{5}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{6}q^{3}+(\beta _{2}-\beta _{6}-\beta _{7})q^{5}+(-1+\cdots)q^{7}+\cdots$
336.4.k.d	$336$	$4$	336.k	21.c	$2$	$8$	$8$	$19.825$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None					84.4.f.a	$4$	$0$	$0$	$0$	$0$	$20$	$2^{7}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{2}q^{3}-\beta _{5}q^{5}+(3+\beta _{7})q^{7}+(-2+\cdots)q^{9}+\cdots$
336.4.k.e	$336$	$4$	336.k	21.c	$2$	$24$	$24$	$19.825$		None			✓		168.4.k.a	$4$	$0$	$0$	$0$	$0$	$12$		$\mathrm{SU}(2)[C_{2}]$

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}$