Space of modular forms of level 324, weight 4, and Character 324.b

• Label: 324.4.b
• Level: $324$
• Weight: $4$
• Character orbit: 324.b
• Rep. character: $\chi_{324}(323,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $4$
• Sturm bound: $216$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$324 = 2^{2} \cdot 3^{4}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	324.b (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$12$
Character field:			$\Q$
Newform subspaces:			$4$
Sturm bound:			$216$
Trace bound:			$1$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(324, [\chi])$.

	Total	New	Old
Modular forms	174	76	98
Cusp forms	150	68	82
Eisenstein series	24	8	16

Trace form

68 * q + 2 * q^4 - 20 * q^10 + 4 * q^13 - 178 * q^16 + 6 * q^22 - 1296 * q^25 - 384 * q^28 - 410 * q^34 - 404 * q^37 - 428 * q^40 - 924 * q^46 - 2152 * q^49 + 664 * q^52 - 704 * q^58 - 464 * q^61 - 958 * q^64 + 2316 * q^70 - 836 * q^73 + 594 * q^76 - 506 * q^82 - 1828 * q^85 + 702 * q^88 + 2976 * q^94 - 32 * q^97

Decomposition of $S_{4}^{\mathrm{new}}(324, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
324.4.b.a	$324$	$4$	324.b	12.b	$2$	$4$	$4$	$19.117$	$\Q(\sqrt{-2}, \sqrt{3})$	$\Q(\sqrt{-1})$		✓			324.4.b.a	$4$	$0$	$0$	$0$	$0$	$0$	$3^{4}$	$\mathrm{U}(1)[D_{2}]$	$q+2\beta _{1}q^{2}-8q^{4}+(-7\beta _{1}+\beta _{2})q^{5}+\cdots$
324.4.b.b	$324$	$4$	324.b	12.b	$2$	$8$	$8$	$19.117$	8.0.$\cdots$.1	None					36.4.h.a	$4$	$0$	$0$	$0$	$0$	$0$	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{1}q^{2}+(-3-\beta _{2}+\beta _{3})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
324.4.b.c	$324$	$4$	324.b	12.b	$2$	$24$	$24$	$19.117$		None					36.4.h.b	$4$	$0$	$0$	$0$	$0$	$0$		$\mathrm{SU}(2)[C_{2}]$
324.4.b.d	$324$	$4$	324.b	12.b	$2$	$32$	$32$	$19.117$		None		✓	✓		324.4.b.d	$4$	$0$	$0$	$0$	$0$	$0$		$\mathrm{SU}(2)[C_{2}]$

Decomposition of $S_{4}^{\mathrm{old}}(324, [\chi])$ into lower level spaces

$S_{4}^{\mathrm{old}}(324, [\chi]) \simeq$ $S_{4}^{\mathrm{new}}(12, [\chi])$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(36, [\chi])$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(108, [\chi])$$^{\oplus 2}$