Space of modular forms of level 72, weight 4, and trivial character

• Label: 72.4.a
• Level: $72$
• Weight: $4$
• Character orbit: 72.a
• Rep. character: $\chi_{72}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$72 = 2^{3} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	72.a (trivial)
Character field:			$\Q$
Newform subspaces:			$4$
Sturm bound:			$48$
Trace bound:			$5$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_0(72))$.

	Total	New	Old
Modular forms	44	4	40
Cusp forms	28	4	24
Eisenstein series	16	0	16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$2$	$3$	Fricke	Dim
$+$	$+$	$+$	$1$
$+$	$-$	$-$	$1$
$-$	$+$	$-$	$1$
$-$	$-$	$+$	$1$
Plus space		$+$	$2$
Minus space		$-$	$2$

Trace form

4 * q - 12 * q^5 - 24 * q^7 + 72 * q^11 + 64 * q^13 - 132 * q^17 - 136 * q^19 + 48 * q^23 + 212 * q^25 - 60 * q^29 - 40 * q^31 + 384 * q^35 - 168 * q^37 - 84 * q^41 - 344 * q^43 - 768 * q^47 + 68 * q^49 + 372 * q^53 + 1744 * q^55 + 72 * q^59 - 584 * q^61 + 1080 * q^65 - 1240 * q^67 - 1584 * q^71 - 2048 * q^73 + 384 * q^77 + 1528 * q^79 + 1272 * q^83 + 2072 * q^85 - 468 * q^89 + 912 * q^91 - 1200 * q^95 + 800 * q^97

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_0(72))$ 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3
72.4.a.a	$72$	$4$	72.a	1.a	$1$	$1$	$1$	$4.248$	$\Q$	None	✓	✓	✓	✓	72.4.a.a	$1$	$1$	$0$	$0$	$-16$	$-12$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2^{4}q^{5}-12q^{7}-2^{6}q^{11}+58q^{13}+\cdots$
72.4.a.b	$72$	$4$	72.a	1.a	$1$	$1$	$1$	$4.248$	$\Q$	None	✓		✓	✓	24.4.a.a	$1$	$1$	$0$	$0$	$-14$	$-24$		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-14q^{5}-24q^{7}+28q^{11}-74q^{13}+\cdots$
72.4.a.c	$72$	$4$	72.a	1.a	$1$	$1$	$1$	$4.248$	$\Q$	None	✓		✓	✓	8.4.a.a	$1$	$0$	$0$	$0$	$2$	$24$		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{5}+24q^{7}+44q^{11}+22q^{13}+\cdots$
72.4.a.d	$72$	$4$	72.a	1.a	$1$	$1$	$1$	$4.248$	$\Q$	None	✓	✓	✓	✓	72.4.a.a	$1$	$0$	$0$	$0$	$16$	$-12$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+2^{4}q^{5}-12q^{7}+2^{6}q^{11}+58q^{13}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_0(72))$ into lower level spaces

$S_{4}^{\mathrm{old}}(\Gamma_0(72)) \simeq$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(9))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(12))$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(18))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(24))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_0(36))$$^{\oplus 2}$