Space of modular forms of level 72, weight 22, and trivial character

• Label: 72.22.a
• Level: $72$
• Weight: $22$
• Character orbit: 72.a
• Rep. character: $\chi_{72}(1,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $8$
• Sturm bound: $264$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(264\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(72))\).

	Total	New	Old
Modular forms	260	26	234
Cusp forms	244	26	218
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
\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(8\)
Plus space		\(+\)	\(13\)
Minus space		\(-\)	\(13\)

Trace form

26 * q - 2472384 * q^5 - 740760072 * q^7 - 121085061120 * q^11 - 565825160812 * q^13 + 12205699937184 * q^17 + 8993580821776 * q^19 - 239408951299584 * q^23 + 2400747347566174 * q^25 - 276620185757184 * q^29 - 4280636573671304 * q^31 - 18538881901537152 * q^35 - 15332418658447524 * q^37 + 83664197687897376 * q^41 + 18293685606040592 * q^43 + 812397314475154176 * q^47 + 2098801974014841898 * q^49 - 939200124166480896 * q^53 + 396614577638321408 * q^55 - 3400058099077423488 * q^59 - 5940602150803336180 * q^61 + 793956267013038336 * q^65 + 29289632598707632480 * q^67 + 46131468848571075840 * q^71 - 58026538794785885956 * q^73 - 56057761951222735488 * q^77 - 119756516804628936328 * q^79 - 385863911102294509056 * q^83 - 126657814473720645632 * q^85 + 170286990164164658016 * q^89 + 606689159467136839536 * q^91 + 950028819334172906496 * q^95 + 1319536760288857279660 * q^97

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(72))\) 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
72.22.a.a	$72$	$22$	72.a	1.a	$1$	$2$	$2$	$201.224$	\(\Q(\sqrt{537541}) \)	None	✓				24.22.a.a	$1$	$0$	\(0\)	\(0\)	\(-21948620\)	\(-659451408\)		$+$	$-$	$-$	$2^{7}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-10974310-5\beta )q^{5}+(-329725704+\cdots)q^{7}+\cdots\)
72.22.a.b	$72$	$22$	72.a	1.a	$1$	$2$	$2$	$201.224$	\(\Q(\sqrt{358549}) \)	None	✓				8.22.a.a	$1$	$1$	\(0\)	\(0\)	\(-2108140\)	\(444771792\)		$-$	$-$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1054070-20\beta )q^{5}+(222385896+\cdots)q^{7}+\cdots\)
72.22.a.c	$72$	$22$	72.a	1.a	$1$	$3$	$3$	$201.224$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				24.22.a.d	$1$	$1$	\(0\)	\(0\)	\(-5280498\)	\(852542376\)		$-$	$-$	$+$	$2^{21}\cdot 3^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-1760166+\beta _{1})q^{5}+(284180792+\cdots)q^{7}+\cdots\)
72.22.a.d	$72$	$22$	72.a	1.a	$1$	$3$	$3$	$201.224$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				24.22.a.c	$1$	$1$	\(0\)	\(0\)	\(-2080026\)	\(-1205282064\)		$-$	$-$	$+$	$2^{21}\cdot 3^{4}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-693342-\beta _{1})q^{5}+(-401760688+\cdots)q^{7}+\cdots\)
72.22.a.e	$72$	$22$	72.a	1.a	$1$	$3$	$3$	$201.224$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				24.22.a.b	$1$	$0$	\(0\)	\(0\)	\(4833126\)	\(271431024\)		$+$	$-$	$-$	$2^{21}\cdot 3^{4}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(1611042+\beta _{1})q^{5}+(90477008+\cdots)q^{7}+\cdots\)
72.22.a.f	$72$	$22$	72.a	1.a	$1$	$3$	$3$	$201.224$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				8.22.a.b	$1$	$0$	\(0\)	\(0\)	\(24111774\)	\(295988280\)		$+$	$-$	$-$	$2^{21}\cdot 3^{5}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(8037258-\beta _{2})q^{5}+(98662760+\cdots)q^{7}+\cdots\)
72.22.a.g	$72$	$22$	72.a	1.a	$1$	$5$	$5$	$201.224$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	72.22.a.g	$1$	$1$	\(0\)	\(0\)	\(-10324496\)	\(-370380036\)		$+$	$+$	$+$	$2^{39}\cdot 3^{13}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-2064899+\beta _{1})q^{5}+(-74076006+\cdots)q^{7}+\cdots\)
72.22.a.h	$72$	$22$	72.a	1.a	$1$	$5$	$5$	$201.224$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	72.22.a.g	$1$	$0$	\(0\)	\(0\)	\(10324496\)	\(-370380036\)		$-$	$+$	$-$	$2^{39}\cdot 3^{13}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(2064899-\beta _{1})q^{5}+(-74076006+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(72))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(72)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)