Space of modular forms of level 72, weight 14, and trivial character

• Label: 72.14.a
• Level: $72$
• Weight: $14$
• Character orbit: 72.a
• Rep. character: $\chi_{72}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $8$
• Sturm bound: $168$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(168\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{14}(\Gamma_0(72))\).

	Total	New	Old
Modular forms	164	16	148
Cusp forms	148	16	132
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(40\)	\(3\)	\(37\)		\(36\)	\(3\)	\(33\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(41\)	\(5\)	\(36\)		\(37\)	\(5\)	\(32\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(42\)	\(3\)	\(39\)		\(38\)	\(3\)	\(35\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(41\)	\(5\)	\(36\)		\(37\)	\(5\)	\(32\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(81\)	\(8\)	\(73\)		\(73\)	\(8\)	\(65\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(83\)	\(8\)	\(75\)		\(75\)	\(8\)	\(67\)		\(8\)	\(0\)	\(8\)

Trace form

16 * q + 45396 * q^5 + 29064 * q^7 - 5029128 * q^11 + 16094584 * q^13 + 1786284 * q^17 + 597489512 * q^19 + 421440912 * q^23 + 5189153624 * q^25 - 1464068988 * q^29 - 4641056104 * q^31 - 16796644992 * q^35 - 37601177856 * q^37 - 12731964228 * q^41 - 47694177032 * q^43 + 3458822400 * q^47 + 383446429808 * q^49 + 109401913620 * q^53 + 222753221488 * q^55 + 344001682296 * q^59 - 601091200064 * q^61 - 513334877544 * q^65 - 1918401530344 * q^67 + 416079972720 * q^71 - 1745725305944 * q^73 - 603188475264 * q^77 + 2862376781944 * q^79 - 587288394360 * q^83 + 7541995342808 * q^85 + 3301156233468 * q^89 + 7073230080144 * q^91 - 9753652546704 * q^95 + 2482183898984 * q^97

Decomposition of \(S_{14}^{\mathrm{new}}(\Gamma_0(72))\) 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
72.14.a.a	$72$	$14$	72.a	1.a	$1$	$1$	$1$	$77.206$	\(\Q\)	None	✓				8.14.a.a	$1$	$1$	\(0\)	\(0\)	\(4330\)	\(-139992\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4330q^{5}-139992q^{7}+6484324q^{11}+\cdots\)
72.14.a.b	$72$	$14$	72.a	1.a	$1$	$1$	$1$	$77.206$	\(\Q\)	None	✓				24.14.a.a	$1$	$0$	\(0\)	\(0\)	\(22490\)	\(181272\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+22490q^{5}+181272q^{7}+9261428q^{11}+\cdots\)
72.14.a.c	$72$	$14$	72.a	1.a	$1$	$2$	$2$	$77.206$	\(\Q(\sqrt{781}) \)	None	✓				8.14.a.b	$1$	$0$	\(0\)	\(0\)	\(-18476\)	\(110928\)		$+$	$-$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-9238-4\beta )q^{5}+(55464-74\beta )q^{7}+\cdots\)
72.14.a.d	$72$	$14$	72.a	1.a	$1$	$2$	$2$	$77.206$	\(\Q(\sqrt{62869}) \)	None	✓				24.14.a.d	$1$	$1$	\(0\)	\(0\)	\(-5068\)	\(104880\)		$-$	$-$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2534-\beta )q^{5}+(52440+5\beta )q^{7}+\cdots\)
72.14.a.e	$72$	$14$	72.a	1.a	$1$	$2$	$2$	$77.206$	\(\Q(\sqrt{1621}) \)	None	✓				24.14.a.c	$1$	$1$	\(0\)	\(0\)	\(11204\)	\(275808\)		$-$	$-$	$+$	$2^{9}\cdot 3$	$\mathrm{SU}(2)$	\(q+(5602-\beta )q^{5}+(137904-13\beta )q^{7}+\cdots\)
72.14.a.f	$72$	$14$	72.a	1.a	$1$	$2$	$2$	$77.206$	\(\Q(\sqrt{406}) \)	None	✓				24.14.a.b	$1$	$0$	\(0\)	\(0\)	\(30916\)	\(-532896\)		$+$	$-$	$-$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(15458+7\beta )q^{5}+(-266448-37\beta )q^{7}+\cdots\)
72.14.a.g	$72$	$14$	72.a	1.a	$1$	$3$	$3$	$77.206$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	72.14.a.g	$1$	$1$	\(0\)	\(0\)	\(-50256\)	\(14532\)		$+$	$+$	$+$	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-16752+\beta _{2})q^{5}+(4844-\beta _{1}+\cdots)q^{7}+\cdots\)
72.14.a.h	$72$	$14$	72.a	1.a	$1$	$3$	$3$	$77.206$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	72.14.a.g	$1$	$0$	\(0\)	\(0\)	\(50256\)	\(14532\)		$-$	$+$	$-$	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(16752-\beta _{2})q^{5}+(4844-\beta _{1}-3\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{14}^{\mathrm{old}}(\Gamma_0(72))\) into lower level spaces

\( S_{14}^{\mathrm{old}}(\Gamma_0(72)) \simeq \) \(S_{14}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)