Space of modular forms of level 36, weight 18, and trivial character

• Label: 36.18.a
• Level: $36$
• Weight: $18$
• Character orbit: 36.a
• Rep. character: $\chi_{36}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $5$
• Sturm bound: $108$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	36.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(108\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(36))\).

	Total	New	Old
Modular forms	108	7	101
Cusp forms	96	7	89
Eisenstein series	12	0	12

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
\(+\)	\(+\)	\(+\)		\(27\)	\(0\)	\(27\)		\(23\)	\(0\)	\(23\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(28\)	\(0\)	\(28\)		\(24\)	\(0\)	\(24\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(27\)	\(3\)	\(24\)		\(25\)	\(3\)	\(22\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(26\)	\(4\)	\(22\)		\(24\)	\(4\)	\(20\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(53\)	\(4\)	\(49\)		\(47\)	\(4\)	\(43\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(55\)	\(3\)	\(52\)		\(49\)	\(3\)	\(46\)		\(6\)	\(0\)	\(6\)

Trace form

7 * q + 873936 * q^5 - 4136404 * q^7 - 227268072 * q^11 - 3300829138 * q^13 - 16236937080 * q^17 - 30255301744 * q^19 - 54240053760 * q^23 + 499925939341 * q^25 - 54604532160 * q^29 + 2846135046908 * q^31 - 6105162153168 * q^35 - 8277847495270 * q^37 - 21907196555832 * q^41 - 4721391086752 * q^43 - 24094093237200 * q^47 + 264540388167999 * q^49 - 599077967207040 * q^53 + 616037278545360 * q^55 - 2653400930298552 * q^59 + 600379425869090 * q^61 - 8098658643163968 * q^65 + 10158232296990104 * q^67 - 15892874070853536 * q^71 + 15595098679731530 * q^73 - 45904425655186080 * q^77 + 35567118931509500 * q^79 - 87279079509107640 * q^83 + 38858193524134656 * q^85 - 150324347604557448 * q^89 + 131296124004824824 * q^91 - 249597769991850864 * q^95 + 219796544123416274 * q^97

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(36))\) 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
36.18.a.a	$36$	$18$	36.a	1.a	$1$	$1$	$1$	$65.960$	\(\Q\)	None	✓				12.18.a.b	$1$	$1$	\(0\)	\(0\)	\(-130950\)	\(-14846776\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-130950q^{5}-14846776q^{7}+845469684q^{11}+\cdots\)
36.18.a.b	$36$	$18$	36.a	1.a	$1$	$1$	$1$	$65.960$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			36.18.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-27688516\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-27688516q^{7}-5869817662q^{13}+\cdots\)
36.18.a.c	$36$	$18$	36.a	1.a	$1$	$1$	$1$	$65.960$	\(\Q\)	None	✓				12.18.a.a	$1$	$1$	\(0\)	\(0\)	\(1608930\)	\(-9417184\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+1608930q^{5}-9417184q^{7}+186910524q^{11}+\cdots\)
36.18.a.d	$36$	$18$	36.a	1.a	$1$	$2$	$2$	$65.960$	\(\Q(\sqrt{9361}) \)	None	✓		✓		4.18.a.a	$1$	$1$	\(0\)	\(0\)	\(-604044\)	\(25350160\)		$-$	$-$	$+$	$2^{7}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-302022-4\beta )q^{5}+(12675080+\cdots)q^{7}+\cdots\)
36.18.a.e	$36$	$18$	36.a	1.a	$1$	$2$	$2$	$65.960$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓		36.18.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(22465912\)		$-$	$+$	$-$	$2^{5}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+11232956q^{7}+460\beta q^{11}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(36))\) into lower level spaces

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