Space of modular forms of level 90, weight 12, and trivial character

• Label: 90.12.a
• Level: $90$
• Weight: $12$
• Character orbit: 90.a
• Rep. character: $\chi_{90}(1,\cdot)$
• Character field: $\Q$
• Dimension: $17$
• Newform subspaces: $14$
• Sturm bound: $216$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(216\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(90))\).

	Total	New	Old
Modular forms	206	17	189
Cusp forms	190	17	173
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.

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(27\)	\(2\)	\(25\)		\(25\)	\(2\)	\(23\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(25\)	\(1\)	\(24\)		\(23\)	\(1\)	\(22\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(25\)	\(3\)	\(22\)		\(23\)	\(3\)	\(20\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(26\)	\(3\)	\(23\)		\(24\)	\(3\)	\(21\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(26\)	\(1\)	\(25\)		\(24\)	\(1\)	\(23\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(26\)	\(2\)	\(24\)		\(24\)	\(2\)	\(22\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(26\)	\(3\)	\(23\)		\(24\)	\(3\)	\(21\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(25\)	\(2\)	\(23\)		\(23\)	\(2\)	\(21\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(105\)	\(10\)	\(95\)		\(97\)	\(10\)	\(87\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(101\)	\(7\)	\(94\)		\(93\)	\(7\)	\(86\)		\(8\)	\(0\)	\(8\)

Trace form

17 * q - 32 * q^2 + 17408 * q^4 - 3125 * q^5 - 22880 * q^7 - 32768 * q^8 + 100000 * q^10 - 1453056 * q^11 + 1463038 * q^13 + 310016 * q^14 + 17825792 * q^16 + 8801982 * q^17 - 40388924 * q^19 - 3200000 * q^20 + 20705664 * q^22 + 53360880 * q^23 + 166015625 * q^25 - 142494016 * q^26 - 23429120 * q^28 + 155715402 * q^29 - 3925184 * q^31 - 33554432 * q^32 + 97712448 * q^34 - 69287500 * q^35 + 363933334 * q^37 - 423641728 * q^38 + 102400000 * q^40 + 2165215722 * q^41 - 448299116 * q^43 - 1487929344 * q^44 + 1837805952 * q^46 - 3239085120 * q^47 - 1670283495 * q^49 - 312500000 * q^50 + 1498150912 * q^52 + 8072182962 * q^53 + 4194337500 * q^55 + 317456384 * q^56 - 8832707904 * q^58 + 12331450320 * q^59 + 23635383598 * q^61 + 2124091904 * q^62 + 18253611008 * q^64 - 18439956250 * q^65 + 3886770124 * q^67 + 9013229568 * q^68 + 12455600000 * q^70 - 6866584416 * q^71 + 58013445034 * q^73 + 48843424448 * q^74 - 41358258176 * q^76 - 56557894800 * q^77 + 62762316640 * q^79 - 3276800000 * q^80 + 59156382528 * q^82 + 55758036396 * q^83 - 61838043750 * q^85 + 56878616576 * q^86 + 21202599936 * q^88 + 26575139754 * q^89 + 140611371224 * q^91 + 54641541120 * q^92 - 12759106944 * q^94 + 66662487500 * q^95 + 254054810578 * q^97 - 274465665312 * q^98

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(90))\) 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	5
90.12.a.a	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓				30.12.a.e	$1$	$1$	\(-32\)	\(0\)	\(-3125\)	\(-5152\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}-5152q^{7}+\cdots\)
90.12.a.b	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓				10.12.a.c	$1$	$0$	\(-32\)	\(0\)	\(3125\)	\(-70714\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}-70714q^{7}+\cdots\)
90.12.a.c	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓	✓	✓	✓	90.12.a.c	$1$	$1$	\(-32\)	\(0\)	\(3125\)	\(-14014\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}-14014q^{7}+\cdots\)
90.12.a.d	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓				30.12.a.f	$1$	$0$	\(-32\)	\(0\)	\(3125\)	\(10556\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}+10556q^{7}+\cdots\)
90.12.a.e	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓				30.12.a.d	$1$	$0$	\(-32\)	\(0\)	\(3125\)	\(29348\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}+29348q^{7}+\cdots\)
90.12.a.f	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓				30.12.a.b	$1$	$0$	\(32\)	\(0\)	\(-3125\)	\(-57376\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}-57376q^{7}+\cdots\)
90.12.a.g	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓				10.12.a.a	$1$	$0$	\(32\)	\(0\)	\(-3125\)	\(-14176\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}-14176q^{7}+\cdots\)
90.12.a.h	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓	✓	✓	✓	90.12.a.c	$1$	$1$	\(32\)	\(0\)	\(-3125\)	\(-14014\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}-14014q^{7}+\cdots\)
90.12.a.i	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓				30.12.a.c	$1$	$0$	\(32\)	\(0\)	\(-3125\)	\(56672\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}+56672q^{7}+\cdots\)
90.12.a.j	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓				30.12.a.a	$1$	$1$	\(32\)	\(0\)	\(3125\)	\(-22876\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}-22876q^{7}+\cdots\)
90.12.a.k	$90$	$12$	90.a	1.a	$1$	$1$	$1$	$69.151$	\(\Q\)	None	✓				10.12.a.b	$1$	$1$	\(32\)	\(0\)	\(3125\)	\(25574\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}+25574q^{7}+\cdots\)
90.12.a.l	$90$	$12$	90.a	1.a	$1$	$2$	$2$	$69.151$	\(\Q(\sqrt{1969}) \)	None	✓		✓		10.12.a.d	$1$	$1$	\(-64\)	\(0\)	\(-6250\)	\(14092\)		$+$	$-$	$+$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}+(7046+\cdots)q^{7}+\cdots\)
90.12.a.m	$90$	$12$	90.a	1.a	$1$	$2$	$2$	$69.151$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	90.12.a.m	$1$	$0$	\(-64\)	\(0\)	\(-6250\)	\(19600\)		$+$	$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+2^{10}q^{4}-5^{5}q^{5}+(9800+\cdots)q^{7}+\cdots\)
90.12.a.n	$90$	$12$	90.a	1.a	$1$	$2$	$2$	$69.151$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	90.12.a.m	$1$	$0$	\(64\)	\(0\)	\(6250\)	\(19600\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+2^{10}q^{4}+5^{5}q^{5}+(9800+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(90)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)