Space of modular forms of level 90, weight 18, and trivial character

• Label: 90.18.a
• Level: $90$
• Weight: $18$
• Character orbit: 90.a
• Rep. character: $\chi_{90}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $16$
• Sturm bound: $324$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(324\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(90))\).

	Total	New	Old
Modular forms	314	27	287
Cusp forms	298	27	271
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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(12\)
Minus space			\(-\)	\(15\)

Trace form

27 * q + 256 * q^2 + 1769472 * q^4 + 390625 * q^5 - 2868240 * q^7 + 16777216 * q^8 - 100000000 * q^10 + 1337372256 * q^11 - 9935909766 * q^13 - 773334016 * q^14 + 115964116992 * q^16 + 66106268874 * q^17 + 62183943060 * q^19 + 25600000000 * q^20 - 445042363392 * q^22 + 1185530966160 * q^23 + 4119873046875 * q^25 - 754045500928 * q^26 - 187972976640 * q^28 - 4017218571690 * q^29 + 8330519875104 * q^31 + 1099511627776 * q^32 - 21118474811904 * q^34 + 1657685937500 * q^35 + 15619718778258 * q^37 + 23823423650816 * q^38 - 6553600000000 * q^40 + 47859340813206 * q^41 + 77419920177612 * q^43 + 87646028169216 * q^44 + 48256029155328 * q^46 + 328254336837840 * q^47 + 358559381571819 * q^49 + 39062500000000 * q^50 - 651159782424576 * q^52 - 1641528812637594 * q^53 + 1097828067187500 * q^55 - 50681218072576 * q^56 - 45494634819072 * q^58 - 3022698977891040 * q^59 + 1887493591011354 * q^61 + 2118765761742848 * q^62 + 7599824371187712 * q^64 + 525159375781250 * q^65 + 424297459943028 * q^67 + 4332340436926464 * q^68 - 693552800000000 * q^70 + 907311148409376 * q^71 - 4679974221893538 * q^73 + 26497802624090624 * q^74 + 4075286892380160 * q^76 - 20001588904984800 * q^77 + 17861018678221440 * q^79 + 1677721600000000 * q^80 - 97117188912413184 * q^82 - 63049292697558732 * q^83 + 12972901867968750 * q^85 + 43038456829893632 * q^86 - 29166296327258112 * q^88 - 182720156439154770 * q^89 + 242046523673103144 * q^91 + 77694957398261760 * q^92 - 56840757232164864 * q^94 - 12811320751562500 * q^95 - 149221540084866474 * q^97 + 258000629801781504 * q^98

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(90))\) 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
90.18.a.a	$90$	$18$	90.a	1.a	$1$	$1$	$1$	$164.900$	\(\Q\)	None	✓				30.18.a.e	$1$	$0$	\(-256\)	\(0\)	\(-390625\)	\(-12193216\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}-12193216q^{7}+\cdots\)
90.18.a.b	$90$	$18$	90.a	1.a	$1$	$1$	$1$	$164.900$	\(\Q\)	None	✓				10.18.a.a	$1$	$0$	\(-256\)	\(0\)	\(-390625\)	\(14808668\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+14808668q^{7}+\cdots\)
90.18.a.c	$90$	$18$	90.a	1.a	$1$	$1$	$1$	$164.900$	\(\Q\)	None	✓				30.18.a.f	$1$	$1$	\(-256\)	\(0\)	\(390625\)	\(-16972732\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}-16972732q^{7}+\cdots\)
90.18.a.d	$90$	$18$	90.a	1.a	$1$	$1$	$1$	$164.900$	\(\Q\)	None	✓				30.18.a.d	$1$	$1$	\(-256\)	\(0\)	\(390625\)	\(-7079716\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}-7079716q^{7}+\cdots\)
90.18.a.e	$90$	$18$	90.a	1.a	$1$	$1$	$1$	$164.900$	\(\Q\)	None	✓				30.18.a.b	$1$	$1$	\(256\)	\(0\)	\(-390625\)	\(-2533888\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}-2533888q^{7}+\cdots\)
90.18.a.f	$90$	$18$	90.a	1.a	$1$	$1$	$1$	$164.900$	\(\Q\)	None	✓				30.18.a.c	$1$	$1$	\(256\)	\(0\)	\(-390625\)	\(1929536\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+1929536q^{7}+\cdots\)
90.18.a.g	$90$	$18$	90.a	1.a	$1$	$1$	$1$	$164.900$	\(\Q\)	None	✓				30.18.a.a	$1$	$0$	\(256\)	\(0\)	\(390625\)	\(2579612\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+2579612q^{7}+\cdots\)
90.18.a.h	$90$	$18$	90.a	1.a	$1$	$2$	$2$	$164.900$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	90.18.a.h	$1$	$1$	\(-512\)	\(0\)	\(-781250\)	\(-12328400\)		$+$	$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+(-6164200+\cdots)q^{7}+\cdots\)
90.18.a.i	$90$	$18$	90.a	1.a	$1$	$2$	$2$	$164.900$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓		✓		30.18.a.h	$1$	$0$	\(-512\)	\(0\)	\(-781250\)	\(6956296\)		$+$	$-$	$+$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+(3478148+\cdots)q^{7}+\cdots\)
90.18.a.j	$90$	$18$	90.a	1.a	$1$	$2$	$2$	$164.900$	\(\Q(\sqrt{2941}) \)	None	✓		✓		10.18.a.d	$1$	$1$	\(-512\)	\(0\)	\(781250\)	\(27684196\)		$+$	$-$	$-$	$+$	$2^{5}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(13842098+\cdots)q^{7}+\cdots\)
90.18.a.k	$90$	$18$	90.a	1.a	$1$	$2$	$2$	$164.900$	\(\Q(\sqrt{83281}) \)	None	✓		✓		10.18.a.c	$1$	$1$	\(512\)	\(0\)	\(-781250\)	\(603844\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+(301922+\cdots)q^{7}+\cdots\)
90.18.a.l	$90$	$18$	90.a	1.a	$1$	$2$	$2$	$164.900$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	90.18.a.h	$1$	$1$	\(512\)	\(0\)	\(781250\)	\(-12328400\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(-6164200+\cdots)q^{7}+\cdots\)
90.18.a.m	$90$	$18$	90.a	1.a	$1$	$2$	$2$	$164.900$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓				30.18.a.g	$1$	$0$	\(512\)	\(0\)	\(781250\)	\(1059712\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(529856+\cdots)q^{7}+\cdots\)
90.18.a.n	$90$	$18$	90.a	1.a	$1$	$2$	$2$	$164.900$	\(\Q(\sqrt{36061}) \)	None	✓				10.18.a.b	$1$	$0$	\(512\)	\(0\)	\(781250\)	\(6543844\)		$-$	$-$	$-$	$-$	$2^{5}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(3271922+\cdots)q^{7}+\cdots\)
90.18.a.o	$90$	$18$	90.a	1.a	$1$	$3$	$3$	$164.900$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	90.18.a.o	$1$	$0$	\(-768\)	\(0\)	\(1171875\)	\(-798798\)		$+$	$+$	$-$	$-$	$2^{5}\cdot 3^{7}\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+2^{16}q^{4}+5^{8}q^{5}+(-266266+\cdots)q^{7}+\cdots\)
90.18.a.p	$90$	$18$	90.a	1.a	$1$	$3$	$3$	$164.900$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	90.18.a.o	$1$	$0$	\(768\)	\(0\)	\(-1171875\)	\(-798798\)		$-$	$+$	$+$	$-$	$2^{5}\cdot 3^{7}\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+2^{16}q^{4}-5^{8}q^{5}+(-266266+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

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