Space of modular forms of level 15, weight 18, and trivial character

• Label: 15.18.a
• Level: $15$
• Weight: $18$
• Character orbit: 15.a
• Rep. character: $\chi_{15}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $36$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(36\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(15))\).

	Total	New	Old
Modular forms	36	12	24
Cusp forms	32	12	20
Eisenstein series	4	0	4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(8\)	\(3\)	\(5\)		\(7\)	\(3\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(10\)	\(4\)	\(6\)		\(9\)	\(4\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(9\)	\(3\)	\(6\)		\(8\)	\(3\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(17\)	\(5\)	\(12\)		\(15\)	\(5\)	\(10\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(19\)	\(7\)	\(12\)		\(17\)	\(7\)	\(10\)		\(2\)	\(0\)	\(2\)

Trace form

12 * q - 1084 * q^2 - 13122 * q^3 + 1081794 * q^4 - 879174 * q^6 - 2541288 * q^7 - 273106212 * q^8 + 516560652 * q^9 + 119531250 * q^10 + 286288304 * q^11 - 2579890176 * q^12 - 4331047296 * q^13 + 18348974268 * q^14 - 5125781250 * q^15 + 59002927122 * q^16 + 54362854304 * q^17 - 46662645564 * q^18 - 185480973840 * q^19 + 195184375000 * q^20 - 312518695824 * q^21 + 557456785668 * q^22 + 1396763817576 * q^23 - 1198994078430 * q^24 + 1831054687500 * q^25 - 2061742763716 * q^26 - 564859072962 * q^27 + 606552179244 * q^28 + 6471038778640 * q^29 - 1312200000000 * q^30 + 10047776977104 * q^31 - 24634141371308 * q^32 - 4346142658848 * q^33 + 40791386610828 * q^34 - 1485003125000 * q^35 + 46567684497474 * q^36 + 27002235566832 * q^37 + 54835208292296 * q^38 + 78415291003428 * q^39 - 9009907031250 * q^40 + 53298973084024 * q^41 - 153687818575764 * q^42 - 193130338051560 * q^43 + 618779521146988 * q^44 - 450503406679536 * q^46 - 197782206835240 * q^47 - 5301521571600 * q^48 + 926282434005324 * q^49 - 165405273437500 * q^50 - 45352736691444 * q^51 - 1798061175079536 * q^52 - 599330313960064 * q^53 - 37845557888454 * q^54 - 1445522850000000 * q^55 + 798394805775780 * q^56 - 2358786225057624 * q^57 + 5222298743948052 * q^58 + 2543399785019840 * q^59 + 556798239843750 * q^60 - 5501461938577896 * q^61 - 16670640114749976 * q^62 - 109394115516648 * q^63 + 15716151365242314 * q^64 - 2602183365625000 * q^65 + 7184602112090652 * q^66 - 6160732815160056 * q^67 + 14397719440248472 * q^68 - 2391598164286152 * q^69 - 15827363423437500 * q^70 + 26840725578517264 * q^71 - 11756326911330852 * q^72 - 5037065372014104 * q^73 - 14823292994303132 * q^74 - 2002258300781250 * q^75 - 14740279423047840 * q^76 + 33502180477922304 * q^77 + 7731079662305376 * q^78 - 22937314042138080 * q^79 + 22697294893750000 * q^80 + 22236242266222092 * q^81 + 66978393501861144 * q^82 - 9067017281333832 * q^83 + 36622398859782732 * q^84 + 30654841978125000 * q^85 - 102945511566785416 * q^86 - 28833146790852348 * q^87 + 90959749979066028 * q^88 - 109845612243112920 * q^89 + 5145428369531250 * q^90 + 1289885958717744 * q^91 + 158137129135880352 * q^92 + 36715116332233728 * q^93 - 19609132621991832 * q^94 + 75998534431250000 * q^95 - 103084611744857814 * q^96 - 9644513071651176 * q^97 - 496322012655584204 * q^98 + 12323772747851184 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(15))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
15.18.a.a	$15$	$18$	15.a	1.a	$1$	$2$	$2$	$27.483$	\(\Q(\sqrt{849}) \)	None	✓	✓	✓	✓	15.18.a.a	$1$	$1$	\(-356\)	\(13122\)	\(781250\)	\(-20754552\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-178-\beta )q^{2}+3^{8}q^{3}+(175688+\cdots)q^{4}+\cdots\)
15.18.a.b	$15$	$18$	15.a	1.a	$1$	$3$	$3$	$27.483$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	15.18.a.b	$1$	$1$	\(-442\)	\(-19683\)	\(-1171875\)	\(4962644\)		$+$	$+$	$+$	$2^{6}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-147+\beta _{1})q^{2}-3^{8}q^{3}+(99318+\cdots)q^{4}+\cdots\)
15.18.a.c	$15$	$18$	15.a	1.a	$1$	$3$	$3$	$27.483$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	15.18.a.c	$1$	$0$	\(-253\)	\(19683\)	\(-1171875\)	\(-4332484\)		$-$	$+$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-84-\beta _{1})q^{2}+3^{8}q^{3}+(-2408+\cdots)q^{4}+\cdots\)
15.18.a.d	$15$	$18$	15.a	1.a	$1$	$4$	$4$	$27.483$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	15.18.a.d	$1$	$0$	\(-33\)	\(-26244\)	\(1562500\)	\(17583104\)		$+$	$-$	$-$	$2^{6}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-8-\beta _{1})q^{2}-3^{8}q^{3}+(109856+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(15)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)