Space of modular forms of level 30, weight 18, and trivial character

• Label: 30.18.a
• Level: $30$
• Weight: $18$
• Character orbit: 30.a
• Rep. character: $\chi_{30}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $8$
• Sturm bound: $108$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	30.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(108\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(30))\).

	Total	New	Old
Modular forms	106	10	96
Cusp forms	98	10	88
Eisenstein series	8	0	8

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
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(6\)

Trace form

10 * q + 13122 * q^3 + 655360 * q^4 - 26254396 * q^7 + 430467210 * q^9 + 200000000 * q^10 - 1034780340 * q^11 + 859963392 * q^12 + 4735309616 * q^13 - 8275031040 * q^14 + 42949672960 * q^16 - 64966043976 * q^17 + 215067724880 * q^19 + 80044331220 * q^21 + 630017559552 * q^22 - 1103183368296 * q^23 + 1525878906250 * q^25 + 277811727360 * q^26 + 564859072962 * q^27 - 1720608096256 * q^28 + 3909817166520 * q^29 + 1312200000000 * q^30 - 17237786980600 * q^31 + 3066650844732 * q^33 - 6986917770240 * q^34 + 5692129687500 * q^35 + 28211099074560 * q^36 + 66330668496680 * q^37 - 49510241396736 * q^38 - 64752781205520 * q^39 + 13107200000000 * q^40 - 5562052531020 * q^41 + 10603368778752 * q^42 - 7050585363160 * q^43 - 67815364362240 * q^44 + 205479011358720 * q^46 + 140654588004528 * q^47 + 56358560858112 * q^48 - 159211865087670 * q^49 + 115188144799320 * q^51 + 310333250994176 * q^52 + 869759872942128 * q^53 + 1097828067187500 * q^55 - 542312434237440 * q^56 + 1593558386413776 * q^57 - 1672374120766464 * q^58 - 283280567367300 * q^59 + 494155496779460 * q^61 + 4556756454610944 * q^62 - 1130165659635516 * q^63 + 2814749767106560 * q^64 + 2215097348437500 * q^65 + 3090883883535360 * q^66 + 10128577865833664 * q^67 - 4257614658011136 * q^68 - 6432890899987080 * q^69 + 2305920400000000 * q^70 + 586492757978760 * q^71 + 9573867737382716 * q^73 - 12454584238955520 * q^74 + 2002258300781250 * q^75 + 14094678417735680 * q^76 + 32355816551067936 * q^77 + 20289865570716672 * q^78 - 22040499897196360 * q^79 + 18530201888518410 * q^81 - 4328417212157952 * q^82 + 40608273935402976 * q^83 + 5245785290833920 * q^84 - 22945785464062500 * q^85 - 36390233423400960 * q^86 - 42016987747953576 * q^87 + 41288830782799872 * q^88 + 266058543361219740 * q^89 + 8609344200000000 * q^90 + 77004433290324520 * q^91 - 72298225224646656 * q^92 - 112219912717715160 * q^93 + 45776311250595840 * q^94 - 92367103115625000 * q^95 - 256623805899256780 * q^97 - 59179740014825472 * q^98 - 44543900592265140 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(30))\) 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
30.18.a.a	$30$	$18$	30.a	1.a	$1$	$1$	$1$	$54.967$	\(\Q\)	None	✓	✓	✓	✓	30.18.a.a	$1$	$1$	\(-256\)	\(-6561\)	\(-390625\)	\(2579612\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}-5^{8}q^{5}+\cdots\)
30.18.a.b	$30$	$18$	30.a	1.a	$1$	$1$	$1$	$54.967$	\(\Q\)	None	✓	✓	✓	✓	30.18.a.b	$1$	$0$	\(-256\)	\(-6561\)	\(390625\)	\(-2533888\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}+5^{8}q^{5}+\cdots\)
30.18.a.c	$30$	$18$	30.a	1.a	$1$	$1$	$1$	$54.967$	\(\Q\)	None	✓	✓	✓	✓	30.18.a.c	$1$	$1$	\(-256\)	\(6561\)	\(390625\)	\(1929536\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+3^{8}q^{3}+2^{16}q^{4}+5^{8}q^{5}+\cdots\)
30.18.a.d	$30$	$18$	30.a	1.a	$1$	$1$	$1$	$54.967$	\(\Q\)	None	✓	✓	✓	✓	30.18.a.d	$1$	$0$	\(256\)	\(-6561\)	\(-390625\)	\(-7079716\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}-5^{8}q^{5}+\cdots\)
30.18.a.e	$30$	$18$	30.a	1.a	$1$	$1$	$1$	$54.967$	\(\Q\)	None	✓	✓	✓	✓	30.18.a.e	$1$	$1$	\(256\)	\(-6561\)	\(390625\)	\(-12193216\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}+5^{8}q^{5}+\cdots\)
30.18.a.f	$30$	$18$	30.a	1.a	$1$	$1$	$1$	$54.967$	\(\Q\)	None	✓	✓	✓	✓	30.18.a.f	$1$	$1$	\(256\)	\(6561\)	\(-390625\)	\(-16972732\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+3^{8}q^{3}+2^{16}q^{4}-5^{8}q^{5}+\cdots\)
30.18.a.g	$30$	$18$	30.a	1.a	$1$	$2$	$2$	$54.967$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	30.18.a.g	$1$	$0$	\(-512\)	\(13122\)	\(-781250\)	\(1059712\)		$+$	$-$	$+$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+3^{8}q^{3}+2^{16}q^{4}-5^{8}q^{5}+\cdots\)
30.18.a.h	$30$	$18$	30.a	1.a	$1$	$2$	$2$	$54.967$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	30.18.a.h	$1$	$0$	\(512\)	\(13122\)	\(781250\)	\(6956296\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}+3^{8}q^{3}+2^{16}q^{4}+5^{8}q^{5}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(30))\) into lower level spaces

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