Space of modular forms of level 3330, weight 2, and trivial character

• Label: 3330.2.a
• Level: $3330$
• Weight: $2$
• Character orbit: 3330.a
• Rep. character: $\chi_{3330}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $38$
• Sturm bound: $1368$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(1368\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3330))\).

	Total	New	Old
Modular forms	700	60	640
Cusp forms	669	60	609
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(5\)	\(37\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(2\)
Plus space				\(+\)	\(21\)
Minus space				\(-\)	\(39\)

Trace form

60 * q - 2 * q^2 + 60 * q^4 - 2 * q^5 - 2 * q^8 - 4 * q^11 + 12 * q^13 + 4 * q^14 + 60 * q^16 + 4 * q^17 + 12 * q^19 - 2 * q^20 + 24 * q^22 + 16 * q^23 + 60 * q^25 - 8 * q^26 - 4 * q^29 + 16 * q^31 - 2 * q^32 + 12 * q^34 - 4 * q^35 + 2 * q^37 + 16 * q^38 + 12 * q^41 + 8 * q^43 - 4 * q^44 + 40 * q^47 + 96 * q^49 - 2 * q^50 + 12 * q^52 - 4 * q^53 + 4 * q^56 - 24 * q^58 - 12 * q^59 + 20 * q^61 + 12 * q^62 + 60 * q^64 + 4 * q^67 + 4 * q^68 + 12 * q^70 + 8 * q^71 + 8 * q^73 + 12 * q^76 + 24 * q^79 - 2 * q^80 + 4 * q^82 + 52 * q^83 + 24 * q^85 + 20 * q^86 + 24 * q^88 + 64 * q^91 + 16 * q^92 + 4 * q^94 + 16 * q^95 + 12 * q^97 - 2 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\) 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	37
3330.2.a.a	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓	✓	✓	✓	3330.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-3q^{7}-q^{8}+q^{10}+\cdots\)
3330.2.a.b	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.j	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(-3\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-3q^{7}-q^{8}+q^{10}+\cdots\)
3330.2.a.c	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.k	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-4q^{11}+\cdots\)
3330.2.a.d	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				370.2.a.d	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+2q^{7}-q^{8}+q^{10}+\cdots\)
3330.2.a.e	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.l	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(3\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+3q^{7}-q^{8}+q^{10}+\cdots\)
3330.2.a.f	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.m	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-5\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-5q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.g	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓	✓			3330.2.a.g	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-3\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-3q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.h	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.n	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(-1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.i	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓	✓			3330.2.a.i	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}-2q^{11}+\cdots\)
3330.2.a.j	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.i	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}+2q^{11}+\cdots\)
3330.2.a.k	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓	✓			3330.2.a.k	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(4\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+4q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.l	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.o	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(4\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+4q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.m	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.h	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.n	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓	✓			3330.2.a.g	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-3q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.o	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓	✓			3330.2.a.i	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+q^{8}-q^{10}+2q^{11}+\cdots\)
3330.2.a.p	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				370.2.a.c	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.q	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.d	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(3\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+3q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.r	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓	✓			3330.2.a.k	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(4\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.s	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(-4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.t	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓	✓	✓	✓	3330.2.a.a	$1$	$1$	\(1\)	\(0\)	\(1\)	\(-3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-3q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.u	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.e	$1$	$1$	\(1\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.v	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				370.2.a.a	$1$	$1$	\(1\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.w	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				370.2.a.b	$1$	$0$	\(1\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}+4q^{11}+\cdots\)
3330.2.a.x	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.b	$1$	$0$	\(1\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.y	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.f	$1$	$0$	\(1\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.z	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.c	$1$	$0$	\(1\)	\(0\)	\(1\)	\(3\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+3q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.ba	$3330$	$2$	3330.a	1.a	$1$	$1$	$1$	$26.590$	\(\Q\)	None	✓				1110.2.a.g	$1$	$0$	\(1\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+4q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.bb	$3330$	$2$	3330.a	1.a	$1$	$2$	$2$	$26.590$	\(\Q(\sqrt{33}) \)	None	✓		✓		370.2.a.f	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(-3\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+(-1-\beta )q^{7}-q^{8}+\cdots\)
3330.2.a.bc	$3330$	$2$	3330.a	1.a	$1$	$2$	$2$	$26.590$	\(\Q(\sqrt{17}) \)	None	✓		✓		1110.2.a.r	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.bd	$3330$	$2$	3330.a	1.a	$1$	$2$	$2$	$26.590$	\(\Q(\sqrt{3}) \)	None	✓				370.2.a.e	$1$	$0$	\(2\)	\(0\)	\(-2\)	\(-6\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+(-3+\beta )q^{7}+q^{8}+\cdots\)
3330.2.a.be	$3330$	$2$	3330.a	1.a	$1$	$2$	$2$	$26.590$	\(\Q(\sqrt{33}) \)	None	✓		✓		1110.2.a.p	$1$	$1$	\(2\)	\(0\)	\(-2\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-\beta q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.bf	$3330$	$2$	3330.a	1.a	$1$	$2$	$2$	$26.590$	\(\Q(\sqrt{113}) \)	None	✓				1110.2.a.q	$1$	$0$	\(2\)	\(0\)	\(-2\)	\(6\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+3q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.bg	$3330$	$2$	3330.a	1.a	$1$	$3$	$3$	$26.590$	3.3.892.1	None	✓		✓		370.2.a.g	$1$	$1$	\(-3\)	\(0\)	\(3\)	\(-1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-\beta _{1}q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.bh	$3330$	$2$	3330.a	1.a	$1$	$3$	$3$	$26.590$	3.3.316.1	None	✓	✓	✓		3330.2.a.bh	$1$	$0$	\(-3\)	\(0\)	\(3\)	\(-1\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+(-1+2\beta _{1})q^{7}+\cdots\)
3330.2.a.bi	$3330$	$2$	3330.a	1.a	$1$	$3$	$3$	$26.590$	3.3.316.1	None	✓	✓	✓		3330.2.a.bh	$1$	$0$	\(3\)	\(0\)	\(-3\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+(-1+2\beta _{1})q^{7}+\cdots\)
3330.2.a.bj	$3330$	$2$	3330.a	1.a	$1$	$4$	$4$	$26.590$	4.4.54764.1	None	✓		✓		1110.2.a.s	$1$	$0$	\(-4\)	\(0\)	\(-4\)	\(4\)		$+$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+(1+\beta _{2})q^{7}-q^{8}+\cdots\)
3330.2.a.bk	$3330$	$2$	3330.a	1.a	$1$	$5$	$5$	$26.590$	5.5.23544108.1	None	✓	✓	✓	✓	3330.2.a.bk	$1$	$0$	\(-5\)	\(0\)	\(-5\)	\(3\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+(1+\beta _{4})q^{7}-q^{8}+\cdots\)
3330.2.a.bl	$3330$	$2$	3330.a	1.a	$1$	$5$	$5$	$26.590$	5.5.23544108.1	None	✓	✓	✓	✓	3330.2.a.bk	$1$	$0$	\(5\)	\(0\)	\(5\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+(1+\beta _{4})q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3330))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3330)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(222))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(333))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(555))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(666))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1110))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1665))\)\(^{\oplus 2}\)