Space of modular forms of level 3381, weight 2, and trivial character

• Label: 3381.2.a
• Level: $3381$
• Weight: $2$
• Character orbit: 3381.a
• Rep. character: $\chi_{3381}(1,\cdot)$
• Character field: $\Q$
• Dimension: $150$
• Newform subspaces: $38$
• Sturm bound: $896$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(896\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3381))\).

	Total	New	Old
Modular forms	464	150	314
Cusp forms	433	150	283
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.

\(3\)	\(7\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(16\)
\(+\)	\(+\)	\(-\)	\(-\)	\(20\)
\(+\)	\(-\)	\(+\)	\(-\)	\(22\)
\(+\)	\(-\)	\(-\)	\(+\)	\(16\)
\(-\)	\(+\)	\(+\)	\(-\)	\(19\)
\(-\)	\(+\)	\(-\)	\(+\)	\(15\)
\(-\)	\(-\)	\(+\)	\(+\)	\(18\)
\(-\)	\(-\)	\(-\)	\(-\)	\(24\)
Plus space			\(+\)	\(65\)
Minus space			\(-\)	\(85\)

Trace form

150 * q - 2 * q^2 + 2 * q^3 + 150 * q^4 - 8 * q^5 + 2 * q^6 + 6 * q^8 + 150 * q^9 + 4 * q^10 - 2 * q^12 + 4 * q^13 + 4 * q^15 + 158 * q^16 - 8 * q^17 - 2 * q^18 + 8 * q^19 + 8 * q^20 + 16 * q^22 + 18 * q^24 + 146 * q^25 + 4 * q^26 + 2 * q^27 - 4 * q^29 - 8 * q^30 + 24 * q^31 - 10 * q^32 + 8 * q^33 - 4 * q^34 + 150 * q^36 - 16 * q^37 - 28 * q^38 - 24 * q^39 - 28 * q^40 - 12 * q^41 - 44 * q^43 - 32 * q^44 - 8 * q^45 - 4 * q^46 + 32 * q^47 - 2 * q^48 - 102 * q^50 - 4 * q^51 - 4 * q^52 - 56 * q^53 + 2 * q^54 - 8 * q^55 - 16 * q^57 - 44 * q^58 + 24 * q^59 + 36 * q^60 + 4 * q^61 + 238 * q^64 - 96 * q^65 + 16 * q^66 + 12 * q^67 + 32 * q^68 + 4 * q^69 + 56 * q^71 + 6 * q^72 - 4 * q^73 - 28 * q^74 - 10 * q^75 + 72 * q^76 + 20 * q^78 + 20 * q^79 + 40 * q^80 + 150 * q^81 - 12 * q^82 - 8 * q^83 - 80 * q^85 + 60 * q^86 + 4 * q^87 + 64 * q^88 - 32 * q^89 + 4 * q^90 - 12 * q^93 - 8 * q^94 + 56 * q^95 + 18 * q^96 + 28 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\) 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	7	23
3381.2.a.a	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.i.c	$1$	$1$	\(-1\)	\(-1\)	\(-1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}-q^{5}+q^{6}+3q^{8}+\cdots\)
3381.2.a.b	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.i.d	$1$	$1$	\(-1\)	\(-1\)	\(3\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+3q^{5}+q^{6}+3q^{8}+\cdots\)
3381.2.a.c	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.i.d	$1$	$1$	\(-1\)	\(1\)	\(-3\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-3q^{5}-q^{6}+3q^{8}+\cdots\)
3381.2.a.d	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.i.c	$1$	$1$	\(-1\)	\(1\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}+3q^{8}+\cdots\)
3381.2.a.e	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.i.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{9}-2q^{11}+2q^{12}+\cdots\)
3381.2.a.f	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓	✓			3381.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{9}+5q^{11}+2q^{12}+\cdots\)
3381.2.a.g	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.i.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{9}+6q^{11}+2q^{12}+\cdots\)
3381.2.a.h	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.i.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+q^{9}-2q^{11}-2q^{12}+\cdots\)
3381.2.a.i	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓	✓			3381.2.a.f	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+q^{9}+5q^{11}-2q^{12}+\cdots\)
3381.2.a.j	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.i.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+q^{9}+6q^{11}-2q^{12}+\cdots\)
3381.2.a.k	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				69.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
3381.2.a.l	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.a.b	$1$	$0$	\(2\)	\(-1\)	\(-4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-4q^{5}-2q^{6}+\cdots\)
3381.2.a.m	$3381$	$2$	3381.a	1.a	$1$	$1$	$1$	$26.997$	\(\Q\)	None	✓				483.2.a.a	$1$	$1$	\(2\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+q^{9}+q^{11}+\cdots\)
3381.2.a.n	$3381$	$2$	3381.a	1.a	$1$	$2$	$2$	$26.997$	\(\Q(\sqrt{5}) \)	None	✓				483.2.a.c	$1$	$0$	\(-3\)	\(-2\)	\(5\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}+(2+\beta )q^{5}+\cdots\)
3381.2.a.o	$3381$	$2$	3381.a	1.a	$1$	$2$	$2$	$26.997$	\(\Q(\sqrt{5}) \)	None	✓				483.2.a.e	$1$	$1$	\(-1\)	\(-2\)	\(-5\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+(-3+\beta )q^{5}+\cdots\)
3381.2.a.p	$3381$	$2$	3381.a	1.a	$1$	$2$	$2$	$26.997$	\(\Q(\sqrt{13}) \)	None	✓				483.2.a.f	$1$	$1$	\(-1\)	\(-2\)	\(5\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(1+\beta )q^{4}+(3-\beta )q^{5}+\cdots\)
3381.2.a.q	$3381$	$2$	3381.a	1.a	$1$	$2$	$2$	$26.997$	\(\Q(\sqrt{17}) \)	None	✓				483.2.i.e	$1$	$1$	\(-1\)	\(-2\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+\beta q^{5}+\beta q^{6}+\cdots\)
3381.2.a.r	$3381$	$2$	3381.a	1.a	$1$	$2$	$2$	$26.997$	\(\Q(\sqrt{5}) \)	None	✓				483.2.a.d	$1$	$1$	\(-1\)	\(2\)	\(-1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}+(-1+\beta )q^{5}+\cdots\)
3381.2.a.s	$3381$	$2$	3381.a	1.a	$1$	$2$	$2$	$26.997$	\(\Q(\sqrt{17}) \)	None	✓				483.2.i.e	$1$	$1$	\(-1\)	\(2\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(2+\beta )q^{4}-\beta q^{5}-\beta q^{6}+\cdots\)
3381.2.a.t	$3381$	$2$	3381.a	1.a	$1$	$2$	$2$	$26.997$	\(\Q(\sqrt{5}) \)	None	✓				69.2.a.b	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+3q^{4}+(1-\beta )q^{5}-\beta q^{6}+\cdots\)
3381.2.a.u	$3381$	$2$	3381.a	1.a	$1$	$2$	$2$	$26.997$	\(\Q(\sqrt{5}) \)	None	✓				483.2.a.g	$1$	$0$	\(1\)	\(2\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+(1+\beta )q^{5}+\cdots\)
3381.2.a.v	$3381$	$2$	3381.a	1.a	$1$	$3$	$3$	$26.997$	3.3.837.1	None	✓				483.2.a.h	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
3381.2.a.w	$3381$	$2$	3381.a	1.a	$1$	$4$	$4$	$26.997$	4.4.24197.1	None	✓				483.2.a.i	$1$	$1$	\(0\)	\(4\)	\(-5\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{3})q^{5}+\cdots\)
3381.2.a.x	$3381$	$2$	3381.a	1.a	$1$	$4$	$4$	$26.997$	4.4.15317.1	None	✓				483.2.a.j	$1$	$0$	\(2\)	\(4\)	\(-5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
3381.2.a.y	$3381$	$2$	3381.a	1.a	$1$	$5$	$5$	$26.997$	5.5.3176240.1	None	✓	✓	✓		3381.2.a.y	$1$	$1$	\(1\)	\(-5\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-\beta _{3}q^{5}+\cdots\)
3381.2.a.z	$3381$	$2$	3381.a	1.a	$1$	$5$	$5$	$26.997$	5.5.3176240.1	None	✓	✓			3381.2.a.y	$1$	$0$	\(1\)	\(5\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{3}q^{5}+\cdots\)
3381.2.a.ba	$3381$	$2$	3381.a	1.a	$1$	$6$	$6$	$26.997$	6.6.62622704.1	None	✓	✓			3381.2.a.ba	$1$	$0$	\(-3\)	\(-6\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
3381.2.a.bb	$3381$	$2$	3381.a	1.a	$1$	$6$	$6$	$26.997$	6.6.62622704.1	None	✓	✓			3381.2.a.ba	$1$	$1$	\(-3\)	\(6\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
3381.2.a.bc	$3381$	$2$	3381.a	1.a	$1$	$6$	$6$	$26.997$	6.6.7997584.1	None	✓				483.2.i.f	$1$	$1$	\(-1\)	\(-6\)	\(3\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(\beta _{3}+\beta _{5})q^{5}+\cdots\)
3381.2.a.bd	$3381$	$2$	3381.a	1.a	$1$	$6$	$6$	$26.997$	6.6.7997584.1	None	✓				483.2.i.f	$1$	$1$	\(-1\)	\(6\)	\(-3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
3381.2.a.be	$3381$	$2$	3381.a	1.a	$1$	$8$	$8$	$26.997$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		483.2.i.g	$1$	$0$	\(1\)	\(-8\)	\(-5\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(-1-\beta _{7})q^{5}+\cdots\)
3381.2.a.bf	$3381$	$2$	3381.a	1.a	$1$	$8$	$8$	$26.997$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				483.2.i.g	$1$	$0$	\(1\)	\(8\)	\(5\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{7})q^{5}+\cdots\)
3381.2.a.bg	$3381$	$2$	3381.a	1.a	$1$	$10$	$10$	$26.997$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			3381.2.a.bg	$1$	$0$	\(-4\)	\(-10\)	\(4\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
3381.2.a.bh	$3381$	$2$	3381.a	1.a	$1$	$10$	$10$	$26.997$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		3381.2.a.bg	$1$	$1$	\(-4\)	\(10\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{3}-\beta _{8}+\cdots)q^{5}+\cdots\)
3381.2.a.bi	$3381$	$2$	3381.a	1.a	$1$	$10$	$10$	$26.997$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				483.2.i.h	$1$	$0$	\(3\)	\(-10\)	\(-5\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(-1-\beta _{8}+\cdots)q^{5}+\cdots\)
3381.2.a.bj	$3381$	$2$	3381.a	1.a	$1$	$10$	$10$	$26.997$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		483.2.i.h	$1$	$0$	\(3\)	\(10\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{8}+\cdots)q^{5}+\cdots\)
3381.2.a.bk	$3381$	$2$	3381.a	1.a	$1$	$10$	$10$	$26.997$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		3381.2.a.bk	$1$	$1$	\(4\)	\(-10\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots\)
3381.2.a.bl	$3381$	$2$	3381.a	1.a	$1$	$10$	$10$	$26.997$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		3381.2.a.bk	$1$	$0$	\(4\)	\(10\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{7}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3381))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3381)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(483))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1127))\)\(^{\oplus 2}\)