Space of modular forms of level 3366, weight 2, and trivial character

• Label: 3366.2.a
• Level: $3366$
• Weight: $2$
• Character orbit: 3366.a
• Rep. character: $\chi_{3366}(1,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $34$
• Sturm bound: $1296$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3366 = 2 \cdot 3^{2} \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3366.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(1296\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(13\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3366))\).

	Total	New	Old
Modular forms	664	64	600
Cusp forms	633	64	569
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.

\(2\)	\(3\)	\(11\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(7\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(7\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(2\)
Plus space				\(+\)	\(25\)
Minus space				\(-\)	\(39\)

Trace form

\( 64 q + 64 q^{4} - 8 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3366))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	11	17
3366.2.a.a	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-4\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{5}-2q^{7}-q^{8}+4q^{10}+\cdots\)
3366.2.a.b	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-2q^{7}-q^{8}+2q^{10}+\cdots\)
3366.2.a.c	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-2q^{7}-q^{8}+2q^{10}+\cdots\)
3366.2.a.d	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}+4q^{7}-q^{8}+2q^{10}+\cdots\)
3366.2.a.e	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}+4q^{7}-q^{8}+2q^{10}+\cdots\)
3366.2.a.f	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}-q^{11}-4q^{13}+\cdots\)
3366.2.a.g	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}-q^{11}+4q^{13}+\cdots\)
3366.2.a.h	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{7}-q^{8}-q^{11}+4q^{13}+\cdots\)
3366.2.a.i	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-4q^{7}-q^{8}-2q^{10}+\cdots\)
3366.2.a.j	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-q^{8}-2q^{10}-q^{11}+\cdots\)
3366.2.a.k	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-q^{8}-2q^{10}+q^{11}+\cdots\)
3366.2.a.l	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}-2q^{7}+q^{8}-2q^{10}+\cdots\)
3366.2.a.m	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}+4q^{7}+q^{8}-2q^{10}+\cdots\)
3366.2.a.n	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}+q^{11}-2q^{13}+\cdots\)
3366.2.a.o	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}+q^{11}+4q^{13}+\cdots\)
3366.2.a.p	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}-2q^{7}+q^{8}+2q^{10}+\cdots\)
3366.2.a.q	$3366$	$2$	3366.a	1.a	$1$	$1$	$1$	$26.878$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+q^{8}+2q^{10}-q^{11}+\cdots\)
3366.2.a.r	$3366$	$2$	3366.a	1.a	$1$	$2$	$2$	$26.878$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-\beta )q^{5}+(1+\beta )q^{7}+\cdots\)
3366.2.a.s	$3366$	$2$	3366.a	1.a	$1$	$2$	$2$	$26.878$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(-4\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta )q^{5}-2q^{7}-q^{8}+\cdots\)
3366.2.a.t	$3366$	$2$	3366.a	1.a	$1$	$2$	$2$	$26.878$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(2\)	\(0\)	\(-2\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta )q^{5}-2q^{7}+q^{8}+\cdots\)
3366.2.a.u	$3366$	$2$	3366.a	1.a	$1$	$2$	$2$	$26.878$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(2\)	\(0\)	\(-2\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta )q^{5}+(-1+\beta )q^{7}+\cdots\)
3366.2.a.v	$3366$	$2$	3366.a	1.a	$1$	$2$	$2$	$26.878$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-2+\beta )q^{7}+q^{8}-q^{11}+\cdots\)
3366.2.a.w	$3366$	$2$	3366.a	1.a	$1$	$2$	$2$	$26.878$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}+(-2+\beta )q^{7}+q^{8}+\cdots\)
3366.2.a.x	$3366$	$2$	3366.a	1.a	$1$	$2$	$2$	$26.878$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta )q^{5}+(1+\beta )q^{7}+\cdots\)
3366.2.a.y	$3366$	$2$	3366.a	1.a	$1$	$2$	$2$	$26.878$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(2\)	\(6\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta )q^{5}+(3+\beta )q^{7}+\cdots\)
3366.2.a.z	$3366$	$2$	3366.a	1.a	$1$	$3$	$3$	$26.878$	3.3.148.1	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{2}q^{5}+(1+\beta _{1})q^{7}-q^{8}+\cdots\)
3366.2.a.ba	$3366$	$2$	3366.a	1.a	$1$	$3$	$3$	$26.878$	3.3.257.1	None	✓	$1$	$0$	\(-3\)	\(0\)	\(1\)	\(7\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(\beta _{1}+\beta _{2})q^{5}+(2+\beta _{1}+\cdots)q^{7}+\cdots\)
3366.2.a.bb	$3366$	$2$	3366.a	1.a	$1$	$3$	$3$	$26.878$	3.3.316.1	None	✓	$1$	$0$	\(-3\)	\(0\)	\(2\)	\(-4\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta _{2})q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
3366.2.a.bc	$3366$	$2$	3366.a	1.a	$1$	$3$	$3$	$26.878$	3.3.316.1	None	✓	$1$	$1$	\(3\)	\(0\)	\(-2\)	\(-4\)		$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta _{2})q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
3366.2.a.bd	$3366$	$2$	3366.a	1.a	$1$	$3$	$3$	$26.878$	3.3.785.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(1\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{2}q^{5}+(1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
3366.2.a.be	$3366$	$2$	3366.a	1.a	$1$	$4$	$4$	$26.878$	4.4.19664.1	None	✓	$1$	$1$	\(-4\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{1}q^{5}+(-1-\beta _{2})q^{7}+\cdots\)
3366.2.a.bf	$3366$	$2$	3366.a	1.a	$1$	$4$	$4$	$26.878$	4.4.17417.1	None	✓	$1$	$0$	\(-4\)	\(0\)	\(1\)	\(1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta _{2}+\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
3366.2.a.bg	$3366$	$2$	3366.a	1.a	$1$	$4$	$4$	$26.878$	4.4.55585.1	None	✓	$1$	$0$	\(4\)	\(0\)	\(-5\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta _{2})q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
3366.2.a.bh	$3366$	$2$	3366.a	1.a	$1$	$4$	$4$	$26.878$	4.4.19664.1	None	✓	$1$	$0$	\(4\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{1}q^{5}+(-1-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3366))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3366)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(187))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(306))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(374))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(561))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1122))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1683))\)\(^{\oplus 2}\)