Space of modular forms of level 6699, weight 2, and trivial character

• Label: 6699.2.a
• Level: $6699$
• Weight: $2$
• Character orbit: 6699.a
• Rep. character: $\chi_{6699}(1,\cdot)$
• Character field: $\Q$
• Dimension: $281$
• Newform subspaces: $24$
• Sturm bound: $1920$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 6699 = 3 \cdot 7 \cdot 11 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6699.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(1920\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6699))\).

	Total	New	Old
Modular forms	968	281	687
Cusp forms	953	281	672
Eisenstein series	15	0	15

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

\(3\)	\(7\)	\(11\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(14\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(21\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(23\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(12\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(16\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(19\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(17\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(18\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(18\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(17\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(15\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(20\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(12\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(23\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(25\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(11\)
Plus space				\(+\)	\(117\)
Minus space				\(-\)	\(164\)

Trace form

\( 281 q + 3 q^{2} + q^{3} + 271 q^{4} + 6 q^{5} - 13 q^{6} + q^{7} + 15 q^{8} + 281 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6699))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7	11	29
6699.2.a.a	$6699$	$2$	6699.a	1.a	$1$	$1$	$1$	$53.492$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-2\)	\(-1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}-q^{7}+\cdots\)
6699.2.a.b	$6699$	$2$	6699.a	1.a	$1$	$1$	$1$	$53.492$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(1\)	\(-1\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
6699.2.a.c	$6699$	$2$	6699.a	1.a	$1$	$1$	$1$	$53.492$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(2\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-q^{7}+\cdots\)
6699.2.a.d	$6699$	$2$	6699.a	1.a	$1$	$1$	$1$	$53.492$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(2\)	\(-1\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-q^{7}+\cdots\)
6699.2.a.e	$6699$	$2$	6699.a	1.a	$1$	$1$	$1$	$53.492$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(3\)	\(1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}+3q^{5}-q^{6}+q^{7}+\cdots\)
6699.2.a.f	$6699$	$2$	6699.a	1.a	$1$	$1$	$1$	$53.492$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}-2q^{5}+q^{6}-q^{7}+\cdots\)
6699.2.a.g	$6699$	$2$	6699.a	1.a	$1$	$2$	$2$	$53.492$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-2\)	\(2\)	\(3\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}+(1+\beta )q^{5}-q^{6}+\cdots\)
6699.2.a.h	$6699$	$2$	6699.a	1.a	$1$	$2$	$2$	$53.492$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(2\)	\(-2\)	\(-3\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}+(-2+\beta )q^{5}-q^{6}+\cdots\)
6699.2.a.i	$6699$	$2$	6699.a	1.a	$1$	$10$	$10$	$53.492$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-10\)	\(1\)	\(-10\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{7}q^{2}-q^{3}+(1+\beta _{8})q^{4}+(\beta _{1}+\beta _{6}+\cdots)q^{5}+\cdots\)
6699.2.a.j	$6699$	$2$	6699.a	1.a	$1$	$11$	$11$	$53.492$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(-11\)	\(-5\)	\(-11\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(3-\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)
6699.2.a.k	$6699$	$2$	6699.a	1.a	$1$	$11$	$11$	$53.492$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(11\)	\(-12\)	\(11\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-1-\beta _{5}+\cdots)q^{5}+\cdots\)
6699.2.a.l	$6699$	$2$	6699.a	1.a	$1$	$12$	$12$	$53.492$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(12\)	\(-4\)	\(12\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+\beta _{10}q^{5}-\beta _{1}q^{6}+\cdots\)
6699.2.a.m	$6699$	$2$	6699.a	1.a	$1$	$15$	$15$	$53.492$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(15\)	\(-10\)	\(-15\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(-1-\beta _{5}+\cdots)q^{5}+\cdots\)
6699.2.a.n	$6699$	$2$	6699.a	1.a	$1$	$15$	$15$	$53.492$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(15\)	\(-3\)	\(-15\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{13}q^{5}+\cdots\)
6699.2.a.o	$6699$	$2$	6699.a	1.a	$1$	$16$	$16$	$53.492$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(-16\)	\(14\)	\(16\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{6}+\cdots)q^{5}+\cdots\)
6699.2.a.p	$6699$	$2$	6699.a	1.a	$1$	$17$	$17$	$53.492$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-17\)	\(-8\)	\(17\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{11}q^{5}+\cdots\)
6699.2.a.q	$6699$	$2$	6699.a	1.a	$1$	$17$	$17$	$53.492$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(17\)	\(-2\)	\(-17\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots\)
6699.2.a.r	$6699$	$2$	6699.a	1.a	$1$	$18$	$18$	$53.492$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(-18\)	\(-13\)	\(18\)		$+$	$-$	$+$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1-\beta _{4}+\cdots)q^{5}+\cdots\)
6699.2.a.s	$6699$	$2$	6699.a	1.a	$1$	$18$	$18$	$53.492$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(-18\)	\(8\)	\(18\)		$+$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{10}q^{5}+\cdots\)
6699.2.a.t	$6699$	$2$	6699.a	1.a	$1$	$20$	$20$	$53.492$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(20\)	\(10\)	\(-20\)		$-$	$+$	$-$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{9}+\cdots)q^{5}+\cdots\)
6699.2.a.u	$6699$	$2$	6699.a	1.a	$1$	$21$	$21$	$53.492$		None	✓	$1$	$0$	\(3\)	\(-21\)	\(6\)	\(-21\)		$+$	$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
6699.2.a.v	$6699$	$2$	6699.a	1.a	$1$	$22$	$22$	$53.492$		None	✓	$1$	$0$	\(2\)	\(-22\)	\(2\)	\(-22\)		$+$	$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
6699.2.a.w	$6699$	$2$	6699.a	1.a	$1$	$23$	$23$	$53.492$		None	✓	$1$	$0$	\(1\)	\(23\)	\(4\)	\(23\)		$-$	$-$	$+$	$-$	$-$		$\mathrm{SU}(2)$
6699.2.a.x	$6699$	$2$	6699.a	1.a	$1$	$25$	$25$	$53.492$		None	✓	$1$	$0$	\(1\)	\(25\)	\(14\)	\(25\)		$-$	$-$	$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6699))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6699)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(203))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(319))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(609))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(957))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2233))\)\(^{\oplus 2}\)