Space of modular forms of level 666, weight 2, and trivial character

• Label: 666.2.a
• Level: $666$
• Weight: $2$
• Character orbit: 666.a
• Rep. character: $\chi_{666}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $11$
• Sturm bound: $228$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(228\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(666))\).

	Total	New	Old
Modular forms	122	15	107
Cusp forms	107	15	92
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.

\(2\)	\(3\)	\(37\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
Plus space			\(+\)	\(5\)
Minus space			\(-\)	\(10\)

Trace form

\( 15 q + q^{2} + 15 q^{4} - 2 q^{5} + q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(666))\) 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	37
666.2.a.a	$666$	$2$	666.a	1.a	$1$	$1$	$1$	$5.318$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{11}-q^{13}+\cdots\)
666.2.a.b	$666$	$2$	666.a	1.a	$1$	$1$	$1$	$5.318$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+3q^{7}-q^{8}-q^{11}+q^{13}+\cdots\)
666.2.a.c	$666$	$2$	666.a	1.a	$1$	$1$	$1$	$5.318$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-3q^{7}-q^{8}-2q^{10}+\cdots\)
666.2.a.d	$666$	$2$	666.a	1.a	$1$	$1$	$1$	$5.318$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-4\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-4q^{5}-q^{7}+q^{8}-4q^{10}+\cdots\)
666.2.a.e	$666$	$2$	666.a	1.a	$1$	$1$	$1$	$5.318$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}-3q^{7}+q^{8}-2q^{10}+\cdots\)
666.2.a.f	$666$	$2$	666.a	1.a	$1$	$1$	$1$	$5.318$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}+q^{8}-2q^{10}+4q^{11}+\cdots\)
666.2.a.g	$666$	$2$	666.a	1.a	$1$	$1$	$1$	$5.318$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(4\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{5}+3q^{7}+q^{8}+4q^{10}+\cdots\)
666.2.a.h	$666$	$2$	666.a	1.a	$1$	$2$	$2$	$5.318$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}+(1-\beta )q^{7}-q^{8}+\cdots\)
666.2.a.i	$666$	$2$	666.a	1.a	$1$	$2$	$2$	$5.318$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1-3\beta )q^{5}-2\beta q^{7}-q^{8}+\cdots\)
666.2.a.j	$666$	$2$	666.a	1.a	$1$	$2$	$2$	$5.318$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}+(2-2\beta )q^{7}+q^{8}+\cdots\)
666.2.a.k	$666$	$2$	666.a	1.a	$1$	$2$	$2$	$5.318$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(4\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{5}+(1-\beta )q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(666))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(666)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(222))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(333))\)\(^{\oplus 2}\)