Space of modular forms of level 609, weight 2, and trivial character

• Label: 609.2.a
• Level: $609$
• Weight: $2$
• Character orbit: 609.a
• Rep. character: $\chi_{609}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $9$
• Sturm bound: $160$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 609 = 3 \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	609.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(160\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(609))\).

	Total	New	Old
Modular forms	84	27	57
Cusp forms	77	27	50
Eisenstein series	7	0	7

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

\(3\)	\(7\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(10\)
Minus space			\(-\)	\(17\)

Trace form

\( 27 q - 3 q^{2} - q^{3} + 29 q^{4} - 6 q^{5} + 5 q^{6} - q^{7} - 15 q^{8} + 27 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(609))\) 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	29
609.2.a.a	$609$	$2$	609.a	1.a	$1$	$1$	$1$	$4.863$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-2\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+q^{7}+\cdots\)
609.2.a.b	$609$	$2$	609.a	1.a	$1$	$1$	$1$	$4.863$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}+q^{7}-3q^{8}+\cdots\)
609.2.a.c	$609$	$2$	609.a	1.a	$1$	$2$	$2$	$4.863$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(2\)	\(-4\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
609.2.a.d	$609$	$2$	609.a	1.a	$1$	$3$	$3$	$4.863$	3.3.148.1	None	✓	$1$	$1$	\(-3\)	\(3\)	\(-6\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
609.2.a.e	$609$	$2$	609.a	1.a	$1$	$3$	$3$	$4.863$	3.3.148.1	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(4\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
609.2.a.f	$609$	$2$	609.a	1.a	$1$	$4$	$4$	$4.863$	4.4.14656.1	None	✓	$1$	$0$	\(-2\)	\(-4\)	\(-2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
609.2.a.g	$609$	$2$	609.a	1.a	$1$	$4$	$4$	$4.863$	4.4.6224.1	None	✓	$1$	$0$	\(2\)	\(4\)	\(4\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{2}+q^{3}+(1-\beta _{1}-\beta _{2})q^{4}+\cdots\)
609.2.a.h	$609$	$2$	609.a	1.a	$1$	$4$	$4$	$4.863$	4.4.4352.1	None	✓	$1$	$0$	\(4\)	\(4\)	\(4\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}-\beta _{3})q^{2}+q^{3}+(2+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
609.2.a.i	$609$	$2$	609.a	1.a	$1$	$5$	$5$	$4.863$	5.5.1280720.1	None	✓	$1$	$0$	\(-1\)	\(-5\)	\(-4\)	\(-5\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(609))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(609)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(203))\)\(^{\oplus 2}\)