Space of modular forms of level 3009, weight 2, and trivial character

• Label: 3009.2.a
• Level: $3009$
• Weight: $2$
• Character orbit: 3009.a
• Rep. character: $\chi_{3009}(1,\cdot)$
• Character field: $\Q$
• Dimension: $155$
• Newform subspaces: $11$
• Sturm bound: $720$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 3009 = 3 \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3009.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(720\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3009))\).

	Total	New	Old
Modular forms	364	155	209
Cusp forms	357	155	202
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.

\(3\)	\(17\)	\(59\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(19\)
\(+\)	\(+\)	\(-\)	$-$	\(21\)
\(+\)	\(-\)	\(+\)	$-$	\(20\)
\(+\)	\(-\)	\(-\)	$+$	\(16\)
\(-\)	\(+\)	\(+\)	$-$	\(24\)
\(-\)	\(+\)	\(-\)	$+$	\(14\)
\(-\)	\(-\)	\(+\)	$+$	\(15\)
\(-\)	\(-\)	\(-\)	$-$	\(26\)
Plus space			\(+\)	\(64\)
Minus space			\(-\)	\(91\)

Trace form

\( 155 q + 5 q^{2} + 3 q^{3} + 161 q^{4} - 6 q^{5} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 155 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3009))\) 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	17	59
3009.2.a.a	$3009$	$2$	3009.a	1.a	$1$	$1$	$1$	$24.027$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-q^{5}+2q^{7}+q^{9}-5q^{11}+\cdots\)
3009.2.a.b	$3009$	$2$	3009.a	1.a	$1$	$2$	$2$	$24.027$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$2$	\(-1\)	\(-2\)	\(-3\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+(-1-\beta )q^{5}+\cdots\)
3009.2.a.c	$3009$	$2$	3009.a	1.a	$1$	$2$	$2$	$24.027$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(-2\)	\(3\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+(1+\beta )q^{5}+\cdots\)
3009.2.a.d	$3009$	$2$	3009.a	1.a	$1$	$14$	$14$	$24.027$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(14\)	\(-3\)	\(-11\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{10}q^{5}+\cdots\)
3009.2.a.e	$3009$	$2$	3009.a	1.a	$1$	$14$	$14$	$24.027$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(14\)	\(-5\)	\(-11\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}-\beta _{11}q^{5}-\beta _{1}q^{6}+\cdots\)
3009.2.a.f	$3009$	$2$	3009.a	1.a	$1$	$16$	$16$	$24.027$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(-16\)	\(-3\)	\(-3\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)
3009.2.a.g	$3009$	$2$	3009.a	1.a	$1$	$17$	$17$	$24.027$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(-17\)	\(-11\)	\(-5\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{6}+\cdots)q^{5}+\cdots\)
3009.2.a.h	$3009$	$2$	3009.a	1.a	$1$	$18$	$18$	$24.027$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(-18\)	\(2\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{15}q^{5}+\cdots\)
3009.2.a.i	$3009$	$2$	3009.a	1.a	$1$	$21$	$21$	$24.027$		None	✓	$1$	$0$	\(2\)	\(-21\)	\(10\)	\(1\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
3009.2.a.j	$3009$	$2$	3009.a	1.a	$1$	$24$	$24$	$24.027$		None	✓	$1$	$0$	\(2\)	\(24\)	\(-3\)	\(19\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
3009.2.a.k	$3009$	$2$	3009.a	1.a	$1$	$26$	$26$	$24.027$		None	✓	$1$	$0$	\(3\)	\(26\)	\(8\)	\(13\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3009))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3009)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\)\(^{\oplus 2}\)