LMFDB - Space of modular forms of level 2009, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 21 $
Sturm bound:			$392$
Trace bound:			$3$
Distinguishing $T_p$:			$2$, $3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(2009))$.

	Total	New	Old
Modular forms	204	136	68
Cusp forms	189	136	53
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.

$7$	$41$	Fricke	Dim
$+$	$+$	$+$	$29$
$+$	$-$	$-$	$37$
$-$	$+$	$-$	$41$
$-$	$-$	$+$	$29$
Plus space		$+$	$58$
Minus space		$-$	$78$

Trace form

$ 136 q - 2 q^{2} + 136 q^{4} + 2 q^{6} - 6 q^{8} + 132 q^{9} + O(q^{10}) $

$ 136 q - 2 q^{2} + 136 q^{4} + 2 q^{6} - 6 q^{8} + 132 q^{9} - 6 q^{11} - 8 q^{12} - 12 q^{13} - 2 q^{15} + 120 q^{16} - 8 q^{17} - 22 q^{18} - 4 q^{19} + 16 q^{20} - 8 q^{22} - 12 q^{23} + 26 q^{24} + 124 q^{25} + 8 q^{26} - 6 q^{27} - 8 q^{29} + 10 q^{30} + 8 q^{31} - 14 q^{32} - 4 q^{33} + 140 q^{36} + 16 q^{37} + 26 q^{38} - 16 q^{39} + 20 q^{40} - 4 q^{41} - 8 q^{43} - 38 q^{44} + 32 q^{45} + 20 q^{46} + 4 q^{48} - 6 q^{50} - 32 q^{51} - 36 q^{52} + 4 q^{53} + 30 q^{54} - 18 q^{55} - 4 q^{57} - 32 q^{58} - 16 q^{59} + 34 q^{60} - 4 q^{61} + 80 q^{64} + 4 q^{65} + 96 q^{66} + 6 q^{67} + 32 q^{68} + 32 q^{69} - 12 q^{71} - 50 q^{72} - 52 q^{73} - 16 q^{74} + 24 q^{75} + 24 q^{76} - 24 q^{78} - 24 q^{79} + 36 q^{80} + 96 q^{81} - 2 q^{82} + 4 q^{83} - 44 q^{85} + 36 q^{86} - 4 q^{87} + 28 q^{88} + 44 q^{89} - 32 q^{90} + 20 q^{92} + 8 q^{93} + 42 q^{94} - 18 q^{95} + 46 q^{96} - 36 q^{97} - 24 q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(2009))$ 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
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	7	41	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
2009.2.a.a	$2009$	$2$	2009.a	1.a	$1$	$2$	$2$	$16.042$	$\Q(\sqrt{5}) $	None	✓	$1$	$1$	$-1$	$1$	$-1$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{2}+(1-\beta )q^{3}+(-1+\beta )q^{4}+(-1+\cdots)q^{5}+\cdots$
2009.2.a.b	$2009$	$2$	2009.a	1.a	$1$	$2$	$2$	$16.042$	$\Q(\sqrt{5}) $	None	✓	$1$	$0$	$-1$	$3$	$1$	$0$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{2}+(1+\beta )q^{3}+(-1+\beta )q^{4}+(1+\cdots)q^{5}+\cdots$
2009.2.a.c	$2009$	$2$	2009.a	1.a	$1$	$2$	$2$	$16.042$	$\Q(\sqrt{13}) $	None	✓	$1$	$1$	$0$	$-1$	$1$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{3}-2q^{4}+(1-\beta )q^{5}+\beta q^{9}+(-2+\cdots)q^{11}+\cdots$
2009.2.a.d	$2009$	$2$	2009.a	1.a	$1$	$2$	$2$	$16.042$	$\Q(\sqrt{13}) $	None	✓	$1$	$1$	$0$	$1$	$-1$	$0$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-2q^{4}+(-1+\beta )q^{5}+\beta q^{9}+\cdots$
2009.2.a.e	$2009$	$2$	2009.a	1.a	$1$	$2$	$2$	$16.042$	$\Q(\sqrt{5}) $	None	✓	$1$	$1$	$1$	$-2$	$2$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}+q^{5}-\beta q^{6}+\cdots$
2009.2.a.f	$2009$	$2$	2009.a	1.a	$1$	$2$	$2$	$16.042$	$\Q(\sqrt{5}) $	None	✓	$1$	$1$	$1$	$2$	$-2$	$0$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}-q^{5}+\beta q^{6}+\cdots$
2009.2.a.g	$2009$	$2$	2009.a	1.a	$1$	$3$	$3$	$16.042$	3.3.148.1	None	✓	$1$	$0$	$-1$	$0$	$2$	$0$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-\beta _{1}-\beta _{2})q^{2}-\beta _{2}q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots$
2009.2.a.h	$2009$	$2$	2009.a	1.a	$1$	$3$	$3$	$16.042$	3.3.257.1	None	✓	$1$	$0$	$0$	$-3$	$0$	$0$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots$
2009.2.a.i	$2009$	$2$	2009.a	1.a	$1$	$3$	$3$	$16.042$	3.3.257.1	None	✓	$1$	$1$	$0$	$3$	$0$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots$
2009.2.a.j	$2009$	$2$	2009.a	1.a	$1$	$3$	$3$	$16.042$	3.3.257.1	None	✓	$1$	$1$	$1$	$1$	$-6$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots$
2009.2.a.k	$2009$	$2$	2009.a	1.a	$1$	$3$	$3$	$16.042$	$\Q(\zeta_{14})^+$	None	✓	$1$	$0$	$4$	$-5$	$-2$	$0$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta _{1})q^{2}+(-2+\beta _{1})q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots$
2009.2.a.l	$2009$	$2$	2009.a	1.a	$1$	$5$	$5$	$16.042$	5.5.233489.1	None	✓	$1$	$1$	$-2$	$-2$	$2$	$0$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}-\beta _{3}q^{3}+(1+\beta _{1})q^{4}+\beta _{3}q^{5}+\cdots$
2009.2.a.m	$2009$	$2$	2009.a	1.a	$1$	$5$	$5$	$16.042$	5.5.233489.1	None	✓	$1$	$1$	$-2$	$2$	$-2$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}+\beta _{3}q^{3}+(1+\beta _{1})q^{4}-\beta _{3}q^{5}+\cdots$
2009.2.a.n	$2009$	$2$	2009.a	1.a	$1$	$5$	$5$	$16.042$	5.5.633117.1	None	✓	$1$	$1$	$-1$	$-4$	$5$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(-1+\beta _{1})q^{3}+(1+\beta _{2})q^{4}+\cdots$
2009.2.a.o	$2009$	$2$	2009.a	1.a	$1$	$6$	$6$	$16.042$	6.6.185257757.1	None	✓	$1$	$0$	$-1$	$4$	$1$	$0$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(1+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots$
2009.2.a.p	$2009$	$2$	2009.a	1.a	$1$	$7$	$7$	$16.042$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	$1$	$1$	$-1$	$-7$	$-4$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{2}+(-1+\beta _{1})q^{3}+(1-\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots$
2009.2.a.q	$2009$	$2$	2009.a	1.a	$1$	$7$	$7$	$16.042$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	$1$	$0$	$-1$	$7$	$4$	$0$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{2}+(1-\beta _{1})q^{3}+(1-\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots$
2009.2.a.r	$2009$	$2$	2009.a	1.a	$1$	$17$	$17$	$16.042$	$\mathbb{Q}[x]/(x^{17} - \cdots)$	None	✓	$1$	$0$	$3$	$-1$	$1$	$0$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{12}q^{3}+(1+\beta _{2})q^{4}+\beta _{13}q^{5}+\cdots$
2009.2.a.s	$2009$	$2$	2009.a	1.a	$1$	$17$	$17$	$16.042$	$\mathbb{Q}[x]/(x^{17} - \cdots)$	None	✓	$1$	$0$	$3$	$1$	$-1$	$0$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-\beta _{12}q^{3}+(1+\beta _{2})q^{4}-\beta _{13}q^{5}+\cdots$
2009.2.a.t	$2009$	$2$	2009.a	1.a	$1$	$20$	$20$	$16.042$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	✓	$1$	$1$	$-2$	$-8$	$-8$	$0$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-\beta _{9}q^{3}+(1+\beta _{2})q^{4}-\beta _{18}q^{5}+\cdots$
2009.2.a.u	$2009$	$2$	2009.a	1.a	$1$	$20$	$20$	$16.042$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	✓	$1$	$0$	$-2$	$8$	$8$	$0$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+\beta _{9}q^{3}+(1+\beta _{2})q^{4}+\beta _{18}q^{5}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(2009))$ into lower level spaces

$ S_{2}^{\mathrm{old}}(\Gamma_0(2009)) \cong $ $S_{2}^{\mathrm{new}}(\Gamma_0(41))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(49))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(287))$$^{\oplus 2}$

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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(392\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

\(7\)	\(41\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(29\)
\(+\)	\(-\)	$-$	\(37\)
\(-\)	\(+\)	$-$	\(41\)
\(-\)	\(-\)	$+$	\(29\)
Plus space		\(+\)	\(58\)
Minus space		\(-\)	\(78\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2009))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2009))\) into lower level spaces

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2009.2.a

Level	$2009$
Weight	$2$
Character orbit	2009.a
Rep. character	$\chi_{2009}(1,\cdot)$
Character field	$\Q$
Dimension	$136$
Newform subspaces	$21$
Sturm bound	$392$
Trace bound	$3$