# Properties

 Label 2009.2.a.j Level $2009$ Weight $2$ Character orbit 2009.a Self dual yes Analytic conductor $16.042$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2009 = 7^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2009.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.0419457661$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 287) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{2} ) q^{4} -2 q^{5} + ( 3 - \beta_{1} ) q^{6} + \beta_{2} q^{8} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{2} ) q^{4} -2 q^{5} + ( 3 - \beta_{1} ) q^{6} + \beta_{2} q^{8} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{9} -2 \beta_{1} q^{10} -2 q^{11} + ( -3 + \beta_{1} + \beta_{2} ) q^{12} + ( -3 + \beta_{2} ) q^{13} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{15} + ( -2 + \beta_{1} - \beta_{2} ) q^{16} + ( -\beta_{1} + 2 \beta_{2} ) q^{17} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{18} + ( -4 - \beta_{1} ) q^{19} + ( -2 - 2 \beta_{2} ) q^{20} -2 \beta_{1} q^{22} + ( -4 + \beta_{1} - \beta_{2} ) q^{23} + ( -3 + 2 \beta_{2} ) q^{24} - q^{25} + ( -2 \beta_{1} + \beta_{2} ) q^{26} + ( 3 + \beta_{1} - 4 \beta_{2} ) q^{27} + ( 4 - 2 \beta_{1} ) q^{29} + ( -6 + 2 \beta_{1} ) q^{30} + 2 \beta_{1} q^{31} + ( 3 - 3 \beta_{1} - 2 \beta_{2} ) q^{32} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{34} + ( -3 - 4 \beta_{1} + 2 \beta_{2} ) q^{36} + ( 1 + 5 \beta_{2} ) q^{37} + ( -3 - 4 \beta_{1} - \beta_{2} ) q^{38} + ( -3 - 3 \beta_{1} + 5 \beta_{2} ) q^{39} -2 \beta_{2} q^{40} + q^{41} + ( 4 - 3 \beta_{1} - 4 \beta_{2} ) q^{43} + ( -2 - 2 \beta_{2} ) q^{44} + ( -6 + 2 \beta_{1} + 4 \beta_{2} ) q^{45} + ( 3 - 5 \beta_{1} ) q^{46} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 6 - 3 \beta_{1} ) q^{48} -\beta_{1} q^{50} + ( -9 + \beta_{1} + 4 \beta_{2} ) q^{51} + ( \beta_{1} - 3 \beta_{2} ) q^{52} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{54} + 4 q^{55} + ( -3 - 3 \beta_{1} + 4 \beta_{2} ) q^{57} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{58} + 4 q^{59} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{60} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( 6 + 2 \beta_{2} ) q^{62} + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{64} + ( 6 - 2 \beta_{2} ) q^{65} + ( -6 + 2 \beta_{1} ) q^{66} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 6 - \beta_{2} ) q^{68} + ( 6 - 5 \beta_{1} + 2 \beta_{2} ) q^{69} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -6 - 3 \beta_{1} + 4 \beta_{2} ) q^{72} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{73} + ( 6 \beta_{1} + 5 \beta_{2} ) q^{74} + ( -\beta_{1} + \beta_{2} ) q^{75} + ( -4 - 2 \beta_{1} - 5 \beta_{2} ) q^{76} + ( -9 + 2 \beta_{1} + 2 \beta_{2} ) q^{78} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{80} + ( 6 + 5 \beta_{1} - 5 \beta_{2} ) q^{81} + \beta_{1} q^{82} + ( 2 - 4 \beta_{1} ) q^{83} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{85} + ( -9 - 7 \beta_{2} ) q^{86} + ( -6 + 6 \beta_{1} - 4 \beta_{2} ) q^{87} -2 \beta_{2} q^{88} + ( 1 - 8 \beta_{1} + \beta_{2} ) q^{89} + ( 6 - 2 \beta_{1} + 6 \beta_{2} ) q^{90} + ( -7 + \beta_{1} - 3 \beta_{2} ) q^{92} + ( 6 - 2 \beta_{1} ) q^{93} + ( -6 + 6 \beta_{1} + \beta_{2} ) q^{94} + ( 8 + 2 \beta_{1} ) q^{95} + ( -3 + 6 \beta_{1} - 7 \beta_{2} ) q^{96} + ( -10 + 3 \beta_{1} - \beta_{2} ) q^{97} + ( -6 + 2 \beta_{1} + 4 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} + q^{3} + 3q^{4} - 6q^{5} + 8q^{6} + 8q^{9} + O(q^{10})$$ $$3q + q^{2} + q^{3} + 3q^{4} - 6q^{5} + 8q^{6} + 8q^{9} - 2q^{10} - 6q^{11} - 8q^{12} - 9q^{13} - 2q^{15} - 5q^{16} - q^{17} - 8q^{18} - 13q^{19} - 6q^{20} - 2q^{22} - 11q^{23} - 9q^{24} - 3q^{25} - 2q^{26} + 10q^{27} + 10q^{29} - 16q^{30} + 2q^{31} + 6q^{32} - 2q^{33} - 7q^{34} - 13q^{36} + 3q^{37} - 13q^{38} - 12q^{39} + 3q^{41} + 9q^{43} - 6q^{44} - 16q^{45} + 4q^{46} + 7q^{47} + 15q^{48} - q^{50} - 26q^{51} + q^{52} + 2q^{53} + 8q^{54} + 12q^{55} - 12q^{57} - 14q^{58} + 12q^{59} + 16q^{60} + 4q^{61} + 18q^{62} - 16q^{64} + 18q^{65} - 16q^{66} + 10q^{67} + 18q^{68} + 13q^{69} - 14q^{71} - 21q^{72} + 4q^{73} + 6q^{74} - q^{75} - 14q^{76} - 25q^{78} + 14q^{79} + 10q^{80} + 23q^{81} + q^{82} + 2q^{83} + 2q^{85} - 27q^{86} - 12q^{87} - 5q^{89} + 16q^{90} - 20q^{92} + 16q^{93} - 12q^{94} + 26q^{95} - 3q^{96} - 27q^{97} - 16q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.91223 0.713538 2.19869
−1.91223 −2.56885 1.65662 −2.00000 4.91223 0 0.656620 3.59899 3.82446
1.2 0.713538 3.20440 −1.49086 −2.00000 2.28646 0 −2.49086 7.26819 −1.42708
1.3 2.19869 0.364448 2.83424 −2.00000 0.801309 0 1.83424 −2.86718 −4.39738
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$41$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2009.2.a.j 3
7.b odd 2 1 287.2.a.c 3
21.c even 2 1 2583.2.a.m 3
28.d even 2 1 4592.2.a.t 3
35.c odd 2 1 7175.2.a.k 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.a.c 3 7.b odd 2 1
2009.2.a.j 3 1.a even 1 1 trivial
2583.2.a.m 3 21.c even 2 1
4592.2.a.t 3 28.d even 2 1
7175.2.a.k 3 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2009))$$:

 $$T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 3$$ $$T_{3}^{3} - T_{3}^{2} - 8 T_{3} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 - 4 T - T^{2} + T^{3}$$
$3$ $$3 - 8 T - T^{2} + T^{3}$$
$5$ $$( 2 + T )^{3}$$
$7$ $$T^{3}$$
$11$ $$( 2 + T )^{3}$$
$13$ $$15 + 22 T + 9 T^{2} + T^{3}$$
$17$ $$27 - 22 T + T^{2} + T^{3}$$
$19$ $$61 + 52 T + 13 T^{2} + T^{3}$$
$23$ $$19 + 32 T + 11 T^{2} + T^{3}$$
$29$ $$8 + 16 T - 10 T^{2} + T^{3}$$
$31$ $$24 - 16 T - 2 T^{2} + T^{3}$$
$37$ $$499 - 122 T - 3 T^{2} + T^{3}$$
$41$ $$( -1 + T )^{3}$$
$43$ $$835 - 104 T - 9 T^{2} + T^{3}$$
$47$ $$213 - 40 T - 7 T^{2} + T^{3}$$
$53$ $$-72 - 40 T - 2 T^{2} + T^{3}$$
$59$ $$( -4 + T )^{3}$$
$61$ $$152 - 36 T - 4 T^{2} + T^{3}$$
$67$ $$200 - 8 T - 10 T^{2} + T^{3}$$
$71$ $$-24 - 16 T + 14 T^{2} + T^{3}$$
$73$ $$392 - 84 T - 4 T^{2} + T^{3}$$
$79$ $$56 + 32 T - 14 T^{2} + T^{3}$$
$83$ $$-56 - 68 T - 2 T^{2} + T^{3}$$
$89$ $$-1801 - 266 T + 5 T^{2} + T^{3}$$
$97$ $$461 + 202 T + 27 T^{2} + T^{3}$$