# Properties

 Label 2009.2.a.h Level $2009$ Weight $2$ Character orbit 2009.a Self dual yes Analytic conductor $16.042$ Analytic rank $0$ 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: $$0$$ 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: yes 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.

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

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
 2.19869 −1.91223 0.713538
−1.83424 −2.83424 1.36445 0 5.19869 0 1.16576 5.03293 0
1.2 −0.656620 −1.65662 −1.56885 0 1.08777 0 2.34338 −0.255609 0
1.3 2.49086 1.49086 4.20440 0 3.71354 0 5.49086 −0.777326 0
 $$n$$: 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.h 3
7.b odd 2 1 2009.2.a.i yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2009.2.a.h 3 1.a even 1 1 trivial
2009.2.a.i yes 3 7.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{3} - 5 T_{2} - 3$$ $$T_{3}^{3} + 3 T_{3}^{2} - 2 T_{3} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 - 5 T + T^{3}$$
$3$ $$-7 - 2 T + 3 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3}$$
$11$ $$-40 + 4 T + 8 T^{2} + T^{3}$$
$13$ $$63 - 18 T - 5 T^{2} + T^{3}$$
$17$ $$7 + 8 T - 7 T^{2} + T^{3}$$
$19$ $$-49 + 46 T - 13 T^{2} + T^{3}$$
$23$ $$25 - 20 T + T^{2} + T^{3}$$
$29$ $$-24 - 20 T + T^{3}$$
$31$ $$56 - 68 T + 2 T^{2} + T^{3}$$
$37$ $$-9 + 30 T + 13 T^{2} + T^{3}$$
$41$ $$( 1 + T )^{3}$$
$43$ $$-3 - 8 T + T^{2} + T^{3}$$
$47$ $$-133 - 12 T + 9 T^{2} + T^{3}$$
$53$ $$-40 + 4 T + 8 T^{2} + T^{3}$$
$59$ $$168 - 92 T + 4 T^{2} + T^{3}$$
$61$ $$-1176 - 200 T + 6 T^{2} + T^{3}$$
$67$ $$536 - 76 T - 8 T^{2} + T^{3}$$
$71$ $$-200 - 8 T + 10 T^{2} + T^{3}$$
$73$ $$-504 - 72 T + 10 T^{2} + T^{3}$$
$79$ $$( 6 + T )^{3}$$
$83$ $$56 + 32 T - 14 T^{2} + T^{3}$$
$89$ $$63 - 18 T - 5 T^{2} + T^{3}$$
$97$ $$-63 + 96 T - 19 T^{2} + T^{3}$$