# Properties

 Label 2009.2.a.a Level $2009$ Weight $2$ Character orbit 2009.a Self dual yes Analytic conductor $16.042$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−1.61803 −0.618034 0.618034 0.618034 1.00000 0 2.23607 −2.61803 −1.00000
1.2 0.618034 1.61803 −1.61803 −1.61803 1.00000 0 −2.23607 −0.381966 −1.00000
 $$n$$: 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.a 2
7.b odd 2 1 287.2.a.b 2
21.c even 2 1 2583.2.a.g 2
28.d even 2 1 4592.2.a.n 2
35.c odd 2 1 7175.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.a.b 2 7.b odd 2 1
2009.2.a.a 2 1.a even 1 1 trivial
2583.2.a.g 2 21.c even 2 1
4592.2.a.n 2 28.d even 2 1
7175.2.a.g 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2009))$$:

 $$T_{2}^{2} + T_{2} - 1$$ $$T_{3}^{2} - T_{3} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T + T^{2}$$
$3$ $$-1 - T + T^{2}$$
$5$ $$-1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$11 - 8 T + T^{2}$$
$17$ $$-1 - 4 T + T^{2}$$
$19$ $$-11 - T + T^{2}$$
$23$ $$5 + 5 T + T^{2}$$
$29$ $$-5 + 5 T + T^{2}$$
$31$ $$-25 + 5 T + T^{2}$$
$37$ $$20 + 10 T + T^{2}$$
$41$ $$( -1 + T )^{2}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$-36 + 6 T + T^{2}$$
$53$ $$11 - 7 T + T^{2}$$
$59$ $$89 + 19 T + T^{2}$$
$61$ $$5 + 10 T + T^{2}$$
$67$ $$11 + 13 T + T^{2}$$
$71$ $$-71 - 6 T + T^{2}$$
$73$ $$-79 - 2 T + T^{2}$$
$79$ $$-20 + 10 T + T^{2}$$
$83$ $$-19 - 2 T + T^{2}$$
$89$ $$-59 - 3 T + T^{2}$$
$97$ $$-45 - 15 T + T^{2}$$