# Properties

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

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

## 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 2.61803 0.618034 −0.618034 −4.23607 0 2.23607 3.85410 1.00000
1.2 0.618034 0.381966 −1.61803 1.61803 0.236068 0 −2.23607 −2.85410 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.b 2
7.b odd 2 1 287.2.a.a 2
21.c even 2 1 2583.2.a.h 2
28.d even 2 1 4592.2.a.p 2
35.c odd 2 1 7175.2.a.h 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T + T^{2}$$
$3$ $$1 - 3 T + T^{2}$$
$5$ $$-1 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-5 + T^{2}$$
$13$ $$( -3 + T )^{2}$$
$17$ $$-11 - 6 T + T^{2}$$
$19$ $$-9 - 3 T + T^{2}$$
$23$ $$1 - 3 T + T^{2}$$
$29$ $$9 - 9 T + T^{2}$$
$31$ $$71 - 17 T + T^{2}$$
$37$ $$-44 - 2 T + T^{2}$$
$41$ $$( 1 + T )^{2}$$
$43$ $$-79 + 2 T + T^{2}$$
$47$ $$-4 - 2 T + T^{2}$$
$53$ $$41 + 17 T + T^{2}$$
$59$ $$41 + 13 T + T^{2}$$
$61$ $$( -11 + T )^{2}$$
$67$ $$29 - 11 T + T^{2}$$
$71$ $$-1 + 4 T + T^{2}$$
$73$ $$( -15 + T )^{2}$$
$79$ $$-44 + 2 T + T^{2}$$
$83$ $$-71 + 6 T + T^{2}$$
$89$ $$-81 - 9 T + T^{2}$$
$97$ $$61 - 17 T + T^{2}$$