# Properties

 Label 2009.2.a.f 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.

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

## 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
 −0.618034 1.61803
−0.618034 1.00000 −1.61803 −1.00000 −0.618034 0 2.23607 −2.00000 0.618034
1.2 1.61803 1.00000 0.618034 −1.00000 1.61803 0 −2.23607 −2.00000 −1.61803
 $$n$$: 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.f 2
7.b odd 2 1 2009.2.a.e 2
7.c even 3 2 287.2.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.a 4 7.c even 3 2
2009.2.a.e 2 7.b odd 2 1
2009.2.a.f 2 1.a even 1 1 trivial

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$-1 - 4 T + T^{2}$$
$19$ $$-11 + 6 T + T^{2}$$
$23$ $$-41 + 4 T + T^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$-1 - 4 T + T^{2}$$
$37$ $$-19 - 2 T + T^{2}$$
$41$ $$( 1 + T )^{2}$$
$43$ $$-16 + 4 T + T^{2}$$
$47$ $$-19 - 2 T + T^{2}$$
$53$ $$59 + 16 T + T^{2}$$
$59$ $$-41 - 4 T + T^{2}$$
$61$ $$5 - 10 T + T^{2}$$
$67$ $$-71 + 6 T + T^{2}$$
$71$ $$80 + 20 T + T^{2}$$
$73$ $$29 + 14 T + T^{2}$$
$79$ $$( 1 + T )^{2}$$
$83$ $$16 + 12 T + T^{2}$$
$89$ $$139 + 24 T + T^{2}$$
$97$ $$-4 - 8 T + T^{2}$$