# Properties

 Label 2009.2.a.c 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: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} -2 q^{4} + ( 1 - \beta ) q^{5} + \beta q^{9} +O(q^{10})$$ $$q -\beta q^{3} -2 q^{4} + ( 1 - \beta ) q^{5} + \beta q^{9} + ( -2 - \beta ) q^{11} + 2 \beta q^{12} + \beta q^{13} + 3 q^{15} + 4 q^{16} + ( -3 + 3 \beta ) q^{17} + ( -1 + 2 \beta ) q^{19} + ( -2 + 2 \beta ) q^{20} + ( -1 - 2 \beta ) q^{23} + ( -1 - \beta ) q^{25} + ( -3 + 2 \beta ) q^{27} + ( 1 + 2 \beta ) q^{29} + \beta q^{31} + ( 3 + 3 \beta ) q^{33} -2 \beta q^{36} + ( -2 + 4 \beta ) q^{37} + ( -3 - \beta ) q^{39} + q^{41} + ( -5 - 2 \beta ) q^{43} + ( 4 + 2 \beta ) q^{44} -3 q^{45} + 9 q^{47} -4 \beta q^{48} -9 q^{51} -2 \beta q^{52} + ( 3 + 3 \beta ) q^{53} + ( 1 + 2 \beta ) q^{55} + ( -6 - \beta ) q^{57} + ( -5 + 2 \beta ) q^{59} -6 q^{60} + ( 2 - 4 \beta ) q^{61} -8 q^{64} -3 q^{65} + ( -2 - 2 \beta ) q^{67} + ( 6 - 6 \beta ) q^{68} + ( 6 + 3 \beta ) q^{69} + ( -3 + 6 \beta ) q^{71} + ( -7 - \beta ) q^{73} + ( 3 + 2 \beta ) q^{75} + ( 2 - 4 \beta ) q^{76} + ( 1 - 8 \beta ) q^{79} + ( 4 - 4 \beta ) q^{80} + ( -6 - 2 \beta ) q^{81} + ( -7 - 2 \beta ) q^{83} + ( -12 + 3 \beta ) q^{85} + ( -6 - 3 \beta ) q^{87} + ( -5 + 2 \beta ) q^{89} + ( 2 + 4 \beta ) q^{92} + ( -3 - \beta ) q^{93} + ( -7 + \beta ) q^{95} + ( -7 - \beta ) q^{97} + ( -3 - 3 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - 4q^{4} + q^{5} + q^{9} + O(q^{10})$$ $$2q - q^{3} - 4q^{4} + q^{5} + q^{9} - 5q^{11} + 2q^{12} + q^{13} + 6q^{15} + 8q^{16} - 3q^{17} - 2q^{20} - 4q^{23} - 3q^{25} - 4q^{27} + 4q^{29} + q^{31} + 9q^{33} - 2q^{36} - 7q^{39} + 2q^{41} - 12q^{43} + 10q^{44} - 6q^{45} + 18q^{47} - 4q^{48} - 18q^{51} - 2q^{52} + 9q^{53} + 4q^{55} - 13q^{57} - 8q^{59} - 12q^{60} - 16q^{64} - 6q^{65} - 6q^{67} + 6q^{68} + 15q^{69} - 15q^{73} + 8q^{75} - 6q^{79} + 4q^{80} - 14q^{81} - 16q^{83} - 21q^{85} - 15q^{87} - 8q^{89} + 8q^{92} - 7q^{93} - 13q^{95} - 15q^{97} - 9q^{99} + O(q^{100})$$

## 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
 2.30278 −1.30278
0 −2.30278 −2.00000 −1.30278 0 0 0 2.30278 0
1.2 0 1.30278 −2.00000 2.30278 0 0 0 −1.30278 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.c 2
7.b odd 2 1 2009.2.a.d 2
7.d odd 6 2 287.2.e.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.b 4 7.d odd 6 2
2009.2.a.c 2 1.a even 1 1 trivial
2009.2.a.d 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2009))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-3 + T + T^{2}$$
$5$ $$-3 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$3 + 5 T + T^{2}$$
$13$ $$-3 - T + T^{2}$$
$17$ $$-27 + 3 T + T^{2}$$
$19$ $$-13 + T^{2}$$
$23$ $$-9 + 4 T + T^{2}$$
$29$ $$-9 - 4 T + T^{2}$$
$31$ $$-3 - T + T^{2}$$
$37$ $$-52 + T^{2}$$
$41$ $$( -1 + T )^{2}$$
$43$ $$23 + 12 T + T^{2}$$
$47$ $$( -9 + T )^{2}$$
$53$ $$-9 - 9 T + T^{2}$$
$59$ $$3 + 8 T + T^{2}$$
$61$ $$-52 + T^{2}$$
$67$ $$-4 + 6 T + T^{2}$$
$71$ $$-117 + T^{2}$$
$73$ $$53 + 15 T + T^{2}$$
$79$ $$-199 + 6 T + T^{2}$$
$83$ $$51 + 16 T + T^{2}$$
$89$ $$3 + 8 T + T^{2}$$
$97$ $$53 + 15 T + T^{2}$$