# Properties

 Label 2009.2.a.k 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: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} ) q^{2} + ( -2 + \beta_{1} ) q^{3} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{4} + 2 \beta_{2} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{6} + ( 4 + \beta_{1} + 4 \beta_{2} ) q^{8} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} ) q^{2} + ( -2 + \beta_{1} ) q^{3} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{4} + 2 \beta_{2} q^{5} + ( -\beta_{1} + \beta_{2} ) q^{6} + ( 4 + \beta_{1} + 4 \beta_{2} ) q^{8} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{9} + ( 2 + 4 \beta_{2} ) q^{10} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{11} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{12} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{13} + ( 2 - 2 \beta_{2} ) q^{15} + ( 8 + \beta_{1} + 7 \beta_{2} ) q^{16} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{17} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{18} + ( 2 + 2 \beta_{1} - 3 \beta_{2} ) q^{19} + ( 6 + 2 \beta_{1} + 4 \beta_{2} ) q^{20} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{22} + ( -6 + 3 \beta_{1} - 2 \beta_{2} ) q^{23} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{24} + ( -1 + 4 \beta_{1} - 4 \beta_{2} ) q^{25} + ( -6 - 4 \beta_{1} - \beta_{2} ) q^{26} + ( -7 + 8 \beta_{1} - 5 \beta_{2} ) q^{27} + ( -2 + 4 \beta_{1} - 4 \beta_{2} ) q^{29} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{30} + ( 4 - 6 \beta_{1} + 6 \beta_{2} ) q^{31} + ( 9 + 7 \beta_{1} + 7 \beta_{2} ) q^{32} + ( -2 + 4 \beta_{1} - 4 \beta_{2} ) q^{33} + ( 7 + 2 \beta_{1} + 6 \beta_{2} ) q^{34} + ( -14 + 3 \beta_{1} - 7 \beta_{2} ) q^{36} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{37} + ( 3 + 4 \beta_{1} - 4 \beta_{2} ) q^{38} + ( -3 + 5 \beta_{1} - 4 \beta_{2} ) q^{39} + ( 10 + 8 \beta_{1} + 2 \beta_{2} ) q^{40} - q^{41} + ( 4 + 2 \beta_{1} + 3 \beta_{2} ) q^{43} + ( -4 - 2 \beta_{2} ) q^{44} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{45} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{46} + ( -5 + \beta_{1} - \beta_{2} ) q^{47} + ( -7 + 6 \beta_{1} - 6 \beta_{2} ) q^{48} + ( 3 + 3 \beta_{1} - 4 \beta_{2} ) q^{50} + ( 13 - 10 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -13 - 4 \beta_{1} - 8 \beta_{2} ) q^{52} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{54} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{55} + ( -3 - 2 \beta_{1} + 5 \beta_{2} ) q^{57} + ( 2 + 2 \beta_{1} - 4 \beta_{2} ) q^{58} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{60} + ( -6 + 4 \beta_{1} - 6 \beta_{2} ) q^{61} + ( -2 - 2 \beta_{1} + 6 \beta_{2} ) q^{62} + ( 14 + 14 \beta_{1} + 7 \beta_{2} ) q^{64} + ( -4 + 2 \beta_{1} - 10 \beta_{2} ) q^{65} + ( 2 + 2 \beta_{1} - 4 \beta_{2} ) q^{66} + ( -10 - 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 21 + \beta_{1} + 12 \beta_{2} ) q^{68} + ( 16 - 12 \beta_{1} + 5 \beta_{2} ) q^{69} + ( -2 - 4 \beta_{2} ) q^{71} + ( -7 - 9 \beta_{1} - 7 \beta_{2} ) q^{72} + ( 2 - 4 \beta_{1} ) q^{73} + 3 \beta_{2} q^{74} + ( 6 - 9 \beta_{1} + 8 \beta_{2} ) q^{75} + ( 3 + 3 \beta_{1} + 2 \beta_{2} ) q^{76} + ( 3 + 2 \beta_{1} - 3 \beta_{2} ) q^{78} + ( 8 - 4 \beta_{1} ) q^{79} + ( 16 + 14 \beta_{1} + 4 \beta_{2} ) q^{80} + ( 16 - 11 \beta_{1} + 10 \beta_{2} ) q^{81} + ( -1 - \beta_{1} ) q^{82} + ( 6 - 4 \beta_{1} + 8 \beta_{2} ) q^{83} + ( 10 + 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 11 + 6 \beta_{1} + 8 \beta_{2} ) q^{86} + ( 8 - 10 \beta_{1} + 8 \beta_{2} ) q^{87} + ( -2 - 8 \beta_{2} ) q^{88} + ( -5 + 5 \beta_{1} - 11 \beta_{2} ) q^{89} + ( -6 - 4 \beta_{1} - 6 \beta_{2} ) q^{90} + ( 3 - 11 \beta_{1} - \beta_{2} ) q^{92} + ( -14 + 16 \beta_{1} - 12 \beta_{2} ) q^{93} + ( -4 - 4 \beta_{1} - \beta_{2} ) q^{94} + ( -2 - 6 \beta_{1} + 14 \beta_{2} ) q^{95} + ( 3 - 5 \beta_{1} ) q^{96} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{97} + ( 8 - 4 \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 4q^{2} - 5q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + 9q^{8} + 4q^{9} + O(q^{10})$$ $$3q + 4q^{2} - 5q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + 9q^{8} + 4q^{9} + 2q^{10} - 4q^{11} + 5q^{12} - 7q^{13} + 8q^{15} + 18q^{16} - 3q^{17} - 11q^{18} + 11q^{19} + 16q^{20} - 10q^{22} - 13q^{23} - q^{24} + 5q^{25} - 21q^{26} - 8q^{27} + 2q^{29} + 6q^{30} + 27q^{32} + 2q^{33} + 17q^{34} - 32q^{36} + 3q^{37} + 17q^{38} + 36q^{40} - 3q^{41} + 11q^{43} - 10q^{44} - 12q^{45} - 8q^{46} - 13q^{47} - 9q^{48} + 16q^{50} + 26q^{51} - 35q^{52} - 20q^{53} + 15q^{54} + 12q^{55} - 16q^{57} + 12q^{58} - 4q^{59} - 8q^{60} - 8q^{61} - 14q^{62} + 49q^{64} + 12q^{66} - 36q^{67} + 52q^{68} + 31q^{69} - 2q^{71} - 23q^{72} + 2q^{73} - 3q^{74} + q^{75} + 10q^{76} + 14q^{78} + 20q^{79} + 58q^{80} + 27q^{81} - 4q^{82} + 6q^{83} + 30q^{85} + 31q^{86} + 6q^{87} + 2q^{88} + q^{89} - 16q^{90} - q^{92} - 14q^{93} - 15q^{94} - 26q^{95} + 4q^{96} - 11q^{97} + 18q^{99} + O(q^{100})$$

## 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
 −1.24698 0.445042 1.80194
−0.246980 −3.24698 −1.93900 −0.890084 0.801938 0 0.972853 7.54288 0.219833
1.2 1.44504 −1.55496 0.0881460 −3.60388 −2.24698 0 −2.76271 −0.582105 −5.20775
1.3 2.80194 −0.198062 5.85086 2.49396 −0.554958 0 10.7899 −2.96077 6.98792
 $$n$$: 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.k 3
7.b odd 2 1 287.2.a.d 3
21.c even 2 1 2583.2.a.l 3
28.d even 2 1 4592.2.a.r 3
35.c odd 2 1 7175.2.a.i 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.a.d 3 7.b odd 2 1
2009.2.a.k 3 1.a even 1 1 trivial
2583.2.a.l 3 21.c even 2 1
4592.2.a.r 3 28.d even 2 1
7175.2.a.i 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2009))$$:

 $$T_{2}^{3} - 4 T_{2}^{2} + 3 T_{2} + 1$$ $$T_{3}^{3} + 5 T_{3}^{2} + 6 T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T - 4 T^{2} + T^{3}$$
$3$ $$1 + 6 T + 5 T^{2} + T^{3}$$
$5$ $$-8 - 8 T + 2 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$-8 - 4 T + 4 T^{2} + T^{3}$$
$13$ $$-49 + 7 T^{2} + T^{3}$$
$17$ $$-97 - 46 T + 3 T^{2} + T^{3}$$
$19$ $$-13 + 24 T - 11 T^{2} + T^{3}$$
$23$ $$29 + 40 T + 13 T^{2} + T^{3}$$
$29$ $$8 - 36 T - 2 T^{2} + T^{3}$$
$31$ $$56 - 84 T + T^{3}$$
$37$ $$27 - 18 T - 3 T^{2} + T^{3}$$
$41$ $$( 1 + T )^{3}$$
$43$ $$1 - 4 T - 11 T^{2} + T^{3}$$
$47$ $$71 + 54 T + 13 T^{2} + T^{3}$$
$53$ $$232 + 124 T + 20 T^{2} + T^{3}$$
$59$ $$104 - 116 T + 4 T^{2} + T^{3}$$
$61$ $$-344 - 44 T + 8 T^{2} + T^{3}$$
$67$ $$1336 + 404 T + 36 T^{2} + T^{3}$$
$71$ $$-8 - 36 T + 2 T^{2} + T^{3}$$
$73$ $$8 - 36 T - 2 T^{2} + T^{3}$$
$79$ $$-64 + 96 T - 20 T^{2} + T^{3}$$
$83$ $$664 - 100 T - 6 T^{2} + T^{3}$$
$89$ $$-1049 - 212 T - T^{2} + T^{3}$$
$97$ $$13 + 24 T + 11 T^{2} + T^{3}$$