# Properties

 Label 2009.2.a.n Level $2009$ Weight $2$ Character orbit 2009.a Self dual yes Analytic conductor $16.042$ Analytic rank $1$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.633117.1 Defining polynomial: $$x^{5} - x^{4} - 6 x^{3} + 4 x^{2} + 6 x - 3$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{1} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{2} - \beta_{3} ) q^{5} + ( -3 + \beta_{1} - \beta_{2} ) q^{6} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{8} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{1} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{2} - \beta_{3} ) q^{5} + ( -3 + \beta_{1} - \beta_{2} ) q^{6} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{8} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{9} + ( 1 - 3 \beta_{1} - \beta_{4} ) q^{10} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{11} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{12} + ( -1 - \beta_{2} - \beta_{4} ) q^{13} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{15} + ( \beta_{1} + \beta_{3} ) q^{16} + ( -3 + \beta_{3} + 2 \beta_{4} ) q^{17} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{18} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{19} + ( 6 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{20} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{22} + ( 1 + \beta_{1} + 2 \beta_{3} ) q^{23} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{24} + ( 5 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{25} + ( 2 \beta_{1} + \beta_{2} ) q^{26} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{27} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( -9 + 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{30} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( -3 + \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{32} + ( 1 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{33} + ( 4 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{34} + ( 4 - 3 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{36} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{37} + ( -1 - 2 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 1 - 2 \beta_{1} + \beta_{4} ) q^{39} + ( -3 - \beta_{2} - 3 \beta_{3} ) q^{40} + q^{41} + ( -1 - 3 \beta_{3} ) q^{43} + ( -11 - 3 \beta_{2} - \beta_{3} ) q^{44} + ( 8 - 7 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{45} + ( -7 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{46} + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{47} + ( 5 - 2 \beta_{1} + 2 \beta_{2} ) q^{48} + ( 3 - 5 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{50} + ( 3 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{51} + ( -5 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{52} + ( 3 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{53} + ( -2 + 6 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{54} + ( -11 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{55} + ( \beta_{1} + \beta_{2} - \beta_{4} ) q^{57} + ( -7 - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{58} + ( -3 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{59} + ( -5 + 7 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{60} + ( -5 + \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{61} + ( 5 + 5 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{62} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{64} + ( -7 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{65} + ( 13 - 5 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{66} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{67} + ( -5 + 2 \beta_{1} - 4 \beta_{2} - \beta_{4} ) q^{68} + ( 6 - 2 \beta_{1} + 3 \beta_{2} ) q^{69} + ( -5 - \beta_{1} - 4 \beta_{3} - 3 \beta_{4} ) q^{71} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{72} + ( -9 + 3 \beta_{1} + \beta_{4} ) q^{73} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{74} + ( -8 + 6 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{75} + ( 4 + 3 \beta_{1} + 2 \beta_{2} ) q^{76} + ( 7 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{78} + ( -8 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{79} + ( -5 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{80} + ( 2 - \beta_{1} - \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{81} -\beta_{1} q^{82} + ( 5 - 3 \beta_{1} - 5 \beta_{4} ) q^{83} + ( -5 + 2 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{85} + ( 6 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{86} + ( 7 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{87} + ( 5 + 7 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{88} + ( -1 - \beta_{1} + \beta_{3} ) q^{89} + ( 15 - 8 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} ) q^{90} + ( 2 + 6 \beta_{1} + 4 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{92} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{93} + ( 4 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{94} + ( -3 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + \beta_{4} ) q^{95} + ( 6 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{96} + ( -3 - 5 \beta_{1} - 2 \beta_{3} ) q^{97} + ( -11 + 6 \beta_{1} - \beta_{2} + 5 \beta_{3} + 6 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + O(q^{10})$$ $$5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + 2q^{11} + 2q^{12} - 5q^{13} - 5q^{15} - q^{16} - 13q^{17} + 21q^{18} + 23q^{20} + q^{22} + 2q^{23} - 2q^{24} + 22q^{25} - 10q^{27} - 5q^{29} - 33q^{30} - 17q^{31} - 12q^{32} - 3q^{33} + 8q^{34} + 15q^{36} - 7q^{37} + 3q^{38} + 5q^{39} - 7q^{40} + 5q^{41} + q^{43} - 47q^{44} + 23q^{45} - 24q^{46} - 9q^{47} + 19q^{48} + 2q^{50} + 5q^{51} - 20q^{52} + 5q^{53} - 2q^{54} - 33q^{55} - 3q^{57} - 27q^{58} - 7q^{59} - 16q^{60} - 22q^{61} + 28q^{62} - 3q^{64} - 31q^{65} + 42q^{66} - 3q^{67} - 17q^{68} + 22q^{69} - 24q^{71} - 12q^{72} - 40q^{73} - 5q^{74} - 24q^{75} + 19q^{76} + 30q^{78} - 42q^{79} - 24q^{80} + 9q^{81} - q^{82} + 12q^{83} - 23q^{85} + 16q^{86} + 32q^{87} + 26q^{88} - 8q^{89} + 59q^{90} + 12q^{92} - 11q^{93} + 23q^{94} - 17q^{95} + 17q^{96} - 16q^{97} - 45q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 6 x^{3} + 4 x^{2} + 6 x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 6 \nu^{2} - \nu + 4$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + \nu^{3} + 5 \nu^{2} - 3 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 6 \beta_{2} + \beta_{1} + 14$$

## 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
 2.45719 1.20098 0.460315 −1.08727 −2.03121
−2.45719 1.45719 4.03778 2.26685 −3.58059 0 −5.00722 −0.876597 −5.57007
1.2 −1.20098 0.200978 −0.557652 3.21704 −0.241370 0 3.07168 −2.95961 −3.86360
1.3 −0.460315 −0.539685 −1.78811 −4.10136 0.248425 0 1.74372 −2.70874 1.88791
1.4 1.08727 −2.08727 −0.817843 −0.209668 −2.26943 0 −3.06376 1.35670 −0.227965
1.5 2.03121 −3.03121 2.12582 3.82713 −6.15703 0 0.255573 6.18825 7.77372
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.n 5
7.b odd 2 1 287.2.a.e 5
21.c even 2 1 2583.2.a.r 5
28.d even 2 1 4592.2.a.bb 5
35.c odd 2 1 7175.2.a.n 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.a.e 5 7.b odd 2 1
2009.2.a.n 5 1.a even 1 1 trivial
2583.2.a.r 5 21.c even 2 1
4592.2.a.bb 5 28.d even 2 1
7175.2.a.n 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2009))$$:

 $$T_{2}^{5} + T_{2}^{4} - 6 T_{2}^{3} - 4 T_{2}^{2} + 6 T_{2} + 3$$ $$T_{3}^{5} + 4 T_{3}^{4} - 10 T_{3}^{2} - 3 T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 6 T - 4 T^{2} - 6 T^{3} + T^{4} + T^{5}$$
$3$ $$1 - 3 T - 10 T^{2} + 4 T^{4} + T^{5}$$
$5$ $$-24 - 96 T + 86 T^{2} - 11 T^{3} - 5 T^{4} + T^{5}$$
$7$ $$T^{5}$$
$11$ $$-2472 + 972 T + 140 T^{2} - 63 T^{3} - 2 T^{4} + T^{5}$$
$13$ $$-49 - 120 T - 80 T^{2} - 9 T^{3} + 5 T^{4} + T^{5}$$
$17$ $$-2049 - 1554 T - 304 T^{2} + 25 T^{3} + 13 T^{4} + T^{5}$$
$19$ $$1 - 23 T + 132 T^{2} - 48 T^{3} + T^{5}$$
$23$ $$1317 + 1149 T + 26 T^{2} - 66 T^{3} - 2 T^{4} + T^{5}$$
$29$ $$1512 + 396 T - 350 T^{2} - 71 T^{3} + 5 T^{4} + T^{5}$$
$31$ $$56 - 148 T - 24 T^{2} + 71 T^{3} + 17 T^{4} + T^{5}$$
$37$ $$4 + 78 T - 157 T^{2} - 36 T^{3} + 7 T^{4} + T^{5}$$
$41$ $$( -1 + T )^{5}$$
$43$ $$-1751 + 2222 T - 26 T^{2} - 113 T^{3} - T^{4} + T^{5}$$
$47$ $$10092 + 318 T - 1039 T^{2} - 120 T^{3} + 9 T^{4} + T^{5}$$
$53$ $$2328 - 2820 T + 1024 T^{2} - 109 T^{3} - 5 T^{4} + T^{5}$$
$59$ $$21000 + 7500 T - 880 T^{2} - 171 T^{3} + 7 T^{4} + T^{5}$$
$61$ $$-2504 - 948 T + 252 T^{2} + 153 T^{3} + 22 T^{4} + T^{5}$$
$67$ $$-472 - 932 T - 576 T^{2} - 105 T^{3} + 3 T^{4} + T^{5}$$
$71$ $$43128 - 3876 T - 1862 T^{2} + 41 T^{3} + 24 T^{4} + T^{5}$$
$73$ $$12184 + 11532 T + 3882 T^{2} + 585 T^{3} + 40 T^{4} + T^{5}$$
$79$ $$-75008 - 13760 T + 1920 T^{2} + 572 T^{3} + 42 T^{4} + T^{5}$$
$83$ $$-24696 + 11676 T + 2542 T^{2} - 259 T^{3} - 12 T^{4} + T^{5}$$
$89$ $$-3 - 9 T - 2 T^{2} + 12 T^{3} + 8 T^{4} + T^{5}$$
$97$ $$10493 + 4957 T - 1814 T^{2} - 162 T^{3} + 16 T^{4} + T^{5}$$