# Properties

 Label 2009.2.a.m 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: 5.5.233489.1 Defining polynomial: $$x^{5} - x^{4} - 6 x^{3} + x^{2} + 5 x - 1$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 6 x^{3} + x^{2} + 5 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5 \nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + \nu^{3} + 6 \nu^{2} - \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{4} + \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 9$$

## 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
 2.80262 0.201995 −1.72658 −1.16000 0.881963
−2.40886 0.896451 3.80262 −0.896451 −2.15943 0 −4.34226 −2.19638 2.15943
1.2 −1.78941 2.32065 1.20200 −2.32065 −4.15260 0 1.42796 2.38542 4.15260
1.3 −1.12846 −2.92944 −0.726576 2.92944 3.30576 0 3.07683 5.58161 −3.30576
1.4 1.35647 2.22790 −0.160001 −2.22790 3.02207 0 −2.92997 1.96354 −3.02207
1.5 1.97027 −0.515563 1.88196 0.515563 −1.01580 0 −0.232565 −2.73420 1.01580
 $$n$$: 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.m 5
7.b odd 2 1 2009.2.a.l 5
7.d odd 6 2 287.2.e.c 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.c 10 7.d odd 6 2
2009.2.a.l 5 7.b odd 2 1
2009.2.a.m 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2009))$$:

 $$T_{2}^{5} + 2 T_{2}^{4} - 6 T_{2}^{3} - 11 T_{2}^{2} + 8 T_{2} + 13$$ $$T_{3}^{5} - 2 T_{3}^{4} - 8 T_{3}^{3} + 19 T_{3}^{2} - 2 T_{3} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$13 + 8 T - 11 T^{2} - 6 T^{3} + 2 T^{4} + T^{5}$$
$3$ $$-7 - 2 T + 19 T^{2} - 8 T^{3} - 2 T^{4} + T^{5}$$
$5$ $$7 - 2 T - 19 T^{2} - 8 T^{3} + 2 T^{4} + T^{5}$$
$7$ $$T^{5}$$
$11$ $$53 + 7 T - 45 T^{2} - 3 T^{3} + 6 T^{4} + T^{5}$$
$13$ $$7 - 2 T - 19 T^{2} - 8 T^{3} + 2 T^{4} + T^{5}$$
$17$ $$-161 + 108 T + 51 T^{2} - 25 T^{3} - 3 T^{4} + T^{5}$$
$19$ $$637 + 41 T - 141 T^{2} - 16 T^{3} + 7 T^{4} + T^{5}$$
$23$ $$-191 + 202 T + 19 T^{2} - 38 T^{3} + T^{5}$$
$29$ $$49 - 294 T - 136 T^{2} + 9 T^{3} + 10 T^{4} + T^{5}$$
$31$ $$301 + 958 T + 229 T^{2} - 62 T^{3} - 6 T^{4} + T^{5}$$
$37$ $$-247 - 714 T - 222 T^{2} + 65 T^{3} + 18 T^{4} + T^{5}$$
$41$ $$( -1 + T )^{5}$$
$43$ $$-2573 - 3928 T - 1178 T^{2} - 53 T^{3} + 14 T^{4} + T^{5}$$
$47$ $$1197 + 858 T - 55 T^{2} - 87 T^{3} + 3 T^{4} + T^{5}$$
$53$ $$-1627 - 2913 T - 1373 T^{2} - 120 T^{3} + 9 T^{4} + T^{5}$$
$59$ $$-18011 - 844 T + 1213 T^{2} - 3 T^{3} - 19 T^{4} + T^{5}$$
$61$ $$-12649 + 893 T + 1551 T^{2} + 16 T^{3} - 23 T^{4} + T^{5}$$
$67$ $$19297 + 2348 T - 1343 T^{2} - 135 T^{3} + 11 T^{4} + T^{5}$$
$71$ $$-7241 + 5192 T + 61 T^{2} - 150 T^{3} + T^{5}$$
$73$ $$-3031 - 5011 T - 1686 T^{2} - 110 T^{3} + 13 T^{4} + T^{5}$$
$79$ $$-55751 - 8520 T + 2125 T^{2} + 553 T^{3} + 41 T^{4} + T^{5}$$
$83$ $$-2317 + 1244 T + 144 T^{2} - 77 T^{3} - 2 T^{4} + T^{5}$$
$89$ $$45619 + 300 T - 3642 T^{2} - 251 T^{3} + 14 T^{4} + T^{5}$$
$97$ $$56763 - 37896 T + 5393 T^{2} - 37 T^{3} - 27 T^{4} + T^{5}$$