# Properties

 Label 2009.2.a.g 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: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 41) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} -\beta_{2} q^{3} + ( 1 + 2 \beta_{1} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( 2 - \beta_{2} ) q^{6} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{8} + ( -\beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} -\beta_{2} q^{3} + ( 1 + 2 \beta_{1} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( 2 - \beta_{2} ) q^{6} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{8} + ( -\beta_{1} - \beta_{2} ) q^{9} + ( 3 + \beta_{1} - \beta_{2} ) q^{10} + ( 2 \beta_{1} - \beta_{2} ) q^{11} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{12} + 2 \beta_{1} q^{13} + ( 2 - 2 \beta_{2} ) q^{15} + ( 3 + 4 \beta_{1} + 4 \beta_{2} ) q^{16} + 2 q^{17} + ( 3 + 2 \beta_{1} ) q^{18} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{19} + ( -1 - 3 \beta_{1} - 3 \beta_{2} ) q^{20} + ( -4 \beta_{1} - 3 \beta_{2} ) q^{22} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{23} + ( 2 \beta_{1} + \beta_{2} ) q^{24} + ( -1 - 2 \beta_{2} ) q^{25} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{26} + ( 2 + 2 \beta_{2} ) q^{27} + ( -2 - 2 \beta_{2} ) q^{29} + ( 4 - 2 \beta_{1} - 4 \beta_{2} ) q^{30} + ( -6 + 2 \beta_{1} + 2 \beta_{2} ) q^{31} + ( -8 - 5 \beta_{1} - \beta_{2} ) q^{32} + ( 5 - 3 \beta_{1} - \beta_{2} ) q^{33} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{34} + ( -2 - 5 \beta_{1} - 3 \beta_{2} ) q^{36} + ( -3 + 3 \beta_{1} + 3 \beta_{2} ) q^{37} + ( -2 \beta_{1} - \beta_{2} ) q^{38} + ( 2 - 2 \beta_{1} ) q^{39} + ( 3 + 5 \beta_{1} + 3 \beta_{2} ) q^{40} - q^{41} + ( -2 + 2 \beta_{1} ) q^{43} + ( 10 + 4 \beta_{1} + 3 \beta_{2} ) q^{44} + ( 3 + \beta_{1} - \beta_{2} ) q^{45} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 2 - 6 \beta_{1} - 3 \beta_{2} ) q^{47} + ( -8 + \beta_{2} ) q^{48} + ( 4 + \beta_{1} - \beta_{2} ) q^{50} -2 \beta_{2} q^{51} + ( 8 + 6 \beta_{1} + 4 \beta_{2} ) q^{52} + ( 2 - 2 \beta_{2} ) q^{53} + ( -4 - 2 \beta_{1} ) q^{54} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{55} + ( 5 - 3 \beta_{1} + \beta_{2} ) q^{57} + ( 4 + 2 \beta_{1} ) q^{58} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 6 - 2 \beta_{2} ) q^{60} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -6 + 2 \beta_{1} + 6 \beta_{2} ) q^{62} + ( 1 + 10 \beta_{1} + 4 \beta_{2} ) q^{64} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{65} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{66} + ( -2 \beta_{1} + \beta_{2} ) q^{67} + ( 2 + 4 \beta_{1} ) q^{68} + ( -8 + 4 \beta_{1} ) q^{69} + ( 8 - 4 \beta_{1} - \beta_{2} ) q^{71} + ( 5 + 8 \beta_{1} + 4 \beta_{2} ) q^{72} + ( 3 - 7 \beta_{1} + \beta_{2} ) q^{73} + ( -9 - 3 \beta_{1} + 3 \beta_{2} ) q^{74} + ( 6 - 2 \beta_{1} - \beta_{2} ) q^{75} + ( 8 + 3 \beta_{2} ) q^{76} + ( 2 + 2 \beta_{1} ) q^{78} + ( 10 + 2 \beta_{1} + \beta_{2} ) q^{79} + ( -9 - 7 \beta_{1} + \beta_{2} ) q^{80} + ( -6 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( \beta_{1} + \beta_{2} ) q^{82} + 4 \beta_{2} q^{83} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -2 - 2 \beta_{1} ) q^{86} + ( 6 - 2 \beta_{1} ) q^{87} + ( -10 - 10 \beta_{1} - 5 \beta_{2} ) q^{88} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 1 - 5 \beta_{1} - 5 \beta_{2} ) q^{90} + ( -10 + 2 \beta_{1} - 2 \beta_{2} ) q^{92} + ( -4 + 8 \beta_{2} ) q^{93} + ( 12 + 10 \beta_{1} + \beta_{2} ) q^{94} + ( -2 - 2 \beta_{2} ) q^{95} + ( -2 + 4 \beta_{1} + 7 \beta_{2} ) q^{96} + ( -2 - 4 \beta_{2} ) q^{97} + ( -4 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} + 5q^{4} + 2q^{5} + 6q^{6} - 9q^{8} - q^{9} + O(q^{10})$$ $$3q - q^{2} + 5q^{4} + 2q^{5} + 6q^{6} - 9q^{8} - q^{9} + 10q^{10} + 2q^{11} + 4q^{12} + 2q^{13} + 6q^{15} + 13q^{16} + 6q^{17} + 11q^{18} - 4q^{19} - 6q^{20} - 4q^{22} + 4q^{23} + 2q^{24} - 3q^{25} - 10q^{26} + 6q^{27} - 6q^{29} + 10q^{30} - 16q^{31} - 29q^{32} + 12q^{33} - 2q^{34} - 11q^{36} - 6q^{37} - 2q^{38} + 4q^{39} + 14q^{40} - 3q^{41} - 4q^{43} + 34q^{44} + 10q^{45} - 4q^{46} - 24q^{48} + 13q^{50} + 30q^{52} + 6q^{53} - 14q^{54} - 2q^{55} + 12q^{57} + 14q^{58} + 8q^{59} + 18q^{60} - 2q^{61} - 16q^{62} + 13q^{64} - 8q^{65} + 16q^{66} - 2q^{67} + 10q^{68} - 20q^{69} + 20q^{71} + 23q^{72} + 2q^{73} - 30q^{74} + 16q^{75} + 24q^{76} + 8q^{78} + 32q^{79} - 34q^{80} - 13q^{81} + q^{82} + 4q^{85} - 8q^{86} + 16q^{87} - 40q^{88} + 6q^{89} - 2q^{90} - 28q^{92} - 12q^{93} + 46q^{94} - 6q^{95} - 2q^{96} - 6q^{97} - 4q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$

## 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
 2.17009 −1.48119 0.311108
−2.70928 −0.539189 5.34017 −1.70928 1.46081 0 −9.04945 −2.70928 4.63090
1.2 −0.193937 −1.67513 −1.96239 0.806063 0.324869 0 0.768452 −0.193937 −0.156325
1.3 1.90321 2.21432 1.62222 2.90321 4.21432 0 −0.719004 1.90321 5.52543
 $$n$$: 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.g 3
7.b odd 2 1 41.2.a.a 3
21.c even 2 1 369.2.a.f 3
28.d even 2 1 656.2.a.f 3
35.c odd 2 1 1025.2.a.j 3
35.f even 4 2 1025.2.b.h 6
56.e even 2 1 2624.2.a.q 3
56.h odd 2 1 2624.2.a.r 3
77.b even 2 1 4961.2.a.d 3
84.h odd 2 1 5904.2.a.bk 3
91.b odd 2 1 6929.2.a.b 3
105.g even 2 1 9225.2.a.bv 3
287.d odd 2 1 1681.2.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.a.a 3 7.b odd 2 1
369.2.a.f 3 21.c even 2 1
656.2.a.f 3 28.d even 2 1
1025.2.a.j 3 35.c odd 2 1
1025.2.b.h 6 35.f even 4 2
1681.2.a.d 3 287.d odd 2 1
2009.2.a.g 3 1.a even 1 1 trivial
2624.2.a.q 3 56.e even 2 1
2624.2.a.r 3 56.h odd 2 1
4961.2.a.d 3 77.b even 2 1
5904.2.a.bk 3 84.h odd 2 1
6929.2.a.b 3 91.b odd 2 1
9225.2.a.bv 3 105.g even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 5 T + T^{2} + T^{3}$$
$3$ $$-2 - 4 T + T^{3}$$
$5$ $$4 - 4 T - 2 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$50 - 20 T - 2 T^{2} + T^{3}$$
$13$ $$8 - 12 T - 2 T^{2} + T^{3}$$
$17$ $$( -2 + T )^{3}$$
$19$ $$10 - 16 T + 4 T^{2} + T^{3}$$
$23$ $$-32 - 32 T - 4 T^{2} + T^{3}$$
$29$ $$-40 - 4 T + 6 T^{2} + T^{3}$$
$31$ $$32 + 64 T + 16 T^{2} + T^{3}$$
$37$ $$-108 - 36 T + 6 T^{2} + T^{3}$$
$41$ $$( 1 + T )^{3}$$
$43$ $$-16 - 8 T + 4 T^{2} + T^{3}$$
$47$ $$502 - 120 T + T^{3}$$
$53$ $$8 - 4 T - 6 T^{2} + T^{3}$$
$59$ $$160 - 16 T - 8 T^{2} + T^{3}$$
$61$ $$-184 - 52 T + 2 T^{2} + T^{3}$$
$67$ $$-50 - 20 T + 2 T^{2} + T^{3}$$
$71$ $$134 + 84 T - 20 T^{2} + T^{3}$$
$73$ $$-244 - 180 T - 2 T^{2} + T^{3}$$
$79$ $$-1090 + 328 T - 32 T^{2} + T^{3}$$
$83$ $$128 - 64 T + T^{3}$$
$89$ $$920 - 148 T - 6 T^{2} + T^{3}$$
$97$ $$-248 - 52 T + 6 T^{2} + T^{3}$$