Properties

Label 2009.2.a.a
Level $2009$
Weight $2$
Character orbit 2009.a
Self dual yes
Analytic conductor $16.042$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−1.61803 −0.618034 0.618034 0.618034 1.00000 0 2.23607 −2.61803 −1.00000
1.2 0.618034 1.61803 −1.61803 −1.61803 1.00000 0 −2.23607 −0.381966 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{3}^{2} - T_{3} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T + T^{2} \)
$3$ \( -1 - T + T^{2} \)
$5$ \( -1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 11 - 8 T + T^{2} \)
$17$ \( -1 - 4 T + T^{2} \)
$19$ \( -11 - T + T^{2} \)
$23$ \( 5 + 5 T + T^{2} \)
$29$ \( -5 + 5 T + T^{2} \)
$31$ \( -25 + 5 T + T^{2} \)
$37$ \( 20 + 10 T + T^{2} \)
$41$ \( ( -1 + T )^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( -36 + 6 T + T^{2} \)
$53$ \( 11 - 7 T + T^{2} \)
$59$ \( 89 + 19 T + T^{2} \)
$61$ \( 5 + 10 T + T^{2} \)
$67$ \( 11 + 13 T + T^{2} \)
$71$ \( -71 - 6 T + T^{2} \)
$73$ \( -79 - 2 T + T^{2} \)
$79$ \( -20 + 10 T + T^{2} \)
$83$ \( -19 - 2 T + T^{2} \)
$89$ \( -59 - 3 T + T^{2} \)
$97$ \( -45 - 15 T + T^{2} \)
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