Properties

Label 2009.2.a.c
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} -2 q^{4} + ( 1 - \beta ) q^{5} + \beta q^{9} +O(q^{10})\) \( q -\beta q^{3} -2 q^{4} + ( 1 - \beta ) q^{5} + \beta q^{9} + ( -2 - \beta ) q^{11} + 2 \beta q^{12} + \beta q^{13} + 3 q^{15} + 4 q^{16} + ( -3 + 3 \beta ) q^{17} + ( -1 + 2 \beta ) q^{19} + ( -2 + 2 \beta ) q^{20} + ( -1 - 2 \beta ) q^{23} + ( -1 - \beta ) q^{25} + ( -3 + 2 \beta ) q^{27} + ( 1 + 2 \beta ) q^{29} + \beta q^{31} + ( 3 + 3 \beta ) q^{33} -2 \beta q^{36} + ( -2 + 4 \beta ) q^{37} + ( -3 - \beta ) q^{39} + q^{41} + ( -5 - 2 \beta ) q^{43} + ( 4 + 2 \beta ) q^{44} -3 q^{45} + 9 q^{47} -4 \beta q^{48} -9 q^{51} -2 \beta q^{52} + ( 3 + 3 \beta ) q^{53} + ( 1 + 2 \beta ) q^{55} + ( -6 - \beta ) q^{57} + ( -5 + 2 \beta ) q^{59} -6 q^{60} + ( 2 - 4 \beta ) q^{61} -8 q^{64} -3 q^{65} + ( -2 - 2 \beta ) q^{67} + ( 6 - 6 \beta ) q^{68} + ( 6 + 3 \beta ) q^{69} + ( -3 + 6 \beta ) q^{71} + ( -7 - \beta ) q^{73} + ( 3 + 2 \beta ) q^{75} + ( 2 - 4 \beta ) q^{76} + ( 1 - 8 \beta ) q^{79} + ( 4 - 4 \beta ) q^{80} + ( -6 - 2 \beta ) q^{81} + ( -7 - 2 \beta ) q^{83} + ( -12 + 3 \beta ) q^{85} + ( -6 - 3 \beta ) q^{87} + ( -5 + 2 \beta ) q^{89} + ( 2 + 4 \beta ) q^{92} + ( -3 - \beta ) q^{93} + ( -7 + \beta ) q^{95} + ( -7 - \beta ) q^{97} + ( -3 - 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 4q^{4} + q^{5} + q^{9} + O(q^{10}) \) \( 2q - q^{3} - 4q^{4} + q^{5} + q^{9} - 5q^{11} + 2q^{12} + q^{13} + 6q^{15} + 8q^{16} - 3q^{17} - 2q^{20} - 4q^{23} - 3q^{25} - 4q^{27} + 4q^{29} + q^{31} + 9q^{33} - 2q^{36} - 7q^{39} + 2q^{41} - 12q^{43} + 10q^{44} - 6q^{45} + 18q^{47} - 4q^{48} - 18q^{51} - 2q^{52} + 9q^{53} + 4q^{55} - 13q^{57} - 8q^{59} - 12q^{60} - 16q^{64} - 6q^{65} - 6q^{67} + 6q^{68} + 15q^{69} - 15q^{73} + 8q^{75} - 6q^{79} + 4q^{80} - 14q^{81} - 16q^{83} - 21q^{85} - 15q^{87} - 8q^{89} + 8q^{92} - 7q^{93} - 13q^{95} - 15q^{97} - 9q^{99} + O(q^{100}) \)

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
2.30278
−1.30278
0 −2.30278 −2.00000 −1.30278 0 0 0 2.30278 0
1.2 0 1.30278 −2.00000 2.30278 0 0 0 −1.30278 0
\(n\): 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.c 2
7.b odd 2 1 2009.2.a.d 2
7.d odd 6 2 287.2.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.b 4 7.d odd 6 2
2009.2.a.c 2 1.a even 1 1 trivial
2009.2.a.d 2 7.b odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -3 + T + T^{2} \)
$5$ \( -3 - T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 3 + 5 T + T^{2} \)
$13$ \( -3 - T + T^{2} \)
$17$ \( -27 + 3 T + T^{2} \)
$19$ \( -13 + T^{2} \)
$23$ \( -9 + 4 T + T^{2} \)
$29$ \( -9 - 4 T + T^{2} \)
$31$ \( -3 - T + T^{2} \)
$37$ \( -52 + T^{2} \)
$41$ \( ( -1 + T )^{2} \)
$43$ \( 23 + 12 T + T^{2} \)
$47$ \( ( -9 + T )^{2} \)
$53$ \( -9 - 9 T + T^{2} \)
$59$ \( 3 + 8 T + T^{2} \)
$61$ \( -52 + T^{2} \)
$67$ \( -4 + 6 T + T^{2} \)
$71$ \( -117 + T^{2} \)
$73$ \( 53 + 15 T + T^{2} \)
$79$ \( -199 + 6 T + T^{2} \)
$83$ \( 51 + 16 T + T^{2} \)
$89$ \( 3 + 8 T + T^{2} \)
$97$ \( 53 + 15 T + T^{2} \)
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