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 $1$

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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: \(0\)
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:

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

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