Properties

Label 2009.2.a.o
Level $2009$
Weight $2$
Character orbit 2009.a
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.185257757.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 10 x^{3} + 23 x^{2} - 24 x + 5\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{2} ) q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{3} + \beta_{4} ) q^{5} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} ) q^{8} + ( 2 + \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{2} ) q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{3} + \beta_{4} ) q^{5} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} ) q^{8} + ( 2 + \beta_{4} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{10} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{12} + ( -1 - 2 \beta_{1} - \beta_{5} ) q^{13} + ( -1 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{15} + ( 1 + \beta_{4} + \beta_{5} ) q^{16} + ( -1 - \beta_{3} ) q^{17} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{18} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{19} + ( -3 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{20} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{22} + ( 3 + \beta_{2} ) q^{23} + ( 5 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{24} + ( 4 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{25} + ( 6 + 3 \beta_{2} + \beta_{3} ) q^{26} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{27} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{29} + ( 6 \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} ) q^{30} + ( 4 + \beta_{3} + \beta_{4} ) q^{31} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{32} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{33} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{34} + ( 3 + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{36} + ( 4 + \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{37} + ( 3 + 2 \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{38} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{39} + ( -2 - 4 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{40} - q^{41} + ( 3 - \beta_{3} - 2 \beta_{5} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{44} + ( 5 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{45} + ( -4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{46} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{47} + ( 3 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{48} + ( -8 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{50} + ( -3 + \beta_{1} - 2 \beta_{3} ) q^{51} + ( 3 - 6 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{52} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{53} + ( 5 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{54} + ( -2 + 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{55} + ( 6 - 4 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{57} + ( 2 + 2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{58} + ( 2 - 3 \beta_{3} - \beta_{4} ) q^{59} + ( -15 - 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{60} + ( 1 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{61} + ( 1 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{62} + ( -6 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{64} + ( -4 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 7 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{66} + ( 4 + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{67} + ( -5 + \beta_{1} - 3 \beta_{3} - \beta_{5} ) q^{68} + ( -1 - 4 \beta_{2} - \beta_{4} ) q^{69} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{71} + ( 3 - 4 \beta_{1} - 5 \beta_{2} - \beta_{5} ) q^{72} + ( -9 + 3 \beta_{1} - \beta_{2} - \beta_{5} ) q^{73} + ( -3 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{74} + ( 10 - \beta_{1} - 8 \beta_{2} - 2 \beta_{3} - 5 \beta_{5} ) q^{75} + ( -2 - 4 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{76} + ( -4 - \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{78} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{79} + ( 3 + 5 \beta_{1} + \beta_{2} - \beta_{5} ) q^{80} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{81} + \beta_{1} q^{82} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{83} + ( 4 + 3 \beta_{3} - \beta_{4} ) q^{85} + ( -1 - 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{86} + ( -7 + \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} ) q^{87} + ( 1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{88} + ( 6 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{89} + ( -12 - 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} ) q^{90} + ( 5 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{92} + ( 7 - 3 \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{93} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{94} + ( 1 + 5 \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{95} + ( 1 + 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{96} + ( -3 + 4 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{97} + ( -2 + 6 \beta_{2} + 5 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} + 4q^{3} + 9q^{4} + q^{5} + 4q^{6} + 3q^{8} + 14q^{9} + O(q^{10}) \) \( 6q - q^{2} + 4q^{3} + 9q^{4} + q^{5} + 4q^{6} + 3q^{8} + 14q^{9} - 10q^{10} + 6q^{11} - 9q^{12} - 7q^{13} - 13q^{15} + 7q^{16} - 7q^{17} - 5q^{18} - 2q^{19} - 11q^{20} + 15q^{22} + 20q^{23} + 36q^{24} + 29q^{25} + 43q^{26} - 2q^{27} - 9q^{29} + 13q^{30} + 27q^{31} - 10q^{32} - 17q^{33} - 6q^{34} + 29q^{36} + 19q^{37} + 23q^{38} + q^{39} - 23q^{40} - 6q^{41} + 19q^{43} + 21q^{44} + 35q^{45} - 8q^{46} + 19q^{47} + 9q^{48} - 58q^{50} - 19q^{51} + 5q^{53} + 37q^{54} - 3q^{55} + 37q^{57} + 13q^{58} + 7q^{59} - 110q^{60} + 12q^{61} - 37q^{64} - 13q^{65} + 54q^{66} + 27q^{67} - 31q^{68} - 16q^{69} - 6q^{71} + 5q^{72} - 52q^{73} - 14q^{74} + 46q^{75} - 13q^{76} - 45q^{78} + 26q^{80} - 22q^{81} + q^{82} - 12q^{83} + 25q^{85} - 10q^{86} - 42q^{87} - 2q^{88} + 38q^{89} - 93q^{90} + 45q^{92} + 33q^{93} + 8q^{94} + q^{95} + 12q^{96} - 8q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 10 x^{4} + 10 x^{3} + 23 x^{2} - 24 x + 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 5 \nu + 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + \nu^{2} + 5 \nu - 4 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 10 \nu^{3} + 2 \nu^{2} + 24 \nu - 10 \)
\(\beta_{5}\)\(=\)\( -\nu^{5} + \nu^{4} + 10 \nu^{3} - 8 \nu^{2} - 24 \nu + 13 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{5} + \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{4} - 2 \beta_{3} + 8 \beta_{2} + 26 \beta_{1} - 6\)

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
2.47904
2.05073
0.644787
0.306800
−2.01956
−2.46179
−2.47904 −2.84004 4.14562 3.76023 7.04056 0 −5.31907 5.06582 −9.32175
1.2 −2.05073 1.62935 2.20548 −4.18004 −3.34135 0 −0.421375 −0.345215 8.57212
1.3 −0.644787 2.95586 −1.58425 4.36552 −1.90590 0 2.31108 5.73713 −2.81483
1.4 −0.306800 1.50512 −1.90587 −0.333855 −0.461772 0 1.19832 −0.734606 0.102427
1.5 2.01956 −1.86075 2.07864 0.244521 −3.75791 0 0.158810 0.462403 0.493825
1.6 2.46179 2.61045 4.06039 −2.85638 6.42638 0 5.07224 3.81447 −7.03179
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.o 6
7.b odd 2 1 287.2.a.f 6
21.c even 2 1 2583.2.a.t 6
28.d even 2 1 4592.2.a.bg 6
35.c odd 2 1 7175.2.a.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.a.f 6 7.b odd 2 1
2009.2.a.o 6 1.a even 1 1 trivial
2583.2.a.t 6 21.c even 2 1
4592.2.a.bg 6 28.d even 2 1
7175.2.a.p 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2009))\):

\( T_{2}^{6} + T_{2}^{5} - 10 T_{2}^{4} - 10 T_{2}^{3} + 23 T_{2}^{2} + 24 T_{2} + 5 \)
\( T_{3}^{6} - 4 T_{3}^{5} - 8 T_{3}^{4} + 46 T_{3}^{3} - 13 T_{3}^{2} - 111 T_{3} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5 + 24 T + 23 T^{2} - 10 T^{3} - 10 T^{4} + T^{5} + T^{6} \)
$3$ \( 100 - 111 T - 13 T^{2} + 46 T^{3} - 8 T^{4} - 4 T^{5} + T^{6} \)
$5$ \( -16 + 16 T + 200 T^{2} + 16 T^{3} - 29 T^{4} - T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 2720 - 1928 T + 28 T^{2} + 218 T^{3} - 29 T^{4} - 6 T^{5} + T^{6} \)
$13$ \( -1546 + 1917 T + 400 T^{2} - 330 T^{3} - 49 T^{4} + 7 T^{5} + T^{6} \)
$17$ \( 2 + 9 T - 26 T^{3} + 3 T^{4} + 7 T^{5} + T^{6} \)
$19$ \( -3212 + 833 T + 925 T^{2} - 148 T^{3} - 68 T^{4} + 2 T^{5} + T^{6} \)
$23$ \( 344 - 969 T + 1051 T^{2} - 558 T^{3} + 152 T^{4} - 20 T^{5} + T^{6} \)
$29$ \( 10448 + 10008 T + 196 T^{2} - 772 T^{3} - 79 T^{4} + 9 T^{5} + T^{6} \)
$31$ \( -1280 + 1624 T + 936 T^{2} - 982 T^{3} + 257 T^{4} - 27 T^{5} + T^{6} \)
$37$ \( -376 - 1080 T - 432 T^{2} + 419 T^{3} + 54 T^{4} - 19 T^{5} + T^{6} \)
$41$ \( ( 1 + T )^{6} \)
$43$ \( 29756 - 13389 T - 2586 T^{2} + 990 T^{3} + 27 T^{4} - 19 T^{5} + T^{6} \)
$47$ \( -512 - 1212 T - 750 T^{2} + 63 T^{3} + 94 T^{4} - 19 T^{5} + T^{6} \)
$53$ \( -28432 - 8488 T + 4036 T^{2} + 464 T^{3} - 127 T^{4} - 5 T^{5} + T^{6} \)
$59$ \( 256 - 1792 T + 208 T^{2} + 1476 T^{3} - 183 T^{4} - 7 T^{5} + T^{6} \)
$61$ \( -55952 - 42784 T - 1028 T^{2} + 2610 T^{3} - 183 T^{4} - 12 T^{5} + T^{6} \)
$67$ \( -26848 + 19288 T - 3320 T^{2} - 562 T^{3} + 243 T^{4} - 27 T^{5} + T^{6} \)
$71$ \( 4672 + 7752 T + 2552 T^{2} - 450 T^{3} - 105 T^{4} + 6 T^{5} + T^{6} \)
$73$ \( 656 + 39472 T + 33460 T^{2} + 8866 T^{3} + 1009 T^{4} + 52 T^{5} + T^{6} \)
$79$ \( 2048 + 8032 T + 4112 T^{2} - 96 T^{3} - 152 T^{4} + T^{6} \)
$83$ \( -19744 - 3448 T + 5380 T^{2} - 570 T^{3} - 127 T^{4} + 12 T^{5} + T^{6} \)
$89$ \( -345082 + 221925 T - 40625 T^{2} + 940 T^{3} + 404 T^{4} - 38 T^{5} + T^{6} \)
$97$ \( 303494 + 169901 T + 16337 T^{2} - 2518 T^{3} - 316 T^{4} + 8 T^{5} + T^{6} \)
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