Properties

Label 2009.2.a.q
Level $2009$
Weight $2$
Character orbit 2009.a
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 10 x^{5} - x^{4} + 30 x^{3} + 7 x^{2} - 25 x - 11\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 10 x^{5} - x^{4} + 30 x^{3} + 7 x^{2} - 25 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 3 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 7 \nu^{3} - \nu^{2} + 10 \nu + 4 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - \nu^{5} - 9 \nu^{4} + 7 \nu^{3} + 23 \nu^{2} - 11 \nu - 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 5 \beta_{2} + 12\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 7 \beta_{3} + \beta_{2} + 18 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(\beta_{6} + \beta_{5} + 9 \beta_{4} + 23 \beta_{2} + \beta_{1} + 52\)

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
−0.522645
−0.907606
1.28109
−1.84648
1.95500
−2.21905
2.25969
−2.72684 1.52265 5.43567 −2.93285 −4.15201 0 −9.36852 −0.681552 7.99743
1.2 −2.17625 1.90761 2.73607 2.13420 −4.15143 0 −1.60187 0.638961 −4.64455
1.3 −1.35880 −0.281095 −0.153673 3.77273 0.381951 0 2.92640 −2.92099 −5.12637
1.4 0.409485 2.84648 −1.83232 −2.40160 1.16559 0 −1.56928 5.10244 −0.983421
1.5 0.822027 −0.955000 −1.32427 −3.05691 −0.785036 0 −2.73264 −2.08797 −2.51286
1.6 1.92419 3.21905 1.70251 2.13009 6.19407 0 −0.572424 7.36229 4.09870
1.7 2.10619 −1.25969 2.43602 4.35435 −2.65313 0 0.918336 −1.41319 9.17108
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.q yes 7
7.b odd 2 1 2009.2.a.p 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2009.2.a.p 7 7.b odd 2 1
2009.2.a.q yes 7 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{7} + T_{2}^{6} - 11 T_{2}^{5} - 6 T_{2}^{4} + 37 T_{2}^{3} + 4 T_{2}^{2} - 34 T_{2} + 11 \)
\( T_{3}^{7} - 7 T_{3}^{6} + 11 T_{3}^{5} + 16 T_{3}^{4} - 39 T_{3}^{3} - 12 T_{3}^{2} + 32 T_{3} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 11 - 34 T + 4 T^{2} + 37 T^{3} - 6 T^{4} - 11 T^{5} + T^{6} + T^{7} \)
$3$ \( 9 + 32 T - 12 T^{2} - 39 T^{3} + 16 T^{4} + 11 T^{5} - 7 T^{6} + T^{7} \)
$5$ \( 1608 - 560 T - 674 T^{2} + 209 T^{3} + 92 T^{4} - 25 T^{5} - 4 T^{6} + T^{7} \)
$7$ \( T^{7} \)
$11$ \( -152 - 364 T + 40 T^{2} + 281 T^{3} + 32 T^{4} - 42 T^{5} + T^{7} \)
$13$ \( 57 - 170 T - 343 T^{2} + 229 T^{3} + 47 T^{4} - 32 T^{5} - T^{6} + T^{7} \)
$17$ \( 1257 + 2510 T + 825 T^{2} - 559 T^{3} - 261 T^{4} + 4 T^{5} + 11 T^{6} + T^{7} \)
$19$ \( 18651 + 14000 T - 12652 T^{2} + 469 T^{3} + 668 T^{4} - 61 T^{5} - 9 T^{6} + T^{7} \)
$23$ \( 17431 - 5030 T - 3148 T^{2} + 933 T^{3} + 178 T^{4} - 55 T^{5} - 3 T^{6} + T^{7} \)
$29$ \( -125048 + 40768 T + 22462 T^{2} - 1903 T^{3} - 1190 T^{4} - 41 T^{5} + 14 T^{6} + T^{7} \)
$31$ \( -354168 - 4636 T + 56248 T^{2} - 8587 T^{3} - 1174 T^{4} + 391 T^{5} - 34 T^{6} + T^{7} \)
$37$ \( 23664 + 17216 T - 28692 T^{2} + 4452 T^{3} + 903 T^{4} - 138 T^{5} - 7 T^{6} + T^{7} \)
$41$ \( ( 1 + T )^{7} \)
$43$ \( -254433 - 243680 T + 29589 T^{2} + 12609 T^{3} - 663 T^{4} - 210 T^{5} + 3 T^{6} + T^{7} \)
$47$ \( 355008 + 17888 T - 41032 T^{2} - 284 T^{3} + 1483 T^{4} - 40 T^{5} - 17 T^{6} + T^{7} \)
$53$ \( 152 - 2340 T + 2492 T^{2} - 319 T^{3} - 468 T^{4} + 187 T^{5} - 24 T^{6} + T^{7} \)
$59$ \( 25224 - 12568 T - 4150 T^{2} + 2161 T^{3} + 200 T^{4} - 93 T^{5} - 4 T^{6} + T^{7} \)
$61$ \( 216 - 3164 T + 9960 T^{2} - 5823 T^{3} + 1008 T^{4} + 10 T^{5} - 16 T^{6} + T^{7} \)
$67$ \( -467392 + 324912 T + 34624 T^{2} - 22631 T^{3} - 3608 T^{4} + 3 T^{5} + 24 T^{6} + T^{7} \)
$71$ \( 13768 + 49556 T - 55256 T^{2} + 16869 T^{3} - 996 T^{4} - 206 T^{5} + 12 T^{6} + T^{7} \)
$73$ \( -26136 - 27976 T + 3714 T^{2} + 7641 T^{3} + 1028 T^{4} - 130 T^{5} - 14 T^{6} + T^{7} \)
$79$ \( 862592 - 546496 T - 9184 T^{2} + 23936 T^{3} - 776 T^{4} - 280 T^{5} + 8 T^{6} + T^{7} \)
$83$ \( -7460904 + 3554048 T - 510094 T^{2} - 1423 T^{3} + 5732 T^{4} - 290 T^{5} - 14 T^{6} + T^{7} \)
$89$ \( -157959 - 10378 T + 38998 T^{2} + 3019 T^{3} - 1922 T^{4} - 87 T^{5} + 19 T^{6} + T^{7} \)
$97$ \( 137463 + 158306 T - 4990 T^{2} - 17753 T^{3} + 2782 T^{4} + 17 T^{5} - 23 T^{6} + T^{7} \)
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