Properties

Label 6032.2.a.bb
Level 6032
Weight 2
Character orbit 6032.a
Self dual yes
Analytic conductor 48.166
Analytic rank 0
Dimension 10
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6032.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1657624992\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 10 x^{8} + 21 x^{7} + 28 x^{6} - 67 x^{5} - 20 x^{4} + 76 x^{3} - 8 x^{2} - 26 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 3016)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{7} q^{5} -\beta_{4} q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{7} q^{5} -\beta_{4} q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{9} ) q^{9} + ( 1 - \beta_{9} ) q^{11} - q^{13} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{15} + ( -\beta_{1} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{17} + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} ) q^{19} + ( -1 - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{21} + ( 1 - \beta_{2} + \beta_{3} ) q^{23} + ( \beta_{2} - 2 \beta_{6} + \beta_{8} ) q^{25} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{9} ) q^{27} + q^{29} + ( -\beta_{3} - \beta_{7} - \beta_{8} ) q^{31} + ( -1 - 3 \beta_{1} + \beta_{4} - \beta_{7} - \beta_{8} ) q^{33} + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{35} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{37} + \beta_{1} q^{39} + ( -\beta_{3} + \beta_{6} - \beta_{7} ) q^{41} + ( 3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{7} + \beta_{8} ) q^{43} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{8} ) q^{45} + ( 3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{47} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{7} - \beta_{8} + \beta_{9} ) q^{49} + ( 3 - 4 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{51} + ( 1 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{53} + ( 1 - \beta_{1} - 2 \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{55} + ( -3 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{57} + ( 3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{59} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} ) q^{61} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{63} + \beta_{7} q^{65} + ( 2 + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{67} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{69} + ( 3 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{71} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{73} + ( 1 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{75} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{7} + \beta_{9} ) q^{77} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{79} + ( 1 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{81} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{9} ) q^{83} + ( 1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} ) q^{85} -\beta_{1} q^{87} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} ) q^{89} + \beta_{4} q^{91} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{93} + ( 3 - 2 \beta_{3} + \beta_{4} + \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} ) q^{95} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{97} + ( 5 - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{3} - 5q^{5} + 9q^{9} + O(q^{10}) \) \( 10q + 3q^{3} - 5q^{5} + 9q^{9} + 13q^{11} - 10q^{13} - 3q^{15} + 9q^{17} + 12q^{19} - 10q^{21} + 5q^{23} + 7q^{25} + 3q^{27} + 10q^{29} - 5q^{31} - 9q^{33} + 23q^{35} - 4q^{37} - 3q^{39} - 3q^{41} + 27q^{43} - 20q^{45} + 16q^{47} + 6q^{49} + 34q^{51} + 11q^{53} + q^{55} + 27q^{59} - 7q^{61} + 6q^{63} + 5q^{65} + 35q^{67} - 22q^{69} + 21q^{71} + 7q^{75} - 18q^{77} + 12q^{79} + 6q^{81} + 24q^{83} - 2q^{85} + 3q^{87} - 23q^{89} - 3q^{93} + 17q^{95} + 2q^{97} + 65q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 10 x^{8} + 21 x^{7} + 28 x^{6} - 67 x^{5} - 20 x^{4} + 76 x^{3} - 8 x^{2} - 26 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{9} - 10 \nu^{7} - \nu^{6} + 28 \nu^{5} + 9 \nu^{4} - 22 \nu^{3} - 22 \nu^{2} + 10 \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{8} - 10 \nu^{6} + 28 \nu^{4} - 22 \nu^{2} - 2 \nu + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{9} - \nu^{8} - 10 \nu^{7} + 10 \nu^{6} + 28 \nu^{5} - 29 \nu^{4} - 22 \nu^{3} + 26 \nu^{2} + 2 \nu - 5 \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{9} - 2 \nu^{8} + 12 \nu^{7} + 21 \nu^{6} - 46 \nu^{5} - 65 \nu^{4} + 62 \nu^{3} + 66 \nu^{2} - 16 \nu - 14 \)\()/2\)
\(\beta_{5}\)\(=\)\( \nu^{9} - \nu^{8} - 10 \nu^{7} + 10 \nu^{6} + 28 \nu^{5} - 29 \nu^{4} - 21 \nu^{3} + 27 \nu^{2} - 2 \nu - 7 \)
\(\beta_{6}\)\(=\)\((\)\( \nu^{9} - 2 \nu^{8} - 12 \nu^{7} + 21 \nu^{6} + 48 \nu^{5} - 67 \nu^{4} - 76 \nu^{3} + 76 \nu^{2} + 36 \nu - 24 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{9} + 2 \nu^{8} + 32 \nu^{7} - 19 \nu^{6} - 104 \nu^{5} + 51 \nu^{4} + 120 \nu^{3} - 44 \nu^{2} - 40 \nu + 14 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{9} - 4 \nu^{8} - 34 \nu^{7} + 43 \nu^{6} + 124 \nu^{5} - 143 \nu^{4} - 176 \nu^{3} + 172 \nu^{2} + 84 \nu - 60 \)\()/2\)
\(\beta_{9}\)\(=\)\( 2 \nu^{9} - 2 \nu^{8} - 22 \nu^{7} + 21 \nu^{6} + 76 \nu^{5} - 67 \nu^{4} - 98 \nu^{3} + 74 \nu^{2} + 40 \nu - 23 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{9} - \beta_{7} - 3 \beta_{6} - \beta_{3} - \beta_{2} - 2 \beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{9} + 3 \beta_{8} - \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 4 \beta_{3} - \beta_{2} + \beta_{1} + 13\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{9} - \beta_{8} - \beta_{7} - 3 \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} - 3 \beta_{1} + 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-5 \beta_{9} + 9 \beta_{8} - \beta_{7} - 9 \beta_{6} + 9 \beta_{5} - 13 \beta_{3} - 4 \beta_{2} + 7 \beta_{1} + 25\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(19 \beta_{9} - 12 \beta_{8} - 7 \beta_{7} - 18 \beta_{6} + 9 \beta_{5} + 3 \beta_{4} - 19 \beta_{3} - 4 \beta_{2} - 20 \beta_{1} + 7\)\()/3\)
\(\nu^{6}\)\(=\)\(-8 \beta_{9} + 17 \beta_{8} + \beta_{7} - 17 \beta_{6} + 17 \beta_{5} - 25 \beta_{3} - 8 \beta_{2} + 17 \beta_{1} + 40\)
\(\nu^{7}\)\(=\)\((\)\(116 \beta_{9} - 78 \beta_{8} - 38 \beta_{7} - 87 \beta_{6} + 51 \beta_{5} + 30 \beta_{4} - 116 \beta_{3} - 8 \beta_{2} - 97 \beta_{1} + 38\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-121 \beta_{9} + 291 \beta_{8} + 46 \beta_{7} - 294 \beta_{6} + 291 \beta_{5} - 431 \beta_{3} - 137 \beta_{2} + 323 \beta_{1} + 638\)\()/3\)
\(\nu^{9}\)\(=\)\(231 \beta_{9} - 158 \beta_{8} - 72 \beta_{7} - 156 \beta_{6} + 98 \beta_{5} + 72 \beta_{4} - 232 \beta_{3} - 168 \beta_{1} + 75\)

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
−2.40534
0.418899
−0.805131
1.57583
0.640377
2.43306
−1.68061
1.22858
−1.12886
1.72319
0 −2.61420 0 −3.25734 0 −1.30647 0 3.83403 0
1.2 0 −2.56651 0 0.728348 0 3.51001 0 3.58699 0
1.3 0 −1.65416 0 −0.127785 0 −0.216876 0 −0.263746 0
1.4 0 −0.0834797 0 3.65561 0 2.67943 0 −2.99303 0
1.5 0 0.381396 0 0.331991 0 −2.55994 0 −2.85454 0
1.6 0 0.619695 0 −3.40954 0 −0.0449896 0 −2.61598 0
1.7 0 1.06017 0 1.60145 0 −0.812633 0 −1.87603 0
1.8 0 2.21819 0 −3.58621 0 −5.12377 0 1.92038 0
1.9 0 2.39322 0 −2.03033 0 4.56146 0 2.72750 0
1.10 0 3.24568 0 1.09381 0 −0.686226 0 7.53443 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(13\) \(1\)
\(29\) \(-1\)

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 6032.2.a.bb 10
4.b odd 2 1 3016.2.a.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3016.2.a.f 10 4.b odd 2 1
6032.2.a.bb 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T + 15 T^{2} - 34 T^{3} + 105 T^{4} - 214 T^{5} + 541 T^{6} - 1036 T^{7} + 2230 T^{8} - 3905 T^{9} + 7372 T^{10} - 11715 T^{11} + 20070 T^{12} - 27972 T^{13} + 43821 T^{14} - 52002 T^{15} + 76545 T^{16} - 74358 T^{17} + 98415 T^{18} - 59049 T^{19} + 59049 T^{20} \)
$5$ \( 1 + 5 T + 34 T^{2} + 128 T^{3} + 530 T^{4} + 1560 T^{5} + 4981 T^{6} + 12262 T^{7} + 33265 T^{8} + 72757 T^{9} + 179834 T^{10} + 363785 T^{11} + 831625 T^{12} + 1532750 T^{13} + 3113125 T^{14} + 4875000 T^{15} + 8281250 T^{16} + 10000000 T^{17} + 13281250 T^{18} + 9765625 T^{19} + 9765625 T^{20} \)
$7$ \( 1 + 32 T^{2} - T^{3} + 456 T^{4} + 189 T^{5} + 3893 T^{6} + 5721 T^{7} + 22739 T^{8} + 71315 T^{9} + 129594 T^{10} + 499205 T^{11} + 1114211 T^{12} + 1962303 T^{13} + 9347093 T^{14} + 3176523 T^{15} + 53647944 T^{16} - 823543 T^{17} + 184473632 T^{18} + 282475249 T^{20} \)
$11$ \( 1 - 13 T + 132 T^{2} - 958 T^{3} + 6048 T^{4} - 32554 T^{5} + 159294 T^{6} - 700316 T^{7} + 2848143 T^{8} - 10589211 T^{9} + 36650396 T^{10} - 116481321 T^{11} + 344625303 T^{12} - 932120596 T^{13} + 2332223454 T^{14} - 5242854254 T^{15} + 10714400928 T^{16} - 18668709818 T^{17} + 28295372292 T^{18} - 30653319983 T^{19} + 25937424601 T^{20} \)
$13$ \( ( 1 + T )^{10} \)
$17$ \( 1 - 9 T + 104 T^{2} - 736 T^{3} + 5442 T^{4} - 31954 T^{5} + 185857 T^{6} - 941432 T^{7} + 4651681 T^{8} - 20715021 T^{9} + 89480294 T^{10} - 352155357 T^{11} + 1344335809 T^{12} - 4625255416 T^{13} + 15522962497 T^{14} - 45370110578 T^{15} + 131356650498 T^{16} - 302009263328 T^{17} + 725478773864 T^{18} - 1067290888473 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 - 12 T + 135 T^{2} - 925 T^{3} + 6474 T^{4} - 33518 T^{5} + 187698 T^{6} - 812364 T^{7} + 4020901 T^{8} - 15703091 T^{9} + 76173486 T^{10} - 298358729 T^{11} + 1451545261 T^{12} - 5572004676 T^{13} + 24460991058 T^{14} - 82993886282 T^{15} + 304575033594 T^{16} - 826831358575 T^{17} + 2292781010535 T^{18} - 3872252373348 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 - 5 T + 129 T^{2} - 428 T^{3} + 7735 T^{4} - 19412 T^{5} + 320982 T^{6} - 666430 T^{7} + 10188256 T^{8} - 17823281 T^{9} + 257700866 T^{10} - 409935463 T^{11} + 5389587424 T^{12} - 8108453810 T^{13} + 89823923862 T^{14} - 124942290316 T^{15} + 1145057601415 T^{16} - 1457265291316 T^{17} + 10102117101249 T^{18} - 9005763307315 T^{19} + 41426511213649 T^{20} \)
$29$ \( ( 1 - T )^{10} \)
$31$ \( 1 + 5 T + 201 T^{2} + 920 T^{3} + 19165 T^{4} + 83000 T^{5} + 1180832 T^{6} + 4903994 T^{7} + 53413138 T^{8} + 207576597 T^{9} + 1869940110 T^{10} + 6434874507 T^{11} + 51330025618 T^{12} + 146094885254 T^{13} + 1090523149472 T^{14} + 2376219533000 T^{15} + 17009008046365 T^{16} + 25311604982120 T^{17} + 171431098525641 T^{18} + 132198110803355 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 + 4 T + 301 T^{2} + 1029 T^{3} + 42399 T^{4} + 124679 T^{5} + 3699099 T^{6} + 9384177 T^{7} + 222137340 T^{8} + 485634215 T^{9} + 9615772984 T^{10} + 17968465955 T^{11} + 304106018460 T^{12} + 475336717581 T^{13} + 6932707080939 T^{14} + 8645735214803 T^{15} + 108784234015191 T^{16} + 97684901569857 T^{17} + 1057256315630221 T^{18} + 519846959180308 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 + 3 T + 297 T^{2} + 793 T^{3} + 42012 T^{4} + 98943 T^{5} + 3756432 T^{6} + 7772583 T^{7} + 237490267 T^{8} + 430723742 T^{9} + 11189177758 T^{10} + 17659673422 T^{11} + 399221138827 T^{12} + 535694192943 T^{13} + 10614779044752 T^{14} + 11463160095543 T^{15} + 199561379372892 T^{16} + 154440139187633 T^{17} + 2371522793048937 T^{18} + 982145803181883 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 - 27 T + 445 T^{2} - 5426 T^{3} + 56717 T^{4} - 542182 T^{5} + 4866833 T^{6} - 40593452 T^{7} + 312088338 T^{8} - 2225744921 T^{9} + 14967849592 T^{10} - 95707031603 T^{11} + 577051336962 T^{12} - 3227463588164 T^{13} + 16638733527233 T^{14} - 79705331642626 T^{15} + 358528748050133 T^{16} - 1474887783866582 T^{17} + 5201249123532445 T^{18} - 13570000522294761 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 - 16 T + 336 T^{2} - 4052 T^{3} + 53538 T^{4} - 517469 T^{5} + 5306435 T^{6} - 43515255 T^{7} + 373458641 T^{8} - 2668443246 T^{9} + 19958078018 T^{10} - 125416832562 T^{11} + 824970137969 T^{12} - 4517884319865 T^{13} + 25893710047235 T^{14} - 118678931427283 T^{15} + 577097630284002 T^{16} - 2052836884116076 T^{17} + 8000592318351696 T^{18} - 17906087569644272 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 - 11 T + 334 T^{2} - 3080 T^{3} + 55686 T^{4} - 445748 T^{5} + 6062538 T^{6} - 42653254 T^{7} + 478604633 T^{8} - 2979608187 T^{9} + 28830947824 T^{10} - 157919233911 T^{11} + 1344400414097 T^{12} - 6350088495758 T^{13} + 47836340900778 T^{14} - 186409804613764 T^{15} + 1234244613829494 T^{16} - 3618110310697960 T^{17} + 20794736597394574 T^{18} - 36297399509823463 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 - 27 T + 760 T^{2} - 13170 T^{3} + 218576 T^{4} - 2830944 T^{5} + 34702002 T^{6} - 359980162 T^{7} + 3536742027 T^{8} - 30432920901 T^{9} + 248533437940 T^{10} - 1795542333159 T^{11} + 12311398995987 T^{12} - 73932365691398 T^{13} + 420496685656722 T^{14} - 2023910654708256 T^{15} + 9219652321115216 T^{16} - 32775540055066230 T^{17} + 111591132579283960 T^{18} - 233900887103683353 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 + 7 T + 313 T^{2} + 2291 T^{3} + 56620 T^{4} + 388131 T^{5} + 6955840 T^{6} + 43736059 T^{7} + 634034651 T^{8} + 3574633568 T^{9} + 43971514174 T^{10} + 218052647648 T^{11} + 2359242936371 T^{12} + 9927254407879 T^{13} + 96309454661440 T^{14} + 327814006903431 T^{15} + 2917083596319820 T^{16} + 7200023837324111 T^{17} + 60004388968148953 T^{18} + 81859022649838987 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 - 35 T + 883 T^{2} - 15652 T^{3} + 236453 T^{4} - 3001792 T^{5} + 34629186 T^{6} - 356957574 T^{7} + 3445576466 T^{8} - 30604141187 T^{9} + 259481144894 T^{10} - 2050477459529 T^{11} + 15467192755874 T^{12} - 107359630828962 T^{13} + 697816917217506 T^{14} - 4052794745191744 T^{15} + 21389155839006557 T^{16} - 94862258046515596 T^{17} + 358557759282514003 T^{18} - 952228703870323145 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 21 T + 657 T^{2} - 9344 T^{3} + 166081 T^{4} - 1741348 T^{5} + 22633861 T^{6} - 184670438 T^{7} + 2019559390 T^{8} - 14050966089 T^{9} + 147684926136 T^{10} - 997618592319 T^{11} + 10180598884990 T^{12} - 66095581135018 T^{13} + 575164455530341 T^{14} - 3141791171905148 T^{15} + 21275023253883601 T^{16} - 84984802760005504 T^{17} + 424260070028464977 T^{18} - 962818515087429651 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 + 442 T^{2} + 587 T^{3} + 94527 T^{4} + 248436 T^{5} + 13098096 T^{6} + 48034634 T^{7} + 1337231680 T^{8} + 5498558095 T^{9} + 108222369484 T^{10} + 401394740935 T^{11} + 7126107622720 T^{12} + 18686289214778 T^{13} + 371962886849136 T^{14} + 515025614278548 T^{15} + 14305170408420303 T^{16} + 6484822930709939 T^{17} + 356455360617183802 T^{18} + 4297625829703557649 T^{20} \)
$79$ \( 1 - 12 T + 451 T^{2} - 3969 T^{3} + 88598 T^{4} - 584362 T^{5} + 10627760 T^{6} - 52783640 T^{7} + 948474525 T^{8} - 3797131351 T^{9} + 75650685594 T^{10} - 299973376729 T^{11} + 5919429510525 T^{12} - 26024393081960 T^{13} + 413952112848560 T^{14} - 1798114831432438 T^{15} + 21537062384249558 T^{16} - 76220314766065071 T^{17} + 684216073267859011 T^{18} - 1438219151791419828 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 - 24 T + 621 T^{2} - 8369 T^{3} + 121778 T^{4} - 1084498 T^{5} + 12238084 T^{6} - 85761784 T^{7} + 1031731777 T^{8} - 7201781607 T^{9} + 90859562382 T^{10} - 597747873381 T^{11} + 7107600211753 T^{12} - 49037473188008 T^{13} + 580798918896964 T^{14} - 4271881699252214 T^{15} + 39814144788130082 T^{16} - 227101610732188363 T^{17} + 1398673476158344461 T^{18} - 4486566126420969672 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 + 23 T + 378 T^{2} + 6021 T^{3} + 83966 T^{4} + 962177 T^{5} + 10132204 T^{6} + 99543185 T^{7} + 967674633 T^{8} + 8653363778 T^{9} + 77786010660 T^{10} + 770149376242 T^{11} + 7664950767993 T^{12} + 70174859586265 T^{13} + 635717185229164 T^{14} + 5372853568460473 T^{15} + 41729531076831326 T^{16} + 266316867405980109 T^{17} + 1488030568555386618 T^{18} + 8058197285272159807 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 - 2 T + 457 T^{2} - 605 T^{3} + 111502 T^{4} - 150070 T^{5} + 19228484 T^{6} - 28520620 T^{7} + 2551972201 T^{8} - 3863633759 T^{9} + 273348922486 T^{10} - 374772474623 T^{11} + 24011506439209 T^{12} - 26029999817260 T^{13} + 1702283863240004 T^{14} - 1288702152367990 T^{15} + 92878044493593358 T^{16} - 48882962109258365 T^{17} + 3581707152630271177 T^{18} - 1520462117309130434 T^{19} + 73742412689492826049 T^{20} \)
show more
show less