Properties

Label 4008.2.a.i
Level 4008
Weight 2
Character orbit 4008.a
Self dual yes
Analytic conductor 32.004
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -2 + \beta_{5} + \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -2 + \beta_{5} + \beta_{7} ) q^{7} + q^{9} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{17} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{19} + ( -2 + \beta_{5} + \beta_{7} ) q^{21} + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{25} + q^{27} + ( -5 + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{29} + ( -3 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{31} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{33} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{35} + ( 1 - \beta_{2} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{37} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{39} + ( -3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{41} + ( -4 - 2 \beta_{3} + \beta_{7} ) q^{43} + ( -1 + \beta_{1} ) q^{45} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{49} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{51} + ( -3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{53} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{55} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{57} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{8} ) q^{59} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{61} + ( -2 + \beta_{5} + \beta_{7} ) q^{63} + ( 2 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{65} + ( 1 - 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{67} + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{69} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{8} ) q^{71} + ( 2 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{73} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{75} + ( -3 + \beta_{1} - \beta_{2} - \beta_{7} + 2 \beta_{8} ) q^{77} + ( -4 + 2 \beta_{2} + 2 \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{79} + q^{81} + ( -2 - 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{83} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{85} + ( -5 + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{87} + ( -6 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{8} ) q^{89} + ( -3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{91} + ( -3 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{93} + ( -5 - 3 \beta_{1} - 4 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} ) q^{95} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{97} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 9q^{3} - 6q^{5} - 11q^{7} + 9q^{9} + O(q^{10}) \) \( 9q + 9q^{3} - 6q^{5} - 11q^{7} + 9q^{9} + q^{11} - 4q^{13} - 6q^{15} - 9q^{17} - 8q^{19} - 11q^{21} - 7q^{23} - q^{25} + 9q^{27} - 9q^{29} - 25q^{31} + q^{33} + 5q^{35} - 6q^{37} - 4q^{39} - 4q^{41} - 24q^{43} - 6q^{45} - 16q^{47} + 4q^{49} - 9q^{51} - 26q^{53} - 29q^{55} - 8q^{57} - 4q^{59} - 20q^{61} - 11q^{63} - 8q^{65} - 25q^{67} - 7q^{69} - 15q^{71} - 10q^{73} - q^{75} - 20q^{77} - 34q^{79} + 9q^{81} - 4q^{83} - 13q^{85} - 9q^{87} + 13q^{89} - 21q^{91} - 25q^{93} - 7q^{95} - 4q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 16 x^{7} + 45 x^{6} + 67 x^{5} - 166 x^{4} - 83 x^{3} + 152 x^{2} + 51 x - 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -253 \nu^{8} + 10903 \nu^{7} - 17171 \nu^{6} - 175482 \nu^{5} + 247955 \nu^{4} + 716568 \nu^{3} - 563660 \nu^{2} - 635601 \nu + 68945 \)\()/102665\)
\(\beta_{3}\)\(=\)\((\)\( -428 \nu^{8} + 9923 \nu^{7} - 2266 \nu^{6} - 178372 \nu^{5} + 67645 \nu^{4} + 878658 \nu^{3} - 38080 \nu^{2} - 988001 \nu - 301330 \)\()/102665\)
\(\beta_{4}\)\(=\)\((\)\( -3031 \nu^{8} + 7666 \nu^{7} + 48718 \nu^{6} - 99334 \nu^{5} - 199070 \nu^{4} + 207921 \nu^{3} + 208150 \nu^{2} + 200063 \nu - 130065 \)\()/102665\)
\(\beta_{5}\)\(=\)\((\)\( 3811 \nu^{8} - 23831 \nu^{7} - 35953 \nu^{6} + 387944 \nu^{5} - 79645 \nu^{4} - 1646101 \nu^{3} + 520415 \nu^{2} + 1605297 \nu + 255125 \)\()/102665\)
\(\beta_{6}\)\(=\)\((\)\( 5684 \nu^{8} - 5129 \nu^{7} - 118627 \nu^{6} + 65121 \nu^{5} + 751965 \nu^{4} - 184819 \nu^{3} - 1469865 \nu^{2} + 296533 \nu + 445920 \)\()/102665\)
\(\beta_{7}\)\(=\)\((\)\( -6589 \nu^{8} + 20594 \nu^{7} + 101842 \nu^{6} - 311796 \nu^{5} - 367380 \nu^{4} + 1137454 \nu^{3} + 148730 \nu^{2} - 769633 \nu + 59190 \)\()/102665\)
\(\beta_{8}\)\(=\)\((\)\( 16128 \nu^{8} - 24668 \nu^{7} - 293609 \nu^{6} + 290982 \nu^{5} + 1500975 \nu^{4} - 446023 \nu^{3} - 2000020 \nu^{2} - 396149 \nu + 286160 \)\()/102665\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-2 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 8 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(-3 \beta_{8} - 12 \beta_{7} + 6 \beta_{6} - 12 \beta_{5} + 7 \beta_{4} + 5 \beta_{3} - 17 \beta_{2} - 2 \beta_{1} + 54\)
\(\nu^{5}\)\(=\)\(-30 \beta_{8} - 45 \beta_{7} + 33 \beta_{6} - 27 \beta_{5} - 36 \beta_{4} + 34 \beta_{3} - 32 \beta_{2} + 73 \beta_{1} + 109\)
\(\nu^{6}\)\(=\)\(-56 \beta_{8} - 145 \beta_{7} + 106 \beta_{6} - 132 \beta_{5} + 54 \beta_{4} + 105 \beta_{3} - 225 \beta_{2} - 33 \beta_{1} + 635\)
\(\nu^{7}\)\(=\)\(-391 \beta_{8} - 576 \beta_{7} + 466 \beta_{6} - 324 \beta_{5} - 395 \beta_{4} + 506 \beta_{3} - 459 \beta_{2} + 691 \beta_{1} + 1496\)
\(\nu^{8}\)\(=\)\(-846 \beta_{8} - 1799 \beta_{7} + 1544 \beta_{6} - 1474 \beta_{5} + 418 \beta_{4} + 1662 \beta_{3} - 2818 \beta_{2} - 429 \beta_{1} + 7652\)

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
−3.17025
−1.99382
−0.985991
−0.482381
0.141931
1.41388
1.58856
2.94484
3.54323
0 1.00000 0 −4.17025 0 −1.78655 0 1.00000 0
1.2 0 1.00000 0 −2.99382 0 −4.65862 0 1.00000 0
1.3 0 1.00000 0 −1.98599 0 −0.0909950 0 1.00000 0
1.4 0 1.00000 0 −1.48238 0 −1.03908 0 1.00000 0
1.5 0 1.00000 0 −0.858069 0 2.33225 0 1.00000 0
1.6 0 1.00000 0 0.413878 0 2.72537 0 1.00000 0
1.7 0 1.00000 0 0.588564 0 −1.23518 0 1.00000 0
1.8 0 1.00000 0 1.94484 0 −3.19705 0 1.00000 0
1.9 0 1.00000 0 2.54323 0 −4.05015 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

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 4008.2.a.i 9
4.b odd 2 1 8016.2.a.ba 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.i 9 1.a even 1 1 trivial
8016.2.a.ba 9 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 - T )^{9} \)
$5$ \( 1 + 6 T + 41 T^{2} + 173 T^{3} + 719 T^{4} + 2390 T^{5} + 7554 T^{6} + 20743 T^{7} + 53605 T^{8} + 123718 T^{9} + 268025 T^{10} + 518575 T^{11} + 944250 T^{12} + 1493750 T^{13} + 2246875 T^{14} + 2703125 T^{15} + 3203125 T^{16} + 2343750 T^{17} + 1953125 T^{18} \)
$7$ \( 1 + 11 T + 90 T^{2} + 528 T^{3} + 2641 T^{4} + 11154 T^{5} + 42250 T^{6} + 141980 T^{7} + 435750 T^{8} + 1204094 T^{9} + 3050250 T^{10} + 6957020 T^{11} + 14491750 T^{12} + 26780754 T^{13} + 44387287 T^{14} + 62118672 T^{15} + 74118870 T^{16} + 63412811 T^{17} + 40353607 T^{18} \)
$11$ \( 1 - T + 52 T^{2} - 70 T^{3} + 1332 T^{4} - 2129 T^{5} + 23490 T^{6} - 38142 T^{7} + 322805 T^{8} - 481364 T^{9} + 3550855 T^{10} - 4615182 T^{11} + 31265190 T^{12} - 31170689 T^{13} + 214519932 T^{14} - 124009270 T^{15} + 1013332892 T^{16} - 214358881 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 + 4 T + 74 T^{2} + 264 T^{3} + 2783 T^{4} + 8611 T^{5} + 67581 T^{6} + 182788 T^{7} + 1176589 T^{8} + 2768138 T^{9} + 15295657 T^{10} + 30891172 T^{11} + 148475457 T^{12} + 245938771 T^{13} + 1033308419 T^{14} + 1274277576 T^{15} + 4643390258 T^{16} + 3262922884 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 + 9 T + 105 T^{2} + 636 T^{3} + 4431 T^{4} + 20141 T^{5} + 107753 T^{6} + 398006 T^{7} + 1917132 T^{8} + 6639284 T^{9} + 32591244 T^{10} + 115023734 T^{11} + 529390489 T^{12} + 1682196461 T^{13} + 6291386367 T^{14} + 15351493884 T^{15} + 43085560665 T^{16} + 62781816969 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 + 8 T + 74 T^{2} + 342 T^{3} + 1683 T^{4} + 3555 T^{5} + 13241 T^{6} - 21770 T^{7} + 26115 T^{8} - 779550 T^{9} + 496185 T^{10} - 7858970 T^{11} + 90820019 T^{12} + 463291155 T^{13} + 4167274617 T^{14} + 16089691302 T^{15} + 66146508686 T^{16} + 135868504328 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 7 T + 122 T^{2} + 616 T^{3} + 6996 T^{4} + 30587 T^{5} + 281820 T^{6} + 1091516 T^{7} + 8438553 T^{8} + 28681556 T^{9} + 194086719 T^{10} + 577411964 T^{11} + 3428903940 T^{12} + 8559496667 T^{13} + 45028655628 T^{14} + 91190107624 T^{15} + 415388704534 T^{16} + 548176896967 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 + 9 T + 160 T^{2} + 954 T^{3} + 10872 T^{4} + 50409 T^{5} + 484084 T^{6} + 1856662 T^{7} + 16472499 T^{8} + 56162028 T^{9} + 477702471 T^{10} + 1561452742 T^{11} + 11806324676 T^{12} + 35653327929 T^{13} + 222997211928 T^{14} + 567461448234 T^{15} + 2759980209440 T^{16} + 4502217716649 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 + 25 T + 431 T^{2} + 5443 T^{3} + 57907 T^{4} + 525407 T^{5} + 4233640 T^{6} + 30306973 T^{7} + 196210743 T^{8} + 1145270768 T^{9} + 6082533033 T^{10} + 29125001053 T^{11} + 126124369240 T^{12} + 485224398047 T^{13} + 1657828246957 T^{14} + 4830682535683 T^{15} + 11857936681841 T^{16} + 21322275936025 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 6 T + 165 T^{2} + 967 T^{3} + 12241 T^{4} + 69142 T^{5} + 530842 T^{6} + 3130351 T^{7} + 16937275 T^{8} + 117767388 T^{9} + 626679175 T^{10} + 4285450519 T^{11} + 26888739826 T^{12} + 129583239862 T^{13} + 848839377637 T^{14} + 2481057437503 T^{15} + 15663759726945 T^{16} + 21074876723526 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 4 T + 257 T^{2} + 1351 T^{3} + 31614 T^{4} + 184863 T^{5} + 2521094 T^{6} + 14274085 T^{7} + 142940352 T^{8} + 712583638 T^{9} + 5860554432 T^{10} + 23994736885 T^{11} + 173756319574 T^{12} + 522378655743 T^{13} + 3662677938414 T^{14} + 6417390829591 T^{15} + 50051848387417 T^{16} + 31939700916484 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 24 T + 515 T^{2} + 7423 T^{3} + 96980 T^{4} + 1029695 T^{5} + 10046614 T^{6} + 84278929 T^{7} + 654510506 T^{8} + 4455064042 T^{9} + 28143951758 T^{10} + 155831739721 T^{11} + 798776139298 T^{12} + 3520322295695 T^{13} + 14256878802140 T^{14} + 46923477912727 T^{15} + 139986584720105 T^{16} + 280516806662424 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 16 T + 443 T^{2} + 5414 T^{3} + 84955 T^{4} + 827727 T^{5} + 9385058 T^{6} + 74534126 T^{7} + 662991021 T^{8} + 4317365814 T^{9} + 31160577987 T^{10} + 164645884334 T^{11} + 974384876734 T^{12} + 4039043715087 T^{13} + 19484005069685 T^{14} + 58358671791206 T^{15} + 224434042365109 T^{16} + 380980586588176 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 + 26 T + 527 T^{2} + 7026 T^{3} + 77435 T^{4} + 638817 T^{5} + 4321678 T^{6} + 20643772 T^{7} + 78929197 T^{8} + 281350616 T^{9} + 4183247441 T^{10} + 57988355548 T^{11} + 643398455606 T^{12} + 5040573400977 T^{13} + 32382968000455 T^{14} + 155726801292354 T^{15} + 619072770694099 T^{16} + 1618751950695386 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 + 4 T + 325 T^{2} + 1542 T^{3} + 55375 T^{4} + 257845 T^{5} + 6265316 T^{6} + 26719662 T^{7} + 503343089 T^{8} + 1894242454 T^{9} + 29697242251 T^{10} + 93011143422 T^{11} + 1286764334764 T^{12} + 3124400947045 T^{13} + 39588933057125 T^{14} + 65042382874422 T^{15} + 808811732566175 T^{16} + 587321750417284 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 + 20 T + 602 T^{2} + 9248 T^{3} + 155243 T^{4} + 1900323 T^{5} + 22802367 T^{6} + 226604280 T^{7} + 2116420563 T^{8} + 17147925490 T^{9} + 129101654343 T^{10} + 843194525880 T^{11} + 5175704064027 T^{12} + 26311570106643 T^{13} + 131117663556143 T^{14} + 476460422090528 T^{15} + 1891931187284642 T^{16} + 3834146259945620 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 25 T + 568 T^{2} + 9209 T^{3} + 132959 T^{4} + 1619633 T^{5} + 18110518 T^{6} + 180254541 T^{7} + 1671954472 T^{8} + 14144938314 T^{9} + 112020949624 T^{10} + 809162634549 T^{11} + 5446973725234 T^{12} + 32637420558593 T^{13} + 179511284101613 T^{14} + 833031241394321 T^{15} + 3442484191823464 T^{16} + 10151691938916025 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 15 T + 449 T^{2} + 5499 T^{3} + 91049 T^{4} + 954797 T^{5} + 11431765 T^{6} + 106092249 T^{7} + 1035484196 T^{8} + 8617953600 T^{9} + 73519377916 T^{10} + 534811027209 T^{11} + 4091554442915 T^{12} + 24262996783757 T^{13} + 164273278179199 T^{14} + 704423461281579 T^{15} + 4083708951117559 T^{16} + 9686302968686415 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 10 T + 276 T^{2} + 3380 T^{3} + 53683 T^{4} + 589381 T^{5} + 6867531 T^{6} + 71234924 T^{7} + 666366469 T^{8} + 5992553762 T^{9} + 48644752237 T^{10} + 379610909996 T^{11} + 2671586307027 T^{12} + 16737383678821 T^{13} + 111288702327019 T^{14} + 511509684856820 T^{15} + 3049081991270772 T^{16} + 8064600918940810 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 34 T + 941 T^{2} + 18473 T^{3} + 313218 T^{4} + 4495933 T^{5} + 57476662 T^{6} + 655683215 T^{7} + 6748904942 T^{8} + 63046688450 T^{9} + 533163490418 T^{10} + 4092118944815 T^{11} + 28338235955818 T^{12} + 175116954520573 T^{13} + 963789451181982 T^{14} + 4490554565839433 T^{15} + 18070878355975619 T^{16} + 51581699536823074 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 + 4 T + 451 T^{2} + 512 T^{3} + 86177 T^{4} - 218705 T^{5} + 9317210 T^{6} - 68434988 T^{7} + 730048475 T^{8} - 8141053038 T^{9} + 60594023425 T^{10} - 471448632332 T^{11} + 5327459554270 T^{12} - 10379372094305 T^{13} + 339454705491811 T^{14} + 167393471164928 T^{15} + 12238358996321777 T^{16} + 9009168928556164 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 - 13 T + 480 T^{2} - 6469 T^{3} + 126713 T^{4} - 1525667 T^{5} + 22163950 T^{6} - 229481013 T^{7} + 2736499528 T^{8} - 24097791692 T^{9} + 243548457992 T^{10} - 1817719103973 T^{11} + 15624897667550 T^{12} - 95723766599747 T^{13} + 707572924961137 T^{14} - 3214971971226709 T^{15} + 21231040749853920 T^{16} - 51175654474127053 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 4 T + 306 T^{2} + 941 T^{3} + 54811 T^{4} + 87444 T^{5} + 7554000 T^{6} + 3292913 T^{7} + 811591682 T^{8} - 7315132 T^{9} + 78724393154 T^{10} + 30983018417 T^{11} + 6894331842000 T^{12} + 7741354447764 T^{13} + 470680706826427 T^{14} + 783826656638189 T^{15} + 24724275050302578 T^{16} + 31349734377507844 T^{17} + 760231058654565217 T^{18} \)
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