Properties

Label 8016.2.a.bb
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 0
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + \beta_{6} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + \beta_{6} q^{7} + q^{9} + ( -1 + \beta_{5} ) q^{11} + ( 1 - \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( 1 - \beta_{2} + \beta_{3} + \beta_{8} ) q^{17} + ( \beta_{2} + \beta_{7} ) q^{19} -\beta_{6} q^{21} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{8} ) q^{23} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{25} - q^{27} + ( \beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{29} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{31} + ( 1 - \beta_{5} ) q^{33} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{7} + \beta_{8} ) q^{37} + ( -1 + \beta_{4} - \beta_{6} - \beta_{7} ) q^{39} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( 2 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -4 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{47} + ( 2 - \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{49} + ( -1 + \beta_{2} - \beta_{3} - \beta_{8} ) q^{51} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{53} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{55} + ( -\beta_{2} - \beta_{7} ) q^{57} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} ) q^{59} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{61} + \beta_{6} q^{63} + ( 2 - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{65} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{67} + ( 2 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{8} ) q^{69} + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{71} + ( -3 + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{75} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{77} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{79} + q^{81} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{83} + ( 1 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{85} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{87} + ( 4 + \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{89} + ( 4 - \beta_{1} + 2 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{91} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{93} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{95} + ( 4 - \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{97} + ( -1 + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} - 7q^{11} + 6q^{13} - 9q^{15} + 7q^{17} - 2q^{19} + 2q^{21} - 19q^{23} + 22q^{25} - 9q^{27} + 13q^{29} - 12q^{31} + 7q^{33} - 4q^{35} + 15q^{37} - 6q^{39} + 18q^{41} + 6q^{43} + 9q^{45} - 25q^{47} + 19q^{49} - 7q^{51} + 17q^{53} + 3q^{55} + 2q^{57} - 3q^{59} + 14q^{61} - 2q^{63} + 14q^{65} + 4q^{67} + 19q^{69} - 17q^{71} - 20q^{73} - 22q^{75} + 14q^{77} + 8q^{79} + 9q^{81} + q^{83} + 5q^{85} - 13q^{87} + 36q^{89} + 41q^{91} + 12q^{93} - 5q^{95} + 31q^{97} - 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -1006 \nu^{8} + 2871 \nu^{7} + 22263 \nu^{6} - 53775 \nu^{5} - 157887 \nu^{4} + 312589 \nu^{3} + 454514 \nu^{2} - 550699 \nu - 575149 \)\()/51607\)
\(\beta_{3}\)\(=\)\((\)\( 1976 \nu^{8} - 9538 \nu^{7} - 25159 \nu^{6} + 160208 \nu^{5} - 1570 \nu^{4} - 711563 \nu^{3} + 508013 \nu^{2} + 557721 \nu - 255465 \)\()/51607\)
\(\beta_{4}\)\(=\)\((\)\( 2012 \nu^{8} - 5742 \nu^{7} - 44526 \nu^{6} + 107550 \nu^{5} + 315774 \nu^{4} - 625178 \nu^{3} - 857421 \nu^{2} + 1101398 \nu + 789049 \)\()/51607\)
\(\beta_{5}\)\(=\)\((\)\( 2980 \nu^{8} - 18354 \nu^{7} - 43479 \nu^{6} + 348793 \nu^{5} + 150155 \nu^{4} - 1846062 \nu^{3} - 203222 \nu^{2} + 2331119 \nu + 1064431 \)\()/51607\)
\(\beta_{6}\)\(=\)\((\)\( 3832 \nu^{8} - 14527 \nu^{7} - 71773 \nu^{6} + 255110 \nu^{5} + 401245 \nu^{4} - 1246613 \nu^{3} - 814593 \nu^{2} + 1505084 \nu + 875617 \)\()/51607\)
\(\beta_{7}\)\(=\)\((\)\( 3879 \nu^{8} - 3837 \nu^{7} - 87023 \nu^{6} + 28674 \nu^{5} + 546052 \nu^{4} + 57429 \nu^{3} - 948686 \nu^{2} - 418506 \nu + 172141 \)\()/51607\)
\(\beta_{8}\)\(=\)\((\)\( -3886 \nu^{8} + 8833 \nu^{7} + 75020 \nu^{6} - 124516 \nu^{5} - 361191 \nu^{4} + 420341 \nu^{3} + 179180 \nu^{2} - 75695 \nu + 292783 \)\()/51607\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + 2 \beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + 4 \beta_{2} + 10 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(5 \beta_{8} + 4 \beta_{7} + \beta_{5} + 14 \beta_{4} + 2 \beta_{3} + 31 \beta_{2} + 5 \beta_{1} + 79\)
\(\nu^{5}\)\(=\)\(22 \beta_{8} + 38 \beta_{7} - 23 \beta_{6} + 36 \beta_{5} + 17 \beta_{4} - 12 \beta_{3} + 91 \beta_{2} + 123 \beta_{1} + 91\)
\(\nu^{6}\)\(=\)\(115 \beta_{8} + 102 \beta_{7} - 21 \beta_{6} + 38 \beta_{5} + 207 \beta_{4} + 46 \beta_{3} + 486 \beta_{2} + 163 \beta_{1} + 1059\)
\(\nu^{7}\)\(=\)\(431 \beta_{8} + 648 \beta_{7} - 397 \beta_{6} + 555 \beta_{5} + 464 \beta_{4} - 97 \beta_{3} + 1703 \beta_{2} + 1714 \beta_{1} + 2087\)
\(\nu^{8}\)\(=\)\(2125 \beta_{8} + 2069 \beta_{7} - 679 \beta_{6} + 965 \beta_{5} + 3251 \beta_{4} + 758 \beta_{3} + 7981 \beta_{2} + 3699 \beta_{1} + 15652\)

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
4.16840
2.69402
1.79204
1.61974
−0.529979
−0.907808
−2.42964
−3.19525
−3.21153
0 −1.00000 0 −3.16840 0 0.230890 0 1.00000 0
1.2 0 −1.00000 0 −1.69402 0 4.12928 0 1.00000 0
1.3 0 −1.00000 0 −0.792043 0 −3.80237 0 1.00000 0
1.4 0 −1.00000 0 −0.619742 0 −1.05844 0 1.00000 0
1.5 0 −1.00000 0 1.52998 0 1.05249 0 1.00000 0
1.6 0 −1.00000 0 1.90781 0 −2.81337 0 1.00000 0
1.7 0 −1.00000 0 3.42964 0 −3.44225 0 1.00000 0
1.8 0 −1.00000 0 4.19525 0 −1.43344 0 1.00000 0
1.9 0 −1.00000 0 4.21153 0 5.13720 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 8016.2.a.bb 9
4.b odd 2 1 2004.2.a.d 9
12.b even 2 1 6012.2.a.h 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.d 9 4.b odd 2 1
6012.2.a.h 9 12.b even 2 1
8016.2.a.bb 9 1.a even 1 1 trivial

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{9} - \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)
\(T_{13}^{9} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{9} \)
$5$ \( 1 - 9 T + 52 T^{2} - 234 T^{3} + 886 T^{4} - 2925 T^{5} + 8664 T^{6} - 23344 T^{7} + 58229 T^{8} - 134906 T^{9} + 291145 T^{10} - 583600 T^{11} + 1083000 T^{12} - 1828125 T^{13} + 2768750 T^{14} - 3656250 T^{15} + 4062500 T^{16} - 3515625 T^{17} + 1953125 T^{18} \)
$7$ \( 1 + 2 T + 24 T^{2} + 3 T^{3} + 204 T^{4} - 469 T^{5} + 1446 T^{6} - 5119 T^{7} + 15489 T^{8} - 32962 T^{9} + 108423 T^{10} - 250831 T^{11} + 495978 T^{12} - 1126069 T^{13} + 3428628 T^{14} + 352947 T^{15} + 19765032 T^{16} + 11529602 T^{17} + 40353607 T^{18} \)
$11$ \( 1 + 7 T + 76 T^{2} + 448 T^{3} + 2798 T^{4} + 13617 T^{5} + 64556 T^{6} + 259240 T^{7} + 1013497 T^{8} + 3395532 T^{9} + 11148467 T^{10} + 31368040 T^{11} + 85924036 T^{12} + 199366497 T^{13} + 450620698 T^{14} + 793659328 T^{15} + 1481024996 T^{16} + 1500512167 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 - 6 T + 72 T^{2} - 290 T^{3} + 2079 T^{4} - 5583 T^{5} + 32513 T^{6} - 50538 T^{7} + 355779 T^{8} - 363982 T^{9} + 4625127 T^{10} - 8540922 T^{11} + 71431061 T^{12} - 159456063 T^{13} + 771918147 T^{14} - 1399774610 T^{15} + 4517893224 T^{16} - 4894384326 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 - 7 T + 95 T^{2} - 412 T^{3} + 3239 T^{4} - 7217 T^{5} + 46207 T^{6} + 42146 T^{7} + 182922 T^{8} + 2723810 T^{9} + 3109674 T^{10} + 12180194 T^{11} + 227014991 T^{12} - 602771057 T^{13} + 4598916823 T^{14} - 9944678428 T^{15} + 38982173935 T^{16} - 48830302087 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 + 2 T + 102 T^{2} + 170 T^{3} + 4503 T^{4} + 4717 T^{5} + 117347 T^{6} + 26098 T^{7} + 2285365 T^{8} - 603238 T^{9} + 43421935 T^{10} + 9421378 T^{11} + 804883073 T^{12} + 614724157 T^{13} + 11149873797 T^{14} + 7997799770 T^{15} + 91174917378 T^{16} + 33967126082 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 19 T + 254 T^{2} + 2284 T^{3} + 17484 T^{4} + 110157 T^{5} + 659522 T^{6} + 3574832 T^{7} + 19162755 T^{8} + 93129576 T^{9} + 440743365 T^{10} + 1891086128 T^{11} + 8024404174 T^{12} + 30826445037 T^{13} + 112533021012 T^{14} + 338113970476 T^{15} + 864825663538 T^{16} + 1487908720339 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 - 13 T + 154 T^{2} - 1330 T^{3} + 10268 T^{4} - 65265 T^{5} + 395422 T^{6} - 2088442 T^{7} + 11323883 T^{8} - 58112060 T^{9} + 328392607 T^{10} - 1756379722 T^{11} + 9643947158 T^{12} - 46160694465 T^{13} + 210608477932 T^{14} - 791115016930 T^{15} + 2656480951586 T^{16} - 6503203368493 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 + 12 T + 237 T^{2} + 2081 T^{3} + 24736 T^{4} + 175357 T^{5} + 1571692 T^{6} + 9333683 T^{7} + 68244056 T^{8} + 343757830 T^{9} + 2115565736 T^{10} + 8969669363 T^{11} + 46822276372 T^{12} + 161945871997 T^{13} + 708170679136 T^{14} + 1846895160161 T^{15} + 6520489544307 T^{16} + 10234692449292 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 - 15 T + 242 T^{2} - 2138 T^{3} + 21014 T^{4} - 145273 T^{5} + 1219856 T^{6} - 7859170 T^{7} + 60067423 T^{8} - 344194216 T^{9} + 2222494651 T^{10} - 10759203730 T^{11} + 61789365968 T^{12} - 272264990953 T^{13} + 1457193912398 T^{14} - 5485523062442 T^{15} + 22973514266186 T^{16} - 52687191808815 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 - 18 T + 391 T^{2} - 4825 T^{3} + 61116 T^{4} - 575535 T^{5} + 5406478 T^{6} - 41258085 T^{7} + 314515424 T^{8} - 2011563376 T^{9} + 12895132384 T^{10} - 69354840885 T^{11} + 372619870238 T^{12} - 1626324357135 T^{13} + 7080667580316 T^{14} - 22919252962825 T^{15} + 76148921087471 T^{16} - 143728654124178 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 - 6 T + 97 T^{2} - 675 T^{3} + 5484 T^{4} - 22849 T^{5} + 166420 T^{6} - 187403 T^{7} + 1595534 T^{8} + 10660100 T^{9} + 68607962 T^{10} - 346508147 T^{11} + 13231554940 T^{12} - 78116184049 T^{13} + 806194301412 T^{14} - 4266920058075 T^{15} + 26366405277379 T^{16} - 70129201665606 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 25 T + 448 T^{2} + 5687 T^{3} + 64249 T^{4} + 630769 T^{5} + 5789701 T^{6} + 47606857 T^{7} + 365856543 T^{8} + 2574158528 T^{9} + 17195257521 T^{10} + 105163547113 T^{11} + 601104126923 T^{12} + 3077951504689 T^{13} + 14735187354743 T^{14} + 61301397576023 T^{15} + 226967157967424 T^{16} + 595282166544025 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 - 17 T + 318 T^{2} - 3079 T^{3} + 33015 T^{4} - 222369 T^{5} + 1931189 T^{6} - 10783739 T^{7} + 96671639 T^{8} - 529914954 T^{9} + 5123596867 T^{10} - 30291522851 T^{11} + 287509624753 T^{12} - 1754598369489 T^{13} + 13806724201395 T^{14} - 68244067916191 T^{15} + 373558142468166 T^{16} - 1058414736993137 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 + 3 T + 248 T^{2} + 587 T^{3} + 35733 T^{4} + 67161 T^{5} + 3580941 T^{6} + 5779345 T^{7} + 272819099 T^{8} + 381917456 T^{9} + 16096326841 T^{10} + 20117899945 T^{11} + 735450081639 T^{12} + 813814082121 T^{13} + 25546389976167 T^{14} + 24759973247267 T^{15} + 617185568235112 T^{16} + 440491312812963 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 - 14 T + 286 T^{2} - 2662 T^{3} + 34623 T^{4} - 221617 T^{5} + 2297933 T^{6} - 9978522 T^{7} + 111413285 T^{8} - 393019054 T^{9} + 6796210385 T^{10} - 37130080362 T^{11} + 521587130273 T^{12} - 3068473744897 T^{13} + 29242457729523 T^{14} - 137147236548982 T^{15} + 898824451102006 T^{16} - 2683902381961934 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 - 4 T + 402 T^{2} - 1172 T^{3} + 77435 T^{4} - 162163 T^{5} + 9543569 T^{6} - 14628924 T^{7} + 845476133 T^{8} - 1048306460 T^{9} + 56646900911 T^{10} - 65669239836 T^{11} + 2870352443147 T^{12} - 3267766234723 T^{13} + 104546937660545 T^{14} - 106017223902068 T^{15} + 2436406065339846 T^{16} - 1624270710226564 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 17 T + 319 T^{2} + 2841 T^{3} + 39999 T^{4} + 355099 T^{5} + 4942665 T^{6} + 38075859 T^{7} + 407671928 T^{8} + 2653132208 T^{9} + 28944706888 T^{10} + 191940405219 T^{11} + 1769034172815 T^{12} + 9023662511419 T^{13} + 72167369810649 T^{14} + 363932906619561 T^{15} + 2901343330526729 T^{16} + 10977810031177937 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 20 T + 498 T^{2} + 7086 T^{3} + 112237 T^{4} + 1284803 T^{5} + 15841175 T^{6} + 152923942 T^{7} + 1573749217 T^{8} + 13055799498 T^{9} + 114883692841 T^{10} + 814931686918 T^{11} + 6162486374975 T^{12} + 36486145231523 T^{13} + 232675336383541 T^{14} + 1072354327483854 T^{15} + 5501604462510306 T^{16} + 16129201837881620 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 - 8 T + 471 T^{2} - 3237 T^{3} + 106842 T^{4} - 664023 T^{5} + 15704132 T^{6} - 88577445 T^{7} + 1658985638 T^{8} - 8277689232 T^{9} + 131059865402 T^{10} - 552811834245 T^{11} + 7742749537148 T^{12} - 25863749635863 T^{13} + 328758859781958 T^{14} - 786874093521477 T^{15} + 9045041132480889 T^{16} - 12136870479252488 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 - T + 488 T^{2} - 1161 T^{3} + 113615 T^{4} - 399973 T^{5} + 16903319 T^{6} - 70850779 T^{7} + 1823276283 T^{8} - 7479201516 T^{9} + 151331931489 T^{10} - 488091016531 T^{11} + 9665098061053 T^{12} - 18982047025333 T^{13} + 447534102654445 T^{14} - 379577773481409 T^{15} + 13242392882937976 T^{16} - 2252292232139041 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 - 36 T + 1226 T^{2} - 26142 T^{3} + 519315 T^{4} - 7977763 T^{5} + 115284093 T^{6} - 1381129790 T^{7} + 15698593261 T^{8} - 151798632666 T^{9} + 1397174800229 T^{10} - 10939929066590 T^{11} + 81271711758117 T^{12} - 500542728786883 T^{13} + 2899885832757435 T^{14} - 12992084908302462 T^{15} + 54227616581918554 T^{16} - 141717197005274916 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 - 31 T + 713 T^{2} - 13605 T^{3} + 225439 T^{4} - 3322243 T^{5} + 44076123 T^{6} - 534128975 T^{7} + 5960297404 T^{8} - 60929995076 T^{9} + 578148848188 T^{10} - 5025619525775 T^{11} + 40227087406779 T^{12} - 294115784097283 T^{13} + 1935921400197823 T^{14} - 11332584127059045 T^{15} + 57609176832894569 T^{16} - 242960441425685791 T^{17} + 760231058654565217 T^{18} \)
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