Properties

Label 8004.2.a.e
Level 8004
Weight 2
Character orbit 8004.a
Self dual yes
Analytic conductor 63.912
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 93 \nu^{8} - 7 \nu^{7} - 1649 \nu^{6} - 1742 \nu^{5} + 8484 \nu^{4} + 13848 \nu^{3} - 21492 \nu^{2} - 13414 \nu + 16720 \)\()/4877\)
\(\beta_{2}\)\(=\)\((\)\( -676 \nu^{8} - 54 \nu^{7} + 14451 \nu^{6} + 5373 \nu^{5} - 80862 \nu^{4} - 26245 \nu^{3} + 118464 \nu^{2} + 29593 \nu - 29868 \)\()/4877\)
\(\beta_{3}\)\(=\)\((\)\( -1248 \nu^{8} + 4027 \nu^{7} + 15049 \nu^{6} - 42227 \nu^{5} - 37863 \nu^{4} + 93731 \nu^{3} + 26624 \nu^{2} - 41031 \nu - 4120 \)\()/4877\)
\(\beta_{4}\)\(=\)\((\)\( 1613 \nu^{8} - 4474 \nu^{7} - 21416 \nu^{6} + 45249 \nu^{5} + 67542 \nu^{4} - 97748 \nu^{3} - 66924 \nu^{2} + 53254 \nu + 16986 \)\()/4877\)
\(\beta_{5}\)\(=\)\((\)\( 1706 \nu^{8} - 4481 \nu^{7} - 23065 \nu^{6} + 43507 \nu^{5} + 76026 \nu^{4} - 83900 \nu^{3} - 83539 \nu^{2} + 34963 \nu + 19075 \)\()/4877\)
\(\beta_{6}\)\(=\)\((\)\( -2082 \nu^{8} + 5663 \nu^{7} + 27005 \nu^{6} - 53822 \nu^{5} - 79649 \nu^{4} + 92100 \nu^{3} + 68801 \nu^{2} - 18907 \nu - 12313 \)\()/4877\)
\(\beta_{7}\)\(=\)\((\)\( 2159 \nu^{8} - 5931 \nu^{7} - 29524 \nu^{6} + 60980 \nu^{5} + 102091 \nu^{4} - 136012 \nu^{3} - 122465 \nu^{2} + 64804 \nu + 35858 \)\()/4877\)
\(\beta_{8}\)\(=\)\((\)\( 2535 \nu^{8} - 7113 \nu^{7} - 33464 \nu^{6} + 71295 \nu^{5} + 105714 \nu^{4} - 144212 \nu^{3} - 107727 \nu^{2} + 58502 \nu + 24219 \)\()/4877\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{1} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\(4 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} + 6 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - \beta_{2} + 7\)
\(\nu^{4}\)\(=\)\((\)\(13 \beta_{8} - 19 \beta_{7} + 23 \beta_{6} + 53 \beta_{5} - 32 \beta_{4} - 14 \beta_{3} - 2 \beta_{2} - 18 \beta_{1} + 75\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(83 \beta_{8} - 129 \beta_{7} + 135 \beta_{6} + 193 \beta_{5} - 100 \beta_{4} - 130 \beta_{3} - 30 \beta_{2} - 14 \beta_{1} + 225\)\()/2\)
\(\nu^{6}\)\(=\)\(96 \beta_{8} - 174 \beta_{7} + 201 \beta_{6} + 413 \beta_{5} - 231 \beta_{4} - 167 \beta_{3} - 29 \beta_{2} - 99 \beta_{1} + 520\)
\(\nu^{7}\)\(=\)\((\)\(1033 \beta_{8} - 1823 \beta_{7} + 1943 \beta_{6} + 3085 \beta_{5} - 1588 \beta_{4} - 1928 \beta_{3} - 424 \beta_{2} - 342 \beta_{1} + 3591\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(3035 \beta_{8} - 5877 \beta_{7} + 6591 \beta_{6} + 12709 \beta_{5} - 6834 \beta_{4} - 6034 \beta_{3} - 1142 \beta_{2} - 2514 \beta_{1} + 15401\)\()/2\)

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
−2.86352
−1.47135
3.93556
2.25739
0.761439
−0.975163
1.39711
0.511914
−0.553378
0 −1.00000 0 −3.70748 0 −1.59469 0 1.00000 0
1.2 0 −1.00000 0 −3.48372 0 2.18462 0 1.00000 0
1.3 0 −1.00000 0 −2.36120 0 2.82157 0 1.00000 0
1.4 0 −1.00000 0 −0.650765 0 −1.33403 0 1.00000 0
1.5 0 −1.00000 0 0.461945 0 0.732054 0 1.00000 0
1.6 0 −1.00000 0 0.900888 0 3.90501 0 1.00000 0
1.7 0 −1.00000 0 1.20115 0 −1.61263 0 1.00000 0
1.8 0 −1.00000 0 1.34732 0 2.91839 0 1.00000 0
1.9 0 −1.00000 0 3.29187 0 −1.02028 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 8004.2.a.e 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.e 9 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{9} \)
$5$ \( 1 + 3 T + 26 T^{2} + 76 T^{3} + 352 T^{4} + 897 T^{5} + 3136 T^{6} + 6906 T^{7} + 20281 T^{8} + 39416 T^{9} + 101405 T^{10} + 172650 T^{11} + 392000 T^{12} + 560625 T^{13} + 1100000 T^{14} + 1187500 T^{15} + 2031250 T^{16} + 1171875 T^{17} + 1953125 T^{18} \)
$7$ \( 1 - 7 T + 65 T^{2} - 327 T^{3} + 1809 T^{4} - 7119 T^{5} + 29153 T^{6} - 93123 T^{7} + 303428 T^{8} - 796948 T^{9} + 2123996 T^{10} - 4563027 T^{11} + 9999479 T^{12} - 17092719 T^{13} + 30403863 T^{14} - 38471223 T^{15} + 53530295 T^{16} - 40353607 T^{17} + 40353607 T^{18} \)
$11$ \( 1 + 2 T + 49 T^{2} + 95 T^{3} + 1244 T^{4} + 2309 T^{5} + 22092 T^{6} + 37857 T^{7} + 303948 T^{8} + 467906 T^{9} + 3343428 T^{10} + 4580697 T^{11} + 29404452 T^{12} + 33806069 T^{13} + 200347444 T^{14} + 168298295 T^{15} + 954871379 T^{16} + 428717762 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 + 7 T + 87 T^{2} + 469 T^{3} + 3480 T^{4} + 15664 T^{5} + 87354 T^{6} + 337444 T^{7} + 1543139 T^{8} + 5151761 T^{9} + 20060807 T^{10} + 57028036 T^{11} + 191916738 T^{12} + 447379504 T^{13} + 1292099640 T^{14} + 2263773421 T^{15} + 5459120979 T^{16} + 5710115047 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 + 98 T^{2} - 115 T^{3} + 4525 T^{4} - 8601 T^{5} + 140632 T^{6} - 281898 T^{7} + 3253471 T^{8} - 5743037 T^{9} + 55309007 T^{10} - 81468522 T^{11} + 690925016 T^{12} - 718364121 T^{13} + 6424852925 T^{14} - 2775820435 T^{15} + 40213189954 T^{16} + 118587876497 T^{18} \)
$19$ \( 1 + T + 97 T^{2} + 114 T^{3} + 4774 T^{4} + 6136 T^{5} + 160280 T^{6} + 198588 T^{7} + 4006913 T^{8} + 4421409 T^{9} + 76131347 T^{10} + 71690268 T^{11} + 1099360520 T^{12} + 799649656 T^{13} + 11820896626 T^{14} + 5363230434 T^{15} + 86705558683 T^{16} + 16983563041 T^{17} + 322687697779 T^{18} \)
$23$ \( ( 1 + T )^{9} \)
$29$ \( ( 1 - T )^{9} \)
$31$ \( 1 - 8 T + 153 T^{2} - 1097 T^{3} + 12326 T^{4} - 78609 T^{5} + 662686 T^{6} - 3844949 T^{7} + 26675000 T^{8} - 137552206 T^{9} + 826925000 T^{10} - 3694995989 T^{11} + 19742078626 T^{12} - 72597062289 T^{13} + 352882915226 T^{14} - 973591538057 T^{15} + 4209429958983 T^{16} - 6823128299528 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 8 T + 253 T^{2} + 1765 T^{3} + 30221 T^{4} + 186187 T^{5} + 2251317 T^{6} + 12189502 T^{7} + 115890110 T^{8} + 541298595 T^{9} + 4287934070 T^{10} + 16687428238 T^{11} + 114035960001 T^{12} + 348944414107 T^{13} + 2095643724497 T^{14} + 4528507111885 T^{15} + 24017764914649 T^{16} + 28099835631368 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 19 T + 400 T^{2} + 4888 T^{3} + 60100 T^{4} + 550873 T^{5} + 5032972 T^{6} + 37539398 T^{7} + 281599251 T^{8} + 1792540472 T^{9} + 11545569291 T^{10} + 63103728038 T^{11} + 346877463212 T^{12} + 1556635439353 T^{13} + 6962957680100 T^{14} + 23218509530008 T^{15} + 77901709552400 T^{16} + 151713579353299 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 3 T + 120 T^{2} + 800 T^{3} + 10631 T^{4} + 70060 T^{5} + 768666 T^{6} + 4485320 T^{7} + 39922183 T^{8} + 230952547 T^{9} + 1716653869 T^{10} + 8293356680 T^{11} + 61114327662 T^{12} + 239521198060 T^{13} + 1562846757533 T^{14} + 5057090439200 T^{15} + 32618233332840 T^{16} + 35064600832803 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 3 T + 278 T^{2} + 977 T^{3} + 36995 T^{4} + 144455 T^{5} + 3167707 T^{6} + 12728693 T^{7} + 196210413 T^{8} + 731862316 T^{9} + 9221889411 T^{10} + 28117682837 T^{11} + 328880843861 T^{12} + 704894318855 T^{13} + 8484618533965 T^{14} + 10531293376433 T^{15} + 140841227488714 T^{16} + 71433859985283 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 + 17 T + 398 T^{2} + 4052 T^{3} + 52736 T^{4} + 348193 T^{5} + 3384098 T^{6} + 14443272 T^{7} + 145879247 T^{8} + 516370164 T^{9} + 7731600091 T^{10} + 40571151048 T^{11} + 503814357946 T^{12} + 2747410250833 T^{13} + 22053957518848 T^{14} + 89809991294708 T^{15} + 467535033655126 T^{16} + 1058414736993137 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 + 10 T + 427 T^{2} + 3704 T^{3} + 86119 T^{4} + 648418 T^{5} + 10694205 T^{6} + 69416192 T^{7} + 899209120 T^{8} + 4957696952 T^{9} + 53053338080 T^{10} + 241637764352 T^{11} + 2196365128695 T^{12} + 7857114984898 T^{13} + 61568565705581 T^{14} + 156236696606264 T^{15} + 1062654184017713 T^{16} + 1468304376043210 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 - T + 415 T^{2} - 281 T^{3} + 79991 T^{4} - 31893 T^{5} + 9569195 T^{6} - 1981755 T^{7} + 797704854 T^{8} - 104297468 T^{9} + 48659996094 T^{10} - 7374110355 T^{11} + 2172025450295 T^{12} - 441585407013 T^{13} + 67560102713291 T^{14} - 14477225195441 T^{15} + 1304238276948715 T^{16} - 191707312997281 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 - 12 T + 274 T^{2} - 3235 T^{3} + 47142 T^{4} - 446605 T^{5} + 5204776 T^{6} - 44226721 T^{7} + 436542763 T^{8} - 3279783782 T^{9} + 29248365121 T^{10} - 198533750569 T^{11} + 1565404044088 T^{12} - 8999591394205 T^{13} + 63647597794194 T^{14} - 292632866316715 T^{15} + 1660634979858502 T^{16} - 4872812130679692 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 7 T + 533 T^{2} + 3329 T^{3} + 132072 T^{4} + 728958 T^{5} + 19985738 T^{6} + 95991266 T^{7} + 2034708327 T^{8} + 8308609245 T^{9} + 144464291217 T^{10} + 483891971906 T^{11} + 7153115473318 T^{12} + 18524048158398 T^{13} + 238288178845272 T^{14} + 426445845173009 T^{15} + 4847699044422403 T^{16} + 4520274718720327 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 - 13 T + 506 T^{2} - 4747 T^{3} + 107167 T^{4} - 734909 T^{5} + 13144645 T^{6} - 67606851 T^{7} + 1145412177 T^{8} - 4992931436 T^{9} + 83615088921 T^{10} - 360276908979 T^{11} + 5113490363965 T^{12} - 20870122895069 T^{13} + 222164863407031 T^{14} - 718383572193883 T^{15} + 5589983650663082 T^{16} - 10483981194623053 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 10 T + 425 T^{2} + 2715 T^{3} + 76957 T^{4} + 321871 T^{5} + 9247137 T^{6} + 29782950 T^{7} + 906879556 T^{8} + 2586333853 T^{9} + 71643484924 T^{10} + 185875390950 T^{11} + 4559199179343 T^{12} + 12536901521551 T^{13} + 236801029297843 T^{14} + 659982441739515 T^{15} + 8161661319117575 T^{16} + 15171088099065610 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 - 9 T + 422 T^{2} - 4342 T^{3} + 86772 T^{4} - 940635 T^{5} + 11854664 T^{6} - 126639668 T^{7} + 1224295309 T^{8} - 12173194872 T^{9} + 101616510647 T^{10} - 872420672852 T^{11} + 6778342764568 T^{12} - 44640957773835 T^{13} + 341798434674396 T^{14} - 1419575101168198 T^{15} + 11451413517622594 T^{16} - 20270630089251369 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 5 T + 291 T^{2} + 2317 T^{3} + 51986 T^{4} + 466616 T^{5} + 7203344 T^{6} + 61936092 T^{7} + 797299301 T^{8} + 6209910283 T^{9} + 70959637789 T^{10} + 490595784732 T^{11} + 5078134216336 T^{12} + 29276533526456 T^{13} + 290292914515714 T^{14} + 1151505651156637 T^{15} + 12871318454598939 T^{16} + 19682944028510405 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 7 T + 269 T^{2} + 2553 T^{3} + 38361 T^{4} + 294733 T^{5} + 3831941 T^{6} + 15532083 T^{7} + 260082212 T^{8} + 872400240 T^{9} + 25227974564 T^{10} + 146141368947 T^{11} + 3497309088293 T^{12} + 26092500576973 T^{13} + 329418959598777 T^{14} + 2126577528583737 T^{15} + 21734738524612397 T^{16} + 54862035160638727 T^{17} + 760231058654565217 T^{18} \)
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