Properties

Label 1336.2.a.c
Level $1336$
Weight $2$
Character orbit 1336.a
Self dual yes
Analytic conductor $10.668$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6680137100\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 13 x^{7} + 8 x^{6} + 56 x^{5} - 15 x^{4} - 81 x^{3} + 2 x^{2} + 13 x + 1\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{2} - \beta_{8} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( \beta_{2} + \beta_{7} + \beta_{8} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{2} - \beta_{8} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( \beta_{2} + \beta_{7} + \beta_{8} ) q^{9} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{11} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{15} + ( -1 + \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{17} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{19} + ( -2 + \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{21} + ( -2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{23} + ( 3 \beta_{2} + \beta_{8} ) q^{25} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{27} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{29} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{31} + ( -2 + \beta_{1} + \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{33} + ( -2 - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{35} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{37} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{39} + ( -1 + \beta_{2} - 2 \beta_{5} - 4 \beta_{6} - \beta_{8} ) q^{41} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{43} + ( -2 - \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} ) q^{45} + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{8} ) q^{47} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{49} + ( -1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{51} + ( -3 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{53} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{55} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{57} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - 4 \beta_{8} ) q^{59} + ( -4 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} ) q^{61} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{63} + ( -3 + \beta_{2} + \beta_{3} - 3 \beta_{5} - 3 \beta_{6} ) q^{65} + ( 4 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 3 \beta_{8} ) q^{67} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} ) q^{69} + ( -3 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{5} - 3 \beta_{7} - 4 \beta_{8} ) q^{71} + ( -1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 7 \beta_{6} + 5 \beta_{7} + 5 \beta_{8} ) q^{73} + ( -5 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{75} + ( -3 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{77} + ( -1 + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{79} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{7} - 4 \beta_{8} ) q^{81} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{83} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{8} ) q^{85} + ( 2 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 6 \beta_{8} ) q^{87} + ( -2 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{89} + ( 3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} ) q^{91} + ( -5 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{7} - 3 \beta_{8} ) q^{93} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 5 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{95} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{6} + 2 \beta_{8} ) q^{97} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - q^{3} - 8q^{5} + 2q^{7} + O(q^{10}) \) \( 9q - q^{3} - 8q^{5} + 2q^{7} - 10q^{11} - 13q^{13} + 2q^{15} - 8q^{17} - q^{19} - 19q^{21} - 3q^{23} + 3q^{25} - 10q^{27} - 25q^{29} - q^{31} - 12q^{33} - 17q^{35} - 35q^{37} - 4q^{39} - 16q^{41} + 9q^{43} - 24q^{45} - q^{47} - q^{49} - 10q^{51} - 29q^{53} + 9q^{55} - 17q^{57} - 14q^{59} - 28q^{61} + 4q^{63} - 31q^{65} + 19q^{67} - 19q^{69} - 9q^{71} - 7q^{75} - 33q^{77} - 18q^{79} - 27q^{81} - 13q^{83} - 36q^{85} + 18q^{87} - 21q^{89} + 20q^{91} - 35q^{93} - 12q^{95} + 2q^{97} - 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 13 x^{7} + 8 x^{6} + 56 x^{5} - 15 x^{4} - 81 x^{3} + 2 x^{2} + 13 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{8} - 2 \nu^{7} - 10 \nu^{6} + 16 \nu^{5} + 32 \nu^{4} - 31 \nu^{3} - 34 \nu^{2} + 3 \nu + 4 \)\()/2\)
\(\beta_{3}\)\(=\)\( -\nu^{8} + 2 \nu^{7} + 10 \nu^{6} - 16 \nu^{5} - 31 \nu^{4} + 31 \nu^{3} + 29 \nu^{2} - 5 \nu - 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{8} + 5 \nu^{7} + 18 \nu^{6} - 42 \nu^{5} - 46 \nu^{4} + 92 \nu^{3} + 27 \nu^{2} - 34 \nu - 1 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{8} - 9 \nu^{7} - 22 \nu^{6} + 72 \nu^{5} + 30 \nu^{4} - 137 \nu^{3} + 31 \nu^{2} + 5 \nu - 1 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{8} + 9 \nu^{7} + 22 \nu^{6} - 72 \nu^{5} - 30 \nu^{4} + 139 \nu^{3} - 33 \nu^{2} - 13 \nu + 5 \)\()/2\)
\(\beta_{7}\)\(=\)\( 2 \nu^{8} - 5 \nu^{7} - 18 \nu^{6} + 41 \nu^{5} + 47 \nu^{4} - 84 \nu^{3} - 31 \nu^{2} + 19 \nu + 2 \)
\(\beta_{8}\)\(=\)\((\)\( -5 \nu^{8} + 12 \nu^{7} + 46 \nu^{6} - 98 \nu^{5} - 126 \nu^{4} + 199 \nu^{3} + 98 \nu^{2} - 41 \nu - 14 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(5 \beta_{8} + 5 \beta_{7} + \beta_{3} + 7 \beta_{2} + 2 \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(9 \beta_{8} + 8 \beta_{7} + 8 \beta_{6} + 8 \beta_{5} - 2 \beta_{4} + \beta_{3} + 11 \beta_{2} + 19 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(29 \beta_{8} + 26 \beta_{7} + \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 9 \beta_{3} + 50 \beta_{2} + 21 \beta_{1} + 75\)
\(\nu^{7}\)\(=\)\(69 \beta_{8} + 53 \beta_{7} + 52 \beta_{6} + 54 \beta_{5} - 24 \beta_{4} + 10 \beta_{3} + 99 \beta_{2} + 104 \beta_{1} + 94\)
\(\nu^{8}\)\(=\)\(189 \beta_{8} + 143 \beta_{7} + 17 \beta_{6} + 31 \beta_{5} - 46 \beta_{4} + 62 \beta_{3} + 365 \beta_{2} + 171 \beta_{1} + 443\)

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
2.70492
2.37109
1.63574
0.460631
−0.0812095
−0.374925
−1.74816
−1.76335
−2.20474
0 −2.70492 0 −4.24392 0 1.27743 0 4.31658 0
1.2 0 −2.37109 0 0.438298 0 3.46684 0 2.62208 0
1.3 0 −1.63574 0 −0.530251 0 −0.325283 0 −0.324343 0
1.4 0 −0.460631 0 0.528623 0 −1.88578 0 −2.78782 0
1.5 0 0.0812095 0 2.29286 0 0.862825 0 −2.99341 0
1.6 0 0.374925 0 −2.61380 0 4.69526 0 −2.85943 0
1.7 0 1.74816 0 1.48670 0 −4.05717 0 0.0560672 0
1.8 0 1.76335 0 −2.14065 0 −2.31472 0 0.109415 0
1.9 0 2.20474 0 −3.21786 0 0.280589 0 1.86086 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

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

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 1336.2.a.c 9
4.b odd 2 1 2672.2.a.m 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1336.2.a.c 9 1.a even 1 1 trivial
2672.2.a.m 9 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{9} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + 14 T^{2} + 16 T^{3} + 107 T^{4} + 123 T^{5} + 570 T^{6} + 610 T^{7} + 2245 T^{8} + 2147 T^{9} + 6735 T^{10} + 5490 T^{11} + 15390 T^{12} + 9963 T^{13} + 26001 T^{14} + 11664 T^{15} + 30618 T^{16} + 6561 T^{17} + 19683 T^{18} \)
$5$ \( 1 + 8 T + 53 T^{2} + 254 T^{3} + 1063 T^{4} + 3774 T^{5} + 12035 T^{6} + 34178 T^{7} + 88336 T^{8} + 206612 T^{9} + 441680 T^{10} + 854450 T^{11} + 1504375 T^{12} + 2358750 T^{13} + 3321875 T^{14} + 3968750 T^{15} + 4140625 T^{16} + 3125000 T^{17} + 1953125 T^{18} \)
$7$ \( 1 - 2 T + 34 T^{2} - 72 T^{3} + 567 T^{4} - 1231 T^{5} + 6391 T^{6} - 13376 T^{7} + 55371 T^{8} - 106443 T^{9} + 387597 T^{10} - 655424 T^{11} + 2192113 T^{12} - 2955631 T^{13} + 9529569 T^{14} - 8470728 T^{15} + 28000462 T^{16} - 11529602 T^{17} + 40353607 T^{18} \)
$11$ \( 1 + 10 T + 93 T^{2} + 590 T^{3} + 3378 T^{4} + 16365 T^{5} + 72894 T^{6} + 292609 T^{7} + 1094660 T^{8} + 3753996 T^{9} + 12041260 T^{10} + 35405689 T^{11} + 97021914 T^{12} + 239599965 T^{13} + 544030278 T^{14} + 1045220990 T^{15} + 1812306903 T^{16} + 2143588810 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 + 13 T + 133 T^{2} + 953 T^{3} + 6225 T^{4} + 33638 T^{5} + 169261 T^{6} + 741887 T^{7} + 3086116 T^{8} + 11414650 T^{9} + 40119508 T^{10} + 125378903 T^{11} + 371866417 T^{12} + 960734918 T^{13} + 2311298925 T^{14} + 4599948977 T^{15} + 8345552761 T^{16} + 10604499373 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 + 8 T + 92 T^{2} + 500 T^{3} + 3438 T^{4} + 14774 T^{5} + 76420 T^{6} + 279916 T^{7} + 1273329 T^{8} + 4593380 T^{9} + 21646593 T^{10} + 80895724 T^{11} + 375451460 T^{12} + 1233939254 T^{13} + 4881468366 T^{14} + 12068784500 T^{15} + 37751157916 T^{16} + 55806059528 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 + T + 74 T^{2} + 124 T^{3} + 3178 T^{4} + 6606 T^{5} + 98399 T^{6} + 201683 T^{7} + 2392182 T^{8} + 4452264 T^{9} + 45451458 T^{10} + 72807563 T^{11} + 674918741 T^{12} + 860900526 T^{13} + 7869042622 T^{14} + 5833689244 T^{15} + 66146508686 T^{16} + 16983563041 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 3 T + 59 T^{2} - 39 T^{3} + 1183 T^{4} - 8810 T^{5} + 24729 T^{6} - 233961 T^{7} + 1236788 T^{8} - 3999570 T^{9} + 28446124 T^{10} - 123765369 T^{11} + 300877743 T^{12} - 2465399210 T^{13} + 7614193769 T^{14} - 5773399671 T^{15} + 200884701373 T^{16} + 234932955843 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 + 25 T + 395 T^{2} + 4479 T^{3} + 40930 T^{4} + 309446 T^{5} + 2033900 T^{6} + 11929004 T^{7} + 65912049 T^{8} + 354500133 T^{9} + 1911449421 T^{10} + 10032292364 T^{11} + 49604787100 T^{12} + 218865276326 T^{13} + 839521328570 T^{14} + 2664213654759 T^{15} + 6813701142055 T^{16} + 12506160324025 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 + T + 156 T^{2} + 46 T^{3} + 12646 T^{4} - 64 T^{5} + 695121 T^{6} - 73599 T^{7} + 28337228 T^{8} - 2988116 T^{9} + 878454068 T^{10} - 70728639 T^{11} + 20708349711 T^{12} - 59105344 T^{13} + 362044243546 T^{14} + 40825169326 T^{15} + 4291967801316 T^{16} + 852891037441 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 + 35 T + 763 T^{2} + 11917 T^{3} + 150667 T^{4} + 1593626 T^{5} + 14655455 T^{6} + 118318739 T^{7} + 851803426 T^{8} + 5469266198 T^{9} + 31516726762 T^{10} + 161978353691 T^{11} + 742342762115 T^{12} + 2986711697786 T^{13} + 10447845969319 T^{14} + 30575761616053 T^{15} + 72433022252479 T^{16} + 122936780887235 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 16 T + 303 T^{2} + 2844 T^{3} + 31747 T^{4} + 213788 T^{5} + 1872807 T^{6} + 10023732 T^{7} + 81015990 T^{8} + 398200168 T^{9} + 3321655590 T^{10} + 16849893492 T^{11} + 129075731247 T^{12} + 604113792668 T^{13} + 3678086813147 T^{14} + 13509296461404 T^{15} + 59010544985943 T^{16} + 127758803665936 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 - 9 T + 277 T^{2} - 1841 T^{3} + 33599 T^{4} - 176736 T^{5} + 2548969 T^{6} - 11255519 T^{7} + 141367302 T^{8} - 545391838 T^{9} + 6078793986 T^{10} - 20811454631 T^{11} + 202660878283 T^{12} - 604225213536 T^{13} + 4939336676357 T^{14} - 11637629373209 T^{15} + 75293755276639 T^{16} - 105193802498409 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + T + 272 T^{2} - 136 T^{3} + 32074 T^{4} - 76640 T^{5} + 2223103 T^{6} - 10058089 T^{7} + 113020750 T^{8} - 644899588 T^{9} + 5311975250 T^{10} - 22218318601 T^{11} + 230809222769 T^{12} - 373978751840 T^{13} + 7356011754518 T^{14} - 1465973284744 T^{15} + 137801488765936 T^{16} + 23811286661761 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 + 29 T + 726 T^{2} + 12562 T^{3} + 188263 T^{4} + 2341888 T^{5} + 25766913 T^{6} + 247956318 T^{7} + 2138364493 T^{8} + 16409384230 T^{9} + 113333318129 T^{10} + 696509297262 T^{11} + 3836100706701 T^{12} + 18478622768128 T^{13} + 78730738098659 T^{14} + 278428704502498 T^{15} + 852840287521662 T^{16} + 1805531021929469 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 + 14 T + 236 T^{2} + 2756 T^{3} + 33441 T^{4} + 327888 T^{5} + 3255193 T^{6} + 28063996 T^{7} + 240319503 T^{8} + 1866179140 T^{9} + 14178850677 T^{10} + 97690770076 T^{11} + 668548283147 T^{12} + 3973137263568 T^{13} + 23907783482859 T^{14} + 116249550714596 T^{15} + 587321750417284 T^{16} + 2055626126460494 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 + 28 T + 488 T^{2} + 6082 T^{3} + 61351 T^{4} + 542901 T^{5} + 4384253 T^{6} + 33042120 T^{7} + 240205165 T^{8} + 1799569925 T^{9} + 14652515065 T^{10} + 122949728520 T^{11} + 995142130193 T^{12} + 7516920924741 T^{13} + 51816827662651 T^{14} + 313346916863602 T^{15} + 1533658503978248 T^{16} + 5367804763923868 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 - 19 T + 373 T^{2} - 4527 T^{3} + 55745 T^{4} - 463082 T^{5} + 4103863 T^{6} - 24884529 T^{7} + 197147226 T^{8} - 1121891190 T^{9} + 13208864142 T^{10} - 111706650681 T^{11} + 1234290147469 T^{12} - 9331621414922 T^{13} + 75262724089715 T^{14} - 409505096079063 T^{15} + 2260645428785479 T^{16} - 7715285873576179 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 9 T + 336 T^{2} + 3512 T^{3} + 59605 T^{4} + 653252 T^{5} + 7286033 T^{6} + 77748744 T^{7} + 665144243 T^{8} + 6510156198 T^{9} + 47225241253 T^{10} + 391931418504 T^{11} + 2607751357063 T^{12} + 16600231436612 T^{13} + 107541090466355 T^{14} + 449888197130552 T^{15} + 3055960373219376 T^{16} + 5811781781211849 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 160 T^{2} - 1178 T^{3} + 13812 T^{4} - 174332 T^{5} + 1312718 T^{6} - 13146518 T^{7} + 138980941 T^{8} - 746538888 T^{9} + 10145608693 T^{10} - 70057794422 T^{11} + 510669618206 T^{12} - 4950722150012 T^{13} + 28633264842516 T^{14} - 178271718568442 T^{15} + 1767583763055520 T^{16} + 58871586708267913 T^{18} \)
$79$ \( 1 + 18 T + 613 T^{2} + 8232 T^{3} + 162407 T^{4} + 1767190 T^{5} + 26081621 T^{6} + 239485944 T^{7} + 2879648274 T^{8} + 22496197200 T^{9} + 227492213646 T^{10} + 1494631776504 T^{11} + 12859256336219 T^{12} + 68832193642390 T^{13} + 499735498592393 T^{14} + 2001095933848872 T^{15} + 11771996208515467 T^{16} + 27307958578318098 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 + 13 T + 673 T^{2} + 7417 T^{3} + 205967 T^{4} + 1935852 T^{5} + 37738433 T^{6} + 301498679 T^{7} + 4568581146 T^{8} + 30624712542 T^{9} + 379192235118 T^{10} + 2077024399631 T^{11} + 21578345389771 T^{12} + 91872285624492 T^{13} + 811312384116781 T^{14} + 2424916749277873 T^{15} + 18262562316018971 T^{16} + 29279799017807533 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 21 T + 709 T^{2} + 11529 T^{3} + 229075 T^{4} + 3006419 T^{5} + 44380307 T^{6} + 482662319 T^{7} + 5716348292 T^{8} + 51903552752 T^{9} + 508754997988 T^{10} + 3823168228799 T^{11} + 31286740645483 T^{12} + 188629465444979 T^{13} + 1279168418279675 T^{14} + 5729697303489369 T^{15} + 31360016440930061 T^{16} + 82668364919743701 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 - 2 T + 615 T^{2} - 1236 T^{3} + 171368 T^{4} - 352849 T^{5} + 29435302 T^{6} - 61397603 T^{7} + 3635999768 T^{8} - 7173227220 T^{9} + 352691977496 T^{10} - 577690046627 T^{11} + 26864805382246 T^{12} - 31237468271569 T^{13} + 1471595325161576 T^{14} - 1029553398092244 T^{15} + 49690944954039495 T^{16} - 15674867188753922 T^{17} + 760231058654565217 T^{18} \)
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