Properties

Label 8034.2.a.p
Level 8034
Weight 2
Character orbit 8034.a
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} - 12 x^{6} + 12 x^{5} + 43 x^{4} - 38 x^{3} - 49 x^{2} + 23 x + 20\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + ( -1 + \beta_{5} + \beta_{7} ) q^{5} + q^{6} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + ( -1 + \beta_{5} + \beta_{7} ) q^{5} + q^{6} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{7} + q^{8} + q^{9} + ( -1 + \beta_{5} + \beta_{7} ) q^{10} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{11} + q^{12} + q^{13} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{14} + ( -1 + \beta_{5} + \beta_{7} ) q^{15} + q^{16} + ( -3 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{17} + q^{18} + ( -2 + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{19} + ( -1 + \beta_{5} + \beta_{7} ) q^{20} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{22} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{23} + q^{24} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{25} + q^{26} + q^{27} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{28} + ( -4 - \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} + ( -1 + \beta_{5} + \beta_{7} ) q^{30} + ( -2 + 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{31} + q^{32} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{33} + ( -3 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{34} + ( -2 - 2 \beta_{4} - \beta_{6} ) q^{35} + q^{36} + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{37} + ( -2 + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{38} + q^{39} + ( -1 + \beta_{5} + \beta_{7} ) q^{40} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{41} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{42} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{43} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{44} + ( -1 + \beta_{5} + \beta_{7} ) q^{45} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{46} + ( -1 + \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{47} + q^{48} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{49} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{50} + ( -3 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{51} + q^{52} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{53} + q^{54} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{55} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{56} + ( -2 + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{57} + ( -4 - \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{58} + ( 1 + \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{59} + ( -1 + \beta_{5} + \beta_{7} ) q^{60} + ( 1 - \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{61} + ( -2 + 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{62} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{63} + q^{64} + ( -1 + \beta_{5} + \beta_{7} ) q^{65} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{66} + ( -1 - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{67} + ( -3 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{68} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{69} + ( -2 - 2 \beta_{4} - \beta_{6} ) q^{70} + ( -3 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{71} + q^{72} + ( -2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} - 6 \beta_{7} ) q^{73} + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{74} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{75} + ( -2 + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{76} + ( -2 + 2 \beta_{1} + \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{77} + q^{78} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{79} + ( -1 + \beta_{5} + \beta_{7} ) q^{80} + q^{81} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{82} + ( -4 \beta_{1} - \beta_{2} - 6 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{83} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{84} + ( 1 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 5 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{85} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{86} + ( -4 - \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{87} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{88} + ( -2 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{6} - \beta_{7} ) q^{89} + ( -1 + \beta_{5} + \beta_{7} ) q^{90} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{91} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{92} + ( -2 + 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{93} + ( -1 + \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{94} + ( 1 + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - \beta_{6} - 5 \beta_{7} ) q^{95} + q^{96} + ( 2 + 6 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{97} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{98} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{3} + 8q^{4} - 8q^{5} + 8q^{6} - 6q^{7} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{3} + 8q^{4} - 8q^{5} + 8q^{6} - 6q^{7} + 8q^{8} + 8q^{9} - 8q^{10} - 7q^{11} + 8q^{12} + 8q^{13} - 6q^{14} - 8q^{15} + 8q^{16} - 20q^{17} + 8q^{18} - 12q^{19} - 8q^{20} - 6q^{21} - 7q^{22} - 14q^{23} + 8q^{24} - 2q^{25} + 8q^{26} + 8q^{27} - 6q^{28} - 25q^{29} - 8q^{30} - 12q^{31} + 8q^{32} - 7q^{33} - 20q^{34} - 18q^{35} + 8q^{36} - 15q^{37} - 12q^{38} + 8q^{39} - 8q^{40} - 18q^{41} - 6q^{42} - 8q^{43} - 7q^{44} - 8q^{45} - 14q^{46} - 12q^{47} + 8q^{48} - 8q^{49} - 2q^{50} - 20q^{51} + 8q^{52} - 25q^{53} + 8q^{54} - 8q^{55} - 6q^{56} - 12q^{57} - 25q^{58} - 9q^{59} - 8q^{60} - 2q^{61} - 12q^{62} - 6q^{63} + 8q^{64} - 8q^{65} - 7q^{66} - 8q^{67} - 20q^{68} - 14q^{69} - 18q^{70} - 13q^{71} + 8q^{72} - 2q^{73} - 15q^{74} - 2q^{75} - 12q^{76} - 5q^{77} + 8q^{78} + q^{79} - 8q^{80} + 8q^{81} - 18q^{82} - 6q^{83} - 6q^{84} + 5q^{85} - 8q^{86} - 25q^{87} - 7q^{88} - 17q^{89} - 8q^{90} - 6q^{91} - 14q^{92} - 12q^{93} - 12q^{94} + 10q^{95} + 8q^{96} + 19q^{97} - 8q^{98} - 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 12 x^{6} + 12 x^{5} + 43 x^{4} - 38 x^{3} - 49 x^{2} + 23 x + 20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{6} - 11 \nu^{4} + \nu^{3} + 32 \nu^{2} - 5 \nu - 17 \)
\(\beta_{3}\)\(=\)\( 3 \nu^{7} + \nu^{6} - 32 \nu^{5} - 6 \nu^{4} + 92 \nu^{3} + 7 \nu^{2} - 53 \nu - 11 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{7} + 2 \nu^{6} - 21 \nu^{5} - 18 \nu^{4} + 60 \nu^{3} + 43 \nu^{2} - 37 \nu - 25 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{7} + 2 \nu^{6} - 21 \nu^{5} - 18 \nu^{4} + 60 \nu^{3} + 44 \nu^{2} - 37 \nu - 28 \)
\(\beta_{6}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 11 \nu^{5} + 33 \nu^{4} + 29 \nu^{3} - 95 \nu^{2} - 2 \nu + 46 \)
\(\beta_{7}\)\(=\)\( 2 \nu^{7} + 3 \nu^{6} - 21 \nu^{5} - 29 \nu^{4} + 62 \nu^{3} + 76 \nu^{2} - 47 \nu - 45 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{5} - \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{7} - \beta_{6} + 6 \beta_{5} - 6 \beta_{4} + \beta_{3} - \beta_{2} - \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(10 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 10 \beta_{2} + 27 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(-12 \beta_{7} - 11 \beta_{6} + 35 \beta_{5} - 34 \beta_{4} + 11 \beta_{3} - 9 \beta_{2} - 11 \beta_{1} + 64\)
\(\nu^{7}\)\(=\)\(78 \beta_{7} + 23 \beta_{6} - 46 \beta_{5} + 23 \beta_{4} - 44 \beta_{3} - 75 \beta_{2} + 154 \beta_{1} - 20\)

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
1.82602
−0.685988
0.965971
2.41269
−2.01106
−0.735807
−2.57192
1.80009
1.00000 1.00000 1.00000 −3.53629 1.00000 0.738654 1.00000 1.00000 −3.53629
1.2 1.00000 1.00000 1.00000 −3.08319 1.00000 1.60046 1.00000 1.00000 −3.08319
1.3 1.00000 1.00000 1.00000 −2.24957 1.00000 3.57309 1.00000 1.00000 −2.24957
1.4 1.00000 1.00000 1.00000 −1.15050 1.00000 −2.62035 1.00000 1.00000 −1.15050
1.5 1.00000 1.00000 1.00000 −0.612396 1.00000 −3.48353 1.00000 1.00000 −0.612396
1.6 1.00000 1.00000 1.00000 −0.216275 1.00000 −0.843680 1.00000 1.00000 −0.216275
1.7 1.00000 1.00000 1.00000 −0.176895 1.00000 −2.27129 1.00000 1.00000 −0.176895
1.8 1.00000 1.00000 1.00000 3.02511 1.00000 −2.69336 1.00000 1.00000 3.02511
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 8034.2.a.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.p 8 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{8} \)
$3$ \( ( 1 - T )^{8} \)
$5$ \( 1 + 8 T + 53 T^{2} + 233 T^{3} + 901 T^{4} + 2788 T^{5} + 7970 T^{6} + 19668 T^{7} + 46648 T^{8} + 98340 T^{9} + 199250 T^{10} + 348500 T^{11} + 563125 T^{12} + 728125 T^{13} + 828125 T^{14} + 625000 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 6 T + 50 T^{2} + 200 T^{3} + 1003 T^{4} + 3112 T^{5} + 11930 T^{6} + 30677 T^{7} + 98025 T^{8} + 214739 T^{9} + 584570 T^{10} + 1067416 T^{11} + 2408203 T^{12} + 3361400 T^{13} + 5882450 T^{14} + 4941258 T^{15} + 5764801 T^{16} \)
$11$ \( 1 + 7 T + 61 T^{2} + 243 T^{3} + 1188 T^{4} + 3238 T^{5} + 13049 T^{6} + 30939 T^{7} + 134247 T^{8} + 340329 T^{9} + 1578929 T^{10} + 4309778 T^{11} + 17393508 T^{12} + 39135393 T^{13} + 108065221 T^{14} + 136410197 T^{15} + 214358881 T^{16} \)
$13$ \( ( 1 - T )^{8} \)
$17$ \( 1 + 20 T + 272 T^{2} + 2619 T^{3} + 20648 T^{4} + 134440 T^{5} + 761559 T^{6} + 3749008 T^{7} + 16461265 T^{8} + 63733136 T^{9} + 220090551 T^{10} + 660503720 T^{11} + 1724541608 T^{12} + 3718605483 T^{13} + 6565418768 T^{14} + 8206773460 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 12 T + 144 T^{2} + 1120 T^{3} + 8653 T^{4} + 51782 T^{5} + 304550 T^{6} + 1478705 T^{7} + 7052218 T^{8} + 28095395 T^{9} + 109942550 T^{10} + 355172738 T^{11} + 1127667613 T^{12} + 2773230880 T^{13} + 6774606864 T^{14} + 10726460868 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 + 14 T + 187 T^{2} + 1683 T^{3} + 13831 T^{4} + 95563 T^{5} + 594575 T^{6} + 3325627 T^{7} + 16676608 T^{8} + 76489421 T^{9} + 314530175 T^{10} + 1162715021 T^{11} + 3870480871 T^{12} + 10832365269 T^{13} + 27682711243 T^{14} + 47667556258 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 + 25 T + 429 T^{2} + 5351 T^{3} + 55139 T^{4} + 473956 T^{5} + 3543400 T^{6} + 23012323 T^{7} + 132287252 T^{8} + 667357367 T^{9} + 2979999400 T^{10} + 11559312884 T^{11} + 38998767059 T^{12} + 109755158299 T^{13} + 255179204709 T^{14} + 431246907725 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 12 T + 160 T^{2} + 1020 T^{3} + 8033 T^{4} + 36594 T^{5} + 275710 T^{6} + 1225551 T^{7} + 9308526 T^{8} + 37992081 T^{9} + 264957310 T^{10} + 1090171854 T^{11} + 7418644193 T^{12} + 29201734020 T^{13} + 142000588960 T^{14} + 330151369332 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 15 T + 305 T^{2} + 3241 T^{3} + 38479 T^{4} + 316856 T^{5} + 2757678 T^{6} + 18281909 T^{7} + 125872348 T^{8} + 676430633 T^{9} + 3775261182 T^{10} + 16049706968 T^{11} + 72115841119 T^{12} + 224743764637 T^{13} + 782546554745 T^{14} + 1423978156995 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 18 T + 365 T^{2} + 4210 T^{3} + 49609 T^{4} + 425338 T^{5} + 3714236 T^{6} + 25641027 T^{7} + 182463800 T^{8} + 1051282107 T^{9} + 6243630716 T^{10} + 29314720298 T^{11} + 140183177449 T^{12} + 487754606210 T^{13} + 1733788047965 T^{14} + 3505576929858 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 8 T + 252 T^{2} + 1772 T^{3} + 30126 T^{4} + 181972 T^{5} + 2232955 T^{6} + 11553825 T^{7} + 113879774 T^{8} + 496814475 T^{9} + 4128733795 T^{10} + 14468047804 T^{11} + 102994798926 T^{12} + 260498960996 T^{13} + 1592983488348 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 12 T + 204 T^{2} + 1420 T^{3} + 16459 T^{4} + 91876 T^{5} + 1020018 T^{6} + 5263987 T^{7} + 53613624 T^{8} + 247407389 T^{9} + 2253219762 T^{10} + 9538841948 T^{11} + 80314669579 T^{12} + 325669909940 T^{13} + 2198959927116 T^{14} + 6079477445556 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 25 T + 540 T^{2} + 7866 T^{3} + 101736 T^{4} + 1073772 T^{5} + 10369999 T^{6} + 86610728 T^{7} + 672940003 T^{8} + 4590368584 T^{9} + 29129327191 T^{10} + 159859954044 T^{11} + 802745975016 T^{12} + 3289525747938 T^{13} + 11968755009660 T^{14} + 29367778495925 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 9 T + 297 T^{2} + 2594 T^{3} + 43882 T^{4} + 347012 T^{5} + 4256737 T^{6} + 29163852 T^{7} + 294660638 T^{8} + 1720667268 T^{9} + 14817701497 T^{10} + 71268977548 T^{11} + 531734035402 T^{12} + 1854513631606 T^{13} + 12527618491377 T^{14} + 22397863363371 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 2 T + 328 T^{2} - 224 T^{3} + 45968 T^{4} - 156822 T^{5} + 3882665 T^{6} - 21401799 T^{7} + 252725776 T^{8} - 1305509739 T^{9} + 14447396465 T^{10} - 35595614382 T^{11} + 636465619088 T^{12} - 189189571424 T^{13} + 16898682790408 T^{14} + 6285485672042 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 8 T + 413 T^{2} + 2652 T^{3} + 77333 T^{4} + 408459 T^{5} + 8869686 T^{6} + 39292041 T^{7} + 702186539 T^{8} + 2632566747 T^{9} + 39816020454 T^{10} + 122849354217 T^{11} + 1558346640293 T^{12} + 3580531783764 T^{13} + 37359311835797 T^{14} + 48485692842584 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + 13 T + 469 T^{2} + 4838 T^{3} + 98597 T^{4} + 833187 T^{5} + 12462947 T^{6} + 87885815 T^{7} + 1060667768 T^{8} + 6239892865 T^{9} + 62825715827 T^{10} + 298206792357 T^{11} + 2505515511557 T^{12} + 8728861600138 T^{13} + 60079033158949 T^{14} + 118236562059083 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 2 T + 158 T^{2} - 884 T^{3} + 11155 T^{4} - 116193 T^{5} + 1328284 T^{6} - 5591553 T^{7} + 124867587 T^{8} - 408183369 T^{9} + 7078425436 T^{10} - 45201052281 T^{11} + 316782378355 T^{12} - 1832595288212 T^{13} + 23910807753662 T^{14} + 22094797038194 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - T + 255 T^{2} - 1231 T^{3} + 29351 T^{4} - 197950 T^{5} + 3103156 T^{6} - 12293277 T^{7} + 296938044 T^{8} - 971168883 T^{9} + 19366796596 T^{10} - 97597070050 T^{11} + 1143223827431 T^{12} - 3787856427169 T^{13} + 61987301157855 T^{14} - 19203908986159 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 6 T + 233 T^{2} - 353 T^{3} + 16627 T^{4} - 206133 T^{5} + 1693193 T^{6} - 14679863 T^{7} + 223476998 T^{8} - 1218428629 T^{9} + 11664406577 T^{10} - 117864169671 T^{11} + 789089503267 T^{12} - 1390481346979 T^{13} + 76177106994977 T^{14} + 162816305937762 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 17 T + 577 T^{2} + 8160 T^{3} + 157580 T^{4} + 1840495 T^{5} + 26062700 T^{6} + 251322627 T^{7} + 2830404494 T^{8} + 22367713803 T^{9} + 206442646700 T^{10} + 1297491919655 T^{11} + 9886922336780 T^{12} + 45565925103840 T^{13} + 286758204884497 T^{14} + 751932693223993 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 - 19 T + 384 T^{2} - 4715 T^{3} + 57951 T^{4} - 602037 T^{5} + 6955159 T^{6} - 76098654 T^{7} + 782771872 T^{8} - 7381569438 T^{9} + 65441091031 T^{10} - 549462914901 T^{11} + 5130360363231 T^{12} - 40489309311755 T^{13} + 319861249892736 T^{14} - 1535167405084147 T^{15} + 7837433594376961 T^{16} \)
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