Properties

Label 36.5.g.a
Level 36
Weight 5
Character orbit 36.g
Analytic conductor 3.721
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 36.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.72131867102\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{3} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} ) q^{5} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{7} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} + 3 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{3} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} ) q^{5} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{7} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} + 3 \beta_{7} ) q^{9} + ( -3 + 9 \beta_{2} - \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{11} + ( 5 + \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 11 \beta_{4} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{13} + ( -6 + 60 \beta_{1} - 6 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{15} + ( 47 - 107 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} - 18 \beta_{4} + 2 \beta_{5} - 6 \beta_{6} ) q^{17} + ( 68 - 17 \beta_{1} - 31 \beta_{2} + 12 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} - 10 \beta_{7} ) q^{19} + ( -178 + 76 \beta_{1} - 12 \beta_{3} + \beta_{4} + 9 \beta_{5} + 15 \beta_{6} ) q^{21} + ( -281 + 128 \beta_{1} - 44 \beta_{2} - 3 \beta_{3} + 53 \beta_{4} - 18 \beta_{6} - 10 \beta_{7} ) q^{23} + ( 77 - 83 \beta_{1} + 42 \beta_{2} + 18 \beta_{3} - 57 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{25} + ( 219 - 288 \beta_{1} - 21 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 18 \beta_{6} - 15 \beta_{7} ) q^{27} + ( 171 + 198 \beta_{1} + 54 \beta_{2} - 9 \beta_{3} - 19 \beta_{4} + 8 \beta_{5} - 9 \beta_{6} + 8 \beta_{7} ) q^{29} + ( 15 + 34 \beta_{1} - 6 \beta_{2} + 21 \beta_{3} + 57 \beta_{4} - 12 \beta_{5} - 6 \beta_{7} ) q^{31} + ( 102 + 654 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 6 \beta_{5} + 18 \beta_{6} + 30 \beta_{7} ) q^{33} + ( 512 - 1073 \beta_{1} + 17 \beta_{2} + 12 \beta_{3} - 63 \beta_{4} - 19 \beta_{5} - 21 \beta_{6} ) q^{35} + ( 41 + 26 \beta_{1} - 38 \beta_{2} + 3 \beta_{3} + 70 \beta_{4} + 29 \beta_{5} - 4 \beta_{6} + 58 \beta_{7} ) q^{37} + ( -1200 + 302 \beta_{1} - 26 \beta_{2} - 6 \beta_{3} - \beta_{4} - 45 \beta_{5} + 3 \beta_{6} + 9 \beta_{7} ) q^{39} + ( -1312 + 691 \beta_{1} + 44 \beta_{2} - 6 \beta_{3} - 26 \beta_{4} + 27 \beta_{6} + 37 \beta_{7} ) q^{41} + ( -8 + 26 \beta_{1} + 27 \beta_{2} - 9 \beta_{3} + 27 \beta_{4} + 18 \beta_{6} ) q^{43} + ( 1653 - 1857 \beta_{1} + 57 \beta_{2} + 42 \beta_{3} - 33 \beta_{4} + 18 \beta_{5} - 21 \beta_{6} + 18 \beta_{7} ) q^{45} + ( 1155 + 1107 \beta_{1} - 63 \beta_{2} + 27 \beta_{3} + 66 \beta_{4} - 15 \beta_{5} - 6 \beta_{6} - 15 \beta_{7} ) q^{47} + ( -77 - 84 \beta_{1} + 31 \beta_{2} - 40 \beta_{3} - 191 \beta_{4} - 40 \beta_{5} - 17 \beta_{6} - 20 \beta_{7} ) q^{49} + ( 510 + 1623 \beta_{1} - 6 \beta_{2} - 9 \beta_{3} + 87 \beta_{4} + 48 \beta_{5} - 48 \beta_{6} - 21 \beta_{7} ) q^{51} + ( 1051 - 1906 \beta_{1} - 74 \beta_{2} - 45 \beta_{3} + 270 \beta_{4} + 61 \beta_{5} + 90 \beta_{6} ) q^{53} + ( -309 + 9 \beta_{1} + 195 \beta_{2} - 51 \beta_{3} - 183 \beta_{4} - 42 \beta_{5} + 33 \beta_{6} - 84 \beta_{7} ) q^{55} + ( -2717 - 93 \beta_{1} + 68 \beta_{2} + 102 \beta_{3} - 27 \beta_{4} + 27 \beta_{5} - 96 \beta_{6} - 63 \beta_{7} ) q^{57} + ( -3398 + 1652 \beta_{1} + 73 \beta_{2} + 57 \beta_{3} - 244 \beta_{4} + 9 \beta_{6} - 46 \beta_{7} ) q^{59} + ( -443 + 410 \beta_{1} - 384 \beta_{2} - 111 \beta_{3} + 309 \beta_{4} + 24 \beta_{5} - 33 \beta_{6} - 24 \beta_{7} ) q^{61} + ( 2494 - 3695 \beta_{1} - 145 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} - 30 \beta_{5} - 71 \beta_{6} ) q^{63} + ( 2139 + 2052 \beta_{1} - 342 \beta_{2} - 27 \beta_{3} - 125 \beta_{4} - 44 \beta_{5} + 141 \beta_{6} - 44 \beta_{7} ) q^{65} + ( 84 + 106 \beta_{1} - 213 \beta_{2} - 111 \beta_{3} + 66 \beta_{4} + 186 \beta_{5} + 102 \beta_{6} + 93 \beta_{7} ) q^{67} + ( 834 + 3387 \beta_{1} - 81 \beta_{2} + 177 \beta_{3} - 213 \beta_{4} - 159 \beta_{5} - 105 \beta_{6} - 102 \beta_{7} ) q^{69} + ( 2723 - 5420 \beta_{1} + 284 \beta_{2} + 105 \beta_{3} - 72 \beta_{4} - 31 \beta_{5} - 24 \beta_{6} ) q^{71} + ( -1015 + 51 \beta_{1} + 321 \beta_{2} - 93 \beta_{3} + 132 \beta_{4} - 42 \beta_{5} - 72 \beta_{6} - 84 \beta_{7} ) q^{73} + ( -4397 + 1490 \beta_{1} - 27 \beta_{2} - 72 \beta_{3} + 95 \beta_{4} + 162 \beta_{5} - 18 \beta_{6} + 135 \beta_{7} ) q^{75} + ( -4325 + 2237 \beta_{1} + 451 \beta_{2} - 96 \beta_{3} - 163 \beta_{4} + 126 \beta_{6} - 73 \beta_{7} ) q^{77} + ( -508 + 601 \beta_{1} + 113 \beta_{2} + 26 \beta_{3} + 44 \beta_{4} - 122 \beta_{5} + 93 \beta_{6} + 122 \beta_{7} ) q^{79} + ( 5052 - 4974 \beta_{1} - 75 \beta_{2} + 120 \beta_{3} + 375 \beta_{4} - 78 \beta_{6} + 45 \beta_{7} ) q^{81} + ( 3279 + 3231 \beta_{1} + 45 \beta_{2} + 63 \beta_{3} + 390 \beta_{4} + 201 \beta_{5} - 78 \beta_{6} + 201 \beta_{7} ) q^{83} + ( -174 - 744 \beta_{1} + 390 \beta_{2} + 48 \beta_{3} - 384 \beta_{4} - 138 \beta_{5} - 153 \beta_{6} - 69 \beta_{7} ) q^{85} + ( 1659 + 2481 \beta_{1} + 381 \beta_{2} - 129 \beta_{3} - 162 \beta_{4} + 33 \beta_{5} + 81 \beta_{6} + 138 \beta_{7} ) q^{87} + ( 1325 - 2798 \beta_{1} - 358 \beta_{2} - 39 \beta_{3} + 198 \beta_{4} - 241 \beta_{5} + 66 \beta_{6} ) q^{89} + ( 2053 + 57 \beta_{1} - 309 \beta_{2} + 63 \beta_{3} - 309 \beta_{4} + 120 \beta_{5} + 183 \beta_{6} + 240 \beta_{7} ) q^{91} + ( -4284 - 625 \beta_{1} - 41 \beta_{2} + 99 \beta_{3} - 67 \beta_{4} - 135 \beta_{5} + 99 \beta_{6} ) q^{93} + ( -5996 + 3008 \beta_{1} - 842 \beta_{2} + 102 \beta_{3} + 536 \beta_{4} - 180 \beta_{6} + 302 \beta_{7} ) q^{95} + ( 1742 - 1853 \beta_{1} + 2 \beta_{2} + 71 \beta_{3} - 274 \beta_{4} + 61 \beta_{5} - 111 \beta_{6} - 61 \beta_{7} ) q^{97} + ( 177 - 3072 \beta_{1} + 678 \beta_{2} - 237 \beta_{3} - 618 \beta_{4} - 18 \beta_{5} + 168 \beta_{6} - 81 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 9q^{3} - 9q^{5} + 13q^{7} + 21q^{9} + O(q^{10}) \) \( 8q - 9q^{3} - 9q^{5} + 13q^{7} + 21q^{9} - 18q^{11} - 5q^{13} + 225q^{15} + 562q^{19} - 1167q^{21} - 1719q^{23} + 353q^{25} + 648q^{27} + 2115q^{29} + 187q^{31} + 3258q^{33} + 16q^{37} - 8265q^{39} - 7920q^{41} - 68q^{43} + 5679q^{45} + 13689q^{47} - 327q^{49} + 10449q^{51} - 1818q^{55} - 21861q^{57} - 20052q^{59} - 1937q^{61} + 5559q^{63} + 25965q^{65} + 154q^{67} + 21645q^{69} - 7802q^{73} - 30297q^{75} - 25641q^{77} - 2195q^{79} + 19701q^{81} + 37017q^{83} - 3042q^{85} + 22455q^{87} + 15830q^{91} - 36489q^{93} - 37116q^{95} + 7282q^{97} - 10035q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 6 x^{6} + 121 x^{5} + 1104 x^{4} - 1647 x^{3} + 6529 x^{2} + 85254 x + 440076\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 115 \nu^{7} + 175 \nu^{6} - 4562 \nu^{5} + 26525 \nu^{4} + 65600 \nu^{3} - 13645 \nu^{2} - 1909485 \nu + 10751676 \)\()/5629338\)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{7} - 67 \nu^{6} + 212 \nu^{5} - 1085 \nu^{4} + 1360 \nu^{3} - 32675 \nu^{2} + 104769 \nu - 757536 \)\()/69498\)
\(\beta_{3}\)\(=\)\((\)\( 236 \nu^{7} + 4388 \nu^{6} + 10178 \nu^{5} - 137441 \nu^{4} + 1153423 \nu^{3} + 3476611 \nu^{2} + 3141495 \nu - 63902904 \)\()/2814669\)
\(\beta_{4}\)\(=\)\((\)\( -20 \nu^{7} + 202 \nu^{6} - 485 \nu^{5} + 2360 \nu^{4} - 6760 \nu^{3} + 102320 \nu^{2} + 396585 \nu + 1729977 \)\()/216513\)
\(\beta_{5}\)\(=\)\((\)\( 314 \nu^{7} - 1033 \nu^{6} + 6278 \nu^{5} + 27100 \nu^{4} + 461641 \nu^{3} - 2980346 \nu^{2} + 7345773 \nu + 22814571 \)\()/2814669\)
\(\beta_{6}\)\(=\)\((\)\( 8 \nu^{7} - 61 \nu^{6} - 4 \nu^{5} + 2323 \nu^{4} + 3100 \nu^{3} - 34196 \nu^{2} + 56691 \nu + 1339014 \)\()/24057\)
\(\beta_{7}\)\(=\)\((\)\( -67 \nu^{7} + 350 \nu^{6} - 1699 \nu^{5} - 9914 \nu^{4} - 21161 \nu^{3} + 100717 \nu^{2} - 1332189 \nu - 5513550 \)\()/72171\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} + 11 \beta_{6} + \beta_{4} + 8 \beta_{3} - 2 \beta_{2} - 64 \beta_{1} - 109\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{7} + 24 \beta_{6} + 3 \beta_{5} - 23 \beta_{4} - 110 \beta_{2} - 260 \beta_{1} - 2108\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-90 \beta_{7} - 110 \beta_{6} + 150 \beta_{5} - 650 \beta_{4} - 20 \beta_{3} - 965 \beta_{2} - 2577 \beta_{1} - 2826\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-465 \beta_{7} - 1975 \beta_{6} + 2025 \beta_{5} - 818 \beta_{4} - 700 \beta_{3} - 2675 \beta_{2} + 8555 \beta_{1} + 3137\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-3882 \beta_{7} - 13050 \beta_{6} + 1821 \beta_{5} - 3083 \beta_{4} - 4173 \beta_{3} + 9025 \beta_{2} + 144331 \beta_{1} + 167638\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−3.05006 3.25531i
−3.41053 + 2.74723i
3.72537 4.42407i
4.23522 + 4.06612i
−3.05006 + 3.25531i
−3.41053 2.74723i
3.72537 + 4.42407i
4.23522 4.06612i
0 −8.89427 1.37550i 0 27.4152 + 15.8282i 0 37.6830 + 65.2688i 0 77.2160 + 24.4682i 0
5.2 0 −4.23662 + 7.94047i 0 −34.8718 20.1332i 0 −7.38688 12.7945i 0 −45.1021 67.2815i 0
5.3 0 0.256692 8.99634i 0 −7.67992 4.43400i 0 −30.9381 53.5864i 0 −80.8682 4.61857i 0
5.4 0 8.37420 + 3.29740i 0 10.6364 + 6.14094i 0 7.14202 + 12.3703i 0 59.2543 + 55.2261i 0
29.1 0 −8.89427 + 1.37550i 0 27.4152 15.8282i 0 37.6830 65.2688i 0 77.2160 24.4682i 0
29.2 0 −4.23662 7.94047i 0 −34.8718 + 20.1332i 0 −7.38688 + 12.7945i 0 −45.1021 + 67.2815i 0
29.3 0 0.256692 + 8.99634i 0 −7.67992 + 4.43400i 0 −30.9381 + 53.5864i 0 −80.8682 + 4.61857i 0
29.4 0 8.37420 3.29740i 0 10.6364 6.14094i 0 7.14202 12.3703i 0 59.2543 55.2261i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.5.g.a 8
3.b odd 2 1 108.5.g.a 8
4.b odd 2 1 144.5.q.c 8
9.c even 3 1 108.5.g.a 8
9.c even 3 1 324.5.c.a 8
9.d odd 6 1 inner 36.5.g.a 8
9.d odd 6 1 324.5.c.a 8
12.b even 2 1 432.5.q.c 8
36.f odd 6 1 432.5.q.c 8
36.f odd 6 1 1296.5.e.g 8
36.h even 6 1 144.5.q.c 8
36.h even 6 1 1296.5.e.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.g.a 8 1.a even 1 1 trivial
36.5.g.a 8 9.d odd 6 1 inner
108.5.g.a 8 3.b odd 2 1
108.5.g.a 8 9.c even 3 1
144.5.q.c 8 4.b odd 2 1
144.5.q.c 8 36.h even 6 1
324.5.c.a 8 9.c even 3 1
324.5.c.a 8 9.d odd 6 1
432.5.q.c 8 12.b even 2 1
432.5.q.c 8 36.f odd 6 1
1296.5.e.g 8 36.f odd 6 1
1296.5.e.g 8 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 9 T + 30 T^{2} - 189 T^{3} - 6966 T^{4} - 15309 T^{5} + 196830 T^{6} + 4782969 T^{7} + 43046721 T^{8} \)
$5$ \( 1 + 9 T + 1114 T^{2} + 9783 T^{3} + 533599 T^{4} + 10528056 T^{5} + 59806456 T^{6} + 10305069192 T^{7} - 6001445444 T^{8} + 6440668245000 T^{9} + 23361896875000 T^{10} + 2570326171875000 T^{11} + 81420745849609375 T^{12} + 932979583740234375 T^{13} + 66399574279785156250 T^{14} + \)\(33\!\cdots\!25\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 - 13 T - 4554 T^{2} + 124753 T^{3} + 8962391 T^{4} - 347580324 T^{5} + 2901735784 T^{6} + 447642835484 T^{7} - 30706182623268 T^{8} + 1074790447997084 T^{9} + 16727929349338984 T^{10} - 4810959089900633124 T^{11} + \)\(29\!\cdots\!91\)\( T^{12} + \)\(99\!\cdots\!53\)\( T^{13} - \)\(87\!\cdots\!54\)\( T^{14} - \)\(59\!\cdots\!13\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 + 18 T + 33850 T^{2} + 607356 T^{3} + 452208721 T^{4} + 17443950048 T^{5} + 9074477968558 T^{6} + 445095012639474 T^{7} + 191968264536523468 T^{8} + 6516636080054538834 T^{9} + \)\(19\!\cdots\!98\)\( T^{10} + \)\(54\!\cdots\!08\)\( T^{11} + \)\(20\!\cdots\!81\)\( T^{12} + \)\(40\!\cdots\!56\)\( T^{13} + \)\(33\!\cdots\!50\)\( T^{14} + \)\(25\!\cdots\!58\)\( T^{15} + \)\(21\!\cdots\!21\)\( T^{16} \)
$13$ \( 1 + 5 T - 42054 T^{2} - 7266665 T^{3} + 352176575 T^{4} + 250092029040 T^{5} + 23191943061904 T^{6} - 3464858176098460 T^{7} - 548440340475170196 T^{8} - 98959814367548116060 T^{9} + \)\(18\!\cdots\!84\)\( T^{10} + \)\(58\!\cdots\!40\)\( T^{11} + \)\(23\!\cdots\!75\)\( T^{12} - \)\(13\!\cdots\!65\)\( T^{13} - \)\(22\!\cdots\!94\)\( T^{14} + \)\(77\!\cdots\!05\)\( T^{15} + \)\(44\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} + \)\(63\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!55\)\( T^{10} + \)\(25\!\cdots\!74\)\( T^{12} - \)\(97\!\cdots\!25\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} \)
$19$ \( ( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 8971693946161 T^{5} + 1871079140226970 T^{6} - 621941492257591241 T^{7} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \)
$23$ \( 1 + 1719 T + 1529458 T^{2} + 935945649 T^{3} + 342642958747 T^{4} - 16333135609500 T^{5} - 118516645434237428 T^{6} - \)\(10\!\cdots\!68\)\( T^{7} - \)\(65\!\cdots\!00\)\( T^{8} - \)\(29\!\cdots\!88\)\( T^{9} - \)\(92\!\cdots\!68\)\( T^{10} - \)\(35\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!67\)\( T^{12} + \)\(16\!\cdots\!49\)\( T^{13} + \)\(73\!\cdots\!78\)\( T^{14} + \)\(23\!\cdots\!39\)\( T^{15} + \)\(37\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 - 2115 T + 4091014 T^{2} - 5498870985 T^{3} + 7048695081595 T^{4} - 8024737206821040 T^{5} + 8297140249856169556 T^{6} - \)\(79\!\cdots\!80\)\( T^{7} + \)\(68\!\cdots\!24\)\( T^{8} - \)\(56\!\cdots\!80\)\( T^{9} + \)\(41\!\cdots\!16\)\( T^{10} - \)\(28\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!95\)\( T^{12} - \)\(97\!\cdots\!85\)\( T^{13} + \)\(51\!\cdots\!34\)\( T^{14} - \)\(18\!\cdots\!15\)\( T^{15} + \)\(62\!\cdots\!41\)\( T^{16} \)
$31$ \( 1 - 187 T - 2516004 T^{2} + 186847537 T^{3} + 3362431719041 T^{4} - 296059350096 T^{5} - 3460282108678916846 T^{6} - 13487262715981377034 T^{7} + \)\(31\!\cdots\!92\)\( T^{8} - \)\(12\!\cdots\!14\)\( T^{9} - \)\(29\!\cdots\!86\)\( T^{10} - \)\(23\!\cdots\!56\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} + \)\(12\!\cdots\!37\)\( T^{13} - \)\(15\!\cdots\!84\)\( T^{14} - \)\(10\!\cdots\!67\)\( T^{15} + \)\(52\!\cdots\!61\)\( T^{16} \)
$37$ \( ( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \)
$41$ \( 1 + 7920 T + 37687894 T^{2} + 132890424480 T^{3} + 385083705354505 T^{4} + 963185727644706960 T^{5} + \)\(21\!\cdots\!66\)\( T^{6} + \)\(41\!\cdots\!20\)\( T^{7} + \)\(74\!\cdots\!64\)\( T^{8} + \)\(11\!\cdots\!20\)\( T^{9} + \)\(16\!\cdots\!86\)\( T^{10} + \)\(21\!\cdots\!60\)\( T^{11} + \)\(24\!\cdots\!05\)\( T^{12} + \)\(23\!\cdots\!80\)\( T^{13} + \)\(19\!\cdots\!34\)\( T^{14} + \)\(11\!\cdots\!20\)\( T^{15} + \)\(40\!\cdots\!81\)\( T^{16} \)
$43$ \( 1 + 68 T - 12950604 T^{2} - 209786648 T^{3} + 102574547445791 T^{4} + 59145034173804 T^{5} - \)\(54\!\cdots\!36\)\( T^{6} + \)\(27\!\cdots\!16\)\( T^{7} + \)\(21\!\cdots\!12\)\( T^{8} + \)\(94\!\cdots\!16\)\( T^{9} - \)\(63\!\cdots\!36\)\( T^{10} + \)\(23\!\cdots\!04\)\( T^{11} + \)\(14\!\cdots\!91\)\( T^{12} - \)\(97\!\cdots\!48\)\( T^{13} - \)\(20\!\cdots\!04\)\( T^{14} + \)\(37\!\cdots\!68\)\( T^{15} + \)\(18\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 13689 T + 103685338 T^{2} - 564293857959 T^{3} + 2435028217967227 T^{4} - 8736748337842042500 T^{5} + \)\(26\!\cdots\!72\)\( T^{6} - \)\(71\!\cdots\!32\)\( T^{7} + \)\(16\!\cdots\!20\)\( T^{8} - \)\(35\!\cdots\!92\)\( T^{9} + \)\(63\!\cdots\!92\)\( T^{10} - \)\(10\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!67\)\( T^{12} - \)\(15\!\cdots\!59\)\( T^{13} + \)\(13\!\cdots\!78\)\( T^{14} - \)\(90\!\cdots\!29\)\( T^{15} + \)\(32\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} + \)\(67\!\cdots\!26\)\( T^{8} - \)\(52\!\cdots\!60\)\( T^{10} + \)\(44\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 + 20052 T + 216711700 T^{2} + 1657982214864 T^{3} + 9931594296358591 T^{4} + 49579528565018409012 T^{5} + \)\(21\!\cdots\!48\)\( T^{6} + \)\(84\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!08\)\( T^{8} + \)\(10\!\cdots\!36\)\( T^{9} + \)\(31\!\cdots\!08\)\( T^{10} + \)\(88\!\cdots\!72\)\( T^{11} + \)\(21\!\cdots\!31\)\( T^{12} + \)\(43\!\cdots\!64\)\( T^{13} + \)\(68\!\cdots\!00\)\( T^{14} + \)\(76\!\cdots\!92\)\( T^{15} + \)\(46\!\cdots\!81\)\( T^{16} \)
$61$ \( 1 + 1937 T - 10529634 T^{2} + 149647181023 T^{3} + 416288373490931 T^{4} - 1350680282380662864 T^{5} + \)\(12\!\cdots\!24\)\( T^{6} + \)\(40\!\cdots\!44\)\( T^{7} - \)\(98\!\cdots\!68\)\( T^{8} + \)\(55\!\cdots\!04\)\( T^{9} + \)\(23\!\cdots\!44\)\( T^{10} - \)\(35\!\cdots\!44\)\( T^{11} + \)\(15\!\cdots\!91\)\( T^{12} + \)\(76\!\cdots\!23\)\( T^{13} - \)\(74\!\cdots\!94\)\( T^{14} + \)\(18\!\cdots\!97\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} \)
$67$ \( 1 - 154 T - 33835854 T^{2} - 25606229228 T^{3} + 539365905411977 T^{4} + 738160924156362336 T^{5} + \)\(70\!\cdots\!02\)\( T^{6} - \)\(11\!\cdots\!78\)\( T^{7} - \)\(26\!\cdots\!64\)\( T^{8} - \)\(22\!\cdots\!38\)\( T^{9} + \)\(28\!\cdots\!82\)\( T^{10} + \)\(60\!\cdots\!96\)\( T^{11} + \)\(88\!\cdots\!37\)\( T^{12} - \)\(85\!\cdots\!28\)\( T^{13} - \)\(22\!\cdots\!34\)\( T^{14} - \)\(20\!\cdots\!14\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \)
$71$ \( 1 - 68871716 T^{2} + 3244147638477940 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!74\)\( T^{8} - \)\(73\!\cdots\!64\)\( T^{10} + \)\(13\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!96\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \)
$73$ \( ( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} + \)\(47\!\cdots\!70\)\( T^{6} + \)\(89\!\cdots\!21\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \)
$79$ \( 1 + 2195 T - 87724914 T^{2} - 187644610415 T^{3} + 3128319215246375 T^{4} + 3711455091635884260 T^{5} - \)\(16\!\cdots\!36\)\( T^{6} + \)\(76\!\cdots\!80\)\( T^{7} + \)\(88\!\cdots\!64\)\( T^{8} + \)\(29\!\cdots\!80\)\( T^{9} - \)\(24\!\cdots\!96\)\( T^{10} + \)\(21\!\cdots\!60\)\( T^{11} + \)\(72\!\cdots\!75\)\( T^{12} - \)\(16\!\cdots\!15\)\( T^{13} - \)\(30\!\cdots\!34\)\( T^{14} + \)\(29\!\cdots\!95\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \)
$83$ \( 1 - 37017 T + 725723290 T^{2} - 9956481997959 T^{3} + 104510585134438411 T^{4} - \)\(87\!\cdots\!72\)\( T^{5} + \)\(62\!\cdots\!28\)\( T^{6} - \)\(40\!\cdots\!76\)\( T^{7} + \)\(26\!\cdots\!48\)\( T^{8} - \)\(19\!\cdots\!96\)\( T^{9} + \)\(14\!\cdots\!48\)\( T^{10} - \)\(93\!\cdots\!92\)\( T^{11} + \)\(53\!\cdots\!91\)\( T^{12} - \)\(23\!\cdots\!59\)\( T^{13} + \)\(82\!\cdots\!90\)\( T^{14} - \)\(20\!\cdots\!97\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - 294759296 T^{2} + 46567064448316540 T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!24\)\( T^{10} + \)\(72\!\cdots\!40\)\( T^{12} - \)\(17\!\cdots\!36\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} \)
$97$ \( 1 - 7282 T - 283226964 T^{2} + 993163976152 T^{3} + 57034963146137471 T^{4} - 97460573991801682656 T^{5} - \)\(75\!\cdots\!16\)\( T^{6} + \)\(33\!\cdots\!26\)\( T^{7} + \)\(75\!\cdots\!52\)\( T^{8} + \)\(29\!\cdots\!06\)\( T^{9} - \)\(58\!\cdots\!76\)\( T^{10} - \)\(67\!\cdots\!96\)\( T^{11} + \)\(35\!\cdots\!91\)\( T^{12} + \)\(54\!\cdots\!52\)\( T^{13} - \)\(13\!\cdots\!84\)\( T^{14} - \)\(31\!\cdots\!02\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \)
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