Properties

Label 804.2.ba.a
Level 804
Weight 2
Character orbit 804.ba
Analytic conductor 6.420
Analytic rank 0
Dimension 20
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.ba (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{66}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{33} - \zeta_{33}^{12} ) q^{3} + ( 1 - \zeta_{33} + \zeta_{33}^{3} - \zeta_{33}^{4} + 2 \zeta_{33}^{6} - \zeta_{33}^{7} - 2 \zeta_{33}^{9} - \zeta_{33}^{10} + \zeta_{33}^{11} - \zeta_{33}^{13} + \zeta_{33}^{14} - \zeta_{33}^{16} + 4 \zeta_{33}^{17} - \zeta_{33}^{19} ) q^{7} + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{33} - \zeta_{33}^{12} ) q^{3} + ( 1 - \zeta_{33} + \zeta_{33}^{3} - \zeta_{33}^{4} + 2 \zeta_{33}^{6} - \zeta_{33}^{7} - 2 \zeta_{33}^{9} - \zeta_{33}^{10} + \zeta_{33}^{11} - \zeta_{33}^{13} + \zeta_{33}^{14} - \zeta_{33}^{16} + 4 \zeta_{33}^{17} - \zeta_{33}^{19} ) q^{7} + ( 3 \zeta_{33}^{2} + 3 \zeta_{33}^{13} ) q^{9} + ( -3 \zeta_{33}^{3} + 4 \zeta_{33}^{7} + \zeta_{33}^{14} + \zeta_{33}^{18} ) q^{13} + ( -3 + 3 \zeta_{33}^{2} - 3 \zeta_{33}^{3} + 3 \zeta_{33}^{5} - 3 \zeta_{33}^{6} - 2 \zeta_{33}^{8} - 3 \zeta_{33}^{9} + 5 \zeta_{33}^{10} - 3 \zeta_{33}^{12} + 3 \zeta_{33}^{13} - 3 \zeta_{33}^{15} + 3 \zeta_{33}^{16} - 3 \zeta_{33}^{18} + \zeta_{33}^{19} ) q^{19} + ( -4 + 4 \zeta_{33}^{2} - 4 \zeta_{33}^{3} + 4 \zeta_{33}^{5} - 4 \zeta_{33}^{6} + \zeta_{33}^{7} + 4 \zeta_{33}^{8} - 4 \zeta_{33}^{9} + 5 \zeta_{33}^{10} - 4 \zeta_{33}^{12} + 4 \zeta_{33}^{13} - 4 \zeta_{33}^{15} + 4 \zeta_{33}^{16} - 8 \zeta_{33}^{18} + 4 \zeta_{33}^{19} ) q^{21} + ( 5 \zeta_{33}^{5} + 5 \zeta_{33}^{16} ) q^{25} + ( -3 \zeta_{33}^{3} - 6 \zeta_{33}^{14} ) q^{27} + ( 6 - 6 \zeta_{33} + 12 \zeta_{33}^{3} - 6 \zeta_{33}^{4} + 6 \zeta_{33}^{6} - 6 \zeta_{33}^{7} + \zeta_{33}^{9} - 6 \zeta_{33}^{10} + 6 \zeta_{33}^{11} - 6 \zeta_{33}^{13} + 11 \zeta_{33}^{14} - 6 \zeta_{33}^{16} + 6 \zeta_{33}^{17} - 6 \zeta_{33}^{19} ) q^{31} + ( 7 \zeta_{33}^{5} + 4 \zeta_{33}^{6} + 4 \zeta_{33}^{16} + 7 \zeta_{33}^{17} ) q^{37} + ( 7 \zeta_{33}^{4} - 7 \zeta_{33}^{8} + 2 \zeta_{33}^{15} - 5 \zeta_{33}^{19} ) q^{39} + ( -\zeta_{33} + 7 \zeta_{33}^{2} + 6 \zeta_{33}^{12} + 6 \zeta_{33}^{13} ) q^{43} + ( 3 \zeta_{33} + 7 \zeta_{33}^{4} - 5 \zeta_{33}^{7} - 5 \zeta_{33}^{12} + 7 \zeta_{33}^{15} + 3 \zeta_{33}^{18} ) q^{49} + ( 1 + 7 \zeta_{33} - 7 \zeta_{33}^{3} + 7 \zeta_{33}^{4} - 7 \zeta_{33}^{6} + 7 \zeta_{33}^{7} + \zeta_{33}^{9} + 7 \zeta_{33}^{10} - 6 \zeta_{33}^{11} + 7 \zeta_{33}^{13} - 7 \zeta_{33}^{14} + 7 \zeta_{33}^{16} - 7 \zeta_{33}^{17} + 7 \zeta_{33}^{19} ) q^{57} + ( 4 \zeta_{33}^{2} + 4 \zeta_{33}^{4} + 9 \zeta_{33}^{13} - 5 \zeta_{33}^{15} ) q^{61} + ( 9 - 6 \zeta_{33}^{8} + 3 \zeta_{33}^{11} + 3 \zeta_{33}^{19} ) q^{63} + ( 2 \zeta_{33}^{5} + 9 \zeta_{33}^{16} ) q^{67} + ( -9 + 9 \zeta_{33}^{6} - \zeta_{33}^{11} + 8 \zeta_{33}^{17} ) q^{73} + ( -5 \zeta_{33}^{6} - 10 \zeta_{33}^{17} ) q^{75} + ( -3 - 7 \zeta_{33}^{2} + 7 \zeta_{33}^{3} - 7 \zeta_{33}^{5} + 7 \zeta_{33}^{6} - 7 \zeta_{33}^{8} + 7 \zeta_{33}^{9} - 10 \zeta_{33}^{10} - 3 \zeta_{33}^{11} + 7 \zeta_{33}^{12} - 7 \zeta_{33}^{13} + 7 \zeta_{33}^{15} - 7 \zeta_{33}^{16} + 7 \zeta_{33}^{18} - 7 \zeta_{33}^{19} ) q^{79} + 9 \zeta_{33}^{15} q^{81} + ( 11 - 12 \zeta_{33} - 10 \zeta_{33}^{2} + 11 \zeta_{33}^{3} - 11 \zeta_{33}^{4} + 6 \zeta_{33}^{5} + 11 \zeta_{33}^{6} - 11 \zeta_{33}^{7} + 5 \zeta_{33}^{9} - 11 \zeta_{33}^{10} + 11 \zeta_{33}^{11} + 9 \zeta_{33}^{12} - 20 \zeta_{33}^{13} + 11 \zeta_{33}^{14} - 16 \zeta_{33}^{16} + 11 \zeta_{33}^{17} - 11 \zeta_{33}^{19} ) q^{91} + ( -11 + 11 \zeta_{33}^{2} - 11 \zeta_{33}^{3} - 7 \zeta_{33}^{4} + 11 \zeta_{33}^{5} - 11 \zeta_{33}^{6} + 11 \zeta_{33}^{8} - 11 \zeta_{33}^{9} + 4 \zeta_{33}^{10} - 11 \zeta_{33}^{12} + 11 \zeta_{33}^{13} - 22 \zeta_{33}^{15} + 11 \zeta_{33}^{16} - 11 \zeta_{33}^{18} + 11 \zeta_{33}^{19} ) q^{93} + ( 3 + 3 \zeta_{33} - 3 \zeta_{33}^{2} + 3 \zeta_{33}^{3} - 3 \zeta_{33}^{5} + 3 \zeta_{33}^{6} - 3 \zeta_{33}^{8} + 3 \zeta_{33}^{9} + 8 \zeta_{33}^{10} - 5 \zeta_{33}^{12} - 3 \zeta_{33}^{13} + 3 \zeta_{33}^{15} - 3 \zeta_{33}^{16} + 3 \zeta_{33}^{18} - 3 \zeta_{33}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{7} + 6q^{9} + 9q^{13} - 8q^{19} + 6q^{21} + 10q^{25} - 3q^{31} + 10q^{37} - 9q^{39} - 5q^{49} + 141q^{57} + 27q^{61} + 147q^{63} + 11q^{67} - 180q^{73} - 166q^{79} - 18q^{81} - 36q^{91} - 3q^{93} + 33q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(-\zeta_{33}^{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
−0.888835 0.458227i
0.723734 + 0.690079i
−0.995472 + 0.0950560i
−0.995472 0.0950560i
0.0475819 0.998867i
0.928368 0.371662i
−0.786053 + 0.618159i
−0.786053 0.618159i
−0.327068 0.945001i
0.580057 + 0.814576i
−0.888835 + 0.458227i
0.580057 0.814576i
0.981929 0.189251i
0.928368 + 0.371662i
0.723734 0.690079i
0.0475819 + 0.998867i
0.235759 + 0.971812i
−0.327068 + 0.945001i
0.981929 + 0.189251i
0.235759 0.971812i
0 0.936417 + 1.45709i 0 0 0 −0.502989 5.26754i 0 −1.24625 + 2.72890i 0
101.1 0 −0.487975 1.66189i 0 0 0 1.00786 1.95498i 0 −2.52376 + 1.62192i 0
113.1 0 1.57553 + 0.719520i 0 0 0 3.27565 + 1.13371i 0 1.96458 + 2.26725i 0
185.1 0 1.57553 0.719520i 0 0 0 3.27565 1.13371i 0 1.96458 2.26725i 0
197.1 0 −0.936417 + 1.45709i 0 0 0 −0.614593 0.437650i 0 −1.24625 2.72890i 0
221.1 0 −1.71442 0.246497i 0 0 0 4.93365 1.19689i 0 2.87848 + 0.845198i 0
233.1 0 1.71442 0.246497i 0 0 0 0.518680 0.543976i 0 2.87848 0.845198i 0
245.1 0 1.71442 + 0.246497i 0 0 0 0.518680 + 0.543976i 0 2.87848 + 0.845198i 0
281.1 0 1.30900 + 1.13425i 0 0 0 1.65416 4.13190i 0 0.426945 + 2.96946i 0
329.1 0 −1.57553 0.719520i 0 0 0 −0.842599 4.37182i 0 1.96458 + 2.26725i 0
353.1 0 0.936417 1.45709i 0 0 0 −0.502989 + 5.26754i 0 −1.24625 2.72890i 0
413.1 0 −1.57553 + 0.719520i 0 0 0 −0.842599 + 4.37182i 0 1.96458 2.26725i 0
497.1 0 −1.30900 + 1.13425i 0 0 0 −1.35800 + 1.72684i 0 0.426945 2.96946i 0
593.1 0 −1.71442 + 0.246497i 0 0 0 4.93365 + 1.19689i 0 2.87848 0.845198i 0
605.1 0 −0.487975 + 1.66189i 0 0 0 1.00786 + 1.95498i 0 −2.52376 1.62192i 0
653.1 0 −0.936417 1.45709i 0 0 0 −0.614593 + 0.437650i 0 −1.24625 + 2.72890i 0
677.1 0 0.487975 1.66189i 0 0 0 −5.07182 0.241600i 0 −2.52376 1.62192i 0
701.1 0 1.30900 1.13425i 0 0 0 1.65416 + 4.13190i 0 0.426945 2.96946i 0
749.1 0 −1.30900 1.13425i 0 0 0 −1.35800 1.72684i 0 0.426945 + 2.96946i 0
785.1 0 0.487975 + 1.66189i 0 0 0 −5.07182 + 0.241600i 0 −2.52376 + 1.62192i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 785.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
67.h odd 66 1 inner
201.p even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.ba.a 20
3.b odd 2 1 CM 804.2.ba.a 20
67.h odd 66 1 inner 804.2.ba.a 20
201.p even 66 1 inner 804.2.ba.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.ba.a 20 1.a even 1 1 trivial
804.2.ba.a 20 3.b odd 2 1 CM
804.2.ba.a 20 67.h odd 66 1 inner
804.2.ba.a 20 201.p even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T^{2} + 9 T^{4} - 27 T^{6} + 81 T^{8} - 243 T^{10} + 729 T^{12} - 2187 T^{14} + 6561 T^{16} - 19683 T^{18} + 59049 T^{20} \)
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} - 125 T^{6} + 625 T^{8} - 3125 T^{10} + 15625 T^{12} - 78125 T^{14} + 390625 T^{16} - 1953125 T^{18} + 9765625 T^{20} )^{2} \)
$7$ \( ( 1 - 5 T + 18 T^{2} - 55 T^{3} + 149 T^{4} - 360 T^{5} + 757 T^{6} - 1265 T^{7} + 1026 T^{8} + 3725 T^{9} - 25807 T^{10} + 26075 T^{11} + 50274 T^{12} - 433895 T^{13} + 1817557 T^{14} - 6050520 T^{15} + 17529701 T^{16} - 45294865 T^{17} + 103766418 T^{18} - 201768035 T^{19} + 282475249 T^{20} )( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} - 120 T^{5} - 83 T^{6} + 923 T^{7} - 342 T^{8} - 6119 T^{9} + 8513 T^{10} - 42833 T^{11} - 16758 T^{12} + 316589 T^{13} - 199283 T^{14} - 2016840 T^{15} + 3411821 T^{16} + 10706059 T^{17} - 34588806 T^{18} - 40353607 T^{19} + 282475249 T^{20} ) \)
$11$ \( ( 1 - 11 T + 66 T^{2} - 363 T^{3} + 1815 T^{4} - 7986 T^{5} + 33275 T^{6} - 131769 T^{7} + 497794 T^{8} - 1771561 T^{9} + 5958887 T^{10} - 19487171 T^{11} + 60233074 T^{12} - 175384539 T^{13} + 487179275 T^{14} - 1286153286 T^{15} + 3215383215 T^{16} - 7073843073 T^{17} + 14147686146 T^{18} - 25937424601 T^{19} + 25937424601 T^{20} )( 1 + 11 T + 66 T^{2} + 363 T^{3} + 1815 T^{4} + 7986 T^{5} + 33275 T^{6} + 131769 T^{7} + 497794 T^{8} + 1771561 T^{9} + 5958887 T^{10} + 19487171 T^{11} + 60233074 T^{12} + 175384539 T^{13} + 487179275 T^{14} + 1286153286 T^{15} + 3215383215 T^{16} + 7073843073 T^{17} + 14147686146 T^{18} + 25937424601 T^{19} + 25937424601 T^{20} ) \)
$13$ \( ( 1 - 7 T + 36 T^{2} - 161 T^{3} + 659 T^{4} - 2520 T^{5} + 9073 T^{6} - 30751 T^{7} + 97308 T^{8} - 281393 T^{9} + 704747 T^{10} - 3658109 T^{11} + 16445052 T^{12} - 67559947 T^{13} + 259133953 T^{14} - 935658360 T^{15} + 3180867131 T^{16} - 10102511237 T^{17} + 29366305956 T^{18} - 74231495611 T^{19} + 137858491849 T^{20} )( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} - 630 T^{5} + 883 T^{6} + 6424 T^{7} - 24327 T^{8} - 34858 T^{9} + 385967 T^{10} - 453154 T^{11} - 4111263 T^{12} + 14113528 T^{13} + 25219363 T^{14} - 233914590 T^{15} + 139977461 T^{16} + 2760934748 T^{17} - 7341576489 T^{18} - 21208998746 T^{19} + 137858491849 T^{20} ) \)
$17$ \( 1 - 17 T^{2} + 4913 T^{6} - 83521 T^{8} + 24137569 T^{12} - 410338673 T^{14} + 118587876497 T^{18} - 2015993900449 T^{20} + 34271896307633 T^{22} - 9904578032905937 T^{26} + 168377826559400929 T^{28} - 48661191875666868481 T^{32} + \)\(82\!\cdots\!77\)\( T^{34} - \)\(23\!\cdots\!53\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( ( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} + 1008 T^{5} - 4787 T^{6} - 23939 T^{7} + 67014 T^{8} + 521855 T^{9} - 751411 T^{10} + 9915245 T^{11} + 24192054 T^{12} - 164197601 T^{13} - 623846627 T^{14} + 2495907792 T^{15} + 14348993705 T^{16} - 33073254343 T^{17} - 305704134738 T^{18} + 322687697779 T^{19} + 6131066257801 T^{20} )( 1 + 7 T + 30 T^{2} + 77 T^{3} - 31 T^{4} - 1680 T^{5} - 11171 T^{6} - 46277 T^{7} - 111690 T^{8} + 97433 T^{9} + 2804141 T^{10} + 1851227 T^{11} - 40320090 T^{12} - 317413943 T^{13} - 1455815891 T^{14} - 4159846320 T^{15} - 1458422311 T^{16} + 68828123903 T^{17} + 509506891230 T^{18} + 2258813884453 T^{19} + 6131066257801 T^{20} ) \)
$23$ \( 1 - 23 T^{2} + 12167 T^{6} - 279841 T^{8} + 148035889 T^{12} - 3404825447 T^{14} + 1801152661463 T^{18} - 41426511213649 T^{20} + 952809757913927 T^{22} - 504036361936467383 T^{26} + 11592836324538749809 T^{28} - \)\(61\!\cdots\!61\)\( T^{32} + \)\(14\!\cdots\!03\)\( T^{34} - \)\(74\!\cdots\!87\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( ( 1 + 29 T^{2} + 841 T^{4} )^{10} \)
$31$ \( ( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} - 4620 T^{5} + 26881 T^{6} + 35696 T^{7} - 976095 T^{8} + 2797804 T^{9} + 19067729 T^{10} + 86731924 T^{11} - 938027295 T^{12} + 1063419536 T^{13} + 24825168001 T^{14} - 132266677620 T^{15} - 240513497551 T^{16} + 5062320996424 T^{17} - 12793365561615 T^{18} - 105758488642684 T^{19} + 819628286980801 T^{20} )( 1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 5544 T^{5} - 1763 T^{6} + 159523 T^{7} + 1171314 T^{8} + 3253985 T^{9} - 13532839 T^{10} + 100873535 T^{11} + 1125632754 T^{12} + 4752349693 T^{13} - 1628167523 T^{14} - 158720013144 T^{15} - 1060566898795 T^{16} - 2503647884101 T^{17} + 15352038673938 T^{18} + 185077355124697 T^{19} + 819628286980801 T^{20} ) \)
$37$ \( ( 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 9240 T^{5} + 6217 T^{6} + 273493 T^{7} - 3238452 T^{8} + 25503731 T^{9} - 160718317 T^{10} + 943638047 T^{11} - 4433440788 T^{12} + 13853240929 T^{13} + 11651658937 T^{14} - 640738162680 T^{15} + 6617008408811 T^{16} - 49079780477761 T^{17} + 295048274129364 T^{18} - 1429579137745847 T^{19} + 4808584372417849 T^{20} )( 1 + T - 36 T^{2} - 73 T^{3} + 1259 T^{4} + 3960 T^{5} - 42623 T^{6} - 189143 T^{7} + 1387908 T^{8} + 8386199 T^{9} - 42966397 T^{10} + 310289363 T^{11} + 1900046052 T^{12} - 9580660379 T^{13} - 79882364303 T^{14} + 274602069720 T^{15} + 3230249548931 T^{16} - 6930027030709 T^{17} - 126449260341156 T^{18} + 129961739795077 T^{19} + 4808584372417849 T^{20} ) \)
$41$ \( 1 + 41 T^{2} - 68921 T^{6} - 2825761 T^{8} + 4750104241 T^{12} + 194754273881 T^{14} - 327381934393961 T^{18} - 13422659310152401 T^{20} - 550329031716248441 T^{22} + \)\(92\!\cdots\!21\)\( T^{26} + \)\(37\!\cdots\!61\)\( T^{28} - \)\(63\!\cdots\!41\)\( T^{32} - \)\(26\!\cdots\!81\)\( T^{34} + \)\(43\!\cdots\!61\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( ( 1 - 13 T + 126 T^{2} - 1079 T^{3} + 8609 T^{4} - 65520 T^{5} + 481573 T^{6} - 3443089 T^{7} + 24052518 T^{8} - 164629907 T^{9} + 1105930517 T^{10} - 7079086001 T^{11} + 44473105782 T^{12} - 273749677123 T^{13} + 1646402253973 T^{14} - 9631993185360 T^{15} + 54420614488841 T^{16} - 293292281384453 T^{17} + 1472713234977726 T^{18} - 6533703955178959 T^{19} + 21611482313284249 T^{20} )( 1 + 13 T + 126 T^{2} + 1079 T^{3} + 8609 T^{4} + 65520 T^{5} + 481573 T^{6} + 3443089 T^{7} + 24052518 T^{8} + 164629907 T^{9} + 1105930517 T^{10} + 7079086001 T^{11} + 44473105782 T^{12} + 273749677123 T^{13} + 1646402253973 T^{14} + 9631993185360 T^{15} + 54420614488841 T^{16} + 293292281384453 T^{17} + 1472713234977726 T^{18} + 6533703955178959 T^{19} + 21611482313284249 T^{20} ) \)
$47$ \( 1 - 47 T^{2} + 103823 T^{6} - 4879681 T^{8} + 10779215329 T^{12} - 506623120463 T^{14} + 1119130473102767 T^{18} - 52599132235830049 T^{20} + 2472159215084012303 T^{22} - \)\(54\!\cdots\!27\)\( T^{26} + \)\(25\!\cdots\!69\)\( T^{28} - \)\(56\!\cdots\!21\)\( T^{32} + \)\(26\!\cdots\!87\)\( T^{34} - \)\(58\!\cdots\!83\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( ( 1 - 53 T^{2} + 2809 T^{4} - 148877 T^{6} + 7890481 T^{8} - 418195493 T^{10} + 22164361129 T^{12} - 1174711139837 T^{14} + 62259690411361 T^{16} - 3299763591802133 T^{18} + 174887470365513049 T^{20} )^{2} \)
$59$ \( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} + 714924299 T^{10} + 42180533641 T^{12} + 2488651484819 T^{14} + 146830437604321 T^{16} + 8662995818654939 T^{18} + 511116753300641401 T^{20} )^{2} \)
$61$ \( ( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 24570 T^{5} - 38429 T^{6} + 2036776 T^{7} - 26170695 T^{8} + 242146394 T^{9} - 1793637121 T^{10} + 14770930034 T^{11} - 97381156095 T^{12} + 462309453256 T^{13} - 532081823789 T^{14} - 20751731115570 T^{15} + 322981226869109 T^{16} - 3255881578117756 T^{17} + 25880487254632935 T^{18} - 163718045299677974 T^{19} + 713342911662882601 T^{20} )( 1 - 13 T + 108 T^{2} - 611 T^{3} + 1355 T^{4} + 19656 T^{5} - 338183 T^{6} + 3197363 T^{7} - 20936556 T^{8} + 77136085 T^{9} + 274360811 T^{10} + 4705301185 T^{11} - 77904924876 T^{12} + 725740651103 T^{13} - 4682428046903 T^{14} + 16601384892456 T^{15} + 69810107259155 T^{16} - 1920215872808831 T^{17} + 20704389803706348 T^{18} - 152023899206843833 T^{19} + 713342911662882601 T^{20} ) \)
$67$ \( 1 - 11 T + 54 T^{2} + 143 T^{3} - 5191 T^{4} + 47520 T^{5} - 174923 T^{6} - 1259687 T^{7} + 25576398 T^{8} - 196941349 T^{9} + 452736173 T^{10} - 13195070383 T^{11} + 114812450622 T^{12} - 378867241181 T^{13} - 3524894538683 T^{14} + 64157945084640 T^{15} - 469569461839279 T^{16} + 866681759561189 T^{17} + 21927654588058614 T^{18} - 299271878359244417 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 71 T^{2} + 357911 T^{6} - 25411681 T^{8} + 128100283921 T^{12} - 9095120158391 T^{14} + 45848500718449031 T^{18} - 3255243551009881201 T^{20} + \)\(23\!\cdots\!71\)\( T^{22} - \)\(11\!\cdots\!11\)\( T^{26} + \)\(82\!\cdots\!81\)\( T^{28} - \)\(41\!\cdots\!21\)\( T^{32} + \)\(29\!\cdots\!91\)\( T^{34} - \)\(14\!\cdots\!31\)\( T^{38} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( ( 1 + 17 T + 73 T^{2} )^{10}( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 32130 T^{5} + 158383 T^{6} + 3929320 T^{7} + 27731241 T^{8} - 9527950 T^{9} - 2119660093 T^{10} - 695540350 T^{11} + 147779783289 T^{12} + 1528572278440 T^{13} + 4497798604303 T^{14} - 66607790283090 T^{15} - 994417200945019 T^{16} - 5081803318784620 T^{17} + 21774422481140187 T^{18} + 588715867082679130 T^{19} + 4297625829703557649 T^{20} ) \)
$79$ \( ( 1 + 17 T + 79 T^{2} )^{10}( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} + 4363376 T^{7} - 18170271 T^{8} - 272025620 T^{9} + 2523553889 T^{10} - 21490023980 T^{11} - 113400661311 T^{12} + 2151314539664 T^{13} + 353394084913 T^{14} - 171367424973108 T^{15} + 657551567184305 T^{16} + 10907820304138312 T^{17} - 95577855024113343 T^{18} - 479406383930473276 T^{19} + 9468276082626847201 T^{20} ) \)
$83$ \( 1 - 83 T^{2} + 571787 T^{6} - 47458321 T^{8} + 326940373369 T^{12} - 27136050989627 T^{14} + 186940255267540403 T^{18} - 15516041187205853449 T^{20} + \)\(12\!\cdots\!67\)\( T^{22} - \)\(88\!\cdots\!63\)\( T^{26} + \)\(73\!\cdots\!29\)\( T^{28} - \)\(50\!\cdots\!81\)\( T^{32} + \)\(42\!\cdots\!23\)\( T^{34} - \)\(29\!\cdots\!47\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 + 89 T^{2} + 7921 T^{4} + 704969 T^{6} + 62742241 T^{8} + 5584059449 T^{10} + 496981290961 T^{12} + 44231334895529 T^{14} + 3936588805702081 T^{16} + 350356403707485209 T^{18} + 31181719929966183601 T^{20} )^{2} \)
$97$ \( ( 1 - 19 T + 264 T^{2} - 3173 T^{3} + 34679 T^{4} - 351120 T^{5} + 3307417 T^{6} - 28782283 T^{7} + 226043928 T^{8} - 1502953181 T^{9} + 6629849423 T^{10} - 145786458557 T^{11} + 2126847318552 T^{12} - 26268812572459 T^{13} + 292803248977177 T^{14} - 3015186911037840 T^{15} + 28886636158932791 T^{16} - 256372956649052549 T^{17} + 2069082468915517704 T^{18} - 14444390114436739123 T^{19} + 73742412689492826049 T^{20} )( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 526792 T^{7} + 84766473 T^{8} - 1237829446 T^{9} + 9107264363 T^{10} - 120069456262 T^{11} + 797567744457 T^{12} + 480788835016 T^{13} - 84095114902553 T^{14} + 1130695091639190 T^{15} - 7672505137401019 T^{16} - 2262351965387164 T^{17} + 775905925843319139 T^{18} - 10643234821163913038 T^{19} + 73742412689492826049 T^{20} ) \)
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