Properties

Label 1050.3.f.e
Level $1050$
Weight $3$
Character orbit 1050.f
Analytic conductor $28.610$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + 259514 x^{8} - 486436 x^{7} + 690168 x^{6} - 725188 x^{5} + \cdots + 33124 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_1 q^{3} + 2 q^{4} - \beta_{8} q^{6} + ( - \beta_{6} - \beta_{4} + \beta_1) q^{7} - 2 \beta_{5} q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_1 q^{3} + 2 q^{4} - \beta_{8} q^{6} + ( - \beta_{6} - \beta_{4} + \beta_1) q^{7} - 2 \beta_{5} q^{8} - 3 q^{9} + ( - \beta_{11} + \beta_{10} + 6) q^{11} - 2 \beta_1 q^{12} + (\beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 3 \beta_1) q^{13} + (\beta_{13} + \beta_{8} + \beta_{2} - 1) q^{14} + 4 q^{16} + ( - 2 \beta_{4} - \beta_{3} - \beta_1) q^{17} + 3 \beta_{5} q^{18} + (\beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{9} - 1) q^{19} + ( - \beta_{13} + \beta_{10} + \beta_{2} + 1) q^{21} + ( - \beta_{15} - \beta_{14} - 6 \beta_{5}) q^{22} + (\beta_{15} - 2 \beta_{14} + \beta_{7} + \beta_{6} + 5 \beta_{5} - \beta_1) q^{23} - 2 \beta_{8} q^{24} + ( - 2 \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{8} + 1) q^{26} + 3 \beta_1 q^{27} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{28} + (2 \beta_{11} + 2 \beta_{9} - 4 \beta_{2} - 4) q^{29} + (4 \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{9} - 4 \beta_{8} + 2) q^{31} - 4 \beta_{5} q^{32} + ( - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} - 6 \beta_1) q^{33} + (3 \beta_{13} + 3 \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{8} + 1) q^{34} - 6 q^{36} + (\beta_{15} - 3 \beta_{14} - 3 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 3 \beta_1) q^{37} + (3 \beta_{7} - 3 \beta_{6} - 4 \beta_{4} + 2 \beta_{3} + \beta_1) q^{38} + ( - 2 \beta_{10} - 2 \beta_{9} + 3 \beta_{2} + 9) q^{39} + (2 \beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} + 5 \beta_{8} + 1) q^{41} + ( - \beta_{14} - 2 \beta_{7} - 3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{42} + (2 \beta_{14} - 6 \beta_{7} - 6 \beta_{6} - 18 \beta_{5} + 6 \beta_1) q^{43} + ( - 2 \beta_{11} + 2 \beta_{10} + 12) q^{44} + ( - \beta_{11} + 4 \beta_{10} + 3 \beta_{9} - 2 \beta_{2} - 11) q^{46} + (5 \beta_{7} - 5 \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - 13 \beta_1) q^{47} - 4 \beta_1 q^{48} + ( - 2 \beta_{12} - 3 \beta_{11} + 2 \beta_{10} - \beta_{9} + 10 \beta_{8} - 2 \beta_{2} + \cdots - 14) q^{49}+ \cdots + (3 \beta_{11} - 3 \beta_{10} - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 48 q^{9} + 96 q^{11} - 16 q^{14} + 64 q^{16} + 24 q^{21} - 64 q^{29} - 96 q^{36} + 144 q^{39} + 192 q^{44} - 176 q^{46} - 224 q^{49} - 48 q^{51} - 32 q^{56} + 128 q^{64} - 384 q^{71} - 224 q^{74} + 608 q^{79} + 144 q^{81} + 48 q^{84} + 416 q^{86} + 224 q^{91} - 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + 259514 x^{8} - 486436 x^{7} + 690168 x^{6} - 725188 x^{5} + \cdots + 33124 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 92936 \nu^{14} - 650552 \nu^{13} + 7629434 \nu^{12} - 37319428 \nu^{11} + 225649370 \nu^{10} - 801656916 \nu^{9} + 3054133421 \nu^{8} + \cdots - 649676357 ) / 5030871765 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4788335 \nu^{14} + 33518345 \nu^{13} - 384267955 \nu^{12} + 1869869245 \nu^{11} - 10965281907 \nu^{10} + 38484795345 \nu^{9} + \cdots + 205883061930 ) / 119428521030 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 116229757 \nu^{14} + 813608299 \nu^{13} - 9262265428 \nu^{12} + 44996684681 \nu^{11} - 261565786159 \nu^{10} + \cdots + 2568347439658 ) / 2746855983690 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1093349498 \nu^{14} + 7653446486 \nu^{13} - 88689952367 \nu^{12} + 432644909884 \nu^{11} - 2578996359596 \nu^{10} + \cdots + 12465433464302 ) / 8240567951070 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 279842740 \nu^{15} - 2098820550 \nu^{14} + 23452089504 \nu^{13} - 120606470101 \nu^{12} + 697811667584 \nu^{11} + \cdots + 20065240923728 ) / 21993425639458 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1064447881686 \nu^{15} - 3209377373879 \nu^{14} + 57853449494147 \nu^{13} - 83261406982037 \nu^{12} + 909308118211537 \nu^{11} + \cdots - 22\!\cdots\!14 ) / 68\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1064447881686 \nu^{15} - 11495817736195 \nu^{14} + 115858532030359 \nu^{13} - 757541135750011 \nu^{12} + \cdots + 63\!\cdots\!20 ) / 68\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12448 \nu^{15} - 93360 \nu^{14} + 1065318 \nu^{13} - 5508607 \nu^{12} + 32749646 \nu^{11} - 122643488 \nu^{10} + 464891353 \nu^{9} - 1256401704 \nu^{8} + \cdots - 179130952 ) / 520340730 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3854937063308 \nu^{15} + 23962636724121 \nu^{14} - 295302862411617 \nu^{13} + \cdots + 74\!\cdots\!24 ) / 68\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3854937063308 \nu^{15} - 30212752287741 \nu^{14} + 339053671356957 \nu^{13} + \cdots + 53\!\cdots\!46 ) / 68\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3854937063308 \nu^{15} - 33861419225499 \nu^{14} + 364594339921263 \nu^{13} + \cdots + 70\!\cdots\!94 ) / 68\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5472163425770 \nu^{15} - 41041225693275 \nu^{14} + 464858064633033 \nu^{13} + \cdots - 62\!\cdots\!92 ) / 68\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7209986841170 \nu^{15} - 54074901308775 \nu^{14} + 610495540452873 \nu^{13} + \cdots - 74\!\cdots\!92 ) / 68\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11652755758 \nu^{15} - 87395668185 \nu^{14} + 980333632691 \nu^{13} - 5046667645019 \nu^{12} + 29328141084055 \nu^{11} + \cdots + 767874169626928 ) / 109967128197290 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 165988958814 \nu^{15} + 1244917191105 \nu^{14} - 14017718745515 \nu^{13} + 72233927780755 \nu^{12} + \cdots - 92\!\cdots\!92 ) / 989704153775610 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{13} - \beta_{12} - 2\beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - \beta_{12} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{2} - 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{14} - 9 \beta_{13} + 17 \beta_{12} - \beta_{11} + 3 \beta_{10} + 4 \beta_{9} - 11 \beta_{8} + 9 \beta_{7} + 3 \beta_{6} + 47 \beta_{5} - 3 \beta_{4} - 6 \beta_{2} - 6 \beta _1 - 42 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{15} - 6 \beta_{14} - 19 \beta_{13} + 35 \beta_{12} - 12 \beta_{11} - 12 \beta_{10} - 20 \beta_{9} - 22 \beta_{8} - 44 \beta_{7} + 68 \beta_{6} + 96 \beta_{5} + 44 \beta_{4} - 4 \beta_{3} + 64 \beta_{2} - 72 \beta _1 + 330 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 90 \beta_{15} + 40 \beta_{14} + 101 \beta_{13} - 185 \beta_{12} - 37 \beta_{11} - 35 \beta_{10} - 48 \beta_{9} + 233 \beta_{8} - 325 \beta_{7} - 35 \beta_{6} - 1083 \beta_{5} + 115 \beta_{4} - 10 \beta_{3} + 170 \beta_{2} + \cdots + 904 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 285 \beta_{15} + 135 \beta_{14} + 351 \beta_{13} - 643 \beta_{12} + 174 \beta_{11} - 20 \beta_{10} + 216 \beta_{9} + 754 \beta_{8} + 572 \beta_{7} - 1712 \beta_{6} - 3490 \beta_{5} - 758 \beta_{4} - 52 \beta_{3} + \cdots - 2742 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2009 \beta_{15} - 224 \beta_{14} - 619 \beta_{13} + 1156 \beta_{12} + 1081 \beta_{11} + 56 \beta_{10} + 585 \beta_{9} - 2444 \beta_{8} + 8211 \beta_{7} - 805 \beta_{6} + 19276 \beta_{5} - 3059 \beta_{4} - 147 \beta_{3} + \cdots - 13154 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9380 \beta_{15} - 1540 \beta_{14} - 4159 \beta_{13} + 7707 \beta_{12} - 186 \beta_{11} + 1340 \beta_{10} - 1334 \beta_{9} - 13346 \beta_{8} - 792 \beta_{7} + 35792 \beta_{6} + 93616 \beta_{5} + 10240 \beta_{4} + \cdots + 272 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 32508 \beta_{15} - 2622 \beta_{14} - 6786 \beta_{13} + 12086 \beta_{12} - 12235 \beta_{11} + 5541 \beta_{10} - 4964 \beta_{9} - 6575 \beta_{8} - 167163 \beta_{7} + 52791 \beta_{6} - 266315 \beta_{5} + \cdots + 88788 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 234915 \beta_{15} - 585 \beta_{14} - 184 \beta_{13} - 2050 \beta_{12} - 42672 \beta_{11} - 8352 \beta_{10} - 23300 \beta_{9} + 72608 \beta_{8} - 206946 \beta_{7} - 635514 \beta_{6} - 2058608 \beta_{5} + \cdots + 569434 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 329714 \beta_{15} + 80366 \beta_{14} + 305101 \beta_{13} - 561877 \beta_{12} - 115963 \beta_{11} - 95711 \beta_{10} - 69948 \beta_{9} + 872117 \beta_{8} + 2794847 \beta_{7} - 1680921 \beta_{6} + \cdots + 2168840 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 2361744 \beta_{15} + 230076 \beta_{14} + 877983 \beta_{13} - 1603409 \beta_{12} + 435912 \beta_{11} - 118765 \beta_{10} + 568668 \beta_{9} + 2098427 \beta_{8} + 3670524 \beta_{7} + \cdots - 6519638 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 667368 \beta_{15} - 692783 \beta_{14} - 5309960 \beta_{13} + 9895207 \beta_{12} + 9475281 \beta_{11} - 171496 \beta_{10} + 5519413 \beta_{9} - 18651514 \beta_{8} - 36490831 \beta_{7} + \cdots - 112895206 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 75505976 \beta_{15} - 11723894 \beta_{14} - 63474043 \beta_{13} + 117216695 \beta_{12} - 249204 \beta_{11} + 6881616 \beta_{10} - 26378810 \beta_{9} - 190758814 \beta_{8} + \cdots + 132095548 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 124496877 \beta_{15} - 11320543 \beta_{14} + 24283611 \beta_{13} - 46833803 \beta_{12} - 276574061 \beta_{11} + 49618467 \beta_{10} - 191575552 \beta_{9} + \cdots + 3156761778 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
−0.207107 4.25887i
−0.207107 0.264184i
−0.207107 + 3.39361i
−0.207107 + 1.12945i
−0.207107 + 4.25887i
−0.207107 + 0.264184i
−0.207107 3.39361i
−0.207107 1.12945i
1.20711 1.12945i
1.20711 3.39361i
1.20711 + 0.264184i
1.20711 + 4.25887i
1.20711 + 1.12945i
1.20711 + 3.39361i
1.20711 0.264184i
1.20711 4.25887i
−1.41421 1.73205i 2.00000 0 2.44949i −4.63050 + 5.24961i −2.82843 −3.00000 0
601.2 −1.41421 1.73205i 2.00000 0 2.44949i −0.853218 + 6.94781i −2.82843 −3.00000 0
601.3 −1.41421 1.73205i 2.00000 0 2.44949i 1.57881 6.81963i −2.82843 −3.00000 0
601.4 −1.41421 1.73205i 2.00000 0 2.44949i 6.73333 1.91369i −2.82843 −3.00000 0
601.5 −1.41421 1.73205i 2.00000 0 2.44949i −4.63050 5.24961i −2.82843 −3.00000 0
601.6 −1.41421 1.73205i 2.00000 0 2.44949i −0.853218 6.94781i −2.82843 −3.00000 0
601.7 −1.41421 1.73205i 2.00000 0 2.44949i 1.57881 + 6.81963i −2.82843 −3.00000 0
601.8 −1.41421 1.73205i 2.00000 0 2.44949i 6.73333 + 1.91369i −2.82843 −3.00000 0
601.9 1.41421 1.73205i 2.00000 0 2.44949i −6.73333 1.91369i 2.82843 −3.00000 0
601.10 1.41421 1.73205i 2.00000 0 2.44949i −1.57881 6.81963i 2.82843 −3.00000 0
601.11 1.41421 1.73205i 2.00000 0 2.44949i 0.853218 + 6.94781i 2.82843 −3.00000 0
601.12 1.41421 1.73205i 2.00000 0 2.44949i 4.63050 + 5.24961i 2.82843 −3.00000 0
601.13 1.41421 1.73205i 2.00000 0 2.44949i −6.73333 + 1.91369i 2.82843 −3.00000 0
601.14 1.41421 1.73205i 2.00000 0 2.44949i −1.57881 + 6.81963i 2.82843 −3.00000 0
601.15 1.41421 1.73205i 2.00000 0 2.44949i 0.853218 6.94781i 2.82843 −3.00000 0
601.16 1.41421 1.73205i 2.00000 0 2.44949i 4.63050 5.24961i 2.82843 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.f.e 16
5.b even 2 1 inner 1050.3.f.e 16
5.c odd 4 2 210.3.h.a 16
7.b odd 2 1 inner 1050.3.f.e 16
15.e even 4 2 630.3.h.e 16
20.e even 4 2 1680.3.bd.a 16
35.c odd 2 1 inner 1050.3.f.e 16
35.f even 4 2 210.3.h.a 16
105.k odd 4 2 630.3.h.e 16
140.j odd 4 2 1680.3.bd.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.h.a 16 5.c odd 4 2
210.3.h.a 16 35.f even 4 2
630.3.h.e 16 15.e even 4 2
630.3.h.e 16 105.k odd 4 2
1050.3.f.e 16 1.a even 1 1 trivial
1050.3.f.e 16 5.b even 2 1 inner
1050.3.f.e 16 7.b odd 2 1 inner
1050.3.f.e 16 35.c odd 2 1 inner
1680.3.bd.a 16 20.e even 4 2
1680.3.bd.a 16 140.j odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{4} - 24T_{11}^{3} + 28T_{11}^{2} + 1776T_{11} - 4844 \) Copy content Toggle raw display
\( T_{23}^{8} - 2932T_{23}^{6} + 2745076T_{23}^{4} - 843688800T_{23}^{2} + 11186869824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 112 T^{14} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{4} - 24 T^{3} + 28 T^{2} + 1776 T - 4844)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 1044 T^{6} + 313732 T^{4} + \cdots + 586802176)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 996 T^{6} + 270436 T^{4} + \cdots + 338560000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1796 T^{6} + \cdots + 2149991424)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2932 T^{6} + \cdots + 11186869824)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 16 T^{3} - 2088 T^{2} + \cdots + 19600)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3740 T^{6} + 1609684 T^{4} + \cdots + 747256896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 8536 T^{6} + \cdots + 3617786594304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 932 T^{6} + 280612 T^{4} + \cdots + 1141899264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 12880 T^{6} + \cdots + 9151544623104)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 10904 T^{6} + \cdots + 14545741676544)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 15604 T^{6} + \cdots + 38389325511744)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 16112 T^{6} + \cdots + 203235065856)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 13824 T^{6} + \cdots + 72666906624)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 17224 T^{6} + \cdots + 404129746944)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 96 T^{3} - 6116 T^{2} + \cdots - 18715436)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 24020 T^{6} + \cdots + 105841627140096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 152 T^{3} + 3952 T^{2} + \cdots - 12612096)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 15480 T^{6} + \cdots + 13129665286144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 24860 T^{6} + \cdots + 135150669186624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 20468 T^{6} + \cdots + 5898136817664)^{2} \) Copy content Toggle raw display
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