Properties

Label 480.3.l.a
Level $480$
Weight $3$
Character orbit 480.l
Analytic conductor $13.079$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,3,Mod(161,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 480.l (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.0790526893\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2434 x^{12} - 8780 x^{11} + 25532 x^{10} - 57568 x^{9} + 104312 x^{8} - 150688 x^{7} + 173858 x^{6} - 157972 x^{5} + \cdots + 486 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{3} + \beta_{5} q^{5} + ( - \beta_{10} - \beta_{9}) q^{7} + ( - \beta_{5} + \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{3} + \beta_{5} q^{5} + ( - \beta_{10} - \beta_{9}) q^{7} + ( - \beta_{5} + \beta_{3} - 2) q^{9} + ( - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{4}) q^{11} + ( - \beta_{3} - \beta_{2} - 1) q^{13} - \beta_{4} q^{15} + (\beta_{15} - \beta_{11}) q^{17} + ( - \beta_{12} + \beta_{10} - \beta_{9} + \beta_1) q^{19} + (\beta_{15} + \beta_{14} - \beta_{11} + \beta_{8} + 5) q^{21} + ( - \beta_{13} + \beta_{7} - \beta_{6} + \beta_{4}) q^{23} - 5 q^{25} + ( - \beta_{13} + \beta_{12} + 2 \beta_{10} - \beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{4} - \beta_1) q^{27} + (\beta_{15} + \beta_{14} - \beta_{11} + 7 \beta_{5}) q^{29} + (\beta_{12} - \beta_{10} + 3 \beta_{9} - 2 \beta_{6} - 2 \beta_{4} + \beta_1) q^{31} + (\beta_{15} - 2 \beta_{14} - \beta_{11} + 6 \beta_{5} + \beta_{3} - \beta_{2} - 5) q^{33} + (\beta_{13} + \beta_{6} - \beta_{4}) q^{35} + ( - 2 \beta_{11} + 2 \beta_{8} - 2 \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{37} + (2 \beta_{13} - 3 \beta_{12} + \beta_{10} - \beta_{9} - 2 \beta_{7} - 2 \beta_{6} - \beta_1) q^{39} + (\beta_{15} - \beta_{14} - 2 \beta_{11} - \beta_{8} - 2 \beta_{5} + 2 \beta_{3} - \beta_{2} - 1) q^{41} + (3 \beta_{12} + 2 \beta_{9} + 4 \beta_{6} + 4 \beta_{4}) q^{43} + ( - \beta_{14} - \beta_{11} - 2 \beta_{5} - \beta_{2} + 2) q^{45} + (\beta_{13} - 2 \beta_{12} - 2 \beta_{10} + \beta_{7} - 5 \beta_{6} + 5 \beta_{4}) q^{47} + (\beta_{15} - \beta_{14} + \beta_{8} - 2 \beta_{5} - 3 \beta_{2} + 18) q^{49} + (\beta_{12} + 3 \beta_{9} + 3 \beta_{7} + 3 \beta_{6} + \beta_{4} - 3 \beta_1) q^{51} + (6 \beta_{14} + 4 \beta_{5}) q^{53} + ( - 2 \beta_{12} + 3 \beta_{10} - \beta_{6} - \beta_{4} + \beta_1) q^{55} + (2 \beta_{15} + 2 \beta_{14} - 2 \beta_{11} + 8 \beta_{5} - 3 \beta_{3} + \beta_{2} - 15) q^{57} + (2 \beta_{13} + 4 \beta_{12} + 13 \beta_{10} - 5 \beta_{7} + 5 \beta_{6} - 5 \beta_{4}) q^{59} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{5} + 2 \beta_{3} - 24) q^{61} + ( - \beta_{13} + 2 \beta_{12} - 4 \beta_{10} + 2 \beta_{9} + \beta_{7} - 5 \beta_{6} + \cdots - 4 \beta_1) q^{63}+ \cdots + (2 \beta_{13} + 2 \beta_{12} + 3 \beta_{10} - 4 \beta_{9} + \beta_{7} + 7 \beta_{6} - 7 \beta_{4} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{9} - 16 q^{13} + 88 q^{21} - 80 q^{25} - 64 q^{33} + 16 q^{37} + 40 q^{45} + 320 q^{49} - 272 q^{57} - 368 q^{61} + 104 q^{69} + 256 q^{73} + 192 q^{81} + 416 q^{93} - 544 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2434 x^{12} - 8780 x^{11} + 25532 x^{10} - 57568 x^{9} + 104312 x^{8} - 150688 x^{7} + 173858 x^{6} - 157972 x^{5} + \cdots + 486 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 231098 \nu^{14} + 1617686 \nu^{13} - 18599526 \nu^{12} + 90567238 \nu^{11} - 462307312 \nu^{10} + 1519891728 \nu^{9} - 4141077862 \nu^{8} + \cdots - 175521384 ) / 2058507 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 997042 \nu^{15} - 1058097 \nu^{14} - 24294277 \nu^{13} - 253966185 \nu^{12} + 1130769125 \nu^{11} - 9404278026 \nu^{10} + 34415887678 \nu^{9} + \cdots - 9776463642 ) / 34807482 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 997042 \nu^{15} - 7971087 \nu^{14} + 87498565 \nu^{13} - 472682805 \nu^{12} + 2407469071 \nu^{11} - 8656182288 \nu^{10} + 24958932626 \nu^{9} + \cdots - 1183644576 ) / 34807482 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 36066962 \nu^{15} - 252771207 \nu^{14} + 2913494285 \nu^{13} - 14209516638 \nu^{12} + 72902822759 \nu^{11} - 240460854195 \nu^{10} + \cdots + 1698468912 ) / 382882302 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 116950 \nu^{15} - 877125 \nu^{14} + 9857083 \nu^{13} - 50767977 \nu^{12} + 259594045 \nu^{11} - 898586238 \nu^{10} + 2544488606 \nu^{9} + \cdots - 121233510 ) / 1200258 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14487080 \nu^{15} - 102235232 \nu^{14} + 1175477618 \nu^{13} - 5765621501 \nu^{12} + 29578100336 \nu^{11} - 98063343419 \nu^{10} + \cdots - 1819222686 ) / 127627434 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 129904 \nu^{15} + 974280 \nu^{14} - 10963756 \nu^{13} + 56487834 \nu^{12} - 289508632 \nu^{11} + 1003439778 \nu^{10} - 2854366820 \nu^{9} + \cdots + 174608460 ) / 801009 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2122398 \nu^{15} + 15288809 \nu^{14} - 174045611 \nu^{13} + 867405085 \nu^{12} - 4427792221 \nu^{11} + 14859790638 \nu^{10} - 41080092030 \nu^{9} + \cdots - 431238924 ) / 11602494 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 79528202 \nu^{15} - 581078013 \nu^{14} + 6591134909 \nu^{13} - 33262336089 \nu^{12} + 170207152445 \nu^{11} - 579154679256 \nu^{10} + \cdots - 74336559138 ) / 382882302 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 79528202 \nu^{15} + 597984111 \nu^{14} - 6709477595 \nu^{13} + 34620925101 \nu^{12} - 176820231599 \nu^{11} + 612816576096 \nu^{10} + \cdots + 76625566650 ) / 382882302 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8761702 \nu^{15} + 75149661 \nu^{14} - 803217547 \nu^{13} + 4555430433 \nu^{12} - 23050484461 \nu^{11} + 85806746574 \nu^{10} + \cdots + 17030226186 ) / 34807482 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 79528202 \nu^{15} + 593416323 \nu^{14} - 6677503079 \nu^{13} + 34246129614 \nu^{12} - 174987127385 \nu^{11} + 603156615690 \nu^{10} + \cdots + 64965778326 ) / 191441151 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 374895190 \nu^{15} - 2813236521 \nu^{14} + 31639994869 \nu^{13} - 163071292497 \nu^{12} + 835190424757 \nu^{11} - 2894814185856 \nu^{10} + \cdots - 430899272910 ) / 382882302 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11787550 \nu^{15} - 88406625 \nu^{14} + 992387359 \nu^{13} - 5109684021 \nu^{12} + 26078689825 \nu^{11} - 90176104194 \nu^{10} + \cdots - 10066168566 ) / 11602494 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 45521818 \nu^{15} - 331976739 \nu^{14} + 3766027165 \nu^{13} - 18969909795 \nu^{12} + 96978712123 \nu^{11} - 329159543010 \nu^{10} + \cdots - 26056975698 ) / 34807482 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4 \beta_{15} - 4 \beta_{14} - 3 \beta_{11} + 12 \beta_{10} + \beta_{8} - 6 \beta_{7} - 12 \beta_{6} + \beta_{5} + 12 \beta_{4} - 2 \beta_{3} + \beta_{2} + 25 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{15} + 4 \beta_{14} - 4 \beta_{12} - 3 \beta_{11} - 28 \beta_{10} - 36 \beta_{9} - 7 \beta_{8} - 6 \beta_{7} - 24 \beta_{6} + 17 \beta_{5} - 34 \beta_{3} - 7 \beta_{2} - 18 \beta _1 - 319 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 116 \beta_{15} + 176 \beta_{14} + 18 \beta_{13} + 36 \beta_{12} + 78 \beta_{11} - 132 \beta_{10} - 54 \beta_{9} - 38 \beta_{8} + 111 \beta_{7} + 240 \beta_{6} - 182 \beta_{5} - 276 \beta_{4} + 4 \beta_{3} - 38 \beta_{2} - 27 \beta _1 - 518 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8 \beta_{15} + 112 \beta_{14} + 36 \beta_{13} - 28 \beta_{12} + 117 \beta_{11} + 1172 \beta_{10} + 912 \beta_{9} + 209 \beta_{8} + 228 \beta_{7} + 720 \beta_{6} - 895 \beta_{5} - 336 \beta_{4} + 902 \beta_{3} + 41 \beta_{2} + \cdots + 7121 ) / 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3178 \beta_{15} - 5134 \beta_{14} - 834 \beta_{13} - 1512 \beta_{12} - 2232 \beta_{11} + 4200 \beta_{10} + 2370 \beta_{9} + 1156 \beta_{8} - 3645 \beta_{7} - 6072 \beta_{6} + 7156 \beta_{5} + 7092 \beta_{4} + \cdots + 19240 ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2606 \beta_{15} - 8774 \beta_{14} - 2592 \beta_{13} + 3296 \beta_{12} - 4335 \beta_{11} - 37552 \beta_{10} - 22512 \beta_{9} - 6619 \beta_{8} - 11508 \beta_{7} - 24384 \beta_{6} + 40181 \beta_{5} + \cdots - 174823 ) / 48 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 87110 \beta_{15} + 144158 \beta_{14} + 28734 \beta_{13} + 60024 \beta_{12} + 66108 \beta_{11} - 155904 \beta_{10} - 87150 \beta_{9} - 38852 \beta_{8} + 112785 \beta_{7} + 153288 \beta_{6} + \cdots - 691424 ) / 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 153724 \beta_{15} + 410980 \beta_{14} + 127116 \beta_{13} - 114628 \beta_{12} + 170745 \beta_{11} + 1111724 \beta_{10} + 546624 \beta_{9} + 197021 \beta_{8} + 505380 \beta_{7} + \cdots + 4249301 ) / 48 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2366866 \beta_{15} - 3913510 \beta_{14} - 854118 \beta_{13} - 2238876 \beta_{12} - 1929630 \beta_{11} + 5792844 \beta_{10} + 2992770 \beta_{9} + 1355218 \beta_{8} - 3174585 \beta_{7} + \cdots + 23582830 ) / 48 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6656930 \beta_{15} - 16363346 \beta_{14} - 5242284 \beta_{13} + 2533868 \beta_{12} - 6655041 \beta_{11} - 30741676 \beta_{10} - 12806088 \beta_{9} - 5402113 \beta_{8} + \cdots - 99691693 ) / 48 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 63144098 \beta_{15} + 101700302 \beta_{14} + 22525446 \beta_{13} + 78307596 \beta_{12} + 54514944 \beta_{11} - 208618320 \beta_{10} - 98852226 \beta_{9} + \cdots - 776690336 ) / 48 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 255584704 \beta_{15} + 598074904 \beta_{14} + 194972832 \beta_{13} - 18430144 \beta_{12} + 249887163 \beta_{11} + 783849344 \beta_{10} + 279158208 \beta_{9} + \cdots + 2177062355 ) / 48 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1632969982 \beta_{15} - 2482933414 \beta_{14} - 516343602 \beta_{13} - 2585548212 \beta_{12} - 1470833718 \beta_{11} + 7231885752 \beta_{10} + 3174598518 \beta_{9} + \cdots + 24914694430 ) / 48 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 9174874982 \beta_{15} - 20661944774 \beta_{14} - 6755761692 \beta_{13} - 1791403060 \beta_{12} - 8997167163 \beta_{11} - 17770036420 \beta_{10} + \cdots - 40707438727 ) / 48 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 40159943810 \beta_{15} + 54788157086 \beta_{14} + 9230988438 \beta_{13} + 81124707744 \beta_{12} + 37227162516 \beta_{11} - 241420340724 \beta_{10} + \cdots - 780594138464 ) / 48 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(421\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.500000 5.50891i
0.500000 + 5.50891i
0.500000 + 1.18553i
0.500000 1.18553i
0.500000 + 1.73577i
0.500000 1.73577i
0.500000 0.0650247i
0.500000 + 0.0650247i
0.500000 + 0.585971i
0.500000 0.585971i
0.500000 + 0.383062i
0.500000 0.383062i
0.500000 + 4.54278i
0.500000 4.54278i
0.500000 1.44994i
0.500000 + 1.44994i
0 −2.87012 0.873151i 0 2.23607i 0 −11.2043 0 7.47522 + 5.01210i 0
161.2 0 −2.87012 + 0.873151i 0 2.23607i 0 −11.2043 0 7.47522 5.01210i 0
161.3 0 −2.37394 1.83423i 0 2.23607i 0 3.76207 0 2.27118 + 8.70872i 0
161.4 0 −2.37394 + 1.83423i 0 2.23607i 0 3.76207 0 2.27118 8.70872i 0
161.5 0 −0.956507 2.84343i 0 2.23607i 0 −3.99659 0 −7.17019 + 5.43952i 0
161.6 0 −0.956507 + 2.84343i 0 2.23607i 0 −3.99659 0 −7.17019 5.43952i 0
161.7 0 −0.460323 2.96447i 0 2.23607i 0 10.9698 0 −8.57621 + 2.72923i 0
161.8 0 −0.460323 + 2.96447i 0 2.23607i 0 10.9698 0 −8.57621 2.72923i 0
161.9 0 0.460323 2.96447i 0 2.23607i 0 −10.9698 0 −8.57621 2.72923i 0
161.10 0 0.460323 + 2.96447i 0 2.23607i 0 −10.9698 0 −8.57621 + 2.72923i 0
161.11 0 0.956507 2.84343i 0 2.23607i 0 3.99659 0 −7.17019 5.43952i 0
161.12 0 0.956507 + 2.84343i 0 2.23607i 0 3.99659 0 −7.17019 + 5.43952i 0
161.13 0 2.37394 1.83423i 0 2.23607i 0 −3.76207 0 2.27118 8.70872i 0
161.14 0 2.37394 + 1.83423i 0 2.23607i 0 −3.76207 0 2.27118 + 8.70872i 0
161.15 0 2.87012 0.873151i 0 2.23607i 0 11.2043 0 7.47522 5.01210i 0
161.16 0 2.87012 + 0.873151i 0 2.23607i 0 11.2043 0 7.47522 + 5.01210i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.3.l.a 16
3.b odd 2 1 inner 480.3.l.a 16
4.b odd 2 1 inner 480.3.l.a 16
8.b even 2 1 960.3.l.i 16
8.d odd 2 1 960.3.l.i 16
12.b even 2 1 inner 480.3.l.a 16
24.f even 2 1 960.3.l.i 16
24.h odd 2 1 960.3.l.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.3.l.a 16 1.a even 1 1 trivial
480.3.l.a 16 3.b odd 2 1 inner
480.3.l.a 16 4.b odd 2 1 inner
480.3.l.a 16 12.b even 2 1 inner
960.3.l.i 16 8.b even 2 1
960.3.l.i 16 8.d odd 2 1
960.3.l.i 16 24.f even 2 1
960.3.l.i 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 276T_{7}^{6} + 22740T_{7}^{4} - 510688T_{7}^{2} + 3415104 \) acting on \(S_{3}^{\mathrm{new}}(480, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 12 T^{14} + 24 T^{12} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 276 T^{6} + 22740 T^{4} + \cdots + 3415104)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 408 T^{6} + 39888 T^{4} + \cdots + 3326976)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} - 468 T^{2} - 2784 T - 864)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 960 T^{6} + 227424 T^{4} + \cdots + 2214144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 1192 T^{6} + 400080 T^{4} + \cdots + 827597824)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1508 T^{6} + 628404 T^{4} + \cdots + 554319936)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 2888 T^{6} + \cdots + 22562443264)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4984 T^{6} + \cdots + 312025022464)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} - 4788 T^{2} + \cdots + 2941856)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 6920 T^{6} + \cdots + 286473293824)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 7324 T^{6} + \cdots + 608824393984)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 9668 T^{6} + \cdots + 6335208456256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 18896 T^{6} + \cdots + 110324940730624)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 18344 T^{6} + \cdots + 2358116499456)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 92 T^{3} + 1380 T^{2} + \cdots + 7424)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 18204 T^{6} + \cdots + 9194285355264)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 10352 T^{6} + \cdots + 71430045696)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 64 T^{3} - 11208 T^{2} + \cdots + 27175696)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 42600 T^{6} + \cdots + 17\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 21332 T^{6} + \cdots + 572716500798016)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 33440 T^{6} + \cdots + 370293356888064)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 136 T^{3} - 21768 T^{2} + \cdots - 178613104)^{4} \) Copy content Toggle raw display
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