Properties

Label 210.3.e.a
Level $210$
Weight $3$
Character orbit 210.e
Analytic conductor $5.722$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,3,Mod(71,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.71");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 210.e (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: \(5.72208555157\)
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: \( x^{16} - 8 x^{15} + 34 x^{14} - 80 x^{13} + 97 x^{12} - 80 x^{11} + 498 x^{10} - 3288 x^{9} + 11844 x^{8} - 29592 x^{7} + 40338 x^{6} - 58320 x^{5} + 636417 x^{4} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_1 q^{3} - 2 q^{4} - \beta_{9} q^{5} + ( - \beta_{13} + 1) q^{6} - \beta_{5} q^{7} + 2 \beta_{4} q^{8} + (\beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{5} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_1 q^{3} - 2 q^{4} - \beta_{9} q^{5} + ( - \beta_{13} + 1) q^{6} - \beta_{5} q^{7} + 2 \beta_{4} q^{8} + (\beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{5} + \beta_1 - 1) q^{9} - \beta_{10} q^{10} + (\beta_{14} - \beta_{11} + \beta_{8} + \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - \beta_1) q^{11} + 2 \beta_1 q^{12} + (2 \beta_{14} - \beta_{13} + \beta_{10} - 2 \beta_{8} + \beta_{6} - 2 \beta_{4} - \beta_1) q^{13} + \beta_{11} q^{14} + (\beta_{14} + \beta_{10} + \beta_{9} + \beta_{6} - 2) q^{15} + 4 q^{16} + (2 \beta_{14} + 2 \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 6 \beta_{4} - \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{17}+ \cdots + ( - 12 \beta_{15} - 2 \beta_{14} - \beta_{13} + 6 \beta_{12} + 12 \beta_{11} + 13 \beta_{10} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} - 32 q^{4} + 16 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} - 32 q^{4} + 16 q^{6} - 4 q^{9} + 16 q^{12} - 20 q^{15} + 64 q^{16} - 32 q^{18} + 48 q^{19} + 28 q^{21} - 96 q^{22} - 32 q^{24} - 80 q^{25} + 64 q^{27} - 88 q^{33} + 160 q^{34} + 8 q^{36} + 80 q^{37} + 156 q^{39} - 336 q^{43} - 80 q^{45} + 32 q^{46} - 32 q^{48} + 112 q^{49} + 84 q^{51} - 32 q^{54} - 80 q^{55} - 264 q^{57} + 96 q^{58} + 40 q^{60} + 112 q^{61} + 112 q^{63} - 128 q^{64} + 240 q^{67} + 8 q^{69} + 64 q^{72} + 48 q^{73} + 40 q^{75} - 96 q^{76} + 208 q^{78} + 8 q^{79} - 124 q^{81} - 608 q^{82} - 56 q^{84} + 120 q^{85} - 120 q^{87} + 192 q^{88} + 160 q^{90} - 56 q^{91} + 104 q^{93} + 32 q^{94} + 64 q^{96} - 192 q^{97} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 34 x^{14} - 80 x^{13} + 97 x^{12} - 80 x^{11} + 498 x^{10} - 3288 x^{9} + 11844 x^{8} - 29592 x^{7} + 40338 x^{6} - 58320 x^{5} + 636417 x^{4} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 52123 \nu^{15} - 1700114 \nu^{14} + 6007501 \nu^{13} - 12749018 \nu^{12} - 1811744 \nu^{11} + 7604824 \nu^{10} + 87057894 \nu^{9} + \cdots - 4939018146594 ) / 586124153136 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 399407 \nu^{15} - 1791193 \nu^{14} - 7693021 \nu^{13} + 32003807 \nu^{12} - 38173168 \nu^{11} - 127960096 \nu^{10} + 562590078 \nu^{9} + \cdots + 15882398605935 ) / 2344496612544 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 242026 \nu^{15} + 942563 \nu^{14} - 1277806 \nu^{13} + 701903 \nu^{12} + 2378336 \nu^{11} + 16831352 \nu^{10} - 49349244 \nu^{9} + \cdots - 348051871161 ) / 1172248306272 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13307 \nu^{15} - 179536 \nu^{14} + 640907 \nu^{13} - 1208056 \nu^{12} + 207224 \nu^{11} + 1262120 \nu^{10} + 11237550 \nu^{9} - 68579808 \nu^{8} + \cdots - 532478372832 ) / 63364773312 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 317857 \nu^{15} + 4721090 \nu^{14} - 19290205 \nu^{13} + 36928814 \nu^{12} - 37149256 \nu^{11} + 4023536 \nu^{10} - 309774954 \nu^{9} + \cdots + 19313925366078 ) / 1172248306272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 220079 \nu^{15} - 1436965 \nu^{14} + 2437835 \nu^{13} - 629917 \nu^{12} - 8683744 \nu^{11} - 35001376 \nu^{10} + 210536430 \nu^{9} + \cdots - 111447960669 ) / 781498870848 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 758476 \nu^{15} + 6985817 \nu^{14} - 22326856 \nu^{13} + 53671265 \nu^{12} - 47350168 \nu^{11} + 123531416 \nu^{10} + \cdots + 20507080029849 ) / 2344496612544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33544 \nu^{15} + 152423 \nu^{14} - 301840 \nu^{13} + 248939 \nu^{12} + 144488 \nu^{11} - 671464 \nu^{10} - 13125864 \nu^{9} + 43804902 \nu^{8} + \cdots + 49412852739 ) / 101934635328 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16127 \nu^{15} - 95230 \nu^{14} + 302735 \nu^{13} - 423640 \nu^{12} - 101032 \nu^{11} - 568144 \nu^{10} + 6767574 \nu^{9} - 30501636 \nu^{8} + \cdots - 173181741552 ) / 31682386656 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29989 \nu^{15} - 210221 \nu^{14} + 754963 \nu^{13} - 1055177 \nu^{12} - 61616 \nu^{11} - 4040 \nu^{10} + 18376986 \nu^{9} - 82340922 \nu^{8} + \cdots - 467281722393 ) / 50967317664 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 343 \nu^{15} + 3788 \nu^{14} - 14263 \nu^{13} + 22760 \nu^{12} - 13840 \nu^{11} - 1864 \nu^{10} - 366438 \nu^{9} + 1648344 \nu^{8} - 3942990 \nu^{7} + \cdots + 11096488080 ) / 459165024 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 110405 \nu^{15} - 772342 \nu^{14} + 2073353 \nu^{13} - 2872762 \nu^{12} + 281192 \nu^{11} - 7908856 \nu^{10} + 48773778 \nu^{9} + \cdots - 1027353043386 ) / 130249811808 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 817513 \nu^{15} + 4462733 \nu^{14} - 10958341 \nu^{13} + 9975341 \nu^{12} + 429008 \nu^{11} + 43919120 \nu^{10} - 369524802 \nu^{9} + \cdots + 3650385855645 ) / 781498870848 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2027164 \nu^{15} + 11413589 \nu^{14} - 32738950 \nu^{13} + 39772031 \nu^{12} - 22732624 \nu^{11} + 144647024 \nu^{10} + \cdots + 17674749277119 ) / 1172248306272 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{5} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + \beta_{11} - 7 \beta_{10} + 5 \beta_{9} - 6 \beta_{7} + 6 \beta_{6} + 11 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 8 \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} + 20 \beta_{14} - 10 \beta_{13} - 2 \beta_{12} - 8 \beta_{11} + 32 \beta_{10} + 10 \beta_{9} + 5 \beta_{8} - 10 \beta_{7} - 9 \beta_{6} + 27 \beta_{5} - 50 \beta_{4} + 21 \beta_{3} - 16 \beta_{2} - 12 \beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 18 \beta_{15} + 40 \beta_{14} + 46 \beta_{13} + 24 \beta_{12} - 56 \beta_{11} + 80 \beta_{10} - 70 \beta_{9} + 30 \beta_{7} + 6 \beta_{6} + 86 \beta_{5} + 294 \beta_{4} - 60 \beta_{2} + 52 \beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 98 \beta_{15} + 8 \beta_{14} - 190 \beta_{13} + 154 \beta_{12} + 40 \beta_{11} - 64 \beta_{10} - 38 \beta_{9} + 56 \beta_{8} + 182 \beta_{7} - 174 \beta_{6} + 222 \beta_{5} + 88 \beta_{4} + 12 \beta_{3} + 8 \beta_{2} + 15 \beta _1 + 536 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 102 \beta_{15} - 221 \beta_{14} - 167 \beta_{13} + 336 \beta_{12} + 175 \beta_{11} + 185 \beta_{10} - 649 \beta_{9} - 72 \beta_{8} - 294 \beta_{7} - 318 \beta_{6} - 829 \beta_{5} + 1806 \beta_{4} + 204 \beta_{3} + 672 \beta_{2} + \cdots - 259 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 157 \beta_{15} + 113 \beta_{14} - 1540 \beta_{13} - 851 \beta_{12} + 1708 \beta_{11} - 628 \beta_{10} - 2975 \beta_{9} - 2071 \beta_{8} - 157 \beta_{7} + 1146 \beta_{6} - 2778 \beta_{5} + 2863 \beta_{4} + 984 \beta_{3} + \cdots - 2524 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3612 \beta_{15} - 1760 \beta_{14} - 3188 \beta_{13} - 1872 \beta_{12} + 1855 \beta_{11} - 3985 \beta_{10} - 19291 \beta_{9} + 1296 \beta_{8} - 3876 \beta_{7} - 444 \beta_{6} + 4763 \beta_{5} - 3603 \beta_{4} + \cdots + 1256 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4196 \beta_{15} - 7504 \beta_{14} + 548 \beta_{13} - 6356 \beta_{12} - 18944 \beta_{11} - 27616 \beta_{10} - 20348 \beta_{9} + 1427 \beta_{8} - 9028 \beta_{7} - 16425 \beta_{6} - 6561 \beta_{5} + \cdots + 68243 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 25596 \beta_{15} - 7736 \beta_{14} - 2996 \beta_{13} - 12000 \beta_{12} - 4304 \beta_{11} - 29440 \beta_{10} + 45740 \beta_{9} + 79920 \beta_{8} - 8124 \beta_{7} - 28236 \beta_{6} + 22100 \beta_{5} + \cdots - 143191 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 60164 \beta_{15} + 72440 \beta_{14} + 121148 \beta_{13} + 6292 \beta_{12} + 187456 \beta_{11} + 35936 \beta_{10} + 120292 \beta_{9} + 114608 \beta_{8} - 28132 \beta_{7} - 156 \beta_{6} + \cdots + 496904 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 101604 \beta_{15} - 179663 \beta_{14} - 520475 \beta_{13} + 117264 \beta_{12} + 646033 \beta_{11} + 329279 \beta_{10} + 697427 \beta_{9} - 250848 \beta_{8} - 30372 \beta_{7} + \cdots + 582167 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 810131 \beta_{15} + 1160807 \beta_{14} - 1263466 \beta_{13} + 90013 \beta_{12} + 589588 \beta_{11} + 1586252 \beta_{10} - 2166683 \beta_{9} - 572833 \beta_{8} - 837229 \beta_{7} + \cdots - 2515036 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
2.97371 + 0.396304i
2.53169 1.60951i
1.86110 2.35293i
1.51079 + 2.59182i
0.812085 2.88800i
−0.650833 2.92855i
−2.18398 + 2.05675i
−2.85457 0.922735i
2.97371 0.396304i
2.53169 + 1.60951i
1.86110 + 2.35293i
1.51079 2.59182i
0.812085 + 2.88800i
−0.650833 + 2.92855i
−2.18398 2.05675i
−2.85457 + 0.922735i
1.41421i −2.97371 0.396304i −2.00000 2.23607i −0.560459 + 4.20546i −2.64575 2.82843i 8.68589 + 2.35699i 3.16228
71.2 1.41421i −2.53169 + 1.60951i −2.00000 2.23607i 2.27619 + 3.58035i 2.64575 2.82843i 3.81894 8.14958i 3.16228
71.3 1.41421i −1.86110 + 2.35293i −2.00000 2.23607i 3.32755 + 2.63200i 2.64575 2.82843i −2.07259 8.75810i −3.16228
71.4 1.41421i −1.51079 2.59182i −2.00000 2.23607i −3.66538 + 2.13658i −2.64575 2.82843i −4.43502 + 7.83139i −3.16228
71.5 1.41421i −0.812085 + 2.88800i −2.00000 2.23607i 4.08424 + 1.14846i −2.64575 2.82843i −7.68104 4.69059i −3.16228
71.6 1.41421i 0.650833 + 2.92855i −2.00000 2.23607i 4.14160 0.920417i −2.64575 2.82843i −8.15283 + 3.81200i 3.16228
71.7 1.41421i 2.18398 2.05675i −2.00000 2.23607i −2.90869 3.08861i 2.64575 2.82843i 0.539533 8.98381i −3.16228
71.8 1.41421i 2.85457 + 0.922735i −2.00000 2.23607i 1.30494 4.03697i 2.64575 2.82843i 7.29712 + 5.26802i 3.16228
71.9 1.41421i −2.97371 + 0.396304i −2.00000 2.23607i −0.560459 4.20546i −2.64575 2.82843i 8.68589 2.35699i 3.16228
71.10 1.41421i −2.53169 1.60951i −2.00000 2.23607i 2.27619 3.58035i 2.64575 2.82843i 3.81894 + 8.14958i 3.16228
71.11 1.41421i −1.86110 2.35293i −2.00000 2.23607i 3.32755 2.63200i 2.64575 2.82843i −2.07259 + 8.75810i −3.16228
71.12 1.41421i −1.51079 + 2.59182i −2.00000 2.23607i −3.66538 2.13658i −2.64575 2.82843i −4.43502 7.83139i −3.16228
71.13 1.41421i −0.812085 2.88800i −2.00000 2.23607i 4.08424 1.14846i −2.64575 2.82843i −7.68104 + 4.69059i −3.16228
71.14 1.41421i 0.650833 2.92855i −2.00000 2.23607i 4.14160 + 0.920417i −2.64575 2.82843i −8.15283 3.81200i 3.16228
71.15 1.41421i 2.18398 + 2.05675i −2.00000 2.23607i −2.90869 + 3.08861i 2.64575 2.82843i 0.539533 + 8.98381i −3.16228
71.16 1.41421i 2.85457 0.922735i −2.00000 2.23607i 1.30494 + 4.03697i 2.64575 2.82843i 7.29712 5.26802i 3.16228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.e.a 16
3.b odd 2 1 inner 210.3.e.a 16
4.b odd 2 1 1680.3.l.c 16
5.b even 2 1 1050.3.e.d 16
5.c odd 4 2 1050.3.c.c 32
12.b even 2 1 1680.3.l.c 16
15.d odd 2 1 1050.3.e.d 16
15.e even 4 2 1050.3.c.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.e.a 16 1.a even 1 1 trivial
210.3.e.a 16 3.b odd 2 1 inner
1050.3.c.c 32 5.c odd 4 2
1050.3.c.c 32 15.e even 4 2
1050.3.e.d 16 5.b even 2 1
1050.3.e.d 16 15.d odd 2 1
1680.3.l.c 16 4.b odd 2 1
1680.3.l.c 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(210, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{15} + 34 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + 1348 T^{14} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{8} - 858 T^{6} - 456 T^{5} + \cdots - 64451196)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 2612 T^{14} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{8} - 24 T^{7} - 1276 T^{6} + \cdots + 1552481856)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 4608 T^{14} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} + 9484 T^{14} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{8} - 4004 T^{6} + \cdots + 212472592896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 40 T^{7} - 3984 T^{6} + \cdots + 74680094976)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 16064 T^{14} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{8} + 168 T^{7} + \cdots + 729854118144)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 16484 T^{14} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} + 29240 T^{14} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + 28928 T^{14} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} - 56 T^{7} + \cdots - 28344025168896)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 120 T^{7} + \cdots - 331910403784704)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 22328 T^{14} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{8} - 24 T^{7} + \cdots - 535924119269376)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 4 T^{7} + \cdots + 148620892321344)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 83472 T^{14} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + 55600 T^{14} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{8} + 96 T^{7} + \cdots - 1371263608636)^{2} \) Copy content Toggle raw display
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