Properties

Label 21.5.h.b
Level $21$
Weight $5$
Character orbit 21.h
Analytic conductor $2.171$
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] = [21,5,Mod(2,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.2");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), 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: \(2.17076922476\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 103 x^{14} + 7227 x^{12} - 270898 x^{10} + 7374256 x^{8} - 115494792 x^{6} + 1245573504 x^{4} - 2908017504 x^{2} + 5639409216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} - \beta_{2}) q^{3} + (\beta_{10} - \beta_{9} - \beta_{7} - 10 \beta_{2}) q^{4} + ( - \beta_{13} + \beta_{4}) q^{5} + ( - \beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{3} + 2) q^{6} + (2 \beta_{8} + 2 \beta_{7} - \beta_{5} + 3 \beta_{4} - 4 \beta_{2} - 12) q^{7} + (\beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots + 1) q^{8}+ \cdots + ( - 3 \beta_{15} + \beta_{11} + 4 \beta_{10} + 3 \beta_{8} + 3 \beta_{5} + 5 \beta_{2} + \cdots + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} - \beta_{2}) q^{3} + (\beta_{10} - \beta_{9} - \beta_{7} - 10 \beta_{2}) q^{4} + ( - \beta_{13} + \beta_{4}) q^{5} + ( - \beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{3} + 2) q^{6} + (2 \beta_{8} + 2 \beta_{7} - \beta_{5} + 3 \beta_{4} - 4 \beta_{2} - 12) q^{7} + (\beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots + 1) q^{8}+ \cdots + (162 \beta_{15} - 126 \beta_{14} - 126 \beta_{13} - 8 \beta_{12} + 290 \beta_{9} + \cdots - 3053) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} + 78 q^{4} + 28 q^{6} - 168 q^{7} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{3} + 78 q^{4} + 28 q^{6} - 168 q^{7} + 62 q^{9} - 98 q^{10} + 436 q^{12} - 492 q^{13} - 856 q^{15} + 150 q^{16} - 884 q^{18} - 1326 q^{19} + 2170 q^{21} + 3244 q^{22} + 630 q^{24} + 2094 q^{25} + 1028 q^{27} - 854 q^{28} + 340 q^{30} - 2504 q^{31} - 616 q^{33} - 7728 q^{34} - 9644 q^{36} + 1342 q^{37} - 2626 q^{39} + 5754 q^{40} - 1582 q^{42} + 1460 q^{43} - 1330 q^{45} + 8844 q^{46} + 25864 q^{48} + 5572 q^{49} + 7794 q^{51} + 9536 q^{52} - 18578 q^{54} - 35140 q^{55} + 11696 q^{57} - 12446 q^{58} - 18890 q^{60} + 15012 q^{61} - 13076 q^{63} + 7660 q^{64} - 17192 q^{66} + 8658 q^{67} + 53676 q^{69} + 13790 q^{70} + 21312 q^{72} + 12322 q^{73} - 19514 q^{75} - 84520 q^{76} - 32120 q^{78} - 40168 q^{79} - 23986 q^{81} + 47348 q^{82} - 5992 q^{84} + 22536 q^{85} - 1162 q^{87} + 56430 q^{88} + 139636 q^{90} - 17878 q^{91} + 8556 q^{93} + 6468 q^{94} - 40894 q^{96} - 28268 q^{97} - 47812 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 103 x^{14} + 7227 x^{12} - 270898 x^{10} + 7374256 x^{8} - 115494792 x^{6} + 1245573504 x^{4} - 2908017504 x^{2} + 5639409216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 805319043514 \nu^{14} + 80361724322821 \nu^{12} + \cdots + 39\!\cdots\!52 ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 805319043514 \nu^{15} + 80361724322821 \nu^{13} + \cdots + 39\!\cdots\!52 \nu ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6321739747609 \nu^{15} - 94449672964612 \nu^{14} + 429246974607973 \nu^{13} + \cdots + 12\!\cdots\!44 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6321739747609 \nu^{15} - 98639052169252 \nu^{14} + 429246974607973 \nu^{13} + \cdots + 45\!\cdots\!48 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6321739747609 \nu^{15} + 110831311084076 \nu^{14} - 429246974607973 \nu^{13} + \cdots - 11\!\cdots\!84 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 77376912306302 \nu^{15} - 18998427328522 \nu^{14} + \cdots + 14\!\cdots\!12 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 161075564360213 \nu^{15} + 127149780827396 \nu^{14} + \cdots - 10\!\cdots\!84 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 161075564360213 \nu^{15} + 215202916713528 \nu^{14} + \cdots + 12\!\cdots\!76 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6321739747609 \nu^{15} - 492819382147164 \nu^{14} - 429246974607973 \nu^{13} + \cdots + 16\!\cdots\!64 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 59901524063219 \nu^{15} - 174880886372968 \nu^{14} + \cdots + 14\!\cdots\!20 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 55949115474713 \nu^{15} + 51619834308255 \nu^{14} + \cdots - 49\!\cdots\!16 ) / 92\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 341327704587575 \nu^{15} - 94449672964612 \nu^{14} + \cdots + 12\!\cdots\!44 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 199118570462318 \nu^{15} - 18998427328522 \nu^{14} + \cdots + 14\!\cdots\!12 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 161075564360213 \nu^{15} - 739692280405892 \nu^{14} + \cdots - 76\!\cdots\!88 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} - \beta_{7} - 26\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + 2\beta_{12} - 2\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} - 36\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{15} + 55\beta_{10} + 8\beta_{8} + 12\beta_{6} - 4\beta_{5} + 71\beta_{4} - 976\beta_{2} - 976 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20 \beta_{15} + 59 \beta_{14} + 142 \beta_{12} + 142 \beta_{11} - 20 \beta_{10} + 20 \beta_{9} + 91 \beta_{8} - 190 \beta_{7} - 91 \beta_{6} + 20 \beta_{5} - 1494 \beta_{3} - 71 \beta_{2} - 1494 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 276\beta_{15} + 2871\beta_{9} + 940\beta_{8} + 4087\beta_{7} + 664\beta_{6} + 664\beta_{5} + 4087\beta_{4} - 41780 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4087 \beta_{15} - 3259 \beta_{13} + 8174 \beta_{11} - 1604 \beta_{10} + 4087 \beta_{8} - 1604 \beta_{6} + 5691 \beta_{5} + 12158 \beta_{4} - 4087 \beta_{2} - 67722 \beta _1 - 4087 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 56236 \beta_{15} - 148427 \beta_{10} + 148427 \beta_{9} + 14900 \beta_{8} + 219563 \beta_{7} - 14900 \beta_{6} + 56236 \beta_{5} + 1934236 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 317135 \beta_{15} - 174863 \beta_{14} - 174863 \beta_{13} - 439126 \beta_{12} - 97572 \beta_{9} - 97572 \beta_{8} + 687142 \beta_{7} + 219563 \beta_{6} + 219563 \beta_{5} + 687142 \beta_{4} + \cdots - 219563 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2321752 \beta_{15} - 7632983 \beta_{10} - 2321752 \beta_{8} - 3077916 \beta_{6} + 756164 \beta_{5} - 11467063 \beta_{4} + 93768476 \beta_{2} + 93768476 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5399668 \beta_{15} - 9198571 \beta_{14} - 22934126 \beta_{12} - 22934126 \beta_{11} + 5399668 \beta_{10} - 5399668 \beta_{9} - 16866731 \beta_{8} + 36864638 \beta_{7} + \cdots + 160001874 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 37880532 \beta_{15} - 391263459 \beta_{9} - 162334604 \beta_{8} - 591478595 \beta_{7} - 124454072 \beta_{6} - 124454072 \beta_{5} - 591478595 \beta_{4} + \cdots + 4661161900 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 591478595 \beta_{15} + 477836999 \beta_{13} - 1182957190 \beta_{11} + 286788676 \beta_{10} - 591478595 \beta_{8} + 286788676 \beta_{6} - 878267271 \beta_{5} + \cdots + 591478595 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 8418044924 \beta_{15} + 20017474303 \beta_{10} - 20017474303 \beta_{9} - 1900609252 \beta_{8} - 30336128479 \beta_{7} + 1900609252 \beta_{6} + \cdots - 234738533852 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 45271609075 \beta_{15} + 24634300723 \beta_{14} + 24634300723 \beta_{13} + 60672256958 \beta_{12} + 14935480596 \beta_{9} + 14935480596 \beta_{8} + \cdots + 30336128479 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−6.18666 3.57187i
−4.46939 2.58040i
−4.14633 2.39389i
−1.34450 0.776250i
1.34450 + 0.776250i
4.14633 + 2.39389i
4.46939 + 2.58040i
6.18666 + 3.57187i
−6.18666 + 3.57187i
−4.46939 + 2.58040i
−4.14633 + 2.39389i
−1.34450 + 0.776250i
1.34450 0.776250i
4.14633 2.39389i
4.46939 2.58040i
6.18666 3.57187i
−6.18666 3.57187i −7.57871 4.85420i 17.5165 + 30.3395i 13.6392 + 7.87461i 29.5483 + 57.1014i −44.8988 + 19.6239i 135.967i 33.8736 + 73.5771i −56.2542 97.4351i
2.2 −4.46939 2.58040i 8.54954 2.81164i 5.31698 + 9.20927i −31.7353 18.3224i −45.4664 9.49493i −18.9163 45.2015i 27.6932i 65.1893 48.0765i 94.5584 + 163.780i
2.3 −4.14633 2.39389i 2.75140 + 8.56912i 3.46138 + 5.99529i 30.3128 + 17.5011i 9.10526 42.1170i 48.8998 3.13209i 43.4597i −65.8596 + 47.1542i −83.7912 145.131i
2.4 −1.34450 0.776250i −7.02686 + 5.62345i −6.79487 11.7691i −23.4142 13.5182i 13.8128 2.10615i −27.0847 + 40.8340i 45.9381i 17.7536 79.0304i 20.9870 + 36.3506i
2.5 1.34450 + 0.776250i 8.38348 3.27372i −6.79487 11.7691i 23.4142 + 13.5182i 13.8128 + 2.10615i −27.0847 + 40.8340i 45.9381i 59.5656 54.8903i 20.9870 + 36.3506i
2.6 4.14633 + 2.39389i 6.04537 + 6.66734i 3.46138 + 5.99529i −30.3128 17.5011i 9.10526 + 42.1170i 48.8998 3.13209i 43.4597i −7.90694 + 80.6132i −83.7912 145.131i
2.7 4.46939 + 2.58040i −6.70973 + 5.99830i 5.31698 + 9.20927i 31.7353 + 18.3224i −45.4664 + 9.49493i −18.9163 45.2015i 27.6932i 9.04086 80.4939i 94.5584 + 163.780i
2.8 6.18666 + 3.57187i −0.414505 8.99045i 17.5165 + 30.3395i −13.6392 7.87461i 29.5483 57.1014i −44.8988 + 19.6239i 135.967i −80.6564 + 7.45317i −56.2542 97.4351i
11.1 −6.18666 + 3.57187i −7.57871 + 4.85420i 17.5165 30.3395i 13.6392 7.87461i 29.5483 57.1014i −44.8988 19.6239i 135.967i 33.8736 73.5771i −56.2542 + 97.4351i
11.2 −4.46939 + 2.58040i 8.54954 + 2.81164i 5.31698 9.20927i −31.7353 + 18.3224i −45.4664 + 9.49493i −18.9163 + 45.2015i 27.6932i 65.1893 + 48.0765i 94.5584 163.780i
11.3 −4.14633 + 2.39389i 2.75140 8.56912i 3.46138 5.99529i 30.3128 17.5011i 9.10526 + 42.1170i 48.8998 + 3.13209i 43.4597i −65.8596 47.1542i −83.7912 + 145.131i
11.4 −1.34450 + 0.776250i −7.02686 5.62345i −6.79487 + 11.7691i −23.4142 + 13.5182i 13.8128 + 2.10615i −27.0847 40.8340i 45.9381i 17.7536 + 79.0304i 20.9870 36.3506i
11.5 1.34450 0.776250i 8.38348 + 3.27372i −6.79487 + 11.7691i 23.4142 13.5182i 13.8128 2.10615i −27.0847 40.8340i 45.9381i 59.5656 + 54.8903i 20.9870 36.3506i
11.6 4.14633 2.39389i 6.04537 6.66734i 3.46138 5.99529i −30.3128 + 17.5011i 9.10526 42.1170i 48.8998 + 3.13209i 43.4597i −7.90694 80.6132i −83.7912 + 145.131i
11.7 4.46939 2.58040i −6.70973 5.99830i 5.31698 9.20927i 31.7353 18.3224i −45.4664 9.49493i −18.9163 + 45.2015i 27.6932i 9.04086 + 80.4939i 94.5584 163.780i
11.8 6.18666 3.57187i −0.414505 + 8.99045i 17.5165 30.3395i −13.6392 + 7.87461i 29.5483 + 57.1014i −44.8988 19.6239i 135.967i −80.6564 7.45317i −56.2542 + 97.4351i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.5.h.b 16
3.b odd 2 1 inner 21.5.h.b 16
7.b odd 2 1 147.5.h.b 16
7.c even 3 1 inner 21.5.h.b 16
7.c even 3 1 147.5.b.c 8
7.d odd 6 1 147.5.b.f 8
7.d odd 6 1 147.5.h.b 16
21.c even 2 1 147.5.h.b 16
21.g even 6 1 147.5.b.f 8
21.g even 6 1 147.5.h.b 16
21.h odd 6 1 inner 21.5.h.b 16
21.h odd 6 1 147.5.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.h.b 16 1.a even 1 1 trivial
21.5.h.b 16 3.b odd 2 1 inner
21.5.h.b 16 7.c even 3 1 inner
21.5.h.b 16 21.h odd 6 1 inner
147.5.b.c 8 7.c even 3 1
147.5.b.c 8 21.h odd 6 1
147.5.b.f 8 7.d odd 6 1
147.5.b.f 8 21.g even 6 1
147.5.h.b 16 7.b odd 2 1
147.5.h.b 16 7.d odd 6 1
147.5.h.b 16 21.c even 2 1
147.5.h.b 16 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 103 T_{2}^{14} + 7227 T_{2}^{12} - 270898 T_{2}^{10} + 7374256 T_{2}^{8} - 115494792 T_{2}^{6} + 1245573504 T_{2}^{4} - 2908017504 T_{2}^{2} + 5639409216 \) acting on \(S_{5}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 103 T^{14} + \cdots + 5639409216 \) Copy content Toggle raw display
$3$ \( T^{16} - 8 T^{15} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} - 3547 T^{14} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( (T^{8} + 84 T^{7} + \cdots + 33232930569601)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 49255 T^{14} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{4} + 123 T^{3} - 25046 T^{2} + \cdots + 114515728)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} - 320190 T^{14} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} + 663 T^{7} + \cdots + 27\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 1456782 T^{14} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{8} + 472213 T^{6} + \cdots + 43\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 1252 T^{7} + \cdots + 42\!\cdots\!69)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 671 T^{7} + \cdots + 66\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 9934876 T^{6} + \cdots + 27\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 365 T^{3} + \cdots + 885324127312)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} - 15501246 T^{14} + \cdots + 77\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} - 30342651 T^{14} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} - 44267415 T^{14} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{8} - 7506 T^{7} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 4329 T^{7} + \cdots + 30\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 155086092 T^{6} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 6161 T^{7} + \cdots + 30\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 20084 T^{7} + \cdots + 45\!\cdots\!49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 116812773 T^{6} + \cdots + 51\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 202431178 T^{14} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} + 7067 T^{3} + \cdots + 368225255164464)^{4} \) Copy content Toggle raw display
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