Properties

Label 2268.2.f.b
Level $2268$
Weight $2$
Character orbit 2268.f
Analytic conductor $18.110$
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] = [2268,2,Mod(1133,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.1133");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.f (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: \(18.1100711784\)
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: \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q + \beta_{15} q^{5} + \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{15} q^{5} + \beta_{9} q^{7} + \beta_1 q^{11} + \beta_{7} q^{13} - \beta_{12} q^{17} + ( - \beta_{13} - \beta_{11}) q^{19} - \beta_{6} q^{23} + (\beta_{8} + 1) q^{25} + (\beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}) q^{29} + (\beta_{13} + \beta_{10} - \beta_{9}) q^{31} + (\beta_{6} - \beta_{3}) q^{35} + \beta_{4} q^{37} + (\beta_{15} + \beta_{14}) q^{41} + ( - \beta_{10} - \beta_{9} - \beta_{4}) q^{43} + (\beta_{14} + \beta_{12} - \beta_{3} - \beta_{2}) q^{47} + ( - \beta_{13} - \beta_{10} + \beta_{7} - \beta_{4} + 1) q^{49} + ( - \beta_{6} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{53} + (\beta_{13} + \beta_{11} - \beta_{7}) q^{55} + (\beta_{15} - \beta_{12}) q^{59} + ( - 2 \beta_{13} + \beta_{11} - \beta_{10} + \beta_{9}) q^{61} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{65} + ( - \beta_{8} + \beta_{4} - 2) q^{67} + ( - \beta_{6} - \beta_{5}) q^{71} + (\beta_{13} - \beta_{11} + 2 \beta_{10} - 2 \beta_{9}) q^{73} + (\beta_{14} - \beta_{6} - \beta_{3}) q^{77} + (\beta_{10} + \beta_{9} - \beta_{8} + \beta_{4} - 3) q^{79} + (2 \beta_{15} - \beta_{12} + \beta_{3} + \beta_{2}) q^{83} + (\beta_{10} + \beta_{9} - 1) q^{85} + (\beta_{15} + \beta_{14} - 2 \beta_{12} + \beta_{3} + \beta_{2}) q^{89} + (2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{7} - 1) q^{91} + (2 \beta_{6} - \beta_{3} + \beta_{2} - \beta_1) q^{95} + (2 \beta_{13} - \beta_{11} - \beta_{7}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{7} + 16 q^{25} + 4 q^{37} - 8 q^{43} + 10 q^{49} - 28 q^{67} - 40 q^{79} - 12 q^{85} - 18 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} + 3\nu^{12} + 9\nu^{10} + 9\nu^{8} - 225\nu^{6} + 81\nu^{4} + 2187 ) / 729 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{15} + 72 \nu^{14} + 12 \nu^{13} + 135 \nu^{12} - 18 \nu^{11} - 486 \nu^{10} - 369 \nu^{9} - 5994 \nu^{8} - 1548 \nu^{7} + 8667 \nu^{6} + 1782 \nu^{5} + 44712 \nu^{4} + 25029 \nu^{3} + \cdots - 347733 ) / 30618 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{15} - 72 \nu^{14} + 12 \nu^{13} - 135 \nu^{12} - 18 \nu^{11} + 486 \nu^{10} - 369 \nu^{9} + 5994 \nu^{8} - 1548 \nu^{7} - 8667 \nu^{6} + 1782 \nu^{5} - 44712 \nu^{4} + 25029 \nu^{3} + \cdots + 347733 ) / 30618 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} - 3\nu^{12} + 27\nu^{10} + 63\nu^{8} - 171\nu^{6} - 783\nu^{4} + 486\nu^{2} + 3888 ) / 243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\nu^{14} + 24\nu^{12} - 36\nu^{10} - 738\nu^{8} + 306\nu^{6} + 3564\nu^{4} + 9234\nu^{2} - 24057 ) / 1701 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 47\nu^{14} + 12\nu^{12} - 396\nu^{10} - 1314\nu^{8} + 3366\nu^{6} + 8019\nu^{4} + 2916\nu^{2} - 34992 ) / 5103 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{15} - 12\nu^{13} + 18\nu^{11} + 99\nu^{9} + 198\nu^{7} - 486\nu^{5} - 486\nu^{3} - 4374\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{14} + \nu^{12} + 12\nu^{10} + 36\nu^{8} - 153\nu^{6} - 261\nu^{4} + 216\nu^{2} + 2430 ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{15} - 18 \nu^{14} - 12 \nu^{13} + 27 \nu^{12} + 72 \nu^{11} + 162 \nu^{10} + 207 \nu^{9} + 648 \nu^{8} - 396 \nu^{7} - 2349 \nu^{6} - 2268 \nu^{5} - 3888 \nu^{4} + 243 \nu^{3} + \cdots + 41553 ) / 4374 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5 \nu^{15} - 18 \nu^{14} + 12 \nu^{13} + 27 \nu^{12} - 72 \nu^{11} + 162 \nu^{10} - 207 \nu^{9} + 648 \nu^{8} + 396 \nu^{7} - 2349 \nu^{6} + 2268 \nu^{5} - 3888 \nu^{4} - 243 \nu^{3} - 2916 \nu^{2} + \cdots + 41553 ) / 4374 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{15} + 3\nu^{13} - 63\nu^{9} - 72\nu^{7} + 297\nu^{5} + 1215\nu^{3} - 1458\nu ) / 729 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\nu^{15} - 51\nu^{13} - 207\nu^{11} - 558\nu^{9} + 3177\nu^{7} + 81\nu^{5} + 9720\nu^{3} - 34992\nu ) / 15309 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4\nu^{15} - 12\nu^{13} - 9\nu^{11} - 117\nu^{9} + 657\nu^{7} + 162\nu^{5} + 972\nu^{3} - 8748\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 55\nu^{15} - 120\nu^{13} - 576\nu^{11} - 90\nu^{9} + 5652\nu^{7} + 3726\nu^{5} - 20655\nu^{3} + 13122\nu ) / 15309 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 61\nu^{15} - 30\nu^{13} - 711\nu^{11} - 2574\nu^{9} + 8217\nu^{7} + 18792\nu^{5} + 1215\nu^{3} - 157464\nu ) / 15309 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{13} - \beta_{12} - \beta_{11} - \beta_{7} + \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} - 2\beta_{9} + 2\beta_{8} - \beta_{4} - 3\beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{15} - \beta_{14} + 4\beta_{12} + 3\beta_{10} - 3\beta_{9} + 3\beta_{7} + 2\beta_{3} + 2\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{5} - 2\beta_{3} + 2\beta_{2} + 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{15} + \beta_{14} + 2 \beta_{13} - 5 \beta_{12} + \beta_{11} + 3 \beta_{10} - 3 \beta_{9} + 4 \beta_{7} + 2 \beta_{3} + 2 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{10} + 3\beta_{9} - \beta_{5} - 3\beta_{3} + 3\beta_{2} - 12\beta _1 + 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7 \beta_{15} - 5 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} - 12 \beta_{11} + 12 \beta_{10} - 12 \beta_{9} - 3 \beta_{7} + 4 \beta_{3} + 4 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6 \beta_{10} - 6 \beta_{9} + 15 \beta_{8} + 15 \beta_{6} - 18 \beta_{5} - 3 \beta_{4} - 15 \beta_{3} + 15 \beta_{2} - 15 \beta _1 - 54 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 21 \beta_{15} + 3 \beta_{14} + 3 \beta_{13} + 12 \beta_{12} + 6 \beta_{11} + 27 \beta_{10} - 27 \beta_{9} + 24 \beta_{7} - 3 \beta_{3} - 3 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 42 \beta_{10} + 42 \beta_{9} - 15 \beta_{8} + 27 \beta_{6} - 36 \beta_{5} + 39 \beta_{4} - 63 \beta_{3} + 63 \beta_{2} + 9 \beta _1 + 108 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 75 \beta_{15} + 3 \beta_{14} + 171 \beta_{13} - 120 \beta_{12} + 18 \beta_{11} + 72 \beta_{10} - 72 \beta_{9} + 9 \beta_{7} - 6 \beta_{3} - 6 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 189 \beta_{10} + 189 \beta_{9} + 72 \beta_{6} + 9 \beta_{5} - 18 \beta_{4} - 63 \beta_{3} + 63 \beta_{2} - 342 \beta _1 - 225 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 333 \beta_{15} - 153 \beta_{14} + 252 \beta_{13} + 297 \beta_{12} - 171 \beta_{11} + 459 \beta_{10} - 459 \beta_{9} - 144 \beta_{7} - 108 \beta_{3} - 108 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 135 \beta_{10} + 135 \beta_{9} + 81 \beta_{8} + 675 \beta_{6} - 396 \beta_{5} + 270 \beta_{4} - 378 \beta_{3} + 378 \beta_{2} + 324 \beta _1 - 2673 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1044 \beta_{15} + 585 \beta_{14} + 594 \beta_{13} + 144 \beta_{12} + 1080 \beta_{11} + 783 \beta_{10} - 783 \beta_{9} + 351 \beta_{7} - 684 \beta_{3} - 684 \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1133.1
1.71965 + 0.206851i
1.71965 0.206851i
−1.69483 + 0.357142i
−1.69483 0.357142i
−0.744857 + 1.56371i
−0.744857 1.56371i
0.604587 1.62311i
0.604587 + 1.62311i
−0.604587 1.62311i
−0.604587 + 1.62311i
0.744857 + 1.56371i
0.744857 1.56371i
1.69483 + 0.357142i
1.69483 0.357142i
−1.71965 + 0.206851i
−1.71965 0.206851i
0 0 0 −4.18671 0 1.27652 2.31743i 0 0 0
1133.2 0 0 0 −4.18671 0 1.27652 + 2.31743i 0 0 0
1133.3 0 0 0 −2.42488 0 −2.62893 0.297883i 0 0 0
1133.4 0 0 0 −2.42488 0 −2.62893 + 0.297883i 0 0 0
1133.5 0 0 0 −0.553827 0 2.50632 0.847573i 0 0 0
1133.6 0 0 0 −0.553827 0 2.50632 + 0.847573i 0 0 0
1133.7 0 0 0 −0.533560 0 −0.653912 2.56367i 0 0 0
1133.8 0 0 0 −0.533560 0 −0.653912 + 2.56367i 0 0 0
1133.9 0 0 0 0.533560 0 −0.653912 2.56367i 0 0 0
1133.10 0 0 0 0.533560 0 −0.653912 + 2.56367i 0 0 0
1133.11 0 0 0 0.553827 0 2.50632 0.847573i 0 0 0
1133.12 0 0 0 0.553827 0 2.50632 + 0.847573i 0 0 0
1133.13 0 0 0 2.42488 0 −2.62893 0.297883i 0 0 0
1133.14 0 0 0 2.42488 0 −2.62893 + 0.297883i 0 0 0
1133.15 0 0 0 4.18671 0 1.27652 2.31743i 0 0 0
1133.16 0 0 0 4.18671 0 1.27652 + 2.31743i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1133.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.f.b 16
3.b odd 2 1 inner 2268.2.f.b 16
7.b odd 2 1 inner 2268.2.f.b 16
9.c even 3 1 252.2.x.a 16
9.c even 3 1 756.2.x.a 16
9.d odd 6 1 252.2.x.a 16
9.d odd 6 1 756.2.x.a 16
21.c even 2 1 inner 2268.2.f.b 16
36.f odd 6 1 1008.2.cc.c 16
36.f odd 6 1 3024.2.cc.c 16
36.h even 6 1 1008.2.cc.c 16
36.h even 6 1 3024.2.cc.c 16
63.g even 3 1 1764.2.bm.b 16
63.g even 3 1 5292.2.bm.b 16
63.h even 3 1 1764.2.w.a 16
63.h even 3 1 5292.2.w.a 16
63.i even 6 1 1764.2.w.a 16
63.i even 6 1 5292.2.w.a 16
63.j odd 6 1 1764.2.w.a 16
63.j odd 6 1 5292.2.w.a 16
63.k odd 6 1 1764.2.bm.b 16
63.k odd 6 1 5292.2.bm.b 16
63.l odd 6 1 252.2.x.a 16
63.l odd 6 1 756.2.x.a 16
63.n odd 6 1 1764.2.bm.b 16
63.n odd 6 1 5292.2.bm.b 16
63.o even 6 1 252.2.x.a 16
63.o even 6 1 756.2.x.a 16
63.s even 6 1 1764.2.bm.b 16
63.s even 6 1 5292.2.bm.b 16
63.t odd 6 1 1764.2.w.a 16
63.t odd 6 1 5292.2.w.a 16
252.s odd 6 1 1008.2.cc.c 16
252.s odd 6 1 3024.2.cc.c 16
252.bi even 6 1 1008.2.cc.c 16
252.bi even 6 1 3024.2.cc.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.x.a 16 9.c even 3 1
252.2.x.a 16 9.d odd 6 1
252.2.x.a 16 63.l odd 6 1
252.2.x.a 16 63.o even 6 1
756.2.x.a 16 9.c even 3 1
756.2.x.a 16 9.d odd 6 1
756.2.x.a 16 63.l odd 6 1
756.2.x.a 16 63.o even 6 1
1008.2.cc.c 16 36.f odd 6 1
1008.2.cc.c 16 36.h even 6 1
1008.2.cc.c 16 252.s odd 6 1
1008.2.cc.c 16 252.bi even 6 1
1764.2.w.a 16 63.h even 3 1
1764.2.w.a 16 63.i even 6 1
1764.2.w.a 16 63.j odd 6 1
1764.2.w.a 16 63.t odd 6 1
1764.2.bm.b 16 63.g even 3 1
1764.2.bm.b 16 63.k odd 6 1
1764.2.bm.b 16 63.n odd 6 1
1764.2.bm.b 16 63.s even 6 1
2268.2.f.b 16 1.a even 1 1 trivial
2268.2.f.b 16 3.b odd 2 1 inner
2268.2.f.b 16 7.b odd 2 1 inner
2268.2.f.b 16 21.c even 2 1 inner
3024.2.cc.c 16 36.f odd 6 1
3024.2.cc.c 16 36.h even 6 1
3024.2.cc.c 16 252.s odd 6 1
3024.2.cc.c 16 252.bi even 6 1
5292.2.w.a 16 63.h even 3 1
5292.2.w.a 16 63.i even 6 1
5292.2.w.a 16 63.j odd 6 1
5292.2.w.a 16 63.t odd 6 1
5292.2.bm.b 16 63.g even 3 1
5292.2.bm.b 16 63.k odd 6 1
5292.2.bm.b 16 63.n odd 6 1
5292.2.bm.b 16 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24T_{5}^{6} + 117T_{5}^{4} - 63T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 24 T^{6} + 117 T^{4} - 63 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - T^{7} - 2 T^{6} + 11 T^{5} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 45 T^{6} + 639 T^{4} + 3078 T^{2} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 48 T^{6} + 558 T^{4} + 504 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 78 T^{6} + 1467 T^{4} - 6741 T^{2} + \cdots + 900)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 75 T^{6} + 1476 T^{4} + 5787 T^{2} + \cdots + 900)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 81 T^{6} + 2151 T^{4} + \cdots + 50625)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 162 T^{6} + 8523 T^{4} + \cdots + 245025)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 72 T^{6} + 1053 T^{4} + 1701 T^{2} + \cdots + 729)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - T^{3} - 66 T^{2} + 23 T + 610)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 177 T^{6} + 9711 T^{4} + \cdots + 576081)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} - 75 T^{2} - 193 T + 679)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 222 T^{6} + 16785 T^{4} + \cdots + 4968441)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 414 T^{6} + 56133 T^{4} + \cdots + 41990400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 96 T^{6} + 2439 T^{4} - 18225 T^{2} + \cdots + 441)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 351 T^{6} + 42741 T^{4} + \cdots + 35319249)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 7 T^{3} - 111 T^{2} - 1004 T - 1985)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 207 T^{6} + 11250 T^{4} + \cdots + 15876)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 243 T^{6} + 19620 T^{4} + \cdots + 76176)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} - 93 T^{2} - 833 T + 565)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 267 T^{6} + 9693 T^{4} + \cdots + 173889)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 648 T^{6} + 133731 T^{4} + \cdots + 211004676)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 372 T^{6} + 34659 T^{4} + \cdots + 9162729)^{2} \) Copy content Toggle raw display
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