Properties

Label 1050.3.e.c
Level $1050$
Weight $3$
Character orbit 1050.e
Analytic conductor $28.610$
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] = [1050,3,Mod(701,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, 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("1050.701");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.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: \(28.6104277578\)
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: \( x^{16} - 4 x^{15} + 4 x^{14} - 4 x^{13} + 4 x^{12} - 364 x^{11} + 972 x^{10} + 1236 x^{9} - 5274 x^{8} + 11124 x^{7} + 78732 x^{6} - 265356 x^{5} + 26244 x^{4} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{2} q^{3} - 2 q^{4} - \beta_{9} q^{6} - \beta_{6} q^{7} + 2 \beta_{3} q^{8} + (\beta_{6} + \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{2} q^{3} - 2 q^{4} - \beta_{9} q^{6} - \beta_{6} q^{7} + 2 \beta_{3} q^{8} + (\beta_{6} + \beta_{3} + \beta_1) q^{9} + ( - \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{3} - \beta_{2}) q^{11} + 2 \beta_{2} q^{12} + ( - \beta_{15} - \beta_{9} + \beta_{8} + 2 \beta_{6} - \beta_1 + 1) q^{13} - \beta_{8} q^{14} + 4 q^{16} + ( - \beta_{15} - \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - 4 \beta_{3} + \beta_{2}) q^{17} + ( - \beta_{13} - \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{18} + ( - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{2} - \beta_1 - 2) q^{19} + ( - \beta_{12} + 2) q^{21} + (\beta_{14} - \beta_{9} - \beta_{6} + \beta_{2} + 4) q^{22} + ( - \beta_{15} + \beta_{13} - 2 \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} + 3 \beta_{3} + \cdots - \beta_1) q^{23}+ \cdots + ( - 4 \beta_{15} + 3 \beta_{13} - 8 \beta_{12} + 4 \beta_{11} - 3 \beta_{10} - 8 \beta_{9} + \cdots + 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} - 32 q^{4} - 8 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{3} - 32 q^{4} - 8 q^{6} + 8 q^{9} - 8 q^{12} + 64 q^{16} + 16 q^{18} - 24 q^{19} + 28 q^{21} + 48 q^{22} + 16 q^{24} + 28 q^{27} + 24 q^{31} - 76 q^{33} - 128 q^{34} - 16 q^{36} + 80 q^{37} - 96 q^{39} - 192 q^{43} + 80 q^{46} + 16 q^{48} + 112 q^{49} + 144 q^{51} + 40 q^{54} + 72 q^{57} - 48 q^{58} - 56 q^{61} - 56 q^{63} - 128 q^{64} - 96 q^{66} - 240 q^{67} - 172 q^{69} - 32 q^{72} + 96 q^{73} + 48 q^{76} + 256 q^{78} + 128 q^{79} + 80 q^{81} + 64 q^{82} - 56 q^{84} - 180 q^{87} - 96 q^{88} - 224 q^{91} - 352 q^{93} - 208 q^{94} - 32 q^{96} - 264 q^{97} + 152 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 4 x^{14} - 4 x^{13} + 4 x^{12} - 364 x^{11} + 972 x^{10} + 1236 x^{9} - 5274 x^{8} + 11124 x^{7} + 78732 x^{6} - 265356 x^{5} + 26244 x^{4} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6037 \nu^{15} + 71792 \nu^{14} + 9343 \nu^{13} - 159760 \nu^{12} + 682453 \nu^{11} - 8189866 \nu^{10} - 20729421 \nu^{9} - 61159362 \nu^{8} + \cdots - 251096306562 ) / 50622943896 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 4 \nu^{14} + 4 \nu^{13} - 4 \nu^{12} + 4 \nu^{11} - 364 \nu^{10} + 972 \nu^{9} + 1236 \nu^{8} - 5274 \nu^{7} + 11124 \nu^{6} + 78732 \nu^{5} - 265356 \nu^{4} + 26244 \nu^{3} + \cdots - 19131876 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12946 \nu^{15} + 7063 \nu^{14} - 5416 \nu^{13} + 150730 \nu^{12} - 994912 \nu^{11} + 6856135 \nu^{10} + 6863814 \nu^{9} - 27687774 \nu^{8} + \cdots + 335066110326 ) / 50622943896 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22481 \nu^{15} - 11830 \nu^{14} + 213169 \nu^{13} - 353380 \nu^{12} + 1519051 \nu^{11} + 9506876 \nu^{10} + 524385 \nu^{9} - 28381218 \nu^{8} + \cdots + 137816468766 ) / 50622943896 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23825 \nu^{15} + 72170 \nu^{14} + 182467 \nu^{13} + 356021 \nu^{12} - 5206787 \nu^{11} + 8449694 \nu^{10} + 9660645 \nu^{9} - 63302055 \nu^{8} + \cdots + 201798245079 ) / 50622943896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1687 \nu^{15} - 227 \nu^{14} + 9893 \nu^{13} - 9650 \nu^{12} + 24959 \nu^{11} + 779191 \nu^{10} + 351819 \nu^{9} - 2669952 \nu^{8} + 4090995 \nu^{7} + \cdots + 24928834428 ) / 2410616376 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 54685 \nu^{15} - 177170 \nu^{14} + 726701 \nu^{13} - 1201037 \nu^{12} - 5377459 \nu^{11} + 36303052 \nu^{10} + 28512171 \nu^{9} + \cdots + 1009163412279 ) / 50622943896 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8249 \nu^{15} - 3521 \nu^{14} + 10757 \nu^{13} - 25094 \nu^{12} + 27281 \nu^{11} - 2681993 \nu^{10} - 2205027 \nu^{9} + 5495982 \nu^{8} - 18498285 \nu^{7} + \cdots - 84897699750 ) / 7231849128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 70054 \nu^{15} + 163702 \nu^{14} - 216649 \nu^{13} + 231472 \nu^{12} + 1076354 \nu^{11} + 16545448 \nu^{10} - 6387273 \nu^{9} + \cdots + 768345706098 ) / 50622943896 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 106637 \nu^{15} + 197813 \nu^{14} - 157664 \nu^{13} - 2894173 \nu^{12} + 6533341 \nu^{11} + 30123875 \nu^{10} - 27672066 \nu^{9} + \cdots + 624230067159 ) / 50622943896 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5183 \nu^{15} + 3542 \nu^{14} - 8510 \nu^{13} - 17977 \nu^{12} + 61717 \nu^{11} + 1902578 \nu^{10} + 425970 \nu^{9} - 3562197 \nu^{8} + 16379595 \nu^{7} + \cdots + 69931789749 ) / 2410616376 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5212 \nu^{15} + 5665 \nu^{14} - 22891 \nu^{13} + 109885 \nu^{12} - 107698 \nu^{11} + 2121799 \nu^{10} + 1946655 \nu^{9} - 3275661 \nu^{8} + \cdots + 106990233561 ) / 2410616376 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 114715 \nu^{15} + 3794 \nu^{14} + 493762 \nu^{13} - 323419 \nu^{12} + 1430041 \nu^{11} - 40857556 \nu^{10} - 54533988 \nu^{9} + 13962855 \nu^{8} + \cdots - 1526068438047 ) / 50622943896 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 159413 \nu^{15} - 294140 \nu^{14} + 454952 \nu^{13} - 1302536 \nu^{12} - 1568995 \nu^{11} - 58870550 \nu^{10} - 27746352 \nu^{9} + \cdots - 2750350664070 ) / 50622943896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 185264 \nu^{15} - 260141 \nu^{14} + 306419 \nu^{13} - 56858 \nu^{12} - 3687286 \nu^{11} - 65231933 \nu^{10} - 3051423 \nu^{9} + 150924822 \nu^{8} + \cdots - 2377106895186 ) / 50622943896 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_{3} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + 2\beta_{11} - \beta_{9} - \beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + 3 \beta_{14} + 6 \beta_{12} - \beta_{11} + 3 \beta_{10} - 4 \beta_{9} - \beta_{7} + 5 \beta_{6} + \beta_{5} - \beta_{3} + 2 \beta_{2} + 6 \beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7 \beta_{15} - 18 \beta_{13} - 4 \beta_{11} - 7 \beta_{9} + 18 \beta_{8} - 7 \beta_{7} + 41 \beta_{6} + 7 \beta_{5} - 34 \beta_{3} - 43 \beta_{2} + 9 \beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 17 \beta_{15} + 3 \beta_{14} - 12 \beta_{12} - 19 \beta_{11} + 3 \beta_{10} - 58 \beta_{9} - 72 \beta_{8} - 19 \beta_{7} - 10 \beta_{6} + 19 \beta_{5} + 15 \beta_{4} + 14 \beta_{3} - 61 \beta_{2} - 12 \beta _1 + 416 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 86 \beta_{15} - 36 \beta_{14} - 18 \beta_{13} + 44 \beta_{11} - 36 \beta_{10} - 76 \beta_{9} + 144 \beta_{8} + 50 \beta_{7} - 253 \beta_{6} + 76 \beta_{5} + 90 \beta_{4} - 535 \beta_{3} + 158 \beta_{2} + 9 \beta _1 + 764 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 466 \beta_{15} - 168 \beta_{14} + 216 \beta_{13} + 276 \beta_{12} + 110 \beta_{11} + 210 \beta_{10} - 226 \beta_{9} - 684 \beta_{8} + 56 \beta_{7} - 88 \beta_{6} - 56 \beta_{5} + 60 \beta_{4} - 496 \beta_{3} - 337 \beta_{2} + 168 \beta _1 - 742 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 370 \beta_{15} + 576 \beta_{14} - 126 \beta_{13} + 1296 \beta_{12} + 164 \beta_{11} + 576 \beta_{10} - 208 \beta_{9} + 1188 \beta_{8} - 226 \beta_{7} + 1604 \beta_{6} + 856 \beta_{5} - 198 \beta_{4} - 4366 \beta_{3} + \cdots + 4079 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1462 \beta_{15} - 96 \beta_{14} - 3456 \beta_{13} + 1212 \beta_{12} - 2650 \beta_{11} + 1794 \beta_{10} - 2446 \beta_{9} + 3888 \beta_{8} - 2488 \beta_{7} + 8519 \beta_{6} + 2272 \beta_{5} + 1221 \beta_{4} + \cdots + 4319 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 9425 \beta_{15} - 972 \beta_{14} - 5094 \beta_{13} - 5184 \beta_{12} + 1166 \beta_{11} - 1620 \beta_{10} - 23461 \beta_{9} + 10440 \beta_{8} - 7099 \beta_{7} + 5666 \beta_{6} + 10501 \beta_{5} + \cdots - 26122 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 22913 \beta_{15} - 11661 \beta_{14} + 7776 \beta_{13} + 8682 \beta_{12} + 3329 \beta_{11} - 1455 \beta_{10} - 43474 \beta_{9} + 2628 \beta_{8} + 3167 \beta_{7} - 163 \beta_{6} + 7201 \beta_{5} + \cdots + 221915 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 33925 \beta_{15} - 29952 \beta_{14} + 39348 \beta_{13} + 103680 \beta_{12} - 70552 \beta_{11} + 1152 \beta_{10} + 1553 \beta_{9} - 107370 \beta_{8} + 60539 \beta_{7} - 90667 \beta_{6} - 40433 \beta_{5} + \cdots - 435439 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 168073 \beta_{15} - 56733 \beta_{14} + 86400 \beta_{13} + 22200 \beta_{12} - 5281 \beta_{11} + 3585 \beta_{10} + 155492 \beta_{9} - 74448 \beta_{8} - 19267 \beta_{7} + 617438 \beta_{6} + \cdots + 1113962 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 497276 \beta_{15} + 35784 \beta_{14} + 159444 \beta_{13} + 295488 \beta_{12} - 549736 \beta_{11} + 416808 \beta_{10} - 112816 \beta_{9} + 1349568 \beta_{8} - 246268 \beta_{7} + \cdots - 144520 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 922892 \beta_{15} - 1476024 \beta_{14} - 1357776 \beta_{13} - 950808 \beta_{12} + 1347980 \beta_{11} - 188988 \beta_{10} - 4567084 \beta_{9} + 9044568 \beta_{8} - 1397272 \beta_{7} + \cdots - 14638204 ) / 3 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
−2.88113 + 0.836130i
−2.07357 2.16802i
−1.98636 + 2.24820i
0.318317 + 2.98306i
1.09006 2.79495i
1.59823 + 2.53883i
2.94382 0.577861i
2.99063 0.236961i
−2.88113 0.836130i
−2.07357 + 2.16802i
−1.98636 2.24820i
0.318317 2.98306i
1.09006 + 2.79495i
1.59823 2.53883i
2.94382 + 0.577861i
2.99063 + 0.236961i
1.41421i −2.88113 0.836130i −2.00000 0 −1.18247 + 4.07453i −2.64575 2.82843i 7.60177 + 4.81799i 0
701.2 1.41421i −2.07357 + 2.16802i −2.00000 0 3.06604 + 2.93247i −2.64575 2.82843i −0.400623 8.99108i 0
701.3 1.41421i −1.98636 2.24820i −2.00000 0 −3.17943 + 2.80913i 2.64575 2.82843i −1.10878 + 8.93144i 0
701.4 1.41421i 0.318317 2.98306i −2.00000 0 −4.21869 0.450168i −2.64575 2.82843i −8.79735 1.89912i 0
701.5 1.41421i 1.09006 + 2.79495i −2.00000 0 3.95266 1.54158i 2.64575 2.82843i −6.62354 + 6.09333i 0
701.6 1.41421i 1.59823 2.53883i −2.00000 0 −3.59045 2.26024i 2.64575 2.82843i −3.89133 8.11526i 0
701.7 1.41421i 2.94382 + 0.577861i −2.00000 0 0.817218 4.16319i 2.64575 2.82843i 8.33215 + 3.40224i 0
701.8 1.41421i 2.99063 + 0.236961i −2.00000 0 0.335113 4.22939i −2.64575 2.82843i 8.88770 + 1.41732i 0
701.9 1.41421i −2.88113 + 0.836130i −2.00000 0 −1.18247 4.07453i −2.64575 2.82843i 7.60177 4.81799i 0
701.10 1.41421i −2.07357 2.16802i −2.00000 0 3.06604 2.93247i −2.64575 2.82843i −0.400623 + 8.99108i 0
701.11 1.41421i −1.98636 + 2.24820i −2.00000 0 −3.17943 2.80913i 2.64575 2.82843i −1.10878 8.93144i 0
701.12 1.41421i 0.318317 + 2.98306i −2.00000 0 −4.21869 + 0.450168i −2.64575 2.82843i −8.79735 + 1.89912i 0
701.13 1.41421i 1.09006 2.79495i −2.00000 0 3.95266 + 1.54158i 2.64575 2.82843i −6.62354 6.09333i 0
701.14 1.41421i 1.59823 + 2.53883i −2.00000 0 −3.59045 + 2.26024i 2.64575 2.82843i −3.89133 + 8.11526i 0
701.15 1.41421i 2.94382 0.577861i −2.00000 0 0.817218 + 4.16319i 2.64575 2.82843i 8.33215 3.40224i 0
701.16 1.41421i 2.99063 0.236961i −2.00000 0 0.335113 + 4.22939i −2.64575 2.82843i 8.88770 1.41732i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.16
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 1050.3.e.c yes 16
3.b odd 2 1 inner 1050.3.e.c yes 16
5.b even 2 1 1050.3.e.b 16
5.c odd 4 2 1050.3.c.b 32
15.d odd 2 1 1050.3.e.b 16
15.e even 4 2 1050.3.c.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.c.b 32 5.c odd 4 2
1050.3.c.b 32 15.e even 4 2
1050.3.e.b 16 5.b even 2 1
1050.3.e.b 16 15.d odd 2 1
1050.3.e.c yes 16 1.a even 1 1 trivial
1050.3.e.c yes 16 3.b odd 2 1 inner

Hecke kernels

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

\( T_{11}^{16} + 1168 T_{11}^{14} + 519460 T_{11}^{12} + 115450208 T_{11}^{10} + 13781338958 T_{11}^{8} + 867204565328 T_{11}^{6} + 25799735722356 T_{11}^{4} + \cdots + 917436908786841 \) Copy content Toggle raw display
\( T_{13}^{8} - 528T_{13}^{6} + 768T_{13}^{5} + 80928T_{13}^{4} - 189312T_{13}^{3} - 3324928T_{13}^{2} + 6834432T_{13} + 31632192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{15} + 4 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 917436908786841 \) Copy content Toggle raw display
$13$ \( (T^{8} - 528 T^{6} + 768 T^{5} + \cdots + 31632192)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 1928 T^{14} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} + 12 T^{7} - 1252 T^{6} + \cdots - 239195376)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 163629200648169 \) Copy content Toggle raw display
$29$ \( T^{16} + 2416 T^{14} + \cdots + 701608939641 \) Copy content Toggle raw display
$31$ \( (T^{8} - 12 T^{7} - 3548 T^{6} + \cdots + 266240016)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 40 T^{7} + \cdots - 5053100628975)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 16568 T^{14} + \cdots + 42\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{8} + 96 T^{7} - 2012 T^{6} + \cdots + 25612666977)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 18440 T^{14} + \cdots + 43\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{16} + 38528 T^{14} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + 31808 T^{14} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} + 28 T^{7} + \cdots - 5582861557488)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 120 T^{7} + \cdots + 16357818284361)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 40160 T^{14} + \cdots + 17\!\cdots\!81 \) Copy content Toggle raw display
$73$ \( (T^{8} - 48 T^{7} - 8384 T^{6} + \cdots + 16825155648)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 64 T^{7} + \cdots + 15508447396353)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 32376 T^{14} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + 70624 T^{14} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{8} + 132 T^{7} + \cdots + 28445845989008)^{2} \) Copy content Toggle raw display
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