Properties

Label 171.2.m.a
Level $171$
Weight $2$
Character orbit 171.m
Analytic conductor $1.365$
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] = [171,2,Mod(8,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.m (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: \(1.36544187456\)
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: \( x^{16} + 16x^{14} + 174x^{12} + 1012x^{10} + 4243x^{8} + 9708x^{6} + 15858x^{4} + 12150x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{10} + 2 \beta_{4} - \beta_{3}) q^{4} - \beta_{11} q^{5} + \beta_{14} q^{7} + ( - 2 \beta_{13} - \beta_{8} + \beta_{7} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{10} + 2 \beta_{4} - \beta_{3}) q^{4} - \beta_{11} q^{5} + \beta_{14} q^{7} + ( - 2 \beta_{13} - \beta_{8} + \beta_{7} + \beta_1) q^{8} + ( - \beta_{14} - \beta_{5} - \beta_{2} + 1) q^{10} + \beta_{8} q^{11} + ( - \beta_{5} - \beta_{4} - 2) q^{13} + ( - 2 \beta_{12} + \beta_{11} - \beta_1) q^{14} + ( - 2 \beta_{10} - \beta_{9} + \beta_{5} - 3 \beta_{4} - 3) q^{16} + (\beta_{13} + \beta_{11} + \beta_{8}) q^{17} + (\beta_{14} + \beta_{10} + \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{19} + (2 \beta_{12} - 2 \beta_{8} + \beta_{6}) q^{20} + ( - \beta_{10} - \beta_{9} - \beta_{5} - \beta_{4} + 2 \beta_{3} + 1) q^{22} + ( - \beta_{15} - \beta_{6}) q^{23} + ( - \beta_{14} - \beta_{10} + 2 \beta_{9} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{25} + (\beta_{12} - 2 \beta_{8} - \beta_{7} + \beta_{6} + \beta_1) q^{26} + ( - \beta_{14} + \beta_{10} - 2 \beta_{9} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{28} - 2 \beta_{7} q^{29} + ( - 2 \beta_{10} + \beta_{9} + \beta_{3}) q^{31} + (\beta_{15} + 2 \beta_{13} - \beta_{12} - \beta_{11} + 4 \beta_{8} - 3 \beta_{7} - \beta_{6}) q^{32} + (\beta_{14} + \beta_{10} + \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{34} + (\beta_{15} + 2 \beta_{13} + 2 \beta_{8} + 4 \beta_{7} + 2 \beta_1) q^{35} + ( - \beta_{14} + 2 \beta_{10} - \beta_{3} + 2 \beta_{2} + 1) q^{37} + ( - \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{7} - \beta_{6} + \beta_1) q^{38} + (\beta_{10} + 4 \beta_{9} + 4 \beta_{5} + \beta_{4} - 2 \beta_{3} - 1) q^{40} + (\beta_{13} + 2 \beta_{12} - \beta_{11} - \beta_{8} + 2 \beta_1) q^{41} + ( - \beta_{10} + \beta_{9} - \beta_{5}) q^{43} + (\beta_{15} + 3 \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{7} + \beta_{6} - 4 \beta_1) q^{44} + (2 \beta_{10} - 2 \beta_{9} + 2 \beta_{4} - \beta_{3} + 1) q^{46} + ( - \beta_{15} - \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{7} - \beta_{6} + 4 \beta_1) q^{47} + (\beta_{9} + 2 \beta_{5}) q^{49} + ( - 2 \beta_{15} - 2 \beta_{13} - \beta_{8} - 3 \beta_{7} - \beta_{6} - 3 \beta_1) q^{50} + (2 \beta_{14} + 3 \beta_{9} + 3 \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{52} + ( - \beta_{13} - 2 \beta_{8}) q^{53} + ( - \beta_{10} - \beta_{9} + \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{55} + (2 \beta_{15} + 2 \beta_{13} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_1) q^{56} + (2 \beta_{3} + 8) q^{58} + ( - \beta_{13} + \beta_{8} - 2 \beta_1) q^{59} + ( - \beta_{14} - \beta_{10} - 2 \beta_{9} - \beta_{5} + 6 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{61} + ( - \beta_{15} + \beta_{13} + \beta_{11} + \beta_{8} - 2 \beta_{7} - \beta_1) q^{62} + ( - 3 \beta_{14} - 3 \beta_{9} - 6 \beta_{5} + 2 \beta_{3} + 12) q^{64} + (2 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} + \beta_{8} - 2 \beta_{7} - 2 \beta_1) q^{65} + (2 \beta_{14} - \beta_{10} - 2 \beta_{5} - \beta_{3} + 2 \beta_{2} - 2) q^{67} + ( - \beta_{12} - \beta_{8} + 2 \beta_{7} - 2 \beta_1) q^{68} + ( - \beta_{10} - 4 \beta_{5} - 7 \beta_{4} - \beta_{3} - 14) q^{70} + (\beta_{15} - \beta_{13} + \beta_{8} + 2 \beta_{6}) q^{71} + (\beta_{10} - 2 \beta_{9} + 2 \beta_{5} - 4 \beta_{4} - 3 \beta_{2} - 4) q^{73} + ( - 3 \beta_{13} - 3 \beta_{11} - 3 \beta_{8} + 2 \beta_{7} + \beta_1) q^{74} + ( - 2 \beta_{14} - 2 \beta_{10} - 3 \beta_{9} - 3 \beta_{4} + 2 \beta_{2} - 6) q^{76} + ( - \beta_{12} - \beta_{8} + \beta_{6}) q^{77} + (\beta_{10} - \beta_{9} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2) q^{79} + ( - 2 \beta_{15} - 7 \beta_{13} + 2 \beta_{7} - 2 \beta_{6} + 4 \beta_1) q^{80} + (3 \beta_{14} + \beta_{10} + 4 \beta_{9} + 2 \beta_{5} - 5 \beta_{4} - \beta_{3} - 3 \beta_{2} + \cdots - 3) q^{82}+ \cdots + ( - \beta_{15} - 2 \beta_{13} - 2 \beta_{12} + \beta_{11} + 2 \beta_{8} - 2 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 8 q^{7} + 12 q^{10} - 24 q^{13} - 24 q^{16} - 12 q^{19} + 24 q^{22} + 20 q^{25} - 12 q^{28} + 12 q^{34} - 24 q^{40} + 12 q^{52} - 4 q^{55} + 128 q^{58} - 44 q^{61} + 168 q^{64} - 24 q^{67} - 168 q^{70} - 20 q^{73} - 96 q^{76} - 48 q^{79} + 28 q^{82} - 56 q^{85} - 24 q^{91} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 16x^{14} + 174x^{12} + 1012x^{10} + 4243x^{8} + 9708x^{6} + 15858x^{4} + 12150x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 481967 \nu^{14} + 30766372 \nu^{12} + 473907666 \nu^{10} + 5119222054 \nu^{8} + 25255513579 \nu^{6} + 88135584891 \nu^{4} + \cdots + 82286048988 ) / 21904094763 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 326606 \nu^{14} + 4547720 \nu^{12} + 47592867 \nu^{10} + 231730940 \nu^{8} + 904757018 \nu^{6} + 962205174 \nu^{4} + 748162737 \nu^{2} + \cdots - 7319943180 ) / 2433788307 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9939136 \nu^{14} + 150207814 \nu^{12} + 1606621224 \nu^{10} + 8773398223 \nu^{8} + 35915018668 \nu^{6} + 72060692802 \nu^{4} + \cdots + 34847824212 ) / 65712284289 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3704024 \nu^{14} + 40373015 \nu^{12} + 381453396 \nu^{10} + 906590177 \nu^{8} + 887860574 \nu^{6} - 24565580241 \nu^{4} + \cdots - 56770896384 ) / 21904094763 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26555897 \nu^{15} + 62210210 \nu^{13} - 770314944 \nu^{11} - 30249558199 \nu^{9} - 191131571350 \nu^{7} - 937524731802 \nu^{5} + \cdots - 2199042758754 \nu ) / 197136852867 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9939136 \nu^{15} - 150207814 \nu^{13} - 1606621224 \nu^{11} - 8773398223 \nu^{9} - 35915018668 \nu^{7} - 72060692802 \nu^{5} + \cdots - 100560108501 \nu ) / 65712284289 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 32058956 \nu^{15} - 505462190 \nu^{13} - 5463091302 \nu^{11} - 31353520355 \nu^{9} - 130718214368 \nu^{7} - 308344384614 \nu^{5} + \cdots - 700940621358 \nu ) / 197136852867 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10005800 \nu^{14} + 163560617 \nu^{12} + 1774630644 \nu^{10} + 10542157166 \nu^{8} + 43490411282 \nu^{6} + 100433630874 \nu^{4} + \cdots + 71014305663 ) / 21904094763 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 30938182 \nu^{14} - 478042816 \nu^{12} - 5141477487 \nu^{10} - 28836857512 \nu^{8} - 119231635186 \nu^{6} - 262263231510 \nu^{4} + \cdots - 337029762708 ) / 65712284289 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17666630 \nu^{15} + 328891601 \nu^{13} + 3662652030 \nu^{11} + 23457056468 \nu^{9} + 96537911477 \nu^{7} + 231158174538 \nu^{5} + \cdots + 151450330770 \nu ) / 65712284289 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 19112531 \nu^{15} - 236592485 \nu^{13} - 2240929032 \nu^{11} - 8099390306 \nu^{9} - 20771370740 \nu^{7} + 33248580135 \nu^{5} + \cdots + 95407816194 \nu ) / 65712284289 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 58514042 \nu^{15} - 873827510 \nu^{13} - 9318113529 \nu^{11} - 50123726495 \nu^{9} - 204003532826 \nu^{7} - 386283003708 \nu^{5} + \cdots + 89111629089 \nu ) / 197136852867 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 6210188 \nu^{14} - 89119619 \nu^{12} - 904945566 \nu^{10} - 4406204120 \nu^{8} - 15342294773 \nu^{6} - 18295668252 \nu^{4} + \cdots + 11675287323 ) / 7301364921 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 324198086 \nu^{15} + 5217299756 \nu^{13} + 56522041548 \nu^{11} + 328205077598 \nu^{9} + 1350470930357 \nu^{7} + \cdots + 2317137727815 \nu ) / 197136852867 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 4\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{13} + \beta_{8} - 5\beta_{7} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{10} - \beta_{9} + \beta_{5} - 23\beta_{4} - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - 10\beta_{13} + \beta_{12} + \beta_{11} - 20\beta_{8} + 31\beta_{7} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{14} - 13\beta_{9} - 26\beta_{5} + 58\beta_{3} + 154 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{15} - 84\beta_{13} - 32\beta_{12} + 16\beta_{11} + 84\beta_{8} - 26\beta_{6} + 209\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 48\beta_{14} + 422\beta_{10} + 252\beta_{9} + 126\beta_{5} + 1049\beta_{4} - 422\beta_{3} - 48\beta_{2} - 48 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 252 \beta_{15} + 1348 \beta_{13} + 174 \beta_{12} - 348 \beta_{11} + 674 \beta_{8} - 1471 \beta_{7} + 126 \beta_{6} - 1471 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -3115\beta_{10} - 1100\beta_{9} + 1100\beta_{5} - 7528\beta_{4} + 522\beta_{2} - 7528 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1100 \beta_{15} - 5315 \beta_{13} + 1622 \beta_{12} + 1622 \beta_{11} - 10630 \beta_{8} + 10643 \beta_{7} + 1100 \beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -4866\beta_{14} - 9137\beta_{9} - 18274\beta_{5} + 23288\beta_{3} + 60083 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9137 \beta_{15} - 41562 \beta_{13} - 28006 \beta_{12} + 14003 \beta_{11} + 41562 \beta_{8} - 18274 \beta_{6} + 78505 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 42009 \beta_{14} + 175780 \beta_{10} + 147678 \beta_{9} + 73839 \beta_{5} + 411295 \beta_{4} - 175780 \beta_{3} - 42009 \beta_{2} - 42009 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 147678 \beta_{15} + 646916 \beta_{13} + 115848 \beta_{12} - 231696 \beta_{11} + 323458 \beta_{8} - 587075 \beta_{7} + 73839 \beta_{6} - 587075 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
1.38883 2.40552i
1.13921 1.97317i
0.733987 1.27130i
0.484374 0.838961i
−0.484374 + 0.838961i
−0.733987 + 1.27130i
−1.13921 + 1.97317i
−1.38883 + 2.40552i
1.38883 + 2.40552i
1.13921 + 1.97317i
0.733987 + 1.27130i
0.484374 + 0.838961i
−0.484374 0.838961i
−0.733987 1.27130i
−1.13921 1.97317i
−1.38883 2.40552i
−1.38883 + 2.40552i 0 −2.85768 4.94964i −2.00395 1.15698i 0 −0.442911 10.3200 0 5.56628 3.21369i
8.2 −1.13921 + 1.97317i 0 −1.59561 2.76368i 2.90154 + 1.67520i 0 3.54697 2.71412 0 −6.61094 + 3.81683i
8.3 −0.733987 + 1.27130i 0 −0.0774748 0.134190i −3.12320 1.80318i 0 −3.25512 −2.70849 0 4.58478 2.64702i
8.4 −0.484374 + 0.838961i 0 0.530763 + 0.919308i 0.557544 + 0.321898i 0 2.15106 −2.96585 0 −0.540120 + 0.311839i
8.5 0.484374 0.838961i 0 0.530763 + 0.919308i −0.557544 0.321898i 0 2.15106 2.96585 0 −0.540120 + 0.311839i
8.6 0.733987 1.27130i 0 −0.0774748 0.134190i 3.12320 + 1.80318i 0 −3.25512 2.70849 0 4.58478 2.64702i
8.7 1.13921 1.97317i 0 −1.59561 2.76368i −2.90154 1.67520i 0 3.54697 −2.71412 0 −6.61094 + 3.81683i
8.8 1.38883 2.40552i 0 −2.85768 4.94964i 2.00395 + 1.15698i 0 −0.442911 −10.3200 0 5.56628 3.21369i
107.1 −1.38883 2.40552i 0 −2.85768 + 4.94964i −2.00395 + 1.15698i 0 −0.442911 10.3200 0 5.56628 + 3.21369i
107.2 −1.13921 1.97317i 0 −1.59561 + 2.76368i 2.90154 1.67520i 0 3.54697 2.71412 0 −6.61094 3.81683i
107.3 −0.733987 1.27130i 0 −0.0774748 + 0.134190i −3.12320 + 1.80318i 0 −3.25512 −2.70849 0 4.58478 + 2.64702i
107.4 −0.484374 0.838961i 0 0.530763 0.919308i 0.557544 0.321898i 0 2.15106 −2.96585 0 −0.540120 0.311839i
107.5 0.484374 + 0.838961i 0 0.530763 0.919308i −0.557544 + 0.321898i 0 2.15106 2.96585 0 −0.540120 0.311839i
107.6 0.733987 + 1.27130i 0 −0.0774748 + 0.134190i 3.12320 1.80318i 0 −3.25512 2.70849 0 4.58478 + 2.64702i
107.7 1.13921 + 1.97317i 0 −1.59561 + 2.76368i −2.90154 + 1.67520i 0 3.54697 −2.71412 0 −6.61094 3.81683i
107.8 1.38883 + 2.40552i 0 −2.85768 + 4.94964i 2.00395 1.15698i 0 −0.442911 −10.3200 0 5.56628 + 3.21369i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.m.a 16
3.b odd 2 1 inner 171.2.m.a 16
4.b odd 2 1 2736.2.dc.c 16
12.b even 2 1 2736.2.dc.c 16
19.d odd 6 1 inner 171.2.m.a 16
57.f even 6 1 inner 171.2.m.a 16
76.f even 6 1 2736.2.dc.c 16
228.n odd 6 1 2736.2.dc.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.m.a 16 1.a even 1 1 trivial
171.2.m.a 16 3.b odd 2 1 inner
171.2.m.a 16 19.d odd 6 1 inner
171.2.m.a 16 57.f even 6 1 inner
2736.2.dc.c 16 4.b odd 2 1
2736.2.dc.c 16 12.b even 2 1
2736.2.dc.c 16 76.f even 6 1
2736.2.dc.c 16 228.n odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(171, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 16 T^{14} + 174 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 30 T^{14} + 612 T^{12} + \cdots + 104976 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} - 12 T^{2} + 20 T + 11)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + 30 T^{6} + 240 T^{4} + 584 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 12 T^{7} + 50 T^{6} + 24 T^{5} + \cdots + 225)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 56 T^{14} + 2320 T^{12} + \cdots + 26873856 \) Copy content Toggle raw display
$19$ \( (T^{8} + 6 T^{7} + 24 T^{6} - 114 T^{5} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 66 T^{14} + 3288 T^{12} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( T^{16} + 64 T^{14} + \cdots + 429981696 \) Copy content Toggle raw display
$31$ \( (T^{8} + 80 T^{6} + 1798 T^{4} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 180 T^{6} + 8910 T^{4} + \cdots + 793881)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 184 T^{14} + \cdots + 16796160000 \) Copy content Toggle raw display
$43$ \( (T^{8} + 44 T^{6} - 84 T^{5} + 1737 T^{4} + \cdots + 39601)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 857889103585536 \) Copy content Toggle raw display
$53$ \( T^{16} + 90 T^{14} + 5940 T^{12} + \cdots + 8503056 \) Copy content Toggle raw display
$59$ \( T^{16} + 106 T^{14} + \cdots + 13680577296 \) Copy content Toggle raw display
$61$ \( (T^{8} + 22 T^{7} + 400 T^{6} + \cdots + 18879025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 12 T^{7} - 130 T^{6} + \cdots + 45369)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 360 T^{14} + \cdots + 892616806656 \) Copy content Toggle raw display
$73$ \( (T^{8} + 10 T^{7} + 312 T^{6} + \cdots + 2152089)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 24 T^{7} + 224 T^{6} + \cdots + 154449)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 456 T^{6} + 50256 T^{4} + \cdots + 627264)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + 438 T^{14} + \cdots + 34867844010000 \) Copy content Toggle raw display
$97$ \( (T^{8} - 24 T^{7} + 116 T^{6} + \cdots + 944784)^{2} \) Copy content Toggle raw display
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