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Properties

Label 1080.2.d.j
Level $1080$
Weight $2$
Character orbit 1080.d
Analytic conductor $8.624$
Analytic rank $0$
Dimension $20$
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] = [1080,2,Mod(109,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.d (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: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3x^{18} + 8x^{16} - 24x^{14} + 56x^{12} - 92x^{10} + 224x^{8} - 384x^{6} + 512x^{4} - 768x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{4} + \beta_{14} q^{5} + (\beta_{15} - \beta_{3} + 1) q^{7} + ( - \beta_{17} - \beta_{13} + \beta_{6}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{4} + \beta_{14} q^{5} + (\beta_{15} - \beta_{3} + 1) q^{7} + ( - \beta_{17} - \beta_{13} + \beta_{6}) q^{8} + \beta_{12} q^{10} + (\beta_{14} - \beta_{8} + \cdots + \beta_{5}) q^{11}+ \cdots + ( - 2 \beta_{19} + \beta_{17} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{4} - 2 q^{10} + 4 q^{13} - 14 q^{16} + 34 q^{22} + 20 q^{25} - 20 q^{28} + 12 q^{31} + 6 q^{34} + 32 q^{37} - 26 q^{40} - 12 q^{43} + 2 q^{46} - 52 q^{49} + 50 q^{52} - 28 q^{55} - 6 q^{58} + 54 q^{64} + 64 q^{70} - 24 q^{76} + 36 q^{79} - 32 q^{82} + 44 q^{85} + 30 q^{88} - 22 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3x^{18} + 8x^{16} - 24x^{14} + 56x^{12} - 92x^{10} + 224x^{8} - 384x^{6} + 512x^{4} - 768x^{2} + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{18} - 5\nu^{16} - 24\nu^{12} + 40\nu^{10} - 164\nu^{8} + 384\nu^{6} - 576\nu^{4} + 640\nu^{2} - 1792 ) / 768 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{18} - \nu^{16} - 6\nu^{14} - 40\nu^{10} + 116\nu^{8} - 24\nu^{6} + 96\nu^{4} - 832\nu^{2} - 128 ) / 384 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{19} - \nu^{17} + 6 \nu^{15} - 36 \nu^{13} + 56 \nu^{11} - 172 \nu^{9} + 456 \nu^{7} + \cdots - 1664 \nu ) / 768 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{19} - 3\nu^{17} - 32\nu^{13} + 24\nu^{11} - 92\nu^{9} + 160\nu^{7} - 416\nu^{5} - 128\nu^{3} - 1536\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{18} - \nu^{16} + 6\nu^{14} - 12\nu^{12} + 32\nu^{10} - 28\nu^{8} + 72\nu^{6} + 48\nu^{4} + 32\nu^{2} + 256 ) / 192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{19} - 2\nu^{17} - 3\nu^{15} + 6\nu^{13} - 8\nu^{11} + 4\nu^{9} - 36\nu^{7} - 72\nu^{5} - 32\nu^{3} - 64\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{19} + 13 \nu^{17} - 18 \nu^{15} + 60 \nu^{13} - 152 \nu^{11} + 268 \nu^{9} - 216 \nu^{7} + \cdots + 896 \nu ) / 768 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{18} - 3 \nu^{16} + 8 \nu^{14} - 24 \nu^{12} + 56 \nu^{10} - 92 \nu^{8} + 224 \nu^{6} - 384 \nu^{4} + \cdots - 768 ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5 \nu^{18} + 11 \nu^{16} - 60 \nu^{14} + 120 \nu^{12} - 184 \nu^{10} + 428 \nu^{8} - 1008 \nu^{6} + \cdots + 2560 ) / 768 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2 \nu^{18} + \nu^{16} + 3 \nu^{14} - 24 \nu^{12} + 16 \nu^{10} - 32 \nu^{8} + 252 \nu^{6} + \cdots - 1024 ) / 192 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -3\nu^{19} + 5\nu^{17} - 12\nu^{15} + 40\nu^{13} - 72\nu^{11} + 52\nu^{9} - 304\nu^{7} + 256\nu^{5} + 256\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - \nu^{19} + \nu^{17} - 9 \nu^{15} + 21 \nu^{13} - 32 \nu^{11} + 76 \nu^{9} - 192 \nu^{7} + \cdots + 608 \nu ) / 192 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{18} - 5 \nu^{16} + 6 \nu^{14} - 32 \nu^{12} + 56 \nu^{10} - 108 \nu^{8} + 216 \nu^{6} + \cdots - 1024 ) / 128 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 5 \nu^{18} - 5 \nu^{16} + 18 \nu^{14} - 48 \nu^{12} + 88 \nu^{10} - 188 \nu^{8} + 456 \nu^{6} + \cdots - 1024 ) / 384 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( \nu^{19} - 2 \nu^{17} + 3 \nu^{15} - 18 \nu^{13} + 24 \nu^{11} - 36 \nu^{9} + 116 \nu^{7} + \cdots - 448 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11 \nu^{19} - 41 \nu^{17} + 96 \nu^{15} - 312 \nu^{13} + 520 \nu^{11} - 1076 \nu^{9} + \cdots - 9472 \nu ) / 1536 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 11 \nu^{19} - 17 \nu^{17} + 60 \nu^{15} - 132 \nu^{13} + 232 \nu^{11} - 500 \nu^{9} + 1152 \nu^{7} + \cdots - 2176 \nu ) / 768 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{13} - \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{7} + \beta_{3} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{18} + \beta_{17} - 2\beta_{14} - \beta_{9} + 2\beta_{8} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{15} + \beta_{10} + \beta_{7} + \beta_{4} + 3\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{17} - 2\beta_{13} + \beta_{9} + 2\beta_{8} + 2\beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -2\beta_{16} - \beta_{15} + 5\beta_{10} + \beta_{7} + 3\beta_{4} - \beta_{3} + 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -2\beta_{19} + 4\beta_{18} + 3\beta_{17} + 2\beta_{14} + 3\beta_{9} - 2\beta_{8} - 6\beta_{6} - 2\beta_{5} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2\beta_{16} - \beta_{15} + 4\beta_{11} + 7\beta_{10} + 7\beta_{7} - \beta_{4} + \beta_{3} + 2\beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2 \beta_{19} + \beta_{17} + 6 \beta_{14} - 4 \beta_{13} - 7 \beta_{9} - 10 \beta_{8} + \cdots - 21 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10 \beta_{16} - 11 \beta_{15} - 8 \beta_{12} + 4 \beta_{11} + 13 \beta_{10} + 5 \beta_{7} + 5 \beta_{4} + \cdots - 39 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10 \beta_{19} + 8 \beta_{18} - 25 \beta_{17} + 26 \beta_{14} - 16 \beta_{13} + 3 \beta_{9} + \cdots - 47 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2 \beta_{16} - 9 \beta_{15} - 8 \beta_{12} - 20 \beta_{11} + 19 \beta_{10} - 13 \beta_{7} + 7 \beta_{4} + \cdots - 21 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2 \beta_{19} + 32 \beta_{18} - 55 \beta_{17} - 10 \beta_{14} - 32 \beta_{13} + 21 \beta_{9} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 22 \beta_{16} + \beta_{15} + 40 \beta_{12} + 4 \beta_{11} + 29 \beta_{10} - 35 \beta_{7} - 55 \beta_{4} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 22 \beta_{19} + 64 \beta_{18} - 41 \beta_{17} + 50 \beta_{14} + 8 \beta_{13} + 19 \beta_{9} + \cdots + 33 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 118 \beta_{16} - \beta_{15} - 8 \beta_{12} + 44 \beta_{11} - 37 \beta_{10} - 21 \beta_{7} + 7 \beta_{4} + \cdots - 101 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 118 \beta_{19} - 16 \beta_{18} + 33 \beta_{17} + 190 \beta_{14} + 56 \beta_{13} + 21 \beta_{9} + \cdots - 217 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.37859 + 0.315404i
1.37859 − 0.315404i
1.35662 + 0.399488i
1.35662 − 0.399488i
0.986501 + 1.01332i
0.986501 − 1.01332i
0.847166 + 1.13239i
0.847166 − 1.13239i
0.564088 + 1.29684i
0.564088 − 1.29684i
−0.564088 + 1.29684i
−0.564088 − 1.29684i
−0.847166 + 1.13239i
−0.847166 − 1.13239i
−0.986501 + 1.01332i
−0.986501 − 1.01332i
−1.35662 + 0.399488i
−1.35662 − 0.399488i
−1.37859 + 0.315404i
−1.37859 − 0.315404i
−1.37859 − 0.315404i 0 1.80104 + 0.869628i 2.03021 − 0.937151i 0 − 3.36920i −2.20862 − 1.76692i 0 −3.09441 + 0.651615i
109.2 −1.37859 + 0.315404i 0 1.80104 − 0.869628i 2.03021 + 0.937151i 0 3.36920i −2.20862 + 1.76692i 0 −3.09441 − 0.651615i
109.3 −1.35662 − 0.399488i 0 1.68082 + 1.08390i −1.49578 − 1.66212i 0 1.43534i −1.84722 − 2.14191i 0 1.36521 + 2.85240i
109.4 −1.35662 + 0.399488i 0 1.68082 − 1.08390i −1.49578 + 1.66212i 0 − 1.43534i −1.84722 + 2.14191i 0 1.36521 − 2.85240i
109.5 −0.986501 − 1.01332i 0 −0.0536316 + 1.99928i −0.569524 + 2.16232i 0 4.64762i 2.07882 − 1.91795i 0 2.75296 − 1.55602i
109.6 −0.986501 + 1.01332i 0 −0.0536316 − 1.99928i −0.569524 − 2.16232i 0 − 4.64762i 2.07882 + 1.91795i 0 2.75296 + 1.55602i
109.7 −0.847166 − 1.13239i 0 −0.564618 + 1.91865i 1.82935 − 1.28588i 0 1.09701i 2.65098 − 0.986045i 0 −3.00588 − 0.982180i
109.8 −0.847166 + 1.13239i 0 −0.564618 − 1.91865i 1.82935 + 1.28588i 0 − 1.09701i 2.65098 + 0.986045i 0 −3.00588 + 0.982180i
109.9 −0.564088 − 1.29684i 0 −1.36361 + 1.46307i −2.22935 + 0.173169i 0 − 3.43285i 2.66657 + 0.943090i 0 1.48212 + 2.79344i
109.10 −0.564088 + 1.29684i 0 −1.36361 − 1.46307i −2.22935 − 0.173169i 0 3.43285i 2.66657 − 0.943090i 0 1.48212 − 2.79344i
109.11 0.564088 − 1.29684i 0 −1.36361 − 1.46307i 2.22935 + 0.173169i 0 3.43285i −2.66657 + 0.943090i 0 1.48212 − 2.79344i
109.12 0.564088 + 1.29684i 0 −1.36361 + 1.46307i 2.22935 − 0.173169i 0 − 3.43285i −2.66657 − 0.943090i 0 1.48212 + 2.79344i
109.13 0.847166 − 1.13239i 0 −0.564618 − 1.91865i −1.82935 − 1.28588i 0 − 1.09701i −2.65098 − 0.986045i 0 −3.00588 + 0.982180i
109.14 0.847166 + 1.13239i 0 −0.564618 + 1.91865i −1.82935 + 1.28588i 0 1.09701i −2.65098 + 0.986045i 0 −3.00588 − 0.982180i
109.15 0.986501 − 1.01332i 0 −0.0536316 − 1.99928i 0.569524 + 2.16232i 0 − 4.64762i −2.07882 − 1.91795i 0 2.75296 + 1.55602i
109.16 0.986501 + 1.01332i 0 −0.0536316 + 1.99928i 0.569524 − 2.16232i 0 4.64762i −2.07882 + 1.91795i 0 2.75296 − 1.55602i
109.17 1.35662 − 0.399488i 0 1.68082 − 1.08390i 1.49578 − 1.66212i 0 − 1.43534i 1.84722 − 2.14191i 0 1.36521 − 2.85240i
109.18 1.35662 + 0.399488i 0 1.68082 + 1.08390i 1.49578 + 1.66212i 0 1.43534i 1.84722 + 2.14191i 0 1.36521 + 2.85240i
109.19 1.37859 − 0.315404i 0 1.80104 − 0.869628i −2.03021 − 0.937151i 0 3.36920i 2.20862 − 1.76692i 0 −3.09441 − 0.651615i
109.20 1.37859 + 0.315404i 0 1.80104 + 0.869628i −2.03021 + 0.937151i 0 − 3.36920i 2.20862 + 1.76692i 0 −3.09441 + 0.651615i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
40.f even 2 1 inner
120.i odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.d.j yes 20
3.b odd 2 1 inner 1080.2.d.j yes 20
4.b odd 2 1 4320.2.d.j 20
5.b even 2 1 1080.2.d.i ✓ 20
8.b even 2 1 1080.2.d.i ✓ 20
8.d odd 2 1 4320.2.d.i 20
12.b even 2 1 4320.2.d.j 20
15.d odd 2 1 1080.2.d.i ✓ 20
20.d odd 2 1 4320.2.d.i 20
24.f even 2 1 4320.2.d.i 20
24.h odd 2 1 1080.2.d.i ✓ 20
40.e odd 2 1 4320.2.d.j 20
40.f even 2 1 inner 1080.2.d.j yes 20
60.h even 2 1 4320.2.d.i 20
120.i odd 2 1 inner 1080.2.d.j yes 20
120.m even 2 1 4320.2.d.j 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.d.i ✓ 20 5.b even 2 1
1080.2.d.i ✓ 20 8.b even 2 1
1080.2.d.i ✓ 20 15.d odd 2 1
1080.2.d.i ✓ 20 24.h odd 2 1
1080.2.d.j yes 20 1.a even 1 1 trivial
1080.2.d.j yes 20 3.b odd 2 1 inner
1080.2.d.j yes 20 40.f even 2 1 inner
1080.2.d.j yes 20 120.i odd 2 1 inner
4320.2.d.i 20 8.d odd 2 1
4320.2.d.i 20 20.d odd 2 1
4320.2.d.i 20 24.f even 2 1
4320.2.d.i 20 60.h even 2 1
4320.2.d.j 20 4.b odd 2 1
4320.2.d.j 20 12.b even 2 1
4320.2.d.j 20 40.e odd 2 1
4320.2.d.j 20 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1080, [\chi])\):

\( T_{7}^{10} + 48T_{7}^{8} + 782T_{7}^{6} + 5068T_{7}^{4} + 11001T_{7}^{2} + 7164 \) Copy content Toggle raw display
\( T_{11}^{10} + 53T_{11}^{8} + 972T_{11}^{6} + 7000T_{11}^{4} + 14928T_{11}^{2} + 144 \) Copy content Toggle raw display
\( T_{13}^{5} - T_{13}^{4} - 34T_{13}^{3} - 22T_{13}^{2} + 189T_{13} + 243 \) Copy content Toggle raw display
\( T_{53}^{10} - 300T_{53}^{8} + 27832T_{53}^{6} - 866704T_{53}^{4} + 9083600T_{53}^{2} - 12239296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 3 T^{18} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 10 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{10} + 48 T^{8} + \cdots + 7164)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 53 T^{8} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} - T^{4} - 34 T^{3} + \cdots + 243)^{4} \) Copy content Toggle raw display
$17$ \( (T^{10} + 69 T^{8} + \cdots + 576)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 132 T^{8} + \cdots + 114624)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + 101 T^{8} + \cdots + 41616)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 81 T^{8} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 3 T^{4} - 40 T^{3} + \cdots + 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{5} - 8 T^{4} - 48 T^{3} + \cdots + 18)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} - 252 T^{8} + \cdots - 83560896)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 3 T^{4} + \cdots - 6192)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + 213 T^{8} + \cdots + 3779136)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 300 T^{8} + \cdots - 12239296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 308 T^{8} + \cdots + 60466176)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 236 T^{8} + \cdots + 114624)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 226 T^{3} + \cdots + 12564)^{4} \) Copy content Toggle raw display
$71$ \( (T^{10} - 464 T^{8} + \cdots - 16505856)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 456 T^{8} + \cdots + 549686556)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 9 T^{4} + \cdots + 547)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} - 352 T^{8} + \cdots - 50944)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 428 T^{8} + \cdots - 16505856)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 708 T^{8} + \cdots + 3391036416)^{2} \) Copy content Toggle raw display
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