Properties

Label 4320.2.k.d
Level $4320$
Weight $2$
Character orbit 4320.k
Analytic conductor $34.495$
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] = [4320,2,Mod(2161,4320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4320.2161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4320.k (of order \(2\), degree \(1\), not 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: \(34.4953736732\)
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: \( x^{20} - x^{18} + 5x^{16} + 28x^{12} - 28x^{10} + 112x^{8} + 320x^{4} - 256x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: no (minimal twist has level 1080)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} - \beta_{5} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} - \beta_{5} q^{7} + \beta_{10} q^{11} - \beta_1 q^{13} - \beta_{16} q^{17} - \beta_{3} q^{19} + \beta_{18} q^{23} - q^{25} + (\beta_{12} - \beta_{9} - \beta_{6}) q^{29} + (\beta_{11} + \beta_{5} - 1) q^{31} + \beta_{12} q^{35} + (\beta_{4} + \beta_{2} + \beta_1) q^{37} + (\beta_{19} + \beta_{8}) q^{41} + (\beta_{3} - \beta_{2} + \beta_1) q^{43} + (\beta_{19} + \beta_{17} + \beta_{8}) q^{47} + (\beta_{14} + \beta_{13} + \beta_{5} + 2) q^{49} + ( - \beta_{15} - \beta_{10} + \beta_{6}) q^{53} + \beta_{14} q^{55} + ( - \beta_{15} + \beta_{10}) q^{59} + (\beta_{3} - \beta_1) q^{61} - \beta_{17} q^{65} + ( - \beta_{7} - \beta_{2} + \beta_1) q^{67} + (\beta_{18} + \beta_{16}) q^{71} + ( - \beta_{14} - \beta_{11} - 2) q^{73} + ( - \beta_{15} - \beta_{12} - \beta_{10} - \beta_{9} - \beta_{6}) q^{77} + ( - \beta_{14} - \beta_{13} + 2 \beta_{5}) q^{79} + ( - \beta_{15} - \beta_{12} - \beta_{10} + \beta_{6}) q^{83} + \beta_{2} q^{85} + ( - \beta_{17} + \beta_{16} - \beta_{8}) q^{89} + ( - \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2}) q^{91} - \beta_{8} q^{95} + ( - 2 \beta_{14} - \beta_{11} - 2 \beta_{5} + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{25} - 20 q^{31} + 36 q^{49} - 40 q^{73} + 4 q^{79} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{18} + 5x^{16} + 28x^{12} - 28x^{10} + 112x^{8} + 320x^{4} - 256x^{2} + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{18} + 13\nu^{16} + 7\nu^{14} + 28\nu^{12} - 28\nu^{10} + 420\nu^{8} + 672\nu^{4} - 96\nu^{2} + 4992 ) / 448 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{18} - 7 \nu^{16} - 5 \nu^{14} - 16 \nu^{12} - 20 \nu^{10} - 116 \nu^{8} + 16 \nu^{6} - 288 \nu^{4} - 256 \nu^{2} - 1408 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9 \nu^{18} + 107 \nu^{16} + 105 \nu^{14} + 196 \nu^{12} + 252 \nu^{10} + 2212 \nu^{8} + 896 \nu^{6} + 2688 \nu^{4} + 6912 \nu^{2} + 26752 ) / 896 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11 \nu^{18} - 31 \nu^{16} + 35 \nu^{14} - 28 \nu^{12} + 140 \nu^{10} - 644 \nu^{8} + 224 \nu^{6} - 224 \nu^{4} + 832 \nu^{2} - 6528 ) / 896 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{18} + \nu^{16} - \nu^{14} + 4\nu^{12} - 16\nu^{10} + 36\nu^{8} - 32\nu^{6} + 16\nu^{4} - 160\nu^{2} + 384 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17 \nu^{19} - 53 \nu^{17} - 7 \nu^{15} - 84 \nu^{13} + 252 \nu^{11} - 1260 \nu^{9} + 224 \nu^{7} - 1344 \nu^{5} + 960 \nu^{3} - 15872 \nu ) / 3584 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13 \nu^{18} + 167 \nu^{16} + 133 \nu^{14} + 364 \nu^{12} + 420 \nu^{10} + 3556 \nu^{8} + 1120 \nu^{6} + 6944 \nu^{4} + 10432 \nu^{2} + 42624 ) / 896 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{19} - \nu^{17} + 5 \nu^{15} - 4 \nu^{13} - 36 \nu^{11} + 132 \nu^{9} - 32 \nu^{7} - 256 \nu^{5} - 320 \nu^{3} + 1536 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{19} + 15 \nu^{17} + 5 \nu^{15} + 60 \nu^{13} - 84 \nu^{11} + 452 \nu^{9} + 96 \nu^{7} + 832 \nu^{5} - 64 \nu^{3} + 6656 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13 \nu^{19} - 29 \nu^{17} - 7 \nu^{15} - 56 \nu^{13} + 308 \nu^{11} - 812 \nu^{9} - 112 \nu^{7} - 224 \nu^{5} + 3264 \nu^{3} - 12032 \nu ) / 1792 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{18} - \nu^{12} - 16\nu^{10} + 8\nu^{8} - 4\nu^{6} - 32\nu^{4} - 224\nu^{2} + 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5 \nu^{19} + 23 \nu^{17} + 7 \nu^{15} + 56 \nu^{13} - 98 \nu^{11} + 532 \nu^{9} - 112 \nu^{7} + 616 \nu^{5} - 480 \nu^{3} + 6592 \nu ) / 448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -3\nu^{18} + \nu^{16} - 5\nu^{14} - 2\nu^{12} - 60\nu^{10} + 44\nu^{8} - 120\nu^{6} - 64\nu^{4} - 512\nu^{2} + 576 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -3\nu^{18} + \nu^{16} - 5\nu^{14} + 6\nu^{12} - 68\nu^{10} + 52\nu^{8} - 88\nu^{6} - 736\nu^{2} + 768 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 33 \nu^{19} - 93 \nu^{17} + 49 \nu^{15} - 252 \nu^{13} + 812 \nu^{11} - 2604 \nu^{9} + 1344 \nu^{7} - 3584 \nu^{5} + 5184 \nu^{3} - 27648 \nu ) / 1792 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 11 \nu^{19} + \nu^{17} + 27 \nu^{15} - 20 \nu^{13} + 196 \nu^{11} - 68 \nu^{9} + 320 \nu^{7} + 128 \nu^{5} + 2624 \nu^{3} - 512 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 3 \nu^{19} + 3 \nu^{17} + 5 \nu^{15} + 2 \nu^{13} + 64 \nu^{11} + 20 \nu^{9} + 88 \nu^{7} + 80 \nu^{5} + 896 \nu^{3} + 128 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 25 \nu^{19} - 19 \nu^{17} - 49 \nu^{15} - 28 \nu^{13} - 508 \nu^{11} - 180 \nu^{9} - 672 \nu^{7} - 1344 \nu^{5} - 5824 \nu^{3} - 2048 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 29 \nu^{19} + 15 \nu^{17} + 69 \nu^{15} + 28 \nu^{13} + 620 \nu^{11} + 68 \nu^{9} + 1120 \nu^{7} + 1344 \nu^{5} + 7104 \nu^{3} - 1024 \nu ) / 512 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{19} - \beta_{18} + \beta_{15} + \beta_{9} - \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - \beta_{11} + \beta_{7} + \beta_{4} + 2\beta_{2} - 2\beta _1 + 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{18} + \beta_{17} + \beta_{16} + 2\beta_{10} + \beta_{9} - \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{13} + \beta_{11} + 3\beta_{7} - 4\beta_{5} + \beta_{4} - 4\beta_{3} - 7 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{19} - \beta_{18} - 6 \beta_{17} + 2 \beta_{16} - \beta_{15} - 4 \beta_{12} + 4 \beta_{10} + \beta_{9} - 4 \beta_{8} - 17 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{14} - 2\beta_{13} + \beta_{11} + \beta_{7} - 4\beta_{5} - 2\beta_{4} + 3\beta_{2} - \beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5 \beta_{19} + 5 \beta_{18} + 4 \beta_{17} - 4 \beta_{16} - \beta_{15} - 12 \beta_{12} - 16 \beta_{10} + 3 \beta_{9} - 4 \beta_{8} - 11 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4 \beta_{14} - \beta_{13} - 3 \beta_{11} - 5 \beta_{7} + 12 \beta_{5} - \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 22 \beta _1 - 65 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4 \beta_{19} + \beta_{18} + \beta_{17} - 3 \beta_{16} - 10 \beta_{15} - 8 \beta_{12} - 6 \beta_{10} - 5 \beta_{9} + 12 \beta_{8} + 9 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 24 \beta_{14} - 9 \beta_{13} + 15 \beta_{11} - 3 \beta_{7} + 52 \beta_{5} - 17 \beta_{4} - 12 \beta_{3} - 16 \beta_{2} + 8 \beta _1 + 79 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5 \beta_{19} - 35 \beta_{18} - 10 \beta_{17} - 58 \beta_{16} + 21 \beta_{15} - 4 \beta_{12} + 4 \beta_{10} - 13 \beta_{9} + 28 \beta_{8} - 67 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 14 \beta_{14} - 18 \beta_{13} - 13 \beta_{11} - \beta_{7} + 52 \beta_{5} + 10 \beta_{4} + 8 \beta_{3} + 5 \beta_{2} - 31 \beta _1 + 42 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 7 \beta_{19} + 23 \beta_{18} + 44 \beta_{17} - 60 \beta_{16} + 37 \beta_{15} + 124 \beta_{12} + 64 \beta_{10} + 49 \beta_{9} + 20 \beta_{8} + 343 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 28 \beta_{14} + 37 \beta_{13} + 47 \beta_{11} - 7 \beta_{7} + 20 \beta_{5} + 133 \beta_{4} + 28 \beta_{3} - 46 \beta_{2} - 14 \beta _1 - 283 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 24 \beta_{19} + 31 \beta_{18} - 77 \beta_{17} + 87 \beta_{16} + 30 \beta_{15} + 120 \beta_{12} + 78 \beta_{10} - 19 \beta_{9} + 4 \beta_{8} + 47 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 56 \beta_{14} + 21 \beta_{13} + 77 \beta_{11} - 9 \beta_{7} - 356 \beta_{5} - 163 \beta_{4} + 28 \beta_{3} - 224 \beta_{2} - 248 \beta _1 - 483 ) / 8 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 9 \beta_{19} + 79 \beta_{18} + 98 \beta_{17} + 50 \beta_{16} + 71 \beta_{15} + 180 \beta_{12} - 212 \beta_{10} - 223 \beta_{9} - 428 \beta_{8} - 337 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 154 \beta_{14} - 34 \beta_{13} - 155 \beta_{11} - 159 \beta_{7} - 340 \beta_{5} + 2 \beta_{4} + 168 \beta_{3} - 89 \beta_{2} + 283 \beta _1 - 142 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 171 \beta_{19} - 139 \beta_{18} + 116 \beta_{17} + 444 \beta_{16} - 497 \beta_{15} - 12 \beta_{12} - 928 \beta_{10} - 157 \beta_{9} - 132 \beta_{8} + 2677 \beta_{6} ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times\).

\(n\) \(2081\) \(2431\) \(3457\) \(3781\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2161.1
−1.19357 + 0.758543i
1.19357 + 0.758543i
1.17425 0.788128i
−1.17425 0.788128i
−0.662801 1.24928i
0.662801 1.24928i
−0.444539 + 1.34253i
0.444539 + 1.34253i
1.34522 + 0.436333i
−1.34522 + 0.436333i
1.19357 0.758543i
−1.19357 0.758543i
−1.17425 + 0.788128i
1.17425 + 0.788128i
0.662801 + 1.24928i
−0.662801 + 1.24928i
0.444539 1.34253i
−0.444539 1.34253i
−1.34522 0.436333i
1.34522 0.436333i
0 0 0 1.00000i 0 −4.63236 0 0 0
2161.2 0 0 0 1.00000i 0 −4.63236 0 0 0
2161.3 0 0 0 1.00000i 0 −2.12726 0 0 0
2161.4 0 0 0 1.00000i 0 −2.12726 0 0 0
2161.5 0 0 0 1.00000i 0 1.52861 0 0 0
2161.6 0 0 0 1.00000i 0 1.52861 0 0 0
2161.7 0 0 0 1.00000i 0 1.61609 0 0 0
2161.8 0 0 0 1.00000i 0 1.61609 0 0 0
2161.9 0 0 0 1.00000i 0 3.61492 0 0 0
2161.10 0 0 0 1.00000i 0 3.61492 0 0 0
2161.11 0 0 0 1.00000i 0 −4.63236 0 0 0
2161.12 0 0 0 1.00000i 0 −4.63236 0 0 0
2161.13 0 0 0 1.00000i 0 −2.12726 0 0 0
2161.14 0 0 0 1.00000i 0 −2.12726 0 0 0
2161.15 0 0 0 1.00000i 0 1.52861 0 0 0
2161.16 0 0 0 1.00000i 0 1.52861 0 0 0
2161.17 0 0 0 1.00000i 0 1.61609 0 0 0
2161.18 0 0 0 1.00000i 0 1.61609 0 0 0
2161.19 0 0 0 1.00000i 0 3.61492 0 0 0
2161.20 0 0 0 1.00000i 0 3.61492 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2161.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4320.2.k.d 20
3.b odd 2 1 inner 4320.2.k.d 20
4.b odd 2 1 1080.2.k.d 20
8.b even 2 1 inner 4320.2.k.d 20
8.d odd 2 1 1080.2.k.d 20
12.b even 2 1 1080.2.k.d 20
24.f even 2 1 1080.2.k.d 20
24.h odd 2 1 inner 4320.2.k.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.k.d 20 4.b odd 2 1
1080.2.k.d 20 8.d odd 2 1
1080.2.k.d 20 12.b even 2 1
1080.2.k.d 20 24.f even 2 1
4320.2.k.d 20 1.a even 1 1 trivial
4320.2.k.d 20 3.b odd 2 1 inner
4320.2.k.d 20 8.b even 2 1 inner
4320.2.k.d 20 24.h 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 \(S_{2}^{\mathrm{new}}(4320, [\chi])\):

\( T_{7}^{5} - 22T_{7}^{3} + 18T_{7}^{2} + 76T_{7} - 88 \) Copy content Toggle raw display
\( T_{17}^{10} - 119T_{17}^{8} + 5122T_{17}^{6} - 94474T_{17}^{4} + 633189T_{17}^{2} - 34839 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} - 22 T^{3} + 18 T^{2} + 76 T - 88)^{4} \) Copy content Toggle raw display
$11$ \( (T^{10} + 72 T^{8} + 1688 T^{6} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 80 T^{8} + 2104 T^{6} + \cdots + 45504)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 119 T^{8} + 5122 T^{6} + \cdots - 34839)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 147 T^{8} + 7646 T^{6} + \cdots + 5464351)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 171 T^{8} + 10898 T^{6} + \cdots - 21608791)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 216 T^{8} + 13572 T^{6} + \cdots + 14400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 5 T^{4} - 96 T^{3} - 466 T^{2} + \cdots + 4309)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 264 T^{8} + 22160 T^{6} + \cdots + 5177344)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 312 T^{8} + 36356 T^{6} + \cdots - 288803776)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 284 T^{8} + 29724 T^{6} + \cdots + 154840000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 316 T^{8} + 36176 T^{6} + \cdots - 12640000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 269 T^{8} + 23902 T^{6} + \cdots + 38626225)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 280 T^{8} + 16040 T^{6} + \cdots + 732736)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 243 T^{8} + 21722 T^{6} + \cdots + 1234375)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 312 T^{8} + 32016 T^{6} + \cdots + 72806400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 176 T^{8} + 8520 T^{6} + \cdots - 126400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 10 T^{4} - 92 T^{3} - 978 T^{2} + \cdots + 6696)^{4} \) Copy content Toggle raw display
$79$ \( (T^{5} - T^{4} - 174 T^{3} + 848 T^{2} + \cdots + 745)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + 269 T^{8} + 27954 T^{6} + \cdots + 268992801)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 284 T^{8} + 29724 T^{6} + \cdots - 154840000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} - 10 T^{4} - 228 T^{3} + \cdots + 12928)^{4} \) Copy content Toggle raw display
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