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Properties

Label 570.2.q.c
Level $570$
Weight $2$
Character orbit 570.q
Analytic conductor $4.551$
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] = [570,2,Mod(49,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.q (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: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 49 x^{16} - 8 x^{15} + 72 x^{13} + 2145 x^{12} - 648 x^{11} + 32 x^{10} - 7056 x^{9} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + \beta_{11} q^{3} + ( - \beta_{12} + 1) q^{4} + ( - \beta_{17} + \beta_{8} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{12} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + \beta_{11} q^{3} + ( - \beta_{12} + 1) q^{4} + ( - \beta_{17} + \beta_{8} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{19} - \beta_{17} + \beta_{16} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 10 q^{4} - 10 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 10 q^{4} - 10 q^{6} + 10 q^{9} - 2 q^{10} + 12 q^{11} + 10 q^{14} + 2 q^{15} - 10 q^{16} + 6 q^{19} - 10 q^{21} + 10 q^{24} + 14 q^{25} + 8 q^{29} + 40 q^{31} + 12 q^{34} + 2 q^{35} - 10 q^{36} + 2 q^{40} - 14 q^{41} + 6 q^{44} + 44 q^{46} - 8 q^{49} - 8 q^{50} - 12 q^{51} + 10 q^{54} + 20 q^{56} + 8 q^{59} - 2 q^{60} + 16 q^{61} - 20 q^{64} + 40 q^{65} - 6 q^{66} - 44 q^{69} + 8 q^{70} - 4 q^{71} + 26 q^{74} + 8 q^{75} + 8 q^{79} - 10 q^{81} - 20 q^{84} - 16 q^{85} - 20 q^{86} - 2 q^{89} + 2 q^{90} - 44 q^{91} - 32 q^{94} - 80 q^{95} + 20 q^{96} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 49 x^{16} - 8 x^{15} + 72 x^{13} + 2145 x^{12} - 648 x^{11} + 32 x^{10} - 7056 x^{9} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!99 \nu^{19} + \cdots + 78\!\cdots\!92 ) / 12\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33\!\cdots\!57 \nu^{19} + \cdots - 93\!\cdots\!36 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 42\!\cdots\!01 \nu^{19} + \cdots + 27\!\cdots\!92 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31\!\cdots\!63 \nu^{19} + \cdots + 22\!\cdots\!44 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!58 \nu^{19} + \cdots - 52\!\cdots\!24 ) / 27\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!81 \nu^{19} + \cdots - 15\!\cdots\!92 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43\!\cdots\!88 \nu^{19} + \cdots - 16\!\cdots\!96 ) / 27\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6169005809303 \nu^{19} + 3083823703934 \nu^{18} - 1600056848304 \nu^{17} + \cdots - 21\!\cdots\!64 ) / 34\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!77 \nu^{19} + \cdots + 22\!\cdots\!04 ) / 27\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 31\!\cdots\!33 \nu^{19} + \cdots + 65\!\cdots\!88 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20734554827161 \nu^{19} + 6169005809303 \nu^{18} - 3083823703934 \nu^{17} + \cdots - 42\!\cdots\!20 ) / 34\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 36\!\cdots\!23 \nu^{19} + \cdots - 17\!\cdots\!48 ) / 49\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 46\!\cdots\!25 \nu^{19} + \cdots - 10\!\cdots\!08 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10\!\cdots\!39 \nu^{19} + \cdots + 14\!\cdots\!28 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 59\!\cdots\!77 \nu^{19} + \cdots - 11\!\cdots\!84 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 15\!\cdots\!31 \nu^{19} + \cdots - 33\!\cdots\!52 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 40\!\cdots\!87 \nu^{19} + \cdots - 82\!\cdots\!84 ) / 27\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 22\!\cdots\!27 \nu^{19} + \cdots + 21\!\cdots\!60 ) / 13\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{16} + \beta_{14} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{17} - 6\beta_{16} - \beta_{15} - \beta_{13} + 7\beta_{10} - \beta_{8} - 6\beta_{7} + \beta_{6} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{19} - 2 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{13} + 13 \beta_{12} - 2 \beta_{10} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{17} + 18 \beta_{16} + 9 \beta_{13} - 4 \beta_{12} + 9 \beta_{11} - 47 \beta_{9} + 9 \beta_{8} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 47 \beta_{18} - 51 \beta_{17} + 22 \beta_{16} + 47 \beta_{14} - 87 \beta_{11} - 22 \beta_{10} + \cdots - 4 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{19} + 4 \beta_{18} + 250 \beta_{17} - 250 \beta_{16} - 65 \beta_{15} - 48 \beta_{12} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 311 \beta_{19} + 319 \beta_{17} - 319 \beta_{16} - 441 \beta_{15} - 189 \beta_{13} + 413 \beta_{12} + \cdots - 94 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 64 \beta_{18} - 8 \beta_{17} + 505 \beta_{16} + 505 \beta_{15} - 64 \beta_{14} + 505 \beta_{13} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2047 \beta_{18} - 2747 \beta_{17} + 3629 \beta_{16} - 3383 \beta_{11} - 1438 \beta_{10} + \cdots + 2191 \beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 700 \beta_{19} - 3629 \beta_{17} + 844 \beta_{15} + 700 \beta_{14} + 3485 \beta_{13} - 3288 \beta_{12} + \cdots - 341 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 25857 \beta_{17} - 25857 \beta_{16} - 15167 \beta_{15} - 15167 \beta_{13} + 25857 \beta_{10} + \cdots - 21751 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6520 \beta_{19} + 6520 \beta_{18} + 24169 \beta_{17} - 25857 \beta_{16} + 17649 \beta_{15} + 1688 \beta_{13} + \cdots + 8208 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 105487 \beta_{16} - 89071 \beta_{14} + 16416 \beta_{11} + 16416 \beta_{10} - 55668 \beta_{9} + \cdots + 55668 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 55668 \beta_{18} - 719327 \beta_{17} + 552330 \beta_{16} + 183413 \beta_{15} + 55668 \beta_{14} + \cdots + 67359 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 591575 \beta_{19} + 561930 \beta_{17} - 561930 \beta_{16} + 111322 \beta_{15} - 255490 \beta_{13} + \cdots - 847065 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 450608 \beta_{19} + 1297673 \beta_{17} - 2595346 \beta_{16} - 594776 \beta_{15} + 450608 \beta_{14} + \cdots + 11541 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 3951199 \beta_{18} + 7472267 \beta_{17} - 4025646 \beta_{16} - 3951199 \beta_{14} + 7347879 \beta_{11} + \cdots + 3521068 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 3521068 \beta_{19} - 3521068 \beta_{18} - 25606226 \beta_{17} + 26795778 \beta_{16} + 4455777 \beta_{15} + \cdots + 4710620 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-1 + \beta_{12}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.78384 − 0.477979i
−1.19457 + 0.320085i
2.34324 − 0.627868i
−2.56046 + 0.686074i
−0.372041 + 0.0996880i
−0.627868 − 2.34324i
0.0996880 + 0.372041i
0.686074 + 2.56046i
−0.477979 − 1.78384i
0.320085 + 1.19457i
1.78384 + 0.477979i
−1.19457 − 0.320085i
2.34324 + 0.627868i
−2.56046 − 0.686074i
−0.372041 − 0.0996880i
−0.627868 + 2.34324i
0.0996880 − 0.372041i
0.686074 − 2.56046i
−0.477979 + 1.78384i
0.320085 − 1.19457i
−0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −2.22489 + 0.223342i −0.500000 − 0.866025i − 1.07560i − 1.00000i 0.500000 + 0.866025i 2.03848 + 0.919023i
49.2 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.34502 − 1.78632i −0.500000 − 0.866025i 4.03495i − 1.00000i 0.500000 + 0.866025i 0.271659 + 2.21950i
49.3 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0.384547 + 2.20275i −0.500000 − 0.866025i 2.51805i − 1.00000i 0.500000 + 0.866025i 0.768349 − 2.09991i
49.4 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 1.99313 − 1.01362i −0.500000 − 0.866025i − 2.79875i − 1.00000i 0.500000 + 0.866025i −2.23291 − 0.118742i
49.5 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 2.05825 + 0.873846i −0.500000 − 0.866025i 2.32136i − 1.00000i 0.500000 + 0.866025i −1.34557 − 1.78590i
49.6 0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 + 0.866025i −2.09991 + 0.768349i −0.500000 − 0.866025i − 2.51805i 1.00000i 0.500000 + 0.866025i −2.20275 − 0.384547i
49.7 0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 + 0.866025i −1.78590 − 1.34557i −0.500000 − 0.866025i − 2.32136i 1.00000i 0.500000 + 0.866025i −0.873846 − 2.05825i
49.8 0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 + 0.866025i −0.118742 − 2.23291i −0.500000 − 0.866025i 2.79875i 1.00000i 0.500000 + 0.866025i 1.01362 − 1.99313i
49.9 0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 + 0.866025i 0.919023 + 2.03848i −0.500000 − 0.866025i 1.07560i 1.00000i 0.500000 + 0.866025i −0.223342 + 2.22489i
49.10 0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 + 0.866025i 2.21950 + 0.271659i −0.500000 − 0.866025i − 4.03495i 1.00000i 0.500000 + 0.866025i 1.78632 + 1.34502i
349.1 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 − 0.866025i −2.22489 − 0.223342i −0.500000 + 0.866025i 1.07560i 1.00000i 0.500000 − 0.866025i 2.03848 − 0.919023i
349.2 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 − 0.866025i −1.34502 + 1.78632i −0.500000 + 0.866025i − 4.03495i 1.00000i 0.500000 − 0.866025i 0.271659 − 2.21950i
349.3 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 − 0.866025i 0.384547 − 2.20275i −0.500000 + 0.866025i − 2.51805i 1.00000i 0.500000 − 0.866025i 0.768349 + 2.09991i
349.4 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 − 0.866025i 1.99313 + 1.01362i −0.500000 + 0.866025i 2.79875i 1.00000i 0.500000 − 0.866025i −2.23291 + 0.118742i
349.5 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 − 0.866025i 2.05825 − 0.873846i −0.500000 + 0.866025i − 2.32136i 1.00000i 0.500000 − 0.866025i −1.34557 + 1.78590i
349.6 0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 − 0.866025i −2.09991 − 0.768349i −0.500000 + 0.866025i 2.51805i − 1.00000i 0.500000 − 0.866025i −2.20275 + 0.384547i
349.7 0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 − 0.866025i −1.78590 + 1.34557i −0.500000 + 0.866025i 2.32136i − 1.00000i 0.500000 − 0.866025i −0.873846 + 2.05825i
349.8 0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 − 0.866025i −0.118742 + 2.23291i −0.500000 + 0.866025i − 2.79875i − 1.00000i 0.500000 − 0.866025i 1.01362 + 1.99313i
349.9 0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 − 0.866025i 0.919023 − 2.03848i −0.500000 + 0.866025i − 1.07560i − 1.00000i 0.500000 − 0.866025i −0.223342 − 2.22489i
349.10 0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 − 0.866025i 2.21950 − 0.271659i −0.500000 + 0.866025i 4.03495i − 1.00000i 0.500000 − 0.866025i 1.78632 − 1.34502i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.q.c ✓ 20
3.b odd 2 1 1710.2.t.c 20
5.b even 2 1 inner 570.2.q.c ✓ 20
15.d odd 2 1 1710.2.t.c 20
19.c even 3 1 inner 570.2.q.c ✓ 20
57.h odd 6 1 1710.2.t.c 20
95.i even 6 1 inner 570.2.q.c ✓ 20
285.n odd 6 1 1710.2.t.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.q.c ✓ 20 1.a even 1 1 trivial
570.2.q.c ✓ 20 5.b even 2 1 inner
570.2.q.c ✓ 20 19.c even 3 1 inner
570.2.q.c ✓ 20 95.i even 6 1 inner
1710.2.t.c 20 3.b odd 2 1
1710.2.t.c 20 15.d odd 2 1
1710.2.t.c 20 57.h odd 6 1
1710.2.t.c 20 285.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 37T_{7}^{8} + 486T_{7}^{6} + 2834T_{7}^{4} + 7041T_{7}^{2} + 5041 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} - 7 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{10} + 37 T^{8} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 3 T^{4} - 13 T^{3} + \cdots - 20)^{4} \) Copy content Toggle raw display
$13$ \( T^{20} - 98 T^{18} + \cdots + 12960000 \) Copy content Toggle raw display
$17$ \( T^{20} - 92 T^{18} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( (T^{10} - 3 T^{9} + \cdots + 2476099)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 283982410000 \) Copy content Toggle raw display
$29$ \( (T^{10} - 4 T^{9} + \cdots + 501264)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 10 T^{4} + \cdots - 1648)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 81 T^{8} + \cdots + 18769)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 7 T^{9} + \cdots + 244421956)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 3110228525056 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 72699496960000 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{10} - 4 T^{9} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 8 T^{9} + \cdots + 313219204)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + 2 T^{9} + \cdots + 593994384)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{10} - 4 T^{9} + \cdots + 263997504)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 356 T^{8} + \cdots + 184090624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + T^{9} + \cdots + 13468900)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
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