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Properties

Label 574.2.h.k
Level $574$
Weight $2$
Character orbit 574.h
Analytic conductor $4.583$
Analytic rank $0$
Dimension $20$
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] = [574,2,Mod(57,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.h (of order \(5\), degree \(4\), 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: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 8 x^{18} + 11 x^{17} + 108 x^{16} + 8 x^{15} + 1537 x^{14} + 1498 x^{13} + \cdots + 2062096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{6} q^{3} + \beta_{3} q^{4} + \beta_{18} q^{5} + \beta_{9} q^{6} - \beta_{12} q^{7} + (\beta_{12} + \beta_{4} + \beta_{3} + 1) q^{8} + (\beta_{16} + 2 \beta_{12} - \beta_{10} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{6} q^{3} + \beta_{3} q^{4} + \beta_{18} q^{5} + \beta_{9} q^{6} - \beta_{12} q^{7} + (\beta_{12} + \beta_{4} + \beta_{3} + 1) q^{8} + (\beta_{16} + 2 \beta_{12} - \beta_{10} + \cdots + 4) q^{9}+ \cdots + (2 \beta_{19} + 2 \beta_{17} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{2} - 6 q^{3} - 5 q^{4} - q^{5} - 4 q^{6} + 5 q^{7} + 5 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 5 q^{2} - 6 q^{3} - 5 q^{4} - q^{5} - 4 q^{6} + 5 q^{7} + 5 q^{8} + 50 q^{9} + q^{10} - 3 q^{11} - q^{12} + 15 q^{13} + 20 q^{14} - 17 q^{15} - 5 q^{16} - 2 q^{17} - 4 q^{19} - q^{20} - 4 q^{21} + 3 q^{22} + 14 q^{23} + q^{24} - 12 q^{25} + 5 q^{26} + 12 q^{27} + 5 q^{28} + 23 q^{29} + 17 q^{30} + 20 q^{31} - 20 q^{32} + 59 q^{33} - 3 q^{34} + q^{35} - 25 q^{36} - 9 q^{37} + 4 q^{38} - 15 q^{39} - 4 q^{40} - 48 q^{41} - 6 q^{42} + 24 q^{43} + 12 q^{44} - 7 q^{45} + 11 q^{46} - 39 q^{47} + 4 q^{48} - 5 q^{49} - 48 q^{50} + 35 q^{51} - 5 q^{52} - 15 q^{53} - 32 q^{54} - 46 q^{55} - 5 q^{56} - 10 q^{57} - 23 q^{58} + 32 q^{59} + 23 q^{60} - 16 q^{61} - 5 q^{64} - q^{65} + 51 q^{66} - 5 q^{67} - 2 q^{68} + 38 q^{69} - q^{70} + 40 q^{71} + 25 q^{72} - 46 q^{73} + 9 q^{74} + 6 q^{75} - 4 q^{76} + 3 q^{77} - 25 q^{78} - 88 q^{79} - q^{80} + 36 q^{81} - 12 q^{82} - 16 q^{83} - 4 q^{84} - 10 q^{85} + 46 q^{86} - 14 q^{87} + 3 q^{88} + 57 q^{89} + 7 q^{90} + 20 q^{91} - 11 q^{92} + 7 q^{93} - 26 q^{94} - 80 q^{95} + 6 q^{96} - 22 q^{97} + 5 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 8 x^{18} + 11 x^{17} + 108 x^{16} + 8 x^{15} + 1537 x^{14} + 1498 x^{13} + \cdots + 2062096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 18\!\cdots\!61 \nu^{19} + \cdots - 79\!\cdots\!60 ) / 26\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23\!\cdots\!57 \nu^{19} + \cdots - 11\!\cdots\!52 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33\!\cdots\!87 \nu^{19} + \cdots + 84\!\cdots\!80 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\!\cdots\!35 \nu^{19} + \cdots - 95\!\cdots\!64 ) / 26\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 34\!\cdots\!24 \nu^{19} + \cdots + 33\!\cdots\!52 ) / 13\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12\!\cdots\!62 \nu^{19} + \cdots + 25\!\cdots\!48 ) / 26\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 90\!\cdots\!33 \nu^{19} + \cdots + 27\!\cdots\!16 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!53 \nu^{19} + \cdots + 42\!\cdots\!72 ) / 26\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47\!\cdots\!40 \nu^{19} + \cdots + 24\!\cdots\!76 ) / 94\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 58\!\cdots\!68 \nu^{19} + \cdots - 13\!\cdots\!60 ) / 94\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!21 \nu^{19} + \cdots - 14\!\cdots\!36 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 90\!\cdots\!43 \nu^{19} + \cdots - 53\!\cdots\!16 ) / 94\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25\!\cdots\!59 \nu^{19} + \cdots - 12\!\cdots\!68 ) / 26\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21\!\cdots\!45 \nu^{19} + \cdots - 63\!\cdots\!60 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!97 \nu^{19} + \cdots + 10\!\cdots\!08 ) / 94\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 85\!\cdots\!44 \nu^{19} + \cdots + 21\!\cdots\!22 ) / 47\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 18\!\cdots\!09 \nu^{19} + \cdots + 22\!\cdots\!36 ) / 94\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 47\!\cdots\!01 \nu^{19} + \cdots + 59\!\cdots\!08 ) / 18\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} - \beta_{17} + \beta_{13} - \beta_{12} + 2\beta_{10} - \beta_{8} - \beta_{6} + 5\beta_{4} + 2\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{19} + \beta_{18} - \beta_{16} + \beta_{15} + \beta_{12} - \beta_{7} + 2 \beta_{6} + 7 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10 \beta_{17} - 4 \beta_{16} - 16 \beta_{12} + 4 \beta_{10} - 11 \beta_{9} - 4 \beta_{8} - \beta_{6} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{18} + 11 \beta_{16} + 4 \beta_{14} - 30 \beta_{12} - 14 \beta_{11} - 11 \beta_{10} + 26 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{19} - 2 \beta_{17} - 97 \beta_{16} - 3 \beta_{15} + 3 \beta_{14} - 90 \beta_{13} + \cdots - 448 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 268 \beta_{19} - 164 \beta_{18} - 264 \beta_{17} + 4 \beta_{16} - 234 \beta_{15} + 164 \beta_{14} + \cdots - 726 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 809 \beta_{19} - 172 \beta_{18} + 172 \beta_{16} - 172 \beta_{15} + 88 \beta_{14} + 40 \beta_{13} + \cdots + 132 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 108 \beta_{19} - 2753 \beta_{18} + 2650 \beta_{17} + 2293 \beta_{16} - 1841 \beta_{15} + 7360 \beta_{12} + \cdots + 7360 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1524 \beta_{18} + 564 \beta_{17} + 18878 \beta_{16} - 1364 \beta_{14} - 564 \beta_{13} + 35606 \beta_{12} + \cdots + 64126 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1872 \beta_{19} + 1872 \beta_{17} + 3833 \beta_{16} + 20430 \beta_{15} - 20430 \beta_{14} - 27937 \beta_{13} + \cdots + 67258 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 66430 \beta_{19} + 23083 \beta_{18} - 73452 \beta_{17} - 7022 \beta_{16} + 41695 \beta_{15} + \cdots + 110671 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 281220 \beta_{19} + 350746 \beta_{18} - 356056 \beta_{16} + 350746 \beta_{15} - 124442 \beta_{14} + \cdots - 329080 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 83112 \beta_{19} + 558168 \beta_{18} + 684009 \beta_{17} - 1492464 \beta_{16} + 318700 \beta_{15} + \cdots - 3609673 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2511529 \beta_{18} - 353880 \beta_{17} - 1610735 \beta_{16} + 1413232 \beta_{14} + 353880 \beta_{13} + \cdots - 8541360 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 965240 \beta_{19} - 965240 \beta_{17} - 12801538 \beta_{16} - 4172664 \beta_{15} + 4172664 \beta_{14} + \cdots - 62343168 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 28474761 \beta_{19} - 27970734 \beta_{18} - 24070089 \beta_{17} + 4404672 \beta_{16} + \cdots - 105785234 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 47649458 \beta_{19} - 89395799 \beta_{18} + 92882311 \beta_{16} - 89395799 \beta_{15} + 36534116 \beta_{14} + \cdots + 81720637 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 53142956 \beta_{19} - 494123818 \beta_{18} + 234752976 \beta_{17} + 573036318 \beta_{16} + \cdots + 1513518060 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(493\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
−2.51711 + 1.82879i
−1.56743 + 1.13880i
−1.03133 + 0.749302i
2.05086 − 1.49004i
2.75599 − 2.00234i
−2.51711 − 1.82879i
−1.56743 − 1.13880i
−1.03133 − 0.749302i
2.05086 + 1.49004i
2.75599 + 2.00234i
0.949877 + 2.92342i
0.680567 + 2.09457i
0.301619 + 0.928288i
−0.510664 − 1.57166i
−0.612382 − 1.88472i
0.949877 − 2.92342i
0.680567 − 2.09457i
0.301619 − 0.928288i
−0.510664 + 1.57166i
−0.612382 + 1.88472i
−0.309017 + 0.951057i −3.11132 −0.809017 − 0.587785i 1.53336 + 1.11405i 0.961450 − 2.95904i −0.309017 − 0.951057i 0.809017 − 0.587785i 6.68030 −1.53336 + 1.11405i
57.2 −0.309017 + 0.951057i −1.93745 −0.809017 − 0.587785i −3.00780 − 2.18530i 0.598704 − 1.84262i −0.309017 − 0.951057i 0.809017 − 0.587785i 0.753706 3.00780 − 2.18530i
57.3 −0.309017 + 0.951057i −1.27479 −0.809017 − 0.587785i 1.37274 + 0.997354i 0.393932 − 1.21240i −0.309017 − 0.951057i 0.809017 − 0.587785i −1.37491 −1.37274 + 0.997354i
57.4 −0.309017 + 0.951057i 2.53500 −0.809017 − 0.587785i 1.95764 + 1.42231i −0.783359 + 2.41093i −0.309017 − 0.951057i 0.809017 − 0.587785i 3.42623 −1.95764 + 1.42231i
57.5 −0.309017 + 0.951057i 3.40659 −0.809017 − 0.587785i −2.66495 − 1.93620i −1.05269 + 3.23986i −0.309017 − 0.951057i 0.809017 − 0.587785i 8.60484 2.66495 − 1.93620i
141.1 −0.309017 − 0.951057i −3.11132 −0.809017 + 0.587785i 1.53336 − 1.11405i 0.961450 + 2.95904i −0.309017 + 0.951057i 0.809017 + 0.587785i 6.68030 −1.53336 − 1.11405i
141.2 −0.309017 − 0.951057i −1.93745 −0.809017 + 0.587785i −3.00780 + 2.18530i 0.598704 + 1.84262i −0.309017 + 0.951057i 0.809017 + 0.587785i 0.753706 3.00780 + 2.18530i
141.3 −0.309017 − 0.951057i −1.27479 −0.809017 + 0.587785i 1.37274 − 0.997354i 0.393932 + 1.21240i −0.309017 + 0.951057i 0.809017 + 0.587785i −1.37491 −1.37274 − 0.997354i
141.4 −0.309017 − 0.951057i 2.53500 −0.809017 + 0.587785i 1.95764 − 1.42231i −0.783359 − 2.41093i −0.309017 + 0.951057i 0.809017 + 0.587785i 3.42623 −1.95764 − 1.42231i
141.5 −0.309017 − 0.951057i 3.40659 −0.809017 + 0.587785i −2.66495 + 1.93620i −1.05269 − 3.23986i −0.309017 + 0.951057i 0.809017 + 0.587785i 8.60484 2.66495 + 1.93620i
365.1 0.809017 − 0.587785i −3.07387 0.309017 − 0.951057i 1.20617 − 3.71222i −2.48681 + 1.80677i 0.809017 + 0.587785i −0.309017 − 0.951057i 6.44866 −1.20617 − 3.71222i
365.2 0.809017 − 0.587785i −2.20236 0.309017 − 0.951057i −0.276370 + 0.850580i −1.78175 + 1.29452i 0.809017 + 0.587785i −0.309017 − 0.951057i 1.85039 0.276370 + 0.850580i
365.3 0.809017 − 0.587785i −0.976060 0.309017 − 0.951057i −0.305515 + 0.940279i −0.789649 + 0.573714i 0.809017 + 0.587785i −0.309017 − 0.951057i −2.04731 0.305515 + 0.940279i
365.4 0.809017 − 0.587785i 1.65254 0.309017 − 0.951057i 0.806447 − 2.48199i 1.33694 − 0.971341i 0.809017 + 0.587785i −0.309017 − 0.951057i −0.269097 −0.806447 − 2.48199i
365.5 0.809017 − 0.587785i 1.98171 0.309017 − 0.951057i −1.12172 + 3.45230i 1.60324 − 1.16482i 0.809017 + 0.587785i −0.309017 − 0.951057i 0.927176 1.12172 + 3.45230i
379.1 0.809017 + 0.587785i −3.07387 0.309017 + 0.951057i 1.20617 + 3.71222i −2.48681 − 1.80677i 0.809017 − 0.587785i −0.309017 + 0.951057i 6.44866 −1.20617 + 3.71222i
379.2 0.809017 + 0.587785i −2.20236 0.309017 + 0.951057i −0.276370 − 0.850580i −1.78175 − 1.29452i 0.809017 − 0.587785i −0.309017 + 0.951057i 1.85039 0.276370 − 0.850580i
379.3 0.809017 + 0.587785i −0.976060 0.309017 + 0.951057i −0.305515 − 0.940279i −0.789649 − 0.573714i 0.809017 − 0.587785i −0.309017 + 0.951057i −2.04731 0.305515 − 0.940279i
379.4 0.809017 + 0.587785i 1.65254 0.309017 + 0.951057i 0.806447 + 2.48199i 1.33694 + 0.971341i 0.809017 − 0.587785i −0.309017 + 0.951057i −0.269097 −0.806447 + 2.48199i
379.5 0.809017 + 0.587785i 1.98171 0.309017 + 0.951057i −1.12172 − 3.45230i 1.60324 + 1.16482i 0.809017 − 0.587785i −0.309017 + 0.951057i 0.927176 1.12172 − 3.45230i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.h.k ✓ 20
41.d even 5 1 inner 574.2.h.k ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.h.k ✓ 20 1.a even 1 1 trivial
574.2.h.k ✓ 20 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\):

\( T_{3}^{10} + 3 T_{3}^{9} - 23 T_{3}^{8} - 74 T_{3}^{7} + 164 T_{3}^{6} + 612 T_{3}^{5} - 341 T_{3}^{4} + \cdots + 1436 \) Copy content Toggle raw display
\( T_{5}^{20} + T_{5}^{19} + 19 T_{5}^{18} + 21 T_{5}^{17} + 260 T_{5}^{16} - 253 T_{5}^{15} + \cdots + 9709456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T^{10} + 3 T^{9} + \cdots + 1436)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + T^{19} + \cdots + 9709456 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 110166016 \) Copy content Toggle raw display
$13$ \( T^{20} - 15 T^{19} + \cdots + 54405376 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 41500208656 \) Copy content Toggle raw display
$19$ \( T^{20} + 4 T^{19} + \cdots + 13808656 \) Copy content Toggle raw display
$23$ \( T^{20} - 14 T^{19} + \cdots + 102400 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 62740230400 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 7409766400 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 5643855462400 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 61501920243856 \) Copy content Toggle raw display
$47$ \( T^{20} + 39 T^{19} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 217487458471936 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 34446569574400 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 501853229056 \) Copy content Toggle raw display
$73$ \( (T^{10} + 23 T^{9} + \cdots + 1786720711)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 44 T^{9} + \cdots - 18326480)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 8 T^{9} + \cdots + 238336)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 137185584390625 \) Copy content Toggle raw display
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