Properties

Label 92.2.e.a
Level $92$
Weight $2$
Character orbit 92.e
Analytic conductor $0.735$
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] = [92,2,Mod(9,92)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(92, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("92.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 92 = 2^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 92.e (of order \(11\), degree \(10\), 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: \(0.734623698596\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 9 x^{19} + 51 x^{18} - 200 x^{17} + 633 x^{16} - 1688 x^{15} + 3957 x^{14} - 8161 x^{13} + 14788 x^{12} - 23925 x^{11} + 35080 x^{10} - 43945 x^{9} + 57269 x^{8} + \cdots + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{18} - \beta_{15} - \beta_{14} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} + \cdots + 1) q^{3}+ \cdots + ( - \beta_{17} - \beta_{14} - \beta_{10} + \beta_{9} - \beta_{6} + \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{18} - \beta_{15} - \beta_{14} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} + \cdots + 1) q^{3}+ \cdots + (\beta_{19} + 2 \beta_{16} - \beta_{15} + 6 \beta_{13} + 2 \beta_{12} + 5 \beta_{11} + \beta_{10} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9} - 2 q^{11} + 6 q^{13} - 17 q^{15} - 9 q^{17} - 11 q^{19} - 47 q^{21} - 22 q^{23} - 16 q^{25} - 19 q^{27} - q^{29} - 13 q^{31} - 5 q^{33} + 14 q^{35} + 34 q^{37} + 30 q^{39} + 28 q^{41} + 44 q^{43} + 78 q^{45} + 26 q^{47} + 60 q^{49} + 62 q^{51} + 14 q^{53} + 26 q^{55} + 3 q^{57} - 10 q^{59} - 56 q^{61} - 27 q^{63} - 87 q^{65} - 44 q^{67} - 51 q^{69} - 37 q^{71} - 12 q^{73} - 53 q^{75} - 47 q^{77} - 6 q^{79} - 10 q^{81} - 25 q^{83} + 8 q^{85} + 48 q^{87} + 10 q^{89} + 26 q^{91} - 14 q^{93} + 29 q^{95} - q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 9 x^{19} + 51 x^{18} - 200 x^{17} + 633 x^{16} - 1688 x^{15} + 3957 x^{14} - 8161 x^{13} + 14788 x^{12} - 23925 x^{11} + 35080 x^{10} - 43945 x^{9} + 57269 x^{8} + \cdots + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26\!\cdots\!24 \nu^{19} + \cdots + 24\!\cdots\!76 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 58\!\cdots\!64 \nu^{19} + \cdots + 73\!\cdots\!27 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 95\!\cdots\!62 \nu^{19} + \cdots - 48\!\cdots\!81 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!69 \nu^{19} + \cdots + 45\!\cdots\!85 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!65 \nu^{19} + \cdots - 35\!\cdots\!17 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28\!\cdots\!41 \nu^{19} + \cdots + 87\!\cdots\!95 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!51 \nu^{19} + \cdots - 28\!\cdots\!69 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 45\!\cdots\!44 \nu^{19} + \cdots + 12\!\cdots\!17 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 86\!\cdots\!65 \nu^{19} + \cdots + 50\!\cdots\!07 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 91\!\cdots\!89 \nu^{19} + \cdots - 46\!\cdots\!37 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!80 \nu^{19} + \cdots + 40\!\cdots\!07 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!63 \nu^{19} + \cdots + 59\!\cdots\!43 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16\!\cdots\!55 \nu^{19} + \cdots - 62\!\cdots\!30 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18\!\cdots\!81 \nu^{19} + \cdots - 37\!\cdots\!88 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 20\!\cdots\!15 \nu^{19} + \cdots - 37\!\cdots\!16 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 22\!\cdots\!50 \nu^{19} + \cdots - 97\!\cdots\!34 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 31\!\cdots\!77 \nu^{19} + \cdots + 90\!\cdots\!66 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 39\!\cdots\!49 \nu^{19} + \cdots - 10\!\cdots\!67 ) / 20\!\cdots\!31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} - \beta_{15} - 4\beta_{12} - \beta_{11} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - 5 \beta_{17} - 2 \beta_{15} - 2 \beta_{14} - 7 \beta_{12} - \beta_{11} - 7 \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} - 2 \beta_{18} - 7 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} - 11 \beta_{14} - 3 \beta_{12} - 25 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} - 11 \beta_{6} + 7 \beta_{5} + \beta_{2} - 3 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{18} + 11 \beta_{16} + 12 \beta_{15} + 23 \beta_{13} + 24 \beta_{12} + 24 \beta_{11} - 8 \beta_{10} - 8 \beta_{9} - 7 \beta_{8} - 12 \beta_{6} + 30 \beta_{5} - 12 \beta_{4} - 12 \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5 \beta_{19} + 18 \beta_{17} + 18 \beta_{16} + 77 \beta_{15} + 98 \beta_{14} + 93 \beta_{13} + 114 \beta_{12} + 112 \beta_{11} + 130 \beta_{10} - 30 \beta_{9} - 72 \beta_{8} + 78 \beta_{6} + 45 \beta_{5} - 36 \beta_{4} - 5 \beta_{3} - 45 \beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 114 \beta_{18} + 19 \beta_{17} - 19 \beta_{16} + 148 \beta_{15} + 227 \beta_{14} - \beta_{13} + 114 \beta_{12} + 50 \beta_{11} + 353 \beta_{10} + 50 \beta_{9} - 227 \beta_{8} - 16 \beta_{7} + 353 \beta_{6} - 8 \beta_{5} + 11 \beta_{4} - 3 \beta_{3} + \cdots - 132 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 82 \beta_{19} + 342 \beta_{18} - 211 \beta_{17} - 131 \beta_{16} - 102 \beta_{15} - 102 \beta_{14} - 787 \beta_{13} - 656 \beta_{12} - 738 \beta_{11} + 283 \beta_{9} - 174 \beta_{8} - 131 \beta_{7} + 414 \beta_{6} - 131 \beta_{5} + \cdots - 305 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 67 \beta_{19} - 86 \beta_{18} - 830 \beta_{17} - 80 \beta_{16} - 1124 \beta_{15} - 2005 \beta_{14} - 2190 \beta_{13} - 2885 \beta_{12} - 2559 \beta_{11} - 3106 \beta_{10} - 166 \beta_{9} + 2055 \beta_{8} - 147 \beta_{7} + \cdots + 977 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1761 \beta_{19} - 4000 \beta_{18} + 1035 \beta_{16} - 1908 \beta_{15} - 3683 \beta_{14} + 615 \beta_{13} - 3669 \beta_{12} - 2965 \beta_{11} - 8810 \beta_{10} - 3385 \beta_{9} + 8810 \beta_{8} + 1848 \beta_{7} + \cdots + 6479 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6957 \beta_{19} - 11824 \beta_{18} + 6957 \beta_{17} + 3994 \beta_{16} + 3237 \beta_{15} + 11652 \beta_{14} + 23795 \beta_{13} + 11652 \beta_{12} + 11971 \beta_{11} + 3237 \beta_{10} - 6092 \beta_{9} + \cdots + 7927 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 939 \beta_{19} + 15349 \beta_{17} + 6292 \beta_{16} + 21527 \beta_{15} + 64536 \beta_{14} + 68375 \beta_{13} + 62706 \beta_{12} + 65099 \beta_{11} + 78946 \beta_{10} + 20588 \beta_{9} - 65099 \beta_{8} + \cdots - 41065 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 69465 \beta_{19} + 94268 \beta_{18} - 22527 \beta_{17} + 3016 \beta_{16} + 37992 \beta_{15} + 61882 \beta_{14} + 3016 \beta_{13} + 98874 \beta_{12} + 107457 \beta_{11} + 178476 \beta_{10} + 131347 \beta_{9} + \cdots - 197987 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 219027 \beta_{19} + 260035 \beta_{18} - 180672 \beta_{17} + 10817 \beta_{15} - 353220 \beta_{14} - 501879 \beta_{13} - 101178 \beta_{12} - 134193 \beta_{11} - 101178 \beta_{10} + 227454 \beta_{9} + \cdots - 227454 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 10817 \beta_{19} + 10817 \beta_{18} - 190828 \beta_{17} - 10817 \beta_{16} + 10817 \beta_{15} - 1270139 \beta_{14} - 1345570 \beta_{13} - 668518 \beta_{12} - 1049152 \beta_{11} + \cdots + 1038335 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2107484 \beta_{19} - 1840420 \beta_{18} + 1359709 \beta_{17} - 267064 \beta_{16} + 292387 \beta_{15} + 25323 \beta_{13} - 672330 \beta_{12} - 2032039 \beta_{11} - 2010725 \beta_{10} + \cdots + 4815018 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 6205628 \beta_{19} - 4662471 \beta_{18} + 5660522 \beta_{17} - 1517834 \beta_{16} - 194412 \beta_{15} + 10748978 \beta_{14} + 9205821 \beta_{13} + 1673060 \beta_{12} + \cdots + 4662471 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 1673060 \beta_{18} + 3385306 \beta_{17} - 3385306 \beta_{16} - 8227854 \beta_{15} + 25425140 \beta_{14} + 20925100 \beta_{13} + 1673060 \beta_{12} + 12040041 \beta_{11} + \cdots - 25983466 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 50776689 \beta_{19} + 35015101 \beta_{18} - 39163529 \beta_{17} + 4148428 \beta_{16} - 32082975 \beta_{15} - 32082975 \beta_{14} - 16208439 \beta_{13} - 20356867 \beta_{12} + \cdots - 104318793 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/92\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(47\)
\(\chi(n)\) \(-\beta_{13}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.858865 + 1.88065i
1.20400 2.63640i
−0.420431 0.123450i
2.26168 + 0.664090i
1.82483 2.10597i
−0.967148 + 1.11615i
0.291382 2.02660i
−0.250875 + 1.74487i
−0.858865 1.88065i
1.20400 + 2.63640i
1.53106 0.983952i
−0.115644 + 0.0743196i
0.291382 + 2.02660i
−0.250875 1.74487i
1.53106 + 0.983952i
−0.115644 0.0743196i
1.82483 + 2.10597i
−0.967148 1.11615i
−0.420431 + 0.123450i
2.26168 0.664090i
0 −1.35392 1.56250i 0 −0.187926 1.30705i 0 1.18148 2.58708i 0 −0.181380 + 1.26153i 0
9.2 0 1.89799 + 2.19040i 0 −0.556197 3.86843i 0 −1.06324 + 2.32817i 0 −0.768534 + 5.34527i 0
13.1 0 −0.368621 0.236898i 0 0.781296 1.71080i 0 4.72847 + 1.38840i 0 −1.16648 2.55425i 0
13.2 0 1.98297 + 1.27438i 0 −0.105327 + 0.230633i 0 −3.93129 1.15433i 0 1.06190 + 2.32523i 0
25.1 0 −0.396574 2.75823i 0 −1.50454 0.441774i 0 −0.107929 + 0.124557i 0 −4.57211 + 1.34249i 0
25.2 0 0.210181 + 1.46184i 0 0.926812 + 0.272136i 0 −1.14874 + 1.32572i 0 0.785668 0.230693i 0
29.1 0 −1.96451 0.576832i 0 3.22799 2.07450i 0 0.267813 1.86268i 0 1.00280 + 0.644459i 0
29.2 0 1.69141 + 0.496642i 0 −0.630070 + 0.404921i 0 −0.0283674 + 0.197300i 0 0.0904475 + 0.0581271i 0
41.1 0 −1.35392 + 1.56250i 0 −0.187926 + 1.30705i 0 1.18148 + 2.58708i 0 −0.181380 1.26153i 0
41.2 0 1.89799 2.19040i 0 −0.556197 + 3.86843i 0 −1.06324 2.32817i 0 −0.768534 5.34527i 0
49.1 0 −0.756044 + 1.65551i 0 −2.72780 + 3.14805i 0 1.70185 1.09371i 0 −0.204514 0.236022i 0
49.2 0 0.0571054 0.125043i 0 1.77577 2.04934i 0 −0.600040 + 0.385622i 0 1.95221 + 2.25297i 0
73.1 0 −1.96451 + 0.576832i 0 3.22799 + 2.07450i 0 0.267813 + 1.86268i 0 1.00280 0.644459i 0
73.2 0 1.69141 0.496642i 0 −0.630070 0.404921i 0 −0.0283674 0.197300i 0 0.0904475 0.0581271i 0
77.1 0 −0.756044 1.65551i 0 −2.72780 3.14805i 0 1.70185 + 1.09371i 0 −0.204514 + 0.236022i 0
77.2 0 0.0571054 + 0.125043i 0 1.77577 + 2.04934i 0 −0.600040 0.385622i 0 1.95221 2.25297i 0
81.1 0 −0.396574 + 2.75823i 0 −1.50454 + 0.441774i 0 −0.107929 0.124557i 0 −4.57211 1.34249i 0
81.2 0 0.210181 1.46184i 0 0.926812 0.272136i 0 −1.14874 1.32572i 0 0.785668 + 0.230693i 0
85.1 0 −0.368621 + 0.236898i 0 0.781296 + 1.71080i 0 4.72847 1.38840i 0 −1.16648 + 2.55425i 0
85.2 0 1.98297 1.27438i 0 −0.105327 0.230633i 0 −3.93129 + 1.15433i 0 1.06190 2.32523i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 92.2.e.a 20
3.b odd 2 1 828.2.q.a 20
4.b odd 2 1 368.2.m.d 20
23.c even 11 1 inner 92.2.e.a 20
23.c even 11 1 2116.2.a.j 10
23.d odd 22 1 2116.2.a.i 10
69.h odd 22 1 828.2.q.a 20
92.g odd 22 1 368.2.m.d 20
92.g odd 22 1 8464.2.a.ce 10
92.h even 22 1 8464.2.a.cd 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.e.a 20 1.a even 1 1 trivial
92.2.e.a 20 23.c even 11 1 inner
368.2.m.d 20 4.b odd 2 1
368.2.m.d 20 92.g odd 22 1
828.2.q.a 20 3.b odd 2 1
828.2.q.a 20 69.h odd 22 1
2116.2.a.i 10 23.d odd 22 1
2116.2.a.j 10 23.c even 11 1
8464.2.a.cd 10 92.h even 22 1
8464.2.a.ce 10 92.g odd 22 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(92, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 2 T^{19} + 7 T^{18} - 9 T^{17} + \cdots + 529 \) Copy content Toggle raw display
$5$ \( T^{20} - 2 T^{19} + 15 T^{18} + \cdots + 14641 \) Copy content Toggle raw display
$7$ \( T^{20} - 2 T^{19} - 21 T^{18} + 26 T^{17} + \cdots + 529 \) Copy content Toggle raw display
$11$ \( T^{20} + 2 T^{19} + 4 T^{18} + \cdots + 25715041 \) Copy content Toggle raw display
$13$ \( T^{20} - 6 T^{19} + \cdots + 14988860041 \) Copy content Toggle raw display
$17$ \( T^{20} + 9 T^{19} + 83 T^{18} + \cdots + 139129 \) Copy content Toggle raw display
$19$ \( T^{20} + 11 T^{19} + 125 T^{18} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{20} + 22 T^{19} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} + T^{19} + \cdots + 10805664116809 \) Copy content Toggle raw display
$31$ \( T^{20} + 13 T^{19} + \cdots + 2079845655889 \) Copy content Toggle raw display
$37$ \( T^{20} - 34 T^{19} + \cdots + 1866439595329 \) Copy content Toggle raw display
$41$ \( T^{20} - 28 T^{19} + \cdots + 65806215082609 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 150230641573321 \) Copy content Toggle raw display
$47$ \( (T^{10} - 13 T^{9} - 139 T^{8} + \cdots - 3761152)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 130889227506489 \) Copy content Toggle raw display
$59$ \( T^{20} + 10 T^{19} + \cdots + 6274789532209 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 791079807551881 \) Copy content Toggle raw display
$67$ \( T^{20} + 44 T^{19} + \cdots + 14858951082361 \) Copy content Toggle raw display
$71$ \( T^{20} + 37 T^{19} + \cdots + 59472089161 \) Copy content Toggle raw display
$73$ \( T^{20} + 12 T^{19} + \cdots + 12\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{20} + 6 T^{19} + \cdots + 1214604572281 \) Copy content Toggle raw display
$83$ \( T^{20} + 25 T^{19} + \cdots + 19165912096321 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 153528281766409 \) Copy content Toggle raw display
$97$ \( T^{20} + T^{19} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
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