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Properties

Label 690.2.m.f
Level $690$
Weight $2$
Character orbit 690.m
Analytic conductor $5.510$
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] = [690,2,Mod(31,690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 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("690.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.m (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: \(5.50967773947\)
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: \( x^{20} - 6 x^{19} + 36 x^{18} - 172 x^{17} + 691 x^{16} - 2342 x^{15} + 7870 x^{14} - 23240 x^{13} + \cdots + 123904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + ( - \beta_{14} + \beta_{13} - \beta_{12} + \cdots + 1) q^{3}+ \cdots + \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + ( - \beta_{14} + \beta_{13} - \beta_{12} + \cdots + 1) q^{3}+ \cdots + (\beta_{19} - \beta_{14} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} - 2 q^{9} - 2 q^{10} + 18 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{14} - 2 q^{15} - 2 q^{16} - 2 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{20} + 13 q^{21} + 26 q^{22} + 20 q^{24} - 2 q^{25} + 9 q^{26} + 2 q^{27} - 2 q^{28} + 6 q^{29} + 2 q^{30} + 8 q^{31} + 2 q^{32} + 4 q^{33} + 13 q^{34} + 13 q^{35} - 2 q^{36} + 16 q^{37} - 15 q^{38} - 2 q^{39} - 2 q^{40} + 3 q^{41} + 9 q^{42} - 16 q^{43} + 18 q^{44} - 20 q^{45} - 34 q^{47} + 2 q^{48} - 6 q^{49} + 2 q^{50} - 9 q^{51} + 2 q^{52} - 4 q^{53} - 2 q^{54} - 7 q^{55} + 2 q^{56} + 29 q^{57} + 16 q^{58} + 25 q^{59} - 2 q^{60} - 19 q^{61} - 8 q^{62} + 9 q^{63} - 2 q^{64} - 2 q^{65} - 15 q^{66} + 20 q^{67} - 2 q^{68} - 22 q^{69} - 2 q^{70} - q^{71} + 2 q^{72} - q^{73} - 16 q^{74} + 2 q^{75} + 4 q^{76} + 53 q^{77} + 2 q^{78} + 28 q^{79} + 2 q^{80} - 2 q^{81} - 25 q^{82} + 9 q^{83} - 9 q^{84} - 9 q^{85} - 28 q^{86} - 6 q^{87} - 7 q^{88} - 31 q^{89} - 2 q^{90} - 26 q^{91} + 11 q^{92} - 30 q^{93} + q^{94} - 15 q^{95} - 2 q^{96} - 35 q^{97} - 5 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} + 36 x^{18} - 172 x^{17} + 691 x^{16} - 2342 x^{15} + 7870 x^{14} - 23240 x^{13} + \cdots + 123904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!51 \nu^{19} + \cdots + 15\!\cdots\!32 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24\!\cdots\!55 \nu^{19} + \cdots + 94\!\cdots\!04 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!19 \nu^{19} + \cdots + 78\!\cdots\!48 ) / 79\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 72\!\cdots\!99 \nu^{19} + \cdots - 77\!\cdots\!28 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 63\!\cdots\!63 \nu^{19} + \cdots + 42\!\cdots\!48 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!03 \nu^{19} + \cdots - 10\!\cdots\!64 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!77 \nu^{19} + \cdots + 43\!\cdots\!20 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 96\!\cdots\!03 \nu^{19} + \cdots + 91\!\cdots\!16 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\!\cdots\!41 \nu^{19} + \cdots + 14\!\cdots\!76 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25\!\cdots\!61 \nu^{19} + \cdots + 13\!\cdots\!24 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 25\!\cdots\!39 \nu^{19} + \cdots - 89\!\cdots\!76 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27\!\cdots\!29 \nu^{19} + \cdots - 11\!\cdots\!64 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 32\!\cdots\!95 \nu^{19} + \cdots + 15\!\cdots\!64 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 59\!\cdots\!91 \nu^{19} + \cdots - 23\!\cdots\!32 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 11\!\cdots\!83 \nu^{19} + \cdots + 47\!\cdots\!92 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 11\!\cdots\!03 \nu^{19} + \cdots - 52\!\cdots\!00 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 12\!\cdots\!11 \nu^{19} + \cdots - 61\!\cdots\!20 ) / 36\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 69\!\cdots\!60 \nu^{19} + \cdots + 38\!\cdots\!60 ) / 18\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} + \beta_{18} - \beta_{15} - 2\beta_{13} - \beta_{10} - \beta_{9} - 5\beta_{8} - \beta_{6} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - 6 \beta_{12} + \beta_{11} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 15 \beta_{19} - 3 \beta_{18} + 4 \beta_{17} + 12 \beta_{16} + 2 \beta_{15} + 22 \beta_{14} + 28 \beta_{13} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{19} + 22 \beta_{18} - 21 \beta_{17} - 19 \beta_{16} - 77 \beta_{15} + 81 \beta_{14} + \cdots - 133 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 71 \beta_{19} + 84 \beta_{18} - 139 \beta_{17} - 33 \beta_{16} - 407 \beta_{14} - 93 \beta_{13} + \cdots - 93 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 344 \beta_{19} - 343 \beta_{18} + 321 \beta_{17} + 834 \beta_{16} + 834 \beta_{15} - 978 \beta_{14} + \cdots + 1185 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1345 \beta_{19} - 1646 \beta_{18} + 464 \beta_{17} - 464 \beta_{15} + 3313 \beta_{14} + 325 \beta_{13} + \cdots + 408 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5122 \beta_{19} + 5039 \beta_{18} - 9782 \beta_{17} - 9782 \beta_{16} - 5039 \beta_{15} + 863 \beta_{14} + \cdots - 14199 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20067 \beta_{19} + 6414 \beta_{18} + 6414 \beta_{16} + 20067 \beta_{15} - 60255 \beta_{14} + \cdots + 43091 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 75281 \beta_{19} - 120846 \beta_{18} + 120846 \beta_{17} + 75281 \beta_{16} + 73162 \beta_{15} + \cdots + 298571 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 89013 \beta_{19} - 89013 \beta_{17} - 251121 \beta_{16} - 276072 \beta_{15} + 823377 \beta_{14} + \cdots - 599767 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1542824 \beta_{19} + 1542824 \beta_{18} - 1088728 \beta_{17} - 1019988 \beta_{16} - 853137 \beta_{15} + \cdots - 2704593 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1239061 \beta_{18} + 3208738 \beta_{17} + 3807616 \beta_{16} + 4094376 \beta_{15} - 9431371 \beta_{14} + \cdots + 10670432 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 20106737 \beta_{19} - 15412851 \beta_{18} + 14027735 \beta_{17} + 11253417 \beta_{16} + \cdots + 14973336 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 17239039 \beta_{19} + 41649927 \beta_{18} - 52022757 \beta_{17} - 57587306 \beta_{16} - 72330500 \beta_{15} + \cdots - 197050067 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 215119523 \beta_{19} + 191441670 \beta_{18} - 148795495 \beta_{17} - 63231234 \beta_{16} + \cdots - 129189437 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 546891998 \beta_{19} - 707249278 \beta_{18} + 800052670 \beta_{17} + 963399658 \beta_{16} + \cdots + 2481503812 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 2601376287 \beta_{19} - 1975124054 \beta_{18} + 882484415 \beta_{17} - 882484415 \beta_{15} + \cdots - 998705527 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.02072 + 1.17798i
−1.91971 − 2.21546i
2.64328 − 1.69874i
−0.626062 + 0.402346i
1.05064 − 2.30058i
−1.08726 + 2.38076i
2.64328 + 1.69874i
−0.626062 − 0.402346i
0.303318 − 2.10963i
−0.523366 + 3.64009i
−0.245245 − 0.0720105i
2.38367 + 0.699910i
0.303318 + 2.10963i
−0.523366 − 3.64009i
1.05064 + 2.30058i
−1.08726 − 2.38076i
−0.245245 + 0.0720105i
2.38367 − 0.699910i
1.02072 − 1.17798i
−1.91971 + 2.21546i
0.142315 − 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i 0.654861 + 0.755750i −0.959493 + 0.281733i −2.22667 − 1.43099i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.841254 − 0.540641i
31.2 0.142315 − 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i 0.654861 + 0.755750i −0.959493 + 0.281733i 1.55070 + 0.996574i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.841254 − 0.540641i
121.1 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 − 0.755750i −0.841254 + 0.540641i −0.654861 + 0.755750i 0.0123285 − 0.0857467i 0.959493 − 0.281733i 0.841254 + 0.540641i −0.142315 − 0.989821i
121.2 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 − 0.755750i −0.841254 + 0.540641i −0.654861 + 0.755750i 0.565404 − 3.93247i 0.959493 − 0.281733i 0.841254 + 0.540641i −0.142315 − 0.989821i
151.1 0.654861 + 0.755750i −0.841254 − 0.540641i −0.142315 + 0.989821i −0.415415 + 0.909632i −0.142315 − 0.989821i −3.76793 − 1.10637i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
151.2 0.654861 + 0.755750i −0.841254 − 0.540641i −0.142315 + 0.989821i −0.415415 + 0.909632i −0.142315 − 0.989821i 1.17001 + 0.343546i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
211.1 −0.415415 − 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i −0.841254 − 0.540641i −0.654861 − 0.755750i 0.0123285 + 0.0857467i 0.959493 + 0.281733i 0.841254 − 0.540641i −0.142315 + 0.989821i
211.2 −0.415415 − 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i −0.841254 − 0.540641i −0.654861 − 0.755750i 0.565404 + 3.93247i 0.959493 + 0.281733i 0.841254 − 0.540641i −0.142315 + 0.989821i
271.1 0.959493 + 0.281733i 0.654861 − 0.755750i 0.841254 + 0.540641i 0.142315 − 0.989821i 0.841254 − 0.540641i −0.730522 − 1.59962i 0.654861 + 0.755750i −0.142315 − 0.989821i 0.415415 − 0.909632i
271.2 0.959493 + 0.281733i 0.654861 − 0.755750i 0.841254 + 0.540641i 0.142315 − 0.989821i 0.841254 − 0.540641i 1.68256 + 3.68429i 0.654861 + 0.755750i −0.142315 − 0.989821i 0.415415 − 0.909632i
301.1 −0.841254 − 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 0.959493 + 0.281733i 0.415415 − 0.909632i −0.525067 + 0.605959i 0.142315 − 0.989821i −0.959493 + 0.281733i −0.654861 − 0.755750i
301.2 −0.841254 − 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 0.959493 + 0.281733i 0.415415 − 0.909632i 1.26919 − 1.46472i 0.142315 − 0.989821i −0.959493 + 0.281733i −0.654861 − 0.755750i
331.1 0.959493 − 0.281733i 0.654861 + 0.755750i 0.841254 − 0.540641i 0.142315 + 0.989821i 0.841254 + 0.540641i −0.730522 + 1.59962i 0.654861 − 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
331.2 0.959493 − 0.281733i 0.654861 + 0.755750i 0.841254 − 0.540641i 0.142315 + 0.989821i 0.841254 + 0.540641i 1.68256 − 3.68429i 0.654861 − 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
361.1 0.654861 − 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i −0.415415 − 0.909632i −0.142315 + 0.989821i −3.76793 + 1.10637i −0.841254 − 0.540641i 0.415415 − 0.909632i −0.959493 − 0.281733i
361.2 0.654861 − 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i −0.415415 − 0.909632i −0.142315 + 0.989821i 1.17001 − 0.343546i −0.841254 − 0.540641i 0.415415 − 0.909632i −0.959493 − 0.281733i
541.1 −0.841254 + 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i 0.959493 − 0.281733i 0.415415 + 0.909632i −0.525067 − 0.605959i 0.142315 + 0.989821i −0.959493 − 0.281733i −0.654861 + 0.755750i
541.2 −0.841254 + 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i 0.959493 − 0.281733i 0.415415 + 0.909632i 1.26919 + 1.46472i 0.142315 + 0.989821i −0.959493 − 0.281733i −0.654861 + 0.755750i
601.1 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.654861 − 0.755750i −0.959493 − 0.281733i −2.22667 + 1.43099i −0.415415 − 0.909632i −0.654861 − 0.755750i 0.841254 + 0.540641i
601.2 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.654861 − 0.755750i −0.959493 − 0.281733i 1.55070 − 0.996574i −0.415415 − 0.909632i −0.654861 − 0.755750i 0.841254 + 0.540641i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.2
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 690.2.m.f ✓ 20
23.c even 11 1 inner 690.2.m.f ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.f ✓ 20 1.a even 1 1 trivial
690.2.m.f ✓ 20 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + 2 T_{7}^{19} + 12 T_{7}^{18} + 106 T_{7}^{17} + 132 T_{7}^{16} + 309 T_{7}^{15} + \cdots + 7921 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + 2 T^{19} + \cdots + 7921 \) Copy content Toggle raw display
$11$ \( T^{20} - 18 T^{19} + \cdots + 7921 \) Copy content Toggle raw display
$13$ \( T^{20} - 2 T^{19} + \cdots + 1771561 \) Copy content Toggle raw display
$17$ \( T^{20} + 2 T^{19} + \cdots + 123904 \) Copy content Toggle raw display
$19$ \( T^{20} - 4 T^{19} + \cdots + 16834609 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 2135900406784 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 1472103424 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 412990025449 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 2145911501449 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 402382160896 \) Copy content Toggle raw display
$47$ \( (T^{10} + 17 T^{9} + \cdots + 1104851)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 3622956481 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12\!\cdots\!69 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 36489718211584 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 2141141533696 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 2364115354624 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 9360858440704 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 1669866496 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 596379329536 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 33\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 61\!\cdots\!24 \) Copy content Toggle raw display
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