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Properties

Label 690.2.m.d
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: \( x^{20} - 5 x^{19} + 8 x^{18} - 32 x^{17} + 277 x^{16} - 1138 x^{15} + 2950 x^{14} - 6404 x^{13} + \cdots + 7921 \) 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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - \beta_{12} q^{3} + \beta_{10} q^{4} - \beta_{6} q^{5} + \beta_{11} q^{6} + (\beta_{19} + \beta_{10} + \cdots - \beta_1) q^{7}+ \cdots + \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - \beta_{12} q^{3} + \beta_{10} q^{4} - \beta_{6} q^{5} + \beta_{11} q^{6} + (\beta_{19} + \beta_{10} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{19} - \beta_{18} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) 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 \( 20 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} - 2 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} - 11 q^{13} + 2 q^{14} - 2 q^{15} - 2 q^{16} + 24 q^{17} - 2 q^{18} + 22 q^{19} + 2 q^{20} - 2 q^{21} + 2 q^{22} - 22 q^{23} - 20 q^{24} - 2 q^{25} + 11 q^{26} + 2 q^{27} + 2 q^{28} + 14 q^{29} - 2 q^{30} - 8 q^{31} - 2 q^{32} - 2 q^{33} - 9 q^{34} - 2 q^{35} - 2 q^{36} + 20 q^{37} + 11 q^{39} + 2 q^{40} - 21 q^{41} - 13 q^{42} + 34 q^{43} + 2 q^{44} - 20 q^{45} + 14 q^{47} + 2 q^{48} + 10 q^{49} - 2 q^{50} + 31 q^{51} + 2 q^{54} - 2 q^{55} + 2 q^{56} + 11 q^{57} - 19 q^{58} - 40 q^{59} - 2 q^{60} - 19 q^{61} - 8 q^{62} + 13 q^{63} - 2 q^{64} + 9 q^{66} + 18 q^{67} - 20 q^{68} - 22 q^{69} + 20 q^{70} - 85 q^{71} - 2 q^{72} + 39 q^{73} + 20 q^{74} + 2 q^{75} - 48 q^{77} - 11 q^{78} - 28 q^{79} + 2 q^{80} - 2 q^{81} + q^{82} + 49 q^{83} + 9 q^{84} - 13 q^{85} - 32 q^{86} + 8 q^{87} + 2 q^{88} + 3 q^{89} + 2 q^{90} - 34 q^{91} - 11 q^{92} - 36 q^{93} + 3 q^{94} + 2 q^{96} + 43 q^{97} + 10 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} + 8 x^{18} - 32 x^{17} + 277 x^{16} - 1138 x^{15} + 2950 x^{14} - 6404 x^{13} + \cdots + 7921 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 18\!\cdots\!21 \nu^{19} + \cdots + 75\!\cdots\!32 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21\!\cdots\!92 \nu^{19} + \cdots + 50\!\cdots\!92 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 82\!\cdots\!12 \nu^{19} + \cdots - 15\!\cdots\!04 ) / 92\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33\!\cdots\!00 \nu^{19} + \cdots - 17\!\cdots\!95 ) / 92\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!00 \nu^{19} + \cdots - 52\!\cdots\!51 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\!\cdots\!79 \nu^{19} + \cdots + 73\!\cdots\!70 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 94\!\cdots\!81 \nu^{19} + \cdots + 17\!\cdots\!37 ) / 92\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 96\!\cdots\!14 \nu^{19} + \cdots - 29\!\cdots\!83 ) / 92\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 34\!\cdots\!00 \nu^{19} + \cdots + 95\!\cdots\!13 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!26 \nu^{19} + \cdots + 24\!\cdots\!06 ) / 92\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 36\!\cdots\!28 \nu^{19} + \cdots - 95\!\cdots\!01 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 37\!\cdots\!77 \nu^{19} + \cdots + 87\!\cdots\!30 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22\!\cdots\!49 \nu^{19} + \cdots - 29\!\cdots\!36 ) / 92\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 22\!\cdots\!73 \nu^{19} + \cdots + 28\!\cdots\!40 ) / 92\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 94\!\cdots\!41 \nu^{19} + \cdots + 26\!\cdots\!33 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 10\!\cdots\!06 \nu^{19} + \cdots + 31\!\cdots\!04 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 10\!\cdots\!38 \nu^{19} + \cdots - 26\!\cdots\!71 ) / 27\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 13\!\cdots\!17 \nu^{19} + \cdots - 30\!\cdots\!77 ) / 27\!\cdots\!41 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{19} - \beta_{14} - 7 \beta_{13} - 8 \beta_{12} + 6 \beta_{11} - 7 \beta_{10} - 6 \beta_{9} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} - \beta_{18} - 2 \beta_{17} + \beta_{16} + 9 \beta_{15} + \beta_{14} - 9 \beta_{13} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} + 2 \beta_{18} + 17 \beta_{17} + 14 \beta_{16} - 4 \beta_{15} - 15 \beta_{14} - 35 \beta_{13} + \cdots - 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 116 \beta_{19} - 20 \beta_{18} + 8 \beta_{17} - 15 \beta_{16} - 30 \beta_{14} - 134 \beta_{13} + \cdots - 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 125 \beta_{19} - 180 \beta_{18} - 38 \beta_{17} + 213 \beta_{16} + 83 \beta_{15} + 38 \beta_{14} + \cdots + 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 330 \beta_{19} + 206 \beta_{18} + 1439 \beta_{17} + 366 \beta_{16} - 1076 \beta_{15} - 1076 \beta_{14} + \cdots - 1129 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3573 \beta_{19} - 2931 \beta_{18} - 2038 \beta_{17} - 577 \beta_{16} + 577 \beta_{15} + 1303 \beta_{14} + \cdots - 157 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 15909 \beta_{19} - 4206 \beta_{18} + 5032 \beta_{17} + 13021 \beta_{16} - 4206 \beta_{15} + 9658 \beta_{13} + \cdots + 4775 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 47812 \beta_{19} + 8147 \beta_{18} + 47243 \beta_{17} - 18689 \beta_{16} - 39665 \beta_{15} + \cdots - 38894 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 162998 \beta_{18} - 204415 \beta_{17} + 41417 \beta_{15} + 162998 \beta_{14} + 359840 \beta_{13} + \cdots + 122122 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 644714 \beta_{19} + 259040 \beta_{18} + 644714 \beta_{17} + 259040 \beta_{16} - 359840 \beta_{15} + \cdots + 359840 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2684859 \beta_{19} - 2084170 \beta_{16} - 1040965 \beta_{15} + 522184 \beta_{14} + 2603639 \beta_{13} + \cdots - 1280520 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8094417 \beta_{19} - 3545157 \beta_{18} - 8662246 \beta_{17} + 1539425 \beta_{16} + 2889314 \beta_{15} + \cdots + 12533051 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 14502816 \beta_{19} + 27022373 \beta_{18} + 35757971 \beta_{17} - 5702188 \beta_{16} - 15422365 \beta_{15} + \cdots + 5133761 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 91422781 \beta_{19} - 21199320 \beta_{18} - 106064565 \beta_{17} - 95153849 \beta_{16} + \cdots - 13390884 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 566784182 \beta_{19} + 61209850 \beta_{18} - 199649430 \beta_{17} + 126798067 \beta_{16} + \cdots + 692774215 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 293860150 \beta_{19} + 1270863889 \beta_{18} + 1290373476 \beta_{17} - 735502047 \beta_{16} + \cdots - 1000275114 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 2287059920 \beta_{19} - 1833231470 \beta_{18} - 7591068386 \beta_{17} - 2726258213 \beta_{16} + \cdots + 2167162524 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\)

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
1.63205 − 1.88348i
−2.42025 + 2.79312i
−2.75487 − 1.77045i
2.31649 + 1.48872i
1.01145 + 2.21477i
−0.764548 − 1.67413i
−2.75487 + 1.77045i
2.31649 − 1.48872i
0.271421 + 1.88778i
−0.162761 − 1.13203i
3.40070 − 0.998536i
−0.0296794 + 0.00871465i
0.271421 − 1.88778i
−0.162761 + 1.13203i
1.01145 − 2.21477i
−0.764548 + 1.67413i
3.40070 + 0.998536i
−0.0296794 − 0.00871465i
1.63205 + 1.88348i
−2.42025 − 2.79312i
−0.142315 + 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i 0.654861 + 0.755750i 0.959493 − 0.281733i −2.73875 − 1.76009i 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 −1.25705 − 0.807858i 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 −0.257432 + 1.79048i −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 0.455355 − 3.16706i −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 −0.783059 − 0.229927i 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 4.58950 + 1.34760i 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 −0.257432 − 1.79048i −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 0.455355 + 3.16706i −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 −1.00994 − 2.21146i −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 1.42213 + 3.11402i −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 −1.67997 + 1.93879i −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 2.25922 − 2.60728i −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 −1.00994 + 2.21146i −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 1.42213 − 3.11402i −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 −0.783059 + 0.229927i 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 4.58950 − 1.34760i 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 −1.67997 − 1.93879i −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 2.25922 + 2.60728i −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 −2.73875 + 1.76009i 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 −1.25705 + 0.807858i 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.d ✓ 20
23.c even 11 1 inner 690.2.m.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.d ✓ 20 1.a even 1 1 trivial
690.2.m.d ✓ 20 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} - 2 T_{7}^{19} + 4 T_{7}^{18} - 19 T_{7}^{17} + 16 T_{7}^{16} - 461 T_{7}^{15} + \cdots + 65545216 \) 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
$7$ \( T^{20} - 2 T^{19} + \cdots + 65545216 \) Copy content Toggle raw display
$11$ \( T^{20} - 2 T^{19} + \cdots + 7745089 \) Copy content Toggle raw display
$13$ \( T^{20} + 11 T^{19} + \cdots + 2374681 \) Copy content Toggle raw display
$17$ \( T^{20} - 24 T^{19} + \cdots + 13980121 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 1814078464 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} - 14 T^{19} + \cdots + 40157569 \) Copy content Toggle raw display
$31$ \( T^{20} + 8 T^{19} + \cdots + 13256881 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 148035889 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 286557184 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 3604441369 \) Copy content Toggle raw display
$47$ \( (T^{10} - 7 T^{9} + \cdots - 10400203)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 474092839936 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 6968743441 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 2318871110656 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 977272622041 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 1143633470464 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 369268184187904 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 188504457241 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 2376925225984 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
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