↑

Properties

Label 690.2.m.e
Level $690$
Weight $2$
Character orbit 690.m
Analytic conductor $5.510$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \( x^{20} - 6 x^{19} + 21 x^{18} - 47 x^{17} + 44 x^{16} + 232 x^{15} - 1084 x^{14} + 1484 x^{13} + \cdots + 1849 \) 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.

\(f(q)\) \(=\) \( q - \beta_{13} q^{2} - \beta_{16} q^{3} - \beta_{2} q^{4} + (\beta_{17} - \beta_{16} + \beta_{15} + \cdots - 1) q^{5}+ \cdots - \beta_{17} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{2} - \beta_{16} q^{3} - \beta_{2} q^{4} + (\beta_{17} - \beta_{16} + \beta_{15} + \cdots - 1) q^{5}+ \cdots + (\beta_{18} + \beta_{17} + \cdots - \beta_{3}) q^{99}+O(q^{100}) \) 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 \( 20 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} - 2 q^{9} + 2 q^{10} - 4 q^{11} + 2 q^{12} - 13 q^{13} + 2 q^{14} + 2 q^{15} - 2 q^{16} + 6 q^{17} + 2 q^{18} - 4 q^{19} - 2 q^{20} + 2 q^{21} + 26 q^{22} + 24 q^{23} + 20 q^{24} - 2 q^{25} + 13 q^{26} + 2 q^{27} - 2 q^{28} + 2 q^{29} - 2 q^{30} + 6 q^{31} + 2 q^{32} + 4 q^{33} + 5 q^{34} - 2 q^{35} - 2 q^{36} - 20 q^{37} + 4 q^{38} + 13 q^{39} + 2 q^{40} - 11 q^{41} - 13 q^{42} + 4 q^{43} - 4 q^{44} + 20 q^{45} - 2 q^{46} + 18 q^{47} + 2 q^{48} - 6 q^{49} + 2 q^{50} + 27 q^{51} - 2 q^{52} - 18 q^{53} - 2 q^{54} + 18 q^{55} + 2 q^{56} - 7 q^{57} + 9 q^{58} - 52 q^{59} + 2 q^{60} - 7 q^{61} + 16 q^{62} - 13 q^{63} - 2 q^{64} - 2 q^{65} + 7 q^{66} - 10 q^{67} + 28 q^{68} + 20 q^{69} - 20 q^{70} + 7 q^{71} + 2 q^{72} - 37 q^{73} + 20 q^{74} + 2 q^{75} - 4 q^{76} - 70 q^{77} + 9 q^{78} + 20 q^{79} - 2 q^{80} - 2 q^{81} - 11 q^{82} - 19 q^{83} - 9 q^{84} + 17 q^{85} - 26 q^{86} - 2 q^{87} + 4 q^{88} + 33 q^{89} + 2 q^{90} - 42 q^{91} - 9 q^{92} + 16 q^{93} - 7 q^{94} - 4 q^{95} - 2 q^{96} + 9 q^{97} + 6 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} + 21 x^{18} - 47 x^{17} + 44 x^{16} + 232 x^{15} - 1084 x^{14} + 1484 x^{13} + \cdots + 1849 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!45 \nu^{19} + \cdots + 12\!\cdots\!48 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 58\!\cdots\!18 \nu^{19} + \cdots + 19\!\cdots\!51 ) / 29\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 44\!\cdots\!88 \nu^{19} + \cdots + 19\!\cdots\!93 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!81 \nu^{19} + \cdots - 45\!\cdots\!35 ) / 29\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!69 \nu^{19} + \cdots + 86\!\cdots\!28 ) / 29\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!52 \nu^{19} + \cdots - 19\!\cdots\!83 ) / 29\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!26 \nu^{19} + \cdots - 11\!\cdots\!71 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 42\!\cdots\!50 \nu^{19} + \cdots - 17\!\cdots\!87 ) / 29\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!96 \nu^{19} + \cdots - 25\!\cdots\!79 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 47\!\cdots\!51 \nu^{19} + \cdots - 50\!\cdots\!18 ) / 29\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25\!\cdots\!42 \nu^{19} + \cdots + 14\!\cdots\!55 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27\!\cdots\!20 \nu^{19} + \cdots - 12\!\cdots\!00 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 34\!\cdots\!55 \nu^{19} + \cdots + 48\!\cdots\!26 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 37\!\cdots\!04 \nu^{19} + \cdots - 17\!\cdots\!84 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 44\!\cdots\!81 \nu^{19} + \cdots - 92\!\cdots\!80 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 46\!\cdots\!57 \nu^{19} + \cdots + 25\!\cdots\!77 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 64\!\cdots\!46 \nu^{19} + \cdots - 53\!\cdots\!64 ) / 12\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 71\!\cdots\!31 \nu^{19} + \cdots + 68\!\cdots\!21 ) / 12\!\cdots\!27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{19} - \beta_{18} + \beta_{16} + \beta_{14} + 4 \beta_{13} + \beta_{10} + \beta_{8} + \cdots - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 6 \beta_{19} - 5 \beta_{18} - 3 \beta_{17} + 6 \beta_{16} + 3 \beta_{15} + 5 \beta_{14} + 8 \beta_{13} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} - 5 \beta_{18} + 12 \beta_{17} - 4 \beta_{16} + 12 \beta_{15} + 8 \beta_{14} + \beta_{13} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{19} + 37 \beta_{17} - 10 \beta_{16} + 20 \beta_{15} - 2 \beta_{14} - 25 \beta_{13} + 2 \beta_{12} + \cdots - 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 49 \beta_{19} + 17 \beta_{18} - 176 \beta_{17} + 59 \beta_{16} - 252 \beta_{15} - 52 \beta_{14} + \cdots + 118 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 149 \beta_{19} + 190 \beta_{18} - 279 \beta_{17} + 126 \beta_{16} - 451 \beta_{15} + 26 \beta_{14} + \cdots + 541 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 958 \beta_{19} + 505 \beta_{18} + 1726 \beta_{17} - 1261 \beta_{16} + 1221 \beta_{15} + 83 \beta_{14} + \cdots - 839 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 709 \beta_{19} - 442 \beta_{18} + 481 \beta_{17} - 6 \beta_{16} + 442 \beta_{15} - 709 \beta_{14} + \cdots - 2434 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 10774 \beta_{19} - 3634 \beta_{18} - 21494 \beta_{17} + 11124 \beta_{16} - 15352 \beta_{15} + \cdots + 9886 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5608 \beta_{18} + 12201 \beta_{17} - 10691 \beta_{16} + 17284 \beta_{15} + 12201 \beta_{14} + \cdots + 10260 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 112497 \beta_{19} + 14864 \beta_{18} + 219206 \beta_{17} - 130827 \beta_{16} + 195024 \beta_{15} + \cdots - 170428 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 93436 \beta_{19} - 93436 \beta_{18} - 297818 \beta_{17} + 174858 \beta_{16} - 278418 \beta_{15} + \cdots - 25597 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1073229 \beta_{19} - 66429 \beta_{18} - 1964912 \beta_{17} + 1073229 \beta_{16} - 1654675 \beta_{15} + \cdots + 1654675 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2034529 \beta_{19} + 1254266 \beta_{18} + 4981939 \beta_{17} - 2952752 \beta_{16} + 4241676 \beta_{15} + \cdots - 1027197 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 9176359 \beta_{19} + 15959038 \beta_{17} - 7906237 \beta_{16} + 13052315 \beta_{15} - 1919605 \beta_{14} + \cdots - 15959038 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 30608973 \beta_{19} - 12926543 \beta_{18} - 67672363 \beta_{17} + 40840915 \beta_{16} + \cdots + 26354328 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 70366964 \beta_{19} + 9689521 \beta_{18} - 103249795 \beta_{17} + 52884467 \beta_{16} + \cdots + 139105414 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 392001502 \beta_{19} + 117472410 \beta_{18} + 821440121 \beta_{17} - 477358238 \beta_{16} + \cdots - 430927863 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{13}\)

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
1.66465 + 1.92111i
−0.205162 − 0.236769i
2.03264 − 1.30630i
−0.877779 + 0.564114i
0.909132 − 1.99072i
−0.266817 + 0.584249i
2.03264 + 1.30630i
−0.877779 − 0.564114i
−0.462319 + 3.21550i
0.121066 − 0.842031i
2.29563 + 0.674057i
−2.21104 − 0.649220i
−0.462319 − 3.21550i
0.121066 + 0.842031i
0.909132 + 1.99072i
−0.266817 − 0.584249i
2.29563 − 0.674057i
−2.21104 + 0.649220i
1.66465 − 1.92111i
−0.205162 + 0.236769i
0.142315 − 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i −0.654861 − 0.755750i −0.959493 + 0.281733i −0.306823 − 0.197183i −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 4.30262 + 2.76513i −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 −0.421385 + 2.93080i 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 0.223463 − 1.55422i 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 −2.28976 − 0.672335i −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 −1.51668 − 0.445336i −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 −0.421385 − 2.93080i 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 0.223463 + 1.55422i 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 −1.63866 − 3.58816i 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 1.22647 + 2.68560i 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 −1.56737 + 1.80884i 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 0.988122 − 1.14035i 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 −1.63866 + 3.58816i 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 1.22647 − 2.68560i 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 −2.28976 + 0.672335i −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 −1.51668 + 0.445336i −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 −1.56737 − 1.80884i 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 0.988122 + 1.14035i 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 −0.306823 + 0.197183i −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 4.30262 − 2.76513i −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.e ✓ 20
23.c even 11 1 inner 690.2.m.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.e ✓ 20 1.a even 1 1 trivial
690.2.m.e ✓ 20 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + 2 T_{7}^{19} + 12 T_{7}^{18} + 7 T_{7}^{17} + 264 T_{7}^{16} + 2025 T_{7}^{15} + \cdots + 1893376 \) 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
$7$ \( T^{20} + 2 T^{19} + \cdots + 1893376 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 707506801 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 144264121 \) Copy content Toggle raw display
$17$ \( T^{20} - 6 T^{19} + \cdots + 33860761 \) Copy content Toggle raw display
$19$ \( T^{20} + 4 T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 179533201 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 16570264694281 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 15338839089289 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 7415295394816 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 11\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( (T^{10} - 9 T^{9} + \cdots + 10197947)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 834348308767744 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 24\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 182035342336 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 488977134361 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 2774329959424 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 91979474091649 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 72\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 118754673664 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
show more
show less