Properties

Label 483.2.q.c
Level $483$
Weight $2$
Character orbit 483.q
Analytic conductor $3.857$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,2,Mod(64,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.q (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: \(3.85677441763\)
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: \( x^{20} - 8 x^{19} + 40 x^{18} - 117 x^{17} + 295 x^{16} - 575 x^{15} + 1777 x^{14} - 1560 x^{13} + \cdots + 2374681 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{2}) q^{2} + \beta_{12} q^{3} + (\beta_{11} - \beta_{5}) q^{4} + ( - \beta_{14} + \beta_{6}) q^{5} + ( - \beta_{13} - \beta_{5}) q^{6} - \beta_{13} q^{7} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_{2}) q^{8}+ \cdots + \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{2}) q^{2} + \beta_{12} q^{3} + (\beta_{11} - \beta_{5}) q^{4} + ( - \beta_{14} + \beta_{6}) q^{5} + ( - \beta_{13} - \beta_{5}) q^{6} - \beta_{13} q^{7} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_{2}) q^{8}+ \cdots + ( - \beta_{16} + \beta_{15} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{2} - 2 q^{3} - 4 q^{4} - q^{5} - 4 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{2} - 2 q^{3} - 4 q^{4} - q^{5} - 4 q^{6} - 2 q^{7} - 2 q^{9} + 9 q^{10} + 3 q^{11} + 18 q^{12} - 2 q^{13} + 18 q^{14} - q^{15} + 8 q^{16} + 8 q^{17} + 18 q^{18} + 6 q^{19} - 2 q^{20} - 2 q^{21} + 6 q^{22} + 11 q^{23} + 9 q^{25} + 7 q^{26} - 2 q^{27} - 4 q^{28} + 23 q^{29} + 9 q^{30} + q^{31} - 28 q^{32} + 14 q^{33} - 28 q^{34} + 10 q^{35} - 4 q^{36} - 9 q^{37} + 34 q^{38} - 2 q^{39} - 15 q^{41} - 4 q^{42} - 23 q^{43} - 16 q^{44} - 12 q^{45} + 11 q^{46} - 66 q^{47} - 36 q^{48} - 2 q^{49} - 26 q^{50} - 14 q^{51} + 7 q^{52} + 9 q^{53} - 4 q^{54} - 62 q^{55} + 22 q^{56} - 27 q^{57} - 20 q^{58} + 49 q^{59} - 2 q^{60} + 46 q^{61} - 9 q^{62} - 2 q^{63} + 16 q^{64} + 11 q^{65} - 16 q^{66} + 14 q^{67} + 38 q^{68} + 11 q^{69} - 2 q^{70} + 36 q^{71} - q^{73} + 4 q^{74} - 2 q^{75} + 34 q^{76} - 8 q^{77} - 15 q^{78} - 22 q^{79} + 15 q^{80} - 2 q^{81} - 30 q^{82} + 8 q^{83} - 4 q^{84} - 32 q^{85} - 68 q^{86} + q^{87} - 11 q^{88} - 2 q^{89} - 2 q^{90} - 24 q^{91} + 11 q^{92} - 32 q^{93} + 33 q^{94} - 107 q^{95} + 16 q^{96} + 18 q^{97} - 4 q^{98} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} + 40 x^{18} - 117 x^{17} + 295 x^{16} - 575 x^{15} + 1777 x^{14} - 1560 x^{13} + \cdots + 2374681 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27\!\cdots\!67 \nu^{19} + \cdots + 93\!\cdots\!23 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 32\!\cdots\!48 \nu^{19} + \cdots - 33\!\cdots\!96 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 64\!\cdots\!61 \nu^{19} + \cdots - 11\!\cdots\!64 ) / 24\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 52\!\cdots\!69 \nu^{19} + \cdots - 13\!\cdots\!60 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 57\!\cdots\!21 \nu^{19} + \cdots + 13\!\cdots\!18 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99\!\cdots\!65 \nu^{19} + \cdots + 25\!\cdots\!14 ) / 24\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 72\!\cdots\!98 \nu^{19} + \cdots + 12\!\cdots\!12 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 86\!\cdots\!19 \nu^{19} + \cdots + 68\!\cdots\!56 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!54 \nu^{19} + \cdots + 72\!\cdots\!74 ) / 24\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!57 \nu^{19} + \cdots - 37\!\cdots\!15 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!15 \nu^{19} + \cdots - 43\!\cdots\!97 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 20\!\cdots\!94 \nu^{19} + \cdots - 27\!\cdots\!67 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 20\!\cdots\!37 \nu^{19} + \cdots + 10\!\cdots\!02 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 22\!\cdots\!47 \nu^{19} + \cdots + 41\!\cdots\!45 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 41\!\cdots\!56 \nu^{19} + \cdots - 79\!\cdots\!21 ) / 24\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 42\!\cdots\!55 \nu^{19} + \cdots - 52\!\cdots\!51 ) / 24\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 49\!\cdots\!62 \nu^{19} + \cdots - 72\!\cdots\!42 ) / 24\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 49\!\cdots\!86 \nu^{19} + \cdots - 39\!\cdots\!62 ) / 16\!\cdots\!33 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{16} - \beta_{15} - \beta_{13} + \beta_{12} - \beta_{5} - \beta_{4} + 5\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{19} - 2 \beta_{18} + 3 \beta_{17} - \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + \cdots - 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 16 \beta_{19} - 19 \beta_{18} + 19 \beta_{17} + 16 \beta_{16} - 2 \beta_{15} - 3 \beta_{14} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 59 \beta_{18} + 42 \beta_{17} + 59 \beta_{16} - 57 \beta_{15} + 36 \beta_{14} - 66 \beta_{13} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 234 \beta_{19} + 27 \beta_{17} + 71 \beta_{16} - 298 \beta_{15} + 170 \beta_{14} - 705 \beta_{13} + \cdots + 298 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 946 \beta_{19} + 946 \beta_{18} - 35 \beta_{17} - 244 \beta_{16} + 122 \beta_{15} + 366 \beta_{14} + \cdots - 591 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1258 \beta_{19} + 4679 \beta_{18} + 167 \beta_{17} - 713 \beta_{16} + 8585 \beta_{15} - 167 \beta_{14} + \cdots - 9549 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3325 \beta_{19} + 11070 \beta_{18} - 2348 \beta_{17} + 2906 \beta_{16} + 46259 \beta_{15} + \cdots - 38188 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 10478 \beta_{19} + 9998 \beta_{18} - 44227 \beta_{17} + 12049 \beta_{16} + 101459 \beta_{15} + \cdots - 44227 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 40344 \beta_{19} - 3725 \beta_{18} - 276261 \beta_{17} - 82394 \beta_{16} - 138853 \beta_{15} + \cdots + 247584 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 158777 \beta_{19} + 10512 \beta_{18} - 835558 \beta_{17} - 835558 \beta_{16} - 1743371 \beta_{15} + \cdots + 1316101 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1286101 \beta_{19} - 728012 \beta_{18} - 2844417 \beta_{16} - 5660088 \beta_{15} - 1299036 \beta_{14} + \cdots + 2422576 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 12421513 \beta_{19} - 10385128 \beta_{18} + 12421513 \beta_{17} - 9310711 \beta_{15} - 7380965 \beta_{14} + \cdots + 2198653 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 41887018 \beta_{19} - 61912028 \beta_{18} + 61912028 \beta_{17} + 41887018 \beta_{16} + \cdots + 33042748 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 184832898 \beta_{18} + 159456397 \beta_{17} + 184832898 \beta_{16} - 187860192 \beta_{15} + \cdots + 267702635 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 617830033 \beta_{19} + 206032295 \beta_{17} + 311251239 \beta_{16} - 716403632 \beta_{15} + \cdots + 716403632 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2752425396 \beta_{19} + 2752425396 \beta_{18} + 83818277 \beta_{17} - 304339042 \beta_{16} + \cdots - 2298968873 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 4830873504 \beta_{19} + 13952446346 \beta_{18} - 398359311 \beta_{17} - 1431428886 \beta_{16} + \cdots - 27461484220 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
2.10485 0.618040i
−1.22309 + 0.359132i
−1.29470 1.49416i
1.49752 + 1.72823i
−1.02673 0.659842i
3.28340 + 2.11011i
0.216617 + 1.50661i
−0.318425 2.21469i
1.58077 + 3.46140i
−0.820216 1.79602i
1.58077 3.46140i
−0.820216 + 1.79602i
−1.02673 + 0.659842i
3.28340 2.11011i
−1.29470 + 1.49416i
1.49752 1.72823i
2.10485 + 0.618040i
−1.22309 0.359132i
0.216617 1.50661i
−0.318425 + 2.21469i
0.273100 + 0.0801894i −0.654861 + 0.755750i −1.61435 1.03748i −0.0390977 + 0.271930i −0.239446 + 0.153882i 0.415415 + 0.909632i −0.730471 0.843008i −0.142315 0.989821i −0.0324835 + 0.0711290i
64.2 0.273100 + 0.0801894i −0.654861 + 0.755750i −1.61435 1.03748i 0.454513 3.16121i −0.239446 + 0.153882i 0.415415 + 0.909632i −0.730471 0.843008i −0.142315 0.989821i 0.377623 0.826879i
85.1 −0.544078 + 0.627899i 0.841254 0.540641i 0.186393 + 1.29639i −1.36538 2.98976i −0.118239 + 0.822373i −0.959493 + 0.281733i −2.31329 1.48666i 0.415415 0.909632i 2.62014 + 0.769343i
85.2 −0.544078 + 0.627899i 0.841254 0.540641i 0.186393 + 1.29639i 0.405886 + 0.888766i −0.118239 + 0.822373i −0.959493 + 0.281733i −2.31329 1.48666i 0.415415 0.909632i −0.778889 0.228702i
127.1 −1.61435 + 1.03748i −0.142315 + 0.989821i 0.698939 1.53046i −2.78540 + 0.817866i −0.797176 1.74557i −0.654861 0.755750i −0.0867074 0.603063i −0.959493 0.281733i 3.64809 4.21012i
127.2 −1.61435 + 1.03748i −0.142315 + 0.989821i 0.698939 1.53046i 2.13054 0.625582i −0.797176 1.74557i −0.654861 0.755750i −0.0867074 0.603063i −0.959493 0.281733i −2.79041 + 3.22030i
169.1 0.186393 1.29639i 0.415415 + 0.909632i 0.273100 + 0.0801894i −0.810370 0.935217i 1.25667 0.368991i 0.841254 + 0.540641i 1.24302 2.72183i −0.654861 + 0.755750i −1.36345 + 0.876238i
169.2 0.186393 1.29639i 0.415415 + 0.909632i 0.273100 + 0.0801894i 1.65162 + 1.90608i 1.25667 0.368991i 0.841254 + 0.540641i 1.24302 2.72183i −0.654861 + 0.755750i 2.77887 1.78587i
190.1 0.698939 1.53046i −0.959493 0.281733i −0.544078 0.627899i −2.50227 + 1.60811i −1.10181 + 1.27155i −0.142315 + 0.989821i 1.88745 0.554206i 0.841254 + 0.540641i 0.712219 + 4.95359i
190.2 0.698939 1.53046i −0.959493 0.281733i −0.544078 0.627899i 2.35995 1.51665i −1.10181 + 1.27155i −0.142315 + 0.989821i 1.88745 0.554206i 0.841254 + 0.540641i −0.671712 4.67186i
211.1 0.698939 + 1.53046i −0.959493 + 0.281733i −0.544078 + 0.627899i −2.50227 1.60811i −1.10181 1.27155i −0.142315 0.989821i 1.88745 + 0.554206i 0.841254 0.540641i 0.712219 4.95359i
211.2 0.698939 + 1.53046i −0.959493 + 0.281733i −0.544078 + 0.627899i 2.35995 + 1.51665i −1.10181 1.27155i −0.142315 0.989821i 1.88745 + 0.554206i 0.841254 0.540641i −0.671712 + 4.67186i
232.1 −1.61435 1.03748i −0.142315 0.989821i 0.698939 + 1.53046i −2.78540 0.817866i −0.797176 + 1.74557i −0.654861 + 0.755750i −0.0867074 + 0.603063i −0.959493 + 0.281733i 3.64809 + 4.21012i
232.2 −1.61435 1.03748i −0.142315 0.989821i 0.698939 + 1.53046i 2.13054 + 0.625582i −0.797176 + 1.74557i −0.654861 + 0.755750i −0.0867074 + 0.603063i −0.959493 + 0.281733i −2.79041 3.22030i
358.1 −0.544078 0.627899i 0.841254 + 0.540641i 0.186393 1.29639i −1.36538 + 2.98976i −0.118239 0.822373i −0.959493 0.281733i −2.31329 + 1.48666i 0.415415 + 0.909632i 2.62014 0.769343i
358.2 −0.544078 0.627899i 0.841254 + 0.540641i 0.186393 1.29639i 0.405886 0.888766i −0.118239 0.822373i −0.959493 0.281733i −2.31329 + 1.48666i 0.415415 + 0.909632i −0.778889 + 0.228702i
400.1 0.273100 0.0801894i −0.654861 0.755750i −1.61435 + 1.03748i −0.0390977 0.271930i −0.239446 0.153882i 0.415415 0.909632i −0.730471 + 0.843008i −0.142315 + 0.989821i −0.0324835 0.0711290i
400.2 0.273100 0.0801894i −0.654861 0.755750i −1.61435 + 1.03748i 0.454513 + 3.16121i −0.239446 0.153882i 0.415415 0.909632i −0.730471 + 0.843008i −0.142315 + 0.989821i 0.377623 + 0.826879i
463.1 0.186393 + 1.29639i 0.415415 0.909632i 0.273100 0.0801894i −0.810370 + 0.935217i 1.25667 + 0.368991i 0.841254 0.540641i 1.24302 + 2.72183i −0.654861 0.755750i −1.36345 0.876238i
463.2 0.186393 + 1.29639i 0.415415 0.909632i 0.273100 0.0801894i 1.65162 1.90608i 1.25667 + 0.368991i 0.841254 0.540641i 1.24302 + 2.72183i −0.654861 0.755750i 2.77887 + 1.78587i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.2
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 483.2.q.c 20
23.c even 11 1 inner 483.2.q.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.q.c 20 1.a even 1 1 trivial
483.2.q.c 20 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 2T_{2}^{9} + 4T_{2}^{8} + 8T_{2}^{7} + 16T_{2}^{6} + 10T_{2}^{5} + 20T_{2}^{4} + 7T_{2}^{3} + 3T_{2}^{2} - 5T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + T^{19} + \cdots + 223729 \) Copy content Toggle raw display
$7$ \( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 2778449521 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 210511081 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 14014061161 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 5727311041 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13615855969 \) Copy content Toggle raw display
$31$ \( T^{20} - T^{19} + \cdots + 1745041 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 6032784241 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 7334352656401 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 744809433692809 \) Copy content Toggle raw display
$47$ \( (T^{10} + 33 T^{9} + \cdots - 22732832)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 7927879291201 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12638970647689 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 571516344169 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 4285227066241 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 52922322201529 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 21\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 211498849 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 34\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 662287188273481 \) Copy content Toggle raw display
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