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Properties

Label 322.2.i.c
Level $322$
Weight $2$
Character orbit 322.i
Analytic conductor $2.571$
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] = [322,2,Mod(29,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.i (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: \(2.57118294509\)
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: \( x^{20} - 9 x^{19} + 41 x^{18} - 119 x^{17} + 245 x^{16} - 404 x^{15} + 623 x^{14} - 898 x^{13} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + \beta_{19} q^{2} + ( - \beta_{16} - \beta_{10} + \cdots + \beta_1) q^{3}+ \cdots + (\beta_{17} - \beta_{13} + \cdots - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{19} q^{2} + ( - \beta_{16} - \beta_{10} + \cdots + \beta_1) q^{3}+ \cdots + (\beta_{19} + 2 \beta_{18} + \beta_{17} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 5 q^{5} + 4 q^{6} - 2 q^{7} - 2 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 5 q^{5} + 4 q^{6} - 2 q^{7} - 2 q^{8} - 12 q^{9} + 6 q^{10} + 6 q^{11} + 4 q^{12} + 14 q^{13} - 2 q^{14} + 13 q^{15} - 2 q^{16} + 7 q^{17} - 12 q^{18} - 2 q^{19} + 6 q^{20} + 4 q^{21} - 16 q^{22} + 9 q^{23} - 18 q^{24} + 23 q^{25} + 3 q^{26} - 47 q^{27} - 2 q^{28} + 2 q^{29} - 20 q^{30} + 28 q^{31} - 2 q^{32} + 26 q^{33} + 18 q^{34} - 5 q^{35} + 10 q^{36} - 17 q^{37} + 9 q^{38} + 36 q^{39} - 5 q^{40} - 7 q^{41} + 4 q^{42} - 17 q^{43} + 6 q^{44} + 90 q^{45} + 9 q^{46} - 68 q^{47} + 4 q^{48} - 2 q^{49} - 43 q^{50} - 15 q^{51} + 14 q^{52} - 35 q^{53} + 19 q^{54} + 8 q^{55} - 2 q^{56} - 20 q^{57} + 24 q^{58} - 60 q^{59} + 13 q^{60} - 48 q^{61} - 5 q^{62} + 10 q^{63} - 2 q^{64} + 40 q^{65} - 29 q^{66} - 57 q^{67} + 18 q^{68} + 4 q^{69} + 6 q^{70} - 39 q^{71} - q^{72} - 30 q^{73} - 17 q^{74} + 57 q^{75} - 2 q^{76} + 6 q^{77} - 19 q^{78} - 18 q^{79} - 5 q^{80} - 90 q^{81} + 26 q^{82} + 28 q^{83} + 4 q^{84} - 13 q^{85} - 6 q^{86} + 13 q^{87} + 6 q^{88} + 34 q^{89} - 20 q^{90} - 30 q^{91} - 2 q^{92} - 16 q^{93} + 9 q^{94} + 110 q^{95} + 4 q^{96} - 2 q^{98} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 9 x^{19} + 41 x^{18} - 119 x^{17} + 245 x^{16} - 404 x^{15} + 623 x^{14} - 898 x^{13} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20\!\cdots\!71 \nu^{19} + \cdots + 25\!\cdots\!26 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 34\!\cdots\!09 \nu^{19} + \cdots + 11\!\cdots\!14 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 54\!\cdots\!91 \nu^{19} + \cdots - 25\!\cdots\!44 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 91\!\cdots\!12 \nu^{19} + \cdots + 97\!\cdots\!18 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!53 \nu^{19} + \cdots + 31\!\cdots\!48 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!80 \nu^{19} + \cdots - 17\!\cdots\!59 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!46 \nu^{19} + \cdots + 23\!\cdots\!14 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!05 \nu^{19} + \cdots + 35\!\cdots\!58 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!01 \nu^{19} + \cdots - 33\!\cdots\!96 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 97\!\cdots\!18 \nu^{19} + \cdots - 56\!\cdots\!41 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!14 \nu^{19} + \cdots - 17\!\cdots\!36 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23\!\cdots\!14 \nu^{19} + \cdots + 41\!\cdots\!58 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 25\!\cdots\!26 \nu^{19} + \cdots - 76\!\cdots\!60 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 30\!\cdots\!29 \nu^{19} + \cdots + 25\!\cdots\!17 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 31\!\cdots\!48 \nu^{19} + \cdots + 77\!\cdots\!85 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 33\!\cdots\!96 \nu^{19} + \cdots - 44\!\cdots\!53 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 35\!\cdots\!58 \nu^{19} + \cdots - 72\!\cdots\!48 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 35\!\cdots\!62 \nu^{19} + \cdots + 60\!\cdots\!93 ) / 38\!\cdots\!89 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} + 2\beta_{17} + \beta_{15} - \beta_{10} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{18} + \beta_{17} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - 3 \beta_{10} - \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{19} - 5 \beta_{18} - 4 \beta_{17} - 5 \beta_{15} + 4 \beta_{14} + 5 \beta_{13} - 4 \beta_{12} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{19} - 8 \beta_{18} - 8 \beta_{17} - 8 \beta_{15} + 7 \beta_{14} + 7 \beta_{12} + \cdots + 20 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8 \beta_{19} - 3 \beta_{15} + 8 \beta_{14} - 26 \beta_{13} + 49 \beta_{12} - 3 \beta_{11} + 7 \beta_{8} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 95 \beta_{19} + 49 \beta_{18} + 50 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 50 \beta_{13} + 95 \beta_{12} + \cdots + 49 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 229 \beta_{19} + 86 \beta_{18} + 86 \beta_{17} - 90 \beta_{16} - 90 \beta_{15} - 56 \beta_{14} + \cdots - 57 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 289 \beta_{18} - 299 \beta_{17} - 590 \beta_{16} - 622 \beta_{15} - 289 \beta_{14} + 577 \beta_{13} + \cdots - 622 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1320 \beta_{19} - 1698 \beta_{18} - 1722 \beta_{17} - 1992 \beta_{16} - 1722 \beta_{15} - 907 \beta_{14} + \cdots - 1992 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3463 \beta_{19} - 3463 \beta_{18} - 3926 \beta_{17} - 3926 \beta_{16} - 1941 \beta_{15} - 1941 \beta_{14} + \cdots - 3834 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2513 \beta_{19} - 4826 \beta_{17} - 2830 \beta_{16} + 1911 \beta_{15} - 2830 \beta_{14} - 7709 \beta_{13} + \cdots - 4826 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8662 \beta_{19} + 20731 \beta_{18} - 4222 \beta_{17} + 8101 \beta_{16} + 8662 \beta_{15} - 4084 \beta_{14} + \cdots - 4084 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 24658 \beta_{19} + 61892 \beta_{18} - 17370 \beta_{17} + 24658 \beta_{16} - 17370 \beta_{14} + \cdots - 3283 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 4267 \beta_{19} + 78204 \beta_{18} - 79211 \beta_{17} - 45997 \beta_{15} - 84351 \beta_{14} + \cdots - 4267 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 166320 \beta_{19} - 27491 \beta_{18} - 166320 \beta_{17} - 136360 \beta_{16} + \cdots - 249762 \beta_{2} \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 462360 \beta_{19} - 205192 \beta_{18} - 205192 \beta_{16} + 540448 \beta_{15} - 462360 \beta_{14} + \cdots + 33354 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 655610 \beta_{19} + 253547 \beta_{18} + 998887 \beta_{17} + 966405 \beta_{16} + 2549581 \beta_{15} + \cdots + 253547 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 497532 \beta_{19} + 3125841 \beta_{18} + 3125841 \beta_{17} + 6415758 \beta_{16} + 6415758 \beta_{15} + \cdots + 1927538 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(1\) \(\beta_{19}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.140215 + 0.975217i
−0.0997080 − 0.693485i
1.61117 + 1.85938i
−0.753480 − 0.869562i
−0.637182 − 0.187094i
2.47844 + 0.727734i
1.61117 − 1.85938i
−0.753480 + 0.869562i
1.45482 + 0.934959i
−0.0394097 − 0.0253271i
1.45482 − 0.934959i
−0.0394097 + 0.0253271i
−0.637182 + 0.187094i
2.47844 − 0.727734i
0.140215 − 0.975217i
−0.0997080 + 0.693485i
−0.643071 + 1.40813i
0.988210 − 2.16388i
−0.643071 − 1.40813i
0.988210 + 2.16388i
0.415415 − 0.909632i 0.0141569 + 0.00415683i −0.654861 − 0.755750i −3.50493 + 2.25248i 0.00966216 − 0.0111507i −0.142315 + 0.989821i −0.959493 + 0.281733i −2.52358 − 1.62180i 0.592928 + 4.12391i
29.2 0.415415 − 0.909632i 1.63173 + 0.479119i −0.654861 − 0.755750i 2.49238 − 1.60176i 1.11367 − 1.28524i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.0907762 − 0.0583384i −0.421636 − 2.93254i
71.1 0.841254 + 0.540641i −0.207825 − 1.44545i 0.415415 + 0.909632i 1.07127 + 0.314552i 0.606638 − 1.32835i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.832335 − 0.244396i 0.731146 + 0.843788i
71.2 0.841254 + 0.540641i 0.306062 + 2.12871i 0.415415 + 0.909632i −0.500990 − 0.147104i −0.893390 + 1.95625i −0.654861 + 0.755750i −0.142315 + 0.989821i −1.55924 + 0.457833i −0.341929 − 0.394607i
85.1 −0.654861 + 0.755750i −1.39992 + 0.899671i −0.142315 − 0.989821i −0.109762 − 0.240346i 0.236824 − 1.64714i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.0958901 + 0.209970i 0.253520 + 0.0744403i
85.2 −0.654861 + 0.755750i 1.33176 − 0.855871i −0.142315 − 0.989821i 0.426940 + 0.934869i −0.225294 + 1.56696i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.205171 + 0.449262i −0.986114 − 0.289549i
127.1 0.841254 − 0.540641i −0.207825 + 1.44545i 0.415415 − 0.909632i 1.07127 − 0.314552i 0.606638 + 1.32835i −0.654861 − 0.755750i −0.142315 − 0.989821i 0.832335 + 0.244396i 0.731146 − 0.843788i
127.2 0.841254 − 0.540641i 0.306062 − 2.12871i 0.415415 − 0.909632i −0.500990 + 0.147104i −0.893390 − 1.95625i −0.654861 − 0.755750i −0.142315 − 0.989821i −1.55924 − 0.457833i −0.341929 + 0.394607i
141.1 −0.142315 − 0.989821i −1.13381 + 2.48271i −0.959493 + 0.281733i −2.36481 + 2.72913i 2.61880 + 0.768948i 0.841254 − 0.540641i 0.415415 + 0.909632i −2.91372 − 3.36261i 3.03790 + 1.95234i
141.2 −0.142315 − 0.989821i −0.395954 + 0.867019i −0.959493 + 0.281733i 0.0640602 − 0.0739294i 0.914544 + 0.268534i 0.841254 − 0.540641i 0.415415 + 0.909632i 1.36964 + 1.58065i −0.0822937 − 0.0528869i
169.1 −0.142315 + 0.989821i −1.13381 − 2.48271i −0.959493 − 0.281733i −2.36481 − 2.72913i 2.61880 − 0.768948i 0.841254 + 0.540641i 0.415415 − 0.909632i −2.91372 + 3.36261i 3.03790 − 1.95234i
169.2 −0.142315 + 0.989821i −0.395954 − 0.867019i −0.959493 − 0.281733i 0.0640602 + 0.0739294i 0.914544 − 0.268534i 0.841254 + 0.540641i 0.415415 − 0.909632i 1.36964 − 1.58065i −0.0822937 + 0.0528869i
197.1 −0.654861 − 0.755750i −1.39992 − 0.899671i −0.142315 + 0.989821i −0.109762 + 0.240346i 0.236824 + 1.64714i −0.959493 − 0.281733i 0.841254 − 0.540641i −0.0958901 − 0.209970i 0.253520 − 0.0744403i
197.2 −0.654861 − 0.755750i 1.33176 + 0.855871i −0.142315 + 0.989821i 0.426940 − 0.934869i −0.225294 − 1.56696i −0.959493 − 0.281733i 0.841254 − 0.540641i −0.205171 − 0.449262i −0.986114 + 0.289549i
211.1 0.415415 + 0.909632i 0.0141569 − 0.00415683i −0.654861 + 0.755750i −3.50493 − 2.25248i 0.00966216 + 0.0111507i −0.142315 − 0.989821i −0.959493 − 0.281733i −2.52358 + 1.62180i 0.592928 − 4.12391i
211.2 0.415415 + 0.909632i 1.63173 − 0.479119i −0.654861 + 0.755750i 2.49238 + 1.60176i 1.11367 + 1.28524i −0.142315 − 0.989821i −0.959493 − 0.281733i −0.0907762 + 0.0583384i −0.421636 + 2.93254i
225.1 −0.959493 − 0.281733i −0.358877 + 0.414166i 0.841254 + 0.540641i 0.138179 − 0.961058i 0.461024 − 0.296282i 0.415415 + 0.909632i −0.654861 − 0.755750i 0.384204 + 2.67220i −0.403343 + 0.883198i
225.2 −0.959493 − 0.281733i 2.21268 − 2.55356i 0.841254 + 0.540641i −0.212341 + 1.47686i −2.84247 + 1.82674i 0.415415 + 0.909632i −0.654861 − 0.755750i −1.19781 − 8.33096i 0.619820 − 1.35722i
239.1 −0.959493 + 0.281733i −0.358877 − 0.414166i 0.841254 − 0.540641i 0.138179 + 0.961058i 0.461024 + 0.296282i 0.415415 − 0.909632i −0.654861 + 0.755750i 0.384204 − 2.67220i −0.403343 − 0.883198i
239.2 −0.959493 + 0.281733i 2.21268 + 2.55356i 0.841254 − 0.540641i −0.212341 − 1.47686i −2.84247 − 1.82674i 0.415415 − 0.909632i −0.654861 + 0.755750i −1.19781 + 8.33096i 0.619820 + 1.35722i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.2
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 322.2.i.c ✓ 20
23.c even 11 1 inner 322.2.i.c ✓ 20
23.c even 11 1 7406.2.a.bp 10
23.d odd 22 1 7406.2.a.bo 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.c ✓ 20 1.a even 1 1 trivial
322.2.i.c ✓ 20 23.c even 11 1 inner
7406.2.a.bo 10 23.d odd 22 1
7406.2.a.bp 10 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 4 T_{3}^{19} + 17 T_{3}^{18} - 39 T_{3}^{17} + 151 T_{3}^{16} - 258 T_{3}^{15} + 479 T_{3}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). 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
$3$ \( T^{20} - 4 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} + 5 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} - 6 T^{19} + \cdots + 192721 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 1804635361 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 1617407089 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 4829833009 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 31841190481 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 1216734745249 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 66617158609 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 214543778119969 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 42\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( (T^{10} + 34 T^{9} + \cdots - 27697)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 1359470641 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 45\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 12\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 266368806414481 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 7137751125649 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 50793658619089 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 726855781029169 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 15\!\cdots\!69 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 19\!\cdots\!49 \) Copy content Toggle raw display
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