↑

Properties

Label 525.2.n.b
Level $525$
Weight $2$
Character orbit 525.n
Analytic conductor $4.192$
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] = [525,2,Mod(106,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.n (of order \(5\), degree \(4\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} + 31 x^{18} - 74 x^{17} + 109 x^{16} - 72 x^{15} - 51 x^{14} + 9 x^{13} + 866 x^{12} + \cdots + 3125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + \beta_{4} q^{3} + (\beta_{19} - \beta_{17} + \cdots + \beta_1) q^{4}+ \cdots + \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + \beta_{4} q^{3} + (\beta_{19} - \beta_{17} + \cdots + \beta_1) q^{4}+ \cdots + (\beta_{17} - \beta_{14} + \cdots + \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 5 q^{3} + 5 q^{5} - 3 q^{6} - 20 q^{7} + 4 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} + 5 q^{3} + 5 q^{5} - 3 q^{6} - 20 q^{7} + 4 q^{8} - 5 q^{9} - 15 q^{10} + 12 q^{11} + 5 q^{12} - 17 q^{13} + 2 q^{14} + 15 q^{15} - 28 q^{16} - 9 q^{17} - 2 q^{18} - 9 q^{19} - 20 q^{20} - 5 q^{21} - 21 q^{22} + 7 q^{23} + 6 q^{24} - 15 q^{25} - 20 q^{26} + 5 q^{27} + 28 q^{29} + 6 q^{31} - 4 q^{32} + 3 q^{33} - 5 q^{35} - 5 q^{36} - 5 q^{37} - 6 q^{38} + 17 q^{39} - 10 q^{40} + 11 q^{41} + 3 q^{42} + 28 q^{43} - 17 q^{44} + 5 q^{45} - 43 q^{46} - 24 q^{47} + 28 q^{48} + 20 q^{49} + 10 q^{50} - 36 q^{51} - 9 q^{52} - 26 q^{53} - 3 q^{54} - 25 q^{55} - 4 q^{56} + 24 q^{57} - 16 q^{58} + 64 q^{59} + 5 q^{60} + 8 q^{61} + 27 q^{62} + 5 q^{63} + 26 q^{64} + 25 q^{65} - 4 q^{66} - 3 q^{67} + 80 q^{68} - 2 q^{69} + 15 q^{70} + 19 q^{71} + 4 q^{72} + 31 q^{73} + 8 q^{74} - 5 q^{75} - 72 q^{76} - 12 q^{77} - 30 q^{78} + 43 q^{79} - 25 q^{80} - 5 q^{81} - 6 q^{82} + 32 q^{83} - 5 q^{84} + 35 q^{85} + 53 q^{86} + 17 q^{87} - 61 q^{88} - 47 q^{89} + 10 q^{90} + 17 q^{91} + 41 q^{92} + 4 q^{93} + 12 q^{94} - 40 q^{95} - 6 q^{96} - 45 q^{97} - 2 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} + 31 x^{18} - 74 x^{17} + 109 x^{16} - 72 x^{15} - 51 x^{14} + 9 x^{13} + 866 x^{12} + \cdots + 3125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 74\!\cdots\!34 \nu^{19} + \cdots - 60\!\cdots\!50 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 89\!\cdots\!95 \nu^{19} + \cdots - 12\!\cdots\!25 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\!\cdots\!31 \nu^{19} + \cdots - 14\!\cdots\!75 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38\!\cdots\!56 \nu^{19} + \cdots + 18\!\cdots\!50 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38\!\cdots\!79 \nu^{19} + \cdots - 32\!\cdots\!25 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!18 \nu^{19} + \cdots + 18\!\cdots\!75 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49\!\cdots\!46 \nu^{19} + \cdots + 23\!\cdots\!00 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 61\!\cdots\!84 \nu^{19} + \cdots + 29\!\cdots\!75 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 67\!\cdots\!65 \nu^{19} + \cdots - 35\!\cdots\!00 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 68\!\cdots\!09 \nu^{19} + \cdots - 20\!\cdots\!50 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 77\!\cdots\!23 \nu^{19} + \cdots + 14\!\cdots\!50 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 80\!\cdots\!31 \nu^{19} + \cdots + 10\!\cdots\!75 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!98 \nu^{19} + \cdots + 20\!\cdots\!25 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 65\!\cdots\!78 \nu^{19} + \cdots + 69\!\cdots\!75 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!59 \nu^{19} + \cdots + 24\!\cdots\!50 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 16\!\cdots\!14 \nu^{19} + \cdots + 19\!\cdots\!75 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 318431387248873 \nu^{19} + \cdots - 30\!\cdots\!75 ) / 982947888438125 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 18\!\cdots\!71 \nu^{19} + \cdots - 16\!\cdots\!75 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 19\!\cdots\!95 \nu^{19} + \cdots - 37\!\cdots\!50 ) / 57\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 3 \beta_{19} - 3 \beta_{18} + \beta_{15} - 4 \beta_{13} - 5 \beta_{12} - 4 \beta_{10} + 8 \beta_{9} + \cdots + 2 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{18} - 3 \beta_{17} - \beta_{16} + 2 \beta_{14} - 5 \beta_{13} - 4 \beta_{12} - \beta_{11} + \cdots - 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 11 \beta_{19} + 7 \beta_{18} + 4 \beta_{17} + 8 \beta_{16} + 2 \beta_{15} + 4 \beta_{14} - 13 \beta_{13} + \cdots - 9 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 22 \beta_{19} + 12 \beta_{18} + 12 \beta_{17} + 9 \beta_{16} + 9 \beta_{15} + 2 \beta_{14} + \cdots + 14 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{19} - 27 \beta_{18} + 12 \beta_{17} + 4 \beta_{16} - 13 \beta_{15} - 23 \beta_{14} + \cdots + 45 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 36 \beta_{19} + 10 \beta_{18} - 32 \beta_{17} - 84 \beta_{16} + 22 \beta_{15} - 27 \beta_{14} + \cdots + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{19} + 61 \beta_{18} - 115 \beta_{17} - 145 \beta_{16} + 8 \beta_{15} + 18 \beta_{13} + 60 \beta_{12} + \cdots + 61 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 240 \beta_{19} - 48 \beta_{18} - 29 \beta_{17} + 92 \beta_{16} - 65 \beta_{15} + 86 \beta_{14} + \cdots - 32 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 147 \beta_{19} - 204 \beta_{18} + 17 \beta_{17} + 279 \beta_{16} - 129 \beta_{15} + 277 \beta_{14} + \cdots + 303 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 176 \beta_{19} - 154 \beta_{18} + 396 \beta_{17} + 567 \beta_{16} - 438 \beta_{15} + 431 \beta_{14} + \cdots + 812 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1708 \beta_{19} + 414 \beta_{18} + 211 \beta_{17} - 838 \beta_{16} + 246 \beta_{15} + 556 \beta_{14} + \cdots + 1095 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1258 \beta_{19} - 2635 \beta_{18} - 721 \beta_{17} - 1222 \beta_{16} - 1959 \beta_{15} - 881 \beta_{14} + \cdots - 1461 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 3827 \beta_{19} + 523 \beta_{18} - 1675 \beta_{17} - 2275 \beta_{16} - \beta_{15} - 615 \beta_{14} + \cdots - 11147 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 5015 \beta_{19} + 10676 \beta_{18} - 3132 \beta_{17} - 3404 \beta_{16} + 1380 \beta_{15} + 1918 \beta_{14} + \cdots + 5379 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 10119 \beta_{19} + 1103 \beta_{18} + 7151 \beta_{17} + 5087 \beta_{16} - 6917 \beta_{15} + \cdots + 34064 ) / 5 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 23357 \beta_{19} - 39752 \beta_{18} - 11277 \beta_{17} - 25364 \beta_{16} - 18454 \beta_{15} + \cdots + 50186 ) / 5 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 46866 \beta_{19} - 67818 \beta_{18} - 34272 \beta_{17} - 30209 \beta_{16} - 70412 \beta_{15} + \cdots - 49695 ) / 5 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 73009 \beta_{19} + 92310 \beta_{18} - 26653 \beta_{17} - 23706 \beta_{16} + 3818 \beta_{15} + \cdots - 187618 ) / 5 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 11459 \beta_{19} - 57676 \beta_{18} + 117505 \beta_{17} + 266970 \beta_{16} - 171368 \beta_{15} + \cdots - 2746 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1
1.14677 + 0.258633i
−1.14371 − 0.271822i
0.891061 − 0.766796i
0.209942 − 1.15667i
0.895943 + 0.761087i
−1.67866 + 0.894495i
1.88955 − 0.218280i
0.670843 + 1.77989i
−0.238506 − 1.88710i
1.35678 + 1.33311i
−1.67866 − 0.894495i
1.88955 + 0.218280i
0.670843 − 1.77989i
−0.238506 + 1.88710i
1.35678 − 1.33311i
1.14677 − 0.258633i
−1.14371 + 0.271822i
0.891061 + 0.766796i
0.209942 + 1.15667i
0.895943 − 0.761087i
−0.724838 − 2.23082i 0.809017 − 0.587785i −2.83314 + 2.05840i 0.884129 + 2.05385i −1.89765 − 1.37872i −1.00000 2.85019 + 2.07078i 0.309017 − 0.951057i 3.94093 − 3.46105i
106.2 −0.401368 − 1.23528i 0.809017 − 0.587785i 0.253205 − 0.183964i −0.860419 − 2.06390i −1.05079 − 0.763447i −1.00000 −2.43047 − 1.76584i 0.309017 − 0.951057i −2.20416 + 1.89124i
106.3 −0.179936 − 0.553787i 0.809017 − 0.587785i 1.34373 − 0.976277i 2.17621 + 0.513897i −0.471080 − 0.342259i −1.00000 −1.72460 − 1.25299i 0.309017 − 0.951057i −0.106991 − 1.29763i
106.4 0.256297 + 0.788802i 0.809017 − 0.587785i 1.06151 − 0.771234i 2.01466 − 0.970131i 0.670995 + 0.487507i −1.00000 2.22241 + 1.61467i 0.309017 − 0.951057i 1.28159 + 1.34052i
106.5 0.549845 + 1.69225i 0.809017 − 0.587785i −0.943341 + 0.685378i −0.169498 + 2.22963i 1.43951 + 1.04587i −1.00000 1.20050 + 0.872217i 0.309017 − 0.951057i −3.86629 + 0.939121i
211.1 −2.02723 + 1.47287i −0.309017 + 0.951057i 1.32228 − 4.06957i 2.20175 − 0.390266i −0.774332 − 2.38315i −1.00000 1.76470 + 5.43119i −0.809017 − 0.587785i −3.88864 + 4.03404i
211.2 −0.870436 + 0.632409i −0.309017 + 0.951057i −0.260316 + 0.801170i −2.19187 − 0.442374i −0.332477 − 1.02326i −1.00000 −0.945033 − 2.90851i −0.809017 − 0.587785i 2.18765 − 1.00110i
211.3 0.140253 − 0.101900i −0.309017 + 0.951057i −0.608747 + 1.87353i −0.103443 − 2.23367i 0.0535718 + 0.164877i −1.00000 0.212677 + 0.654553i −0.809017 − 0.587785i −0.242119 − 0.302738i
211.4 0.269002 − 0.195442i −0.309017 + 0.951057i −0.583869 + 1.79696i −0.418871 + 2.19649i 0.102750 + 0.316231i −1.00000 0.399639 + 1.22996i −0.809017 − 0.587785i 0.316607 + 0.672725i
211.5 1.98841 − 1.44466i −0.309017 + 0.951057i 1.24868 − 3.84305i −1.03265 − 1.98334i 0.759505 + 2.33752i −1.00000 −1.55002 − 4.77046i −0.809017 − 0.587785i −4.91858 − 2.45187i
316.1 −2.02723 − 1.47287i −0.309017 − 0.951057i 1.32228 + 4.06957i 2.20175 + 0.390266i −0.774332 + 2.38315i −1.00000 1.76470 − 5.43119i −0.809017 + 0.587785i −3.88864 − 4.03404i
316.2 −0.870436 − 0.632409i −0.309017 − 0.951057i −0.260316 − 0.801170i −2.19187 + 0.442374i −0.332477 + 1.02326i −1.00000 −0.945033 + 2.90851i −0.809017 + 0.587785i 2.18765 + 1.00110i
316.3 0.140253 + 0.101900i −0.309017 − 0.951057i −0.608747 − 1.87353i −0.103443 + 2.23367i 0.0535718 − 0.164877i −1.00000 0.212677 − 0.654553i −0.809017 + 0.587785i −0.242119 + 0.302738i
316.4 0.269002 + 0.195442i −0.309017 − 0.951057i −0.583869 − 1.79696i −0.418871 − 2.19649i 0.102750 − 0.316231i −1.00000 0.399639 − 1.22996i −0.809017 + 0.587785i 0.316607 − 0.672725i
316.5 1.98841 + 1.44466i −0.309017 − 0.951057i 1.24868 + 3.84305i −1.03265 + 1.98334i 0.759505 − 2.33752i −1.00000 −1.55002 + 4.77046i −0.809017 + 0.587785i −4.91858 + 2.45187i
421.1 −0.724838 + 2.23082i 0.809017 + 0.587785i −2.83314 − 2.05840i 0.884129 − 2.05385i −1.89765 + 1.37872i −1.00000 2.85019 − 2.07078i 0.309017 + 0.951057i 3.94093 + 3.46105i
421.2 −0.401368 + 1.23528i 0.809017 + 0.587785i 0.253205 + 0.183964i −0.860419 + 2.06390i −1.05079 + 0.763447i −1.00000 −2.43047 + 1.76584i 0.309017 + 0.951057i −2.20416 − 1.89124i
421.3 −0.179936 + 0.553787i 0.809017 + 0.587785i 1.34373 + 0.976277i 2.17621 − 0.513897i −0.471080 + 0.342259i −1.00000 −1.72460 + 1.25299i 0.309017 + 0.951057i −0.106991 + 1.29763i
421.4 0.256297 − 0.788802i 0.809017 + 0.587785i 1.06151 + 0.771234i 2.01466 + 0.970131i 0.670995 − 0.487507i −1.00000 2.22241 − 1.61467i 0.309017 + 0.951057i 1.28159 − 1.34052i
421.5 0.549845 − 1.69225i 0.809017 + 0.587785i −0.943341 − 0.685378i −0.169498 − 2.22963i 1.43951 − 1.04587i −1.00000 1.20050 − 0.872217i 0.309017 + 0.951057i −3.86629 − 0.939121i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.n.b ✓ 20
25.d even 5 1 inner 525.2.n.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.n.b ✓ 20 1.a even 1 1 trivial
525.2.n.b ✓ 20 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 2 T_{2}^{19} + 7 T_{2}^{18} + 6 T_{2}^{17} + 37 T_{2}^{16} + 62 T_{2}^{15} + 317 T_{2}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} - 5 T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T + 1)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} - 12 T^{19} + \cdots + 4644025 \) Copy content Toggle raw display
$13$ \( T^{20} + 17 T^{19} + \cdots + 1661521 \) Copy content Toggle raw display
$17$ \( T^{20} + 9 T^{19} + \cdots + 7789681 \) Copy content Toggle raw display
$19$ \( T^{20} + 9 T^{19} + \cdots + 279841 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 217002361 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 190830059281 \) Copy content Toggle raw display
$31$ \( T^{20} - 6 T^{19} + \cdots + 1062961 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 125238481 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 135771434410000 \) Copy content Toggle raw display
$43$ \( (T^{10} - 14 T^{9} + \cdots - 247249)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 436113631321 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 924193744308025 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 188712065037001 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 27543053521 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 77400950523961 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 43\!\cdots\!41 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 75\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
show more
show less