↑

Properties

Label 506.2.e.h
Level $506$
Weight $2$
Character orbit 506.e
Analytic conductor $4.040$
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] = [506,2,Mod(47,506)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(506, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("506.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 506 = 2 \cdot 11 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 506.e (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: \(4.04043034228\)
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: \( x^{20} - 2 x^{19} + 13 x^{18} - 32 x^{17} + 171 x^{16} + 76 x^{15} + 1273 x^{14} + 1050 x^{13} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + (\beta_{8} + \beta_{4}) q^{3} + ( - \beta_{11} - \beta_{8} - \beta_{6} - 1) q^{4} + ( - \beta_{19} - \beta_{17} + \beta_{13} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{18} + \beta_{17} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + (\beta_{8} + \beta_{4}) q^{3} + ( - \beta_{11} - \beta_{8} - \beta_{6} - 1) q^{4} + ( - \beta_{19} - \beta_{17} + \beta_{13} + \cdots + 1) q^{5}+ \cdots + ( - 3 \beta_{19} + \beta_{18} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{2} - 3 q^{3} - 5 q^{4} - q^{5} - 3 q^{6} + 3 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{2} - 3 q^{3} - 5 q^{4} - q^{5} - 3 q^{6} + 3 q^{7} - 5 q^{8} - 8 q^{9} + 24 q^{10} + 8 q^{11} + 12 q^{12} + 4 q^{13} + 3 q^{14} - 3 q^{15} - 5 q^{16} + 7 q^{17} - 8 q^{18} - 9 q^{19} - q^{20} + 16 q^{21} - 7 q^{22} - 20 q^{23} - 3 q^{24} - 22 q^{25} - q^{26} - 21 q^{27} - 7 q^{28} - 20 q^{29} - 3 q^{30} + 10 q^{31} + 20 q^{32} - 23 q^{33} - 8 q^{34} - 39 q^{35} - 8 q^{36} + q^{37} + q^{38} + 8 q^{39} - 11 q^{40} + 31 q^{41} + 21 q^{42} + 8 q^{43} + 3 q^{44} + 144 q^{45} + 5 q^{46} - 36 q^{47} - 3 q^{48} - 36 q^{49} - 12 q^{50} - 19 q^{51} - q^{52} - 8 q^{53} + 24 q^{54} - 7 q^{55} + 8 q^{56} - 43 q^{57} - 20 q^{58} - 37 q^{59} - 3 q^{60} - 3 q^{61} - 10 q^{62} + 40 q^{63} - 5 q^{64} - 62 q^{65} - 33 q^{66} + 48 q^{67} + 7 q^{68} + 3 q^{69} + 31 q^{70} - 56 q^{71} - 8 q^{72} + 51 q^{73} + q^{74} + 86 q^{75} + 16 q^{76} + 19 q^{77} + 58 q^{78} - q^{79} - 11 q^{80} - 25 q^{81} - 34 q^{82} + 12 q^{83} - 29 q^{84} + 35 q^{85} - 42 q^{86} + 30 q^{87} + 3 q^{88} + 126 q^{89} - 46 q^{90} - 93 q^{91} + 5 q^{92} - 74 q^{93} - 6 q^{94} + 5 q^{95} - 3 q^{96} + 29 q^{97} + 24 q^{98} + 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 13 x^{18} - 32 x^{17} + 171 x^{16} + 76 x^{15} + 1273 x^{14} + 1050 x^{13} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20\!\cdots\!30 \nu^{19} + \cdots - 63\!\cdots\!72 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43\!\cdots\!45 \nu^{19} + \cdots - 25\!\cdots\!96 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 56\!\cdots\!45 \nu^{19} + \cdots - 57\!\cdots\!76 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 82\!\cdots\!40 \nu^{19} + \cdots + 60\!\cdots\!36 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!75 \nu^{19} + \cdots - 90\!\cdots\!72 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!15 \nu^{19} + \cdots - 19\!\cdots\!96 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!81 \nu^{19} + \cdots + 71\!\cdots\!48 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!78 \nu^{19} + \cdots + 18\!\cdots\!04 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!57 \nu^{19} + \cdots + 21\!\cdots\!56 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 35\!\cdots\!86 \nu^{19} + \cdots - 27\!\cdots\!76 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 55\!\cdots\!85 \nu^{19} + \cdots + 18\!\cdots\!40 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 55\!\cdots\!60 \nu^{19} + \cdots + 18\!\cdots\!84 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29\!\cdots\!69 \nu^{19} + \cdots + 89\!\cdots\!32 ) / 40\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 72\!\cdots\!77 \nu^{19} + \cdots - 68\!\cdots\!88 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 73\!\cdots\!49 \nu^{19} + \cdots + 59\!\cdots\!68 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 77\!\cdots\!04 \nu^{19} + \cdots + 71\!\cdots\!72 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 83\!\cdots\!61 \nu^{19} + \cdots + 70\!\cdots\!32 ) / 80\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 94\!\cdots\!49 \nu^{19} + \cdots + 78\!\cdots\!40 ) / 80\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} + \beta_{18} - \beta_{17} - \beta_{15} + \beta_{13} + \beta_{8} - \beta_{7} + 5\beta_{6} - \beta_{5} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + 2 \beta_{18} - \beta_{17} - 4 \beta_{15} - 5 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + \cdots - 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} + 12 \beta_{16} - 12 \beta_{15} + 6 \beta_{14} - 10 \beta_{13} - 17 \beta_{11} - 12 \beta_{9} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{19} - 36 \beta_{18} + 36 \beta_{14} + 9 \beta_{13} - 36 \beta_{12} - 61 \beta_{11} + \cdots - 142 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 101 \beta_{19} - 268 \beta_{18} - 13 \beta_{17} - 142 \beta_{16} + 158 \beta_{15} + 158 \beta_{14} + \cdots - 626 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 315 \beta_{19} - 1012 \beta_{18} + 315 \beta_{17} - 72 \beta_{16} + 1012 \beta_{15} + 454 \beta_{14} + \cdots - 1019 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 73 \beta_{19} - 2234 \beta_{18} + 1265 \beta_{17} - 352 \beta_{16} + 3884 \beta_{15} + \cdots + 2818 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 161 \beta_{17} - 5826 \beta_{16} + 8350 \beta_{15} - 6366 \beta_{14} + 4548 \beta_{13} + 1648 \beta_{12} + \cdots + 15907 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 735 \beta_{19} + 32458 \beta_{18} - 32458 \beta_{14} + 735 \beta_{13} + 28286 \beta_{12} + \cdots + 120778 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 60231 \beta_{19} + 213028 \beta_{18} - 5291 \beta_{17} + 73498 \beta_{16} - 123430 \beta_{15} + \cdots + 453076 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 225221 \beta_{19} + 810816 \beta_{18} - 225221 \beta_{17} + 130516 \beta_{16} - 810816 \beta_{15} + \cdots + 936497 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 158975 \beta_{19} + 1814038 \beta_{18} - 829603 \beta_{17} + 536376 \beta_{16} - 3082316 \beta_{15} + \cdots - 1694970 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 737735 \beta_{17} + 3362362 \beta_{16} - 6951658 \beta_{15} + 4777254 \beta_{14} - 2358946 \beta_{13} + \cdots - 13914885 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3205035 \beta_{19} - 26570142 \beta_{18} + 26570142 \beta_{14} - 3205035 \beta_{13} - 20806474 \beta_{12} + \cdots - 94218252 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 43093565 \beta_{19} - 169805312 \beta_{18} + 13623459 \beta_{17} - 43743930 \beta_{16} + \cdots - 358161922 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 161246059 \beta_{19} - 646232972 \beta_{18} + 161246059 \beta_{17} - 136961304 \beta_{16} + \cdots - 788624811 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 230234639 \beta_{19} - 1483498986 \beta_{18} + 605258833 \beta_{17} - 537946836 \beta_{16} + \cdots + 1101789830 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 927366419 \beta_{17} - 2119368978 \beta_{16} + 5664899406 \beta_{15} - 3699800958 \beta_{14} + \cdots + 11581037639 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/506\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(419\)
\(\chi(n)\) \(-1 - \beta_{6} - \beta_{8} - \beta_{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−1.02731 + 3.16174i
−0.658008 + 2.02514i
−0.0182022 + 0.0560208i
0.435510 − 1.34036i
0.649977 − 2.00042i
−1.75777 + 1.27709i
−0.576897 + 0.419140i
−0.305241 + 0.221771i
1.17544 − 0.854006i
3.08250 − 2.23957i
−1.75777 − 1.27709i
−0.576897 − 0.419140i
−0.305241 − 0.221771i
1.17544 + 0.854006i
3.08250 + 2.23957i
−1.02731 − 3.16174i
−0.658008 − 2.02514i
−0.0182022 − 0.0560208i
0.435510 + 1.34036i
0.649977 + 2.00042i
−0.809017 + 0.587785i −0.718294 − 2.21068i 0.309017 − 0.951057i −3.56886 − 2.59293i 1.88052 + 1.36628i 1.12196 − 3.45305i 0.309017 + 0.951057i −1.94412 + 1.41249i 4.41136
47.2 −0.809017 + 0.587785i −0.348991 − 1.07408i 0.309017 − 0.951057i 2.43018 + 1.76563i 0.913669 + 0.663820i 0.436107 − 1.34220i 0.309017 + 0.951057i 1.39519 − 1.01367i −3.00387
47.3 −0.809017 + 0.587785i 0.290815 + 0.895036i 0.309017 − 0.951057i 1.27011 + 0.922789i −0.761363 − 0.553163i −1.22394 + 3.76689i 0.309017 + 0.951057i 1.71054 − 1.24278i −1.56994
47.4 −0.809017 + 0.587785i 0.744527 + 2.29142i 0.309017 − 0.951057i −0.986183 − 0.716504i −1.94920 − 1.41617i 1.42927 − 4.39885i 0.309017 + 0.951057i −2.26923 + 1.64869i 1.21899
47.5 −0.809017 + 0.587785i 0.958994 + 2.95148i 0.309017 − 0.951057i −2.19033 − 1.59137i −2.51068 − 1.82412i −0.454388 + 1.39846i 0.309017 + 0.951057i −5.36452 + 3.89755i 2.70739
93.1 0.309017 + 0.951057i −2.56678 − 1.86488i −0.809017 + 0.587785i 1.17873 − 3.62777i 0.980424 − 3.01743i 0.575378 − 0.418036i −0.809017 − 0.587785i 2.18356 + 6.72030i 3.81446
93.2 0.309017 + 0.951057i −1.38591 − 1.00693i −0.809017 + 0.587785i 0.545406 − 1.67859i 0.529372 − 1.62924i −3.03287 + 2.20351i −0.809017 − 0.587785i −0.0201918 − 0.0621439i 1.76497
93.3 0.309017 + 0.951057i −1.11426 − 0.809556i −0.809017 + 0.587785i −0.948607 + 2.91951i 0.425609 − 1.30989i −0.985540 + 0.716037i −0.809017 − 0.587785i −0.340860 − 1.04906i −3.06976
93.4 0.309017 + 0.951057i 0.366421 + 0.266221i −0.809017 + 0.587785i 1.07422 − 3.30610i −0.139960 + 0.430754i 1.12780 − 0.819395i −0.809017 − 0.587785i −0.863660 − 2.65807i 3.47624
93.5 0.309017 + 0.951057i 2.27348 + 1.65178i −0.809017 + 0.587785i 0.695333 − 2.14002i −0.868393 + 2.67264i 2.50621 − 1.82087i −0.809017 − 0.587785i 1.51329 + 4.65743i 2.25015
185.1 0.309017 − 0.951057i −2.56678 + 1.86488i −0.809017 − 0.587785i 1.17873 + 3.62777i 0.980424 + 3.01743i 0.575378 + 0.418036i −0.809017 + 0.587785i 2.18356 − 6.72030i 3.81446
185.2 0.309017 − 0.951057i −1.38591 + 1.00693i −0.809017 − 0.587785i 0.545406 + 1.67859i 0.529372 + 1.62924i −3.03287 − 2.20351i −0.809017 + 0.587785i −0.0201918 + 0.0621439i 1.76497
185.3 0.309017 − 0.951057i −1.11426 + 0.809556i −0.809017 − 0.587785i −0.948607 − 2.91951i 0.425609 + 1.30989i −0.985540 − 0.716037i −0.809017 + 0.587785i −0.340860 + 1.04906i −3.06976
185.4 0.309017 − 0.951057i 0.366421 − 0.266221i −0.809017 − 0.587785i 1.07422 + 3.30610i −0.139960 − 0.430754i 1.12780 + 0.819395i −0.809017 + 0.587785i −0.863660 + 2.65807i 3.47624
185.5 0.309017 − 0.951057i 2.27348 − 1.65178i −0.809017 − 0.587785i 0.695333 + 2.14002i −0.868393 − 2.67264i 2.50621 + 1.82087i −0.809017 + 0.587785i 1.51329 − 4.65743i 2.25015
323.1 −0.809017 − 0.587785i −0.718294 + 2.21068i 0.309017 + 0.951057i −3.56886 + 2.59293i 1.88052 − 1.36628i 1.12196 + 3.45305i 0.309017 − 0.951057i −1.94412 − 1.41249i 4.41136
323.2 −0.809017 − 0.587785i −0.348991 + 1.07408i 0.309017 + 0.951057i 2.43018 − 1.76563i 0.913669 − 0.663820i 0.436107 + 1.34220i 0.309017 − 0.951057i 1.39519 + 1.01367i −3.00387
323.3 −0.809017 − 0.587785i 0.290815 − 0.895036i 0.309017 + 0.951057i 1.27011 − 0.922789i −0.761363 + 0.553163i −1.22394 − 3.76689i 0.309017 − 0.951057i 1.71054 + 1.24278i −1.56994
323.4 −0.809017 − 0.587785i 0.744527 − 2.29142i 0.309017 + 0.951057i −0.986183 + 0.716504i −1.94920 + 1.41617i 1.42927 + 4.39885i 0.309017 − 0.951057i −2.26923 − 1.64869i 1.21899
323.5 −0.809017 − 0.587785i 0.958994 − 2.95148i 0.309017 + 0.951057i −2.19033 + 1.59137i −2.51068 + 1.82412i −0.454388 − 1.39846i 0.309017 − 0.951057i −5.36452 − 3.89755i 2.70739
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 506.2.e.h ✓ 20
11.c even 5 1 inner 506.2.e.h ✓ 20
11.c even 5 1 5566.2.a.bu 10
11.d odd 10 1 5566.2.a.bt 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
506.2.e.h ✓ 20 1.a even 1 1 trivial
506.2.e.h ✓ 20 11.c even 5 1 inner
5566.2.a.bt 10 11.d odd 10 1
5566.2.a.bu 10 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(506, [\chi])\):

\( T_{3}^{20} + 3 T_{3}^{19} + 16 T_{3}^{18} + 52 T_{3}^{17} + 207 T_{3}^{16} + 414 T_{3}^{15} + \cdots + 30976 \) Copy content Toggle raw display
\( T_{5}^{20} + T_{5}^{19} + 24 T_{5}^{18} + 54 T_{5}^{17} + 455 T_{5}^{16} + 72 T_{5}^{15} + \cdots + 123187801 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + 3 T^{19} + \cdots + 30976 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 123187801 \) Copy content Toggle raw display
$7$ \( T^{20} - 3 T^{19} + \cdots + 3748096 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( T^{20} - 4 T^{19} + \cdots + 3748096 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2342560000 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 17455958641 \) Copy content Toggle raw display
$23$ \( (T + 1)^{20} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10084841840896 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 191900068096 \) Copy content Toggle raw display
$37$ \( T^{20} - T^{19} + \cdots + 65286400 \) Copy content Toggle raw display
$41$ \( T^{20} - 31 T^{19} + \cdots + 17547721 \) Copy content Toggle raw display
$43$ \( (T^{10} - 4 T^{9} + \cdots - 402684496)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 164545679831296 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 93642567794281 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 4196966656 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 12999046165561 \) Copy content Toggle raw display
$67$ \( (T^{10} - 24 T^{9} + \cdots + 84428096)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 5617652401 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{10} - 63 T^{9} + \cdots + 2135104144)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
show more
show less