↑

Properties

Label 665.2.i.h
Level $665$
Weight $2$
Character orbit 665.i
Analytic conductor $5.310$
Analytic rank $0$
Dimension $20$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [665,2,Mod(106,665)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(665, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("665.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 665 = 5 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 665.i (of order \(3\), degree \(2\), 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: \(5.31005173442\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} + 20 x^{18} - 43 x^{17} + 207 x^{16} - 401 x^{15} + 1351 x^{14} - 2135 x^{13} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{2} q^{3} + ( - \beta_{12} - \beta_{8} + \beta_{6}) q^{4} + ( - \beta_{8} + 1) q^{5} + ( - \beta_{16} - \beta_{15} + \cdots - \beta_1) q^{6}+ \cdots + ( - \beta_{18} + \beta_{15} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{2} q^{3} + ( - \beta_{12} - \beta_{8} + \beta_{6}) q^{4} + ( - \beta_{8} + 1) q^{5} + ( - \beta_{16} - \beta_{15} + \cdots - \beta_1) q^{6}+ \cdots + ( - 3 \beta_{19} + 3 \beta_{18} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 3 q^{2} - q^{3} - 11 q^{4} + 10 q^{5} - 6 q^{6} - 20 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 3 q^{2} - q^{3} - 11 q^{4} + 10 q^{5} - 6 q^{6} - 20 q^{7} - 9 q^{9} - 3 q^{10} + 2 q^{11} - 4 q^{12} + 6 q^{13} - 3 q^{14} + q^{15} - 5 q^{16} + 8 q^{17} - 76 q^{18} + 17 q^{19} - 22 q^{20} + q^{21} - 4 q^{22} + 3 q^{23} - 6 q^{24} - 10 q^{25} + 18 q^{26} + 20 q^{27} + 11 q^{28} + q^{29} - 12 q^{30} + 10 q^{31} + 18 q^{32} + q^{33} - 26 q^{34} - 10 q^{35} - 19 q^{36} - 66 q^{37} + 19 q^{38} + 34 q^{39} - 29 q^{41} + 6 q^{42} + q^{43} - 28 q^{44} - 18 q^{45} + 6 q^{46} - 7 q^{47} + 15 q^{48} + 20 q^{49} - 6 q^{50} + 16 q^{51} - 6 q^{52} + 40 q^{53} + 35 q^{54} + q^{55} + 24 q^{57} - 18 q^{58} - 9 q^{59} - 2 q^{60} + 10 q^{61} - 8 q^{62} + 9 q^{63} + 44 q^{64} + 12 q^{65} - 19 q^{66} + 22 q^{67} - 42 q^{68} + 60 q^{69} + 3 q^{70} - 8 q^{71} + 18 q^{72} + 14 q^{73} - 24 q^{74} + 2 q^{75} - 49 q^{76} - 2 q^{77} + 23 q^{78} + 9 q^{79} + 5 q^{80} - 42 q^{81} - 5 q^{82} - 54 q^{83} + 4 q^{84} - 8 q^{85} + 37 q^{86} + 18 q^{87} - 28 q^{88} - 25 q^{89} - 38 q^{90} - 6 q^{91} + 68 q^{92} + 11 q^{93} - 118 q^{94} + 10 q^{95} + 102 q^{96} + 29 q^{97} + 3 q^{98} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} + 20 x^{18} - 43 x^{17} + 207 x^{16} - 401 x^{15} + 1351 x^{14} - 2135 x^{13} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19\!\cdots\!91 \nu^{19} + \cdots - 10\!\cdots\!68 ) / 37\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33\!\cdots\!89 \nu^{19} + \cdots + 23\!\cdots\!16 ) / 47\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 70\!\cdots\!61 \nu^{19} + \cdots - 28\!\cdots\!64 ) / 75\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 20\!\cdots\!11 \nu^{19} + \cdots - 41\!\cdots\!76 ) / 18\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25\!\cdots\!87 \nu^{19} + \cdots + 61\!\cdots\!04 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32\!\cdots\!00 \nu^{19} + \cdots - 10\!\cdots\!40 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 32\!\cdots\!45 \nu^{19} + \cdots + 40\!\cdots\!76 ) / 69\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23\!\cdots\!81 \nu^{19} + \cdots + 15\!\cdots\!04 ) / 37\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 71\!\cdots\!03 \nu^{19} + \cdots + 29\!\cdots\!64 ) / 75\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20\!\cdots\!79 \nu^{19} + \cdots - 25\!\cdots\!56 ) / 18\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 88\!\cdots\!51 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 69\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 97\!\cdots\!23 \nu^{19} + \cdots + 40\!\cdots\!84 ) / 75\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!12 \nu^{19} + \cdots + 59\!\cdots\!36 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21\!\cdots\!55 \nu^{19} + \cdots + 27\!\cdots\!76 ) / 94\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 22\!\cdots\!43 \nu^{19} + \cdots + 38\!\cdots\!96 ) / 94\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 19\!\cdots\!87 \nu^{19} + \cdots + 63\!\cdots\!68 ) / 75\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 10\!\cdots\!19 \nu^{19} + \cdots + 20\!\cdots\!28 ) / 37\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 40\!\cdots\!27 \nu^{19} + \cdots - 64\!\cdots\!56 ) / 94\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} - 3\beta_{8} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} - \beta_{16} - \beta_{14} + \beta_{11} + 5\beta_{7} + \beta_{6} - \beta_{5} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{18} + 7\beta_{12} + \beta_{10} - \beta_{9} + 14\beta_{8} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{19} + 2 \beta_{17} + 9 \beta_{16} + \beta_{15} + 9 \beta_{14} - 9 \beta_{13} + 8 \beta_{12} + \cdots - 29 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} + 11 \beta_{17} + 13 \beta_{16} + \beta_{15} - 2 \beta_{14} - \beta_{11} - 11 \beta_{10} + \cdots + 78 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{18} + 66 \beta_{13} - 59 \beta_{12} - 24 \beta_{10} + 69 \beta_{9} + 40 \beta_{8} + 79 \beta_{5} + \cdots - 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 17 \beta_{19} + 79 \beta_{18} - 93 \beta_{17} - 123 \beta_{16} - 19 \beta_{15} + 25 \beta_{14} + \cdots + 4 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 424 \beta_{19} - 216 \beta_{17} - 511 \beta_{16} - 167 \beta_{15} - 453 \beta_{14} + 582 \beta_{11} + \cdots + 208 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 582 \beta_{18} + 210 \beta_{13} + 2072 \beta_{12} + 727 \beta_{10} - 1032 \beta_{9} + 2999 \beta_{8} + \cdots - 2999 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3009 \beta_{19} - 234 \beta_{18} + 1759 \beta_{17} + 3751 \beta_{16} + 1467 \beta_{15} + \cdots - 7304 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1918 \beta_{19} + 5510 \beta_{17} + 8175 \beta_{16} + 2187 \beta_{15} - 1474 \beta_{14} - 2386 \beta_{11} + \cdots + 19511 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2386 \beta_{18} + 20102 \beta_{13} - 23591 \beta_{12} - 13685 \beta_{10} + 27452 \beta_{9} + 3537 \beta_{8} + \cdots - 3537 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 17174 \beta_{19} + 29575 \beta_{18} - 41137 \beta_{17} - 62797 \beta_{16} - 19186 \beta_{15} + \cdots + 12966 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 149460 \beta_{19} - 103934 \beta_{17} - 200572 \beta_{16} - 91194 \beta_{15} - 133178 \beta_{14} + \cdots + 3367 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 208610 \beta_{18} + 51871 \beta_{13} + 695956 \beta_{12} + 304506 \beta_{10} - 473937 \beta_{9} + \cdots - 864144 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1052333 \beta_{19} - 189523 \beta_{18} + 778443 \beta_{17} + 1463556 \beta_{16} + 684396 \beta_{15} + \cdots - 2160447 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 1191250 \beta_{19} + 2241999 \beta_{17} + 3538850 \beta_{16} + 1263191 \beta_{15} - 255635 \beta_{14} + \cdots + 5840221 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 1570296 \beta_{18} + 5905237 \beta_{13} - 9627314 \beta_{12} - 5780849 \beta_{10} + 10668760 \beta_{9} + \cdots + 1788185 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/665\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(267\) \(381\)
\(\chi(n)\) \(-\beta_{8}\) \(1\) \(1\)

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
−1.26008 + 2.18252i
−0.994744 + 1.72295i
−0.731711 + 1.26736i
−0.278420 + 0.482238i
0.302407 − 0.523784i
0.449475 − 0.778513i
0.745668 − 1.29153i
0.801658 − 1.38851i
1.11529 − 1.93174i
1.35046 − 2.33906i
−1.26008 − 2.18252i
−0.994744 − 1.72295i
−0.731711 − 1.26736i
−0.278420 − 0.482238i
0.302407 + 0.523784i
0.449475 + 0.778513i
0.745668 + 1.29153i
0.801658 + 1.38851i
1.11529 + 1.93174i
1.35046 + 2.33906i
−1.26008 + 2.18252i −0.424874 + 0.735903i −2.17561 − 3.76826i 0.500000 − 0.866025i −1.07075 − 1.85460i −1.00000 5.92545 1.13896 + 1.97274i 1.26008 + 2.18252i
106.2 −0.994744 + 1.72295i 0.457864 − 0.793043i −0.979030 − 1.69573i 0.500000 − 0.866025i 0.910914 + 1.57775i −1.00000 −0.0834388 1.08072 + 1.87186i 0.994744 + 1.72295i
106.3 −0.731711 + 1.26736i −1.07446 + 1.86101i −0.0708008 − 0.122631i 0.500000 − 0.866025i −1.57238 − 2.72345i −1.00000 −2.71962 −0.808916 − 1.40108i 0.731711 + 1.26736i
106.4 −0.278420 + 0.482238i 0.0391618 − 0.0678302i 0.844964 + 1.46352i 0.500000 − 0.866025i 0.0218069 + 0.0377706i −1.00000 −2.05470 1.49693 + 2.59276i 0.278420 + 0.482238i
106.5 0.302407 − 0.523784i 0.354617 − 0.614215i 0.817100 + 1.41526i 0.500000 − 0.866025i −0.214477 − 0.371485i −1.00000 2.19801 1.24849 + 2.16245i −0.302407 − 0.523784i
106.6 0.449475 − 0.778513i −1.28912 + 2.23283i 0.595945 + 1.03221i 0.500000 − 0.866025i 1.15886 + 2.00720i −1.00000 2.86935 −1.82367 − 3.15869i −0.449475 − 0.778513i
106.7 0.745668 − 1.29153i 1.62892 − 2.82136i −0.112041 − 0.194061i 0.500000 − 0.866025i −2.42926 − 4.20760i −1.00000 2.64849 −3.80673 − 6.59345i −0.745668 − 1.29153i
106.8 0.801658 − 1.38851i 0.392253 − 0.679403i −0.285310 − 0.494171i 0.500000 − 0.866025i −0.628906 − 1.08930i −1.00000 2.29175 1.19227 + 2.06508i −0.801658 − 1.38851i
106.9 1.11529 − 1.93174i −1.60334 + 2.77707i −1.48775 − 2.57686i 0.500000 − 0.866025i 3.57639 + 6.19449i −1.00000 −2.17594 −3.64141 − 6.30711i −1.11529 − 1.93174i
106.10 1.35046 − 2.33906i 1.01898 − 1.76493i −2.64747 − 4.58555i 0.500000 − 0.866025i −2.75219 − 4.76693i −1.00000 −8.89934 −0.576655 − 0.998797i −1.35046 − 2.33906i
596.1 −1.26008 − 2.18252i −0.424874 − 0.735903i −2.17561 + 3.76826i 0.500000 + 0.866025i −1.07075 + 1.85460i −1.00000 5.92545 1.13896 − 1.97274i 1.26008 − 2.18252i
596.2 −0.994744 − 1.72295i 0.457864 + 0.793043i −0.979030 + 1.69573i 0.500000 + 0.866025i 0.910914 − 1.57775i −1.00000 −0.0834388 1.08072 − 1.87186i 0.994744 − 1.72295i
596.3 −0.731711 − 1.26736i −1.07446 − 1.86101i −0.0708008 + 0.122631i 0.500000 + 0.866025i −1.57238 + 2.72345i −1.00000 −2.71962 −0.808916 + 1.40108i 0.731711 − 1.26736i
596.4 −0.278420 − 0.482238i 0.0391618 + 0.0678302i 0.844964 − 1.46352i 0.500000 + 0.866025i 0.0218069 − 0.0377706i −1.00000 −2.05470 1.49693 − 2.59276i 0.278420 − 0.482238i
596.5 0.302407 + 0.523784i 0.354617 + 0.614215i 0.817100 − 1.41526i 0.500000 + 0.866025i −0.214477 + 0.371485i −1.00000 2.19801 1.24849 − 2.16245i −0.302407 + 0.523784i
596.6 0.449475 + 0.778513i −1.28912 − 2.23283i 0.595945 − 1.03221i 0.500000 + 0.866025i 1.15886 − 2.00720i −1.00000 2.86935 −1.82367 + 3.15869i −0.449475 + 0.778513i
596.7 0.745668 + 1.29153i 1.62892 + 2.82136i −0.112041 + 0.194061i 0.500000 + 0.866025i −2.42926 + 4.20760i −1.00000 2.64849 −3.80673 + 6.59345i −0.745668 + 1.29153i
596.8 0.801658 + 1.38851i 0.392253 + 0.679403i −0.285310 + 0.494171i 0.500000 + 0.866025i −0.628906 + 1.08930i −1.00000 2.29175 1.19227 − 2.06508i −0.801658 + 1.38851i
596.9 1.11529 + 1.93174i −1.60334 − 2.77707i −1.48775 + 2.57686i 0.500000 + 0.866025i 3.57639 − 6.19449i −1.00000 −2.17594 −3.64141 + 6.30711i −1.11529 + 1.93174i
596.10 1.35046 + 2.33906i 1.01898 + 1.76493i −2.64747 + 4.58555i 0.500000 + 0.866025i −2.75219 + 4.76693i −1.00000 −8.89934 −0.576655 + 0.998797i −1.35046 + 2.33906i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 665.2.i.h ✓ 20
19.c even 3 1 inner 665.2.i.h ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
665.2.i.h ✓ 20 1.a even 1 1 trivial
665.2.i.h ✓ 20 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(665, [\chi])\):

\( T_{2}^{20} - 3 T_{2}^{19} + 20 T_{2}^{18} - 43 T_{2}^{17} + 207 T_{2}^{16} - 401 T_{2}^{15} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{3}^{20} + T_{3}^{19} + 20 T_{3}^{18} + 9 T_{3}^{17} + 265 T_{3}^{16} + 75 T_{3}^{15} + 1797 T_{3}^{14} + \cdots + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 3 T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( T^{20} + T^{19} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{10} \) Copy content Toggle raw display
$7$ \( (T + 1)^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} - T^{9} - 69 T^{8} + \cdots - 479)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} - 6 T^{19} + \cdots + 12559936 \) Copy content Toggle raw display
$17$ \( T^{20} - 8 T^{19} + \cdots + 21307456 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} - 3 T^{19} + \cdots + 1628176 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 128247448480321 \) Copy content Toggle raw display
$31$ \( (T^{10} - 5 T^{9} + \cdots - 1901)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 33 T^{9} + \cdots + 3060676)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 250275184327744 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 357737964544 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 245805829814416 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 129037263334849 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 12031457344 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 150675172561 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( (T^{10} + 27 T^{9} + \cdots + 36896)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
show more
show less