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Properties

Label 170.2.n.a
Level $170$
Weight $2$
Character orbit 170.n
Analytic conductor $1.357$
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] = [170,2,Mod(9,170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(170, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("170.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.n (of order \(8\), 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: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} + \cdots + 2048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 16 x^{15} + 52 x^{14} + 992 x^{13} + 6181 x^{12} + 8952 x^{11} + 6244 x^{10} - 11448 x^{9} + \cdots + 2048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24\!\cdots\!67 \nu^{19} + \cdots - 24\!\cdots\!36 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 47\!\cdots\!67 \nu^{19} + \cdots - 14\!\cdots\!04 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 42\!\cdots\!91 \nu^{19} + \cdots - 49\!\cdots\!32 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!03 \nu^{19} + \cdots + 17\!\cdots\!44 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\!\cdots\!43 \nu^{19} + \cdots - 19\!\cdots\!44 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!32 \nu^{19} + \cdots - 17\!\cdots\!56 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 74\!\cdots\!99 \nu^{19} + \cdots - 33\!\cdots\!32 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!25 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!31 \nu^{19} + \cdots - 12\!\cdots\!56 ) / 30\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 48\!\cdots\!23 \nu^{19} + \cdots + 48\!\cdots\!24 ) / 68\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!29 \nu^{19} + \cdots + 44\!\cdots\!56 ) / 13\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 96\!\cdots\!58 \nu^{19} + \cdots + 87\!\cdots\!56 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 48\!\cdots\!91 \nu^{19} + \cdots + 37\!\cdots\!48 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 42\!\cdots\!55 \nu^{19} + \cdots - 27\!\cdots\!84 ) / 41\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 24\!\cdots\!51 \nu^{19} + \cdots + 18\!\cdots\!16 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 10\!\cdots\!13 \nu^{19} + \cdots - 14\!\cdots\!64 ) / 85\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 49\!\cdots\!77 \nu^{19} + \cdots + 22\!\cdots\!96 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 13\!\cdots\!01 \nu^{19} + \cdots + 10\!\cdots\!68 ) / 51\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{18} - \beta_{16} + \beta_{11} + 4\beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} - \beta_{16} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \cdots + \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{19} - \beta_{16} + \beta_{15} + 28 \beta_{10} - 10 \beta_{9} + \beta_{8} + \beta_{7} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{19} - 13 \beta_{17} + 13 \beta_{15} - 11 \beta_{14} + 13 \beta_{12} - 11 \beta_{11} + 23 \beta_{10} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{19} - 83 \beta_{17} - 2 \beta_{16} + 94 \beta_{15} + \beta_{14} - 14 \beta_{13} + 94 \beta_{12} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{19} + 3 \beta_{18} - 146 \beta_{17} - 104 \beta_{16} + 147 \beta_{15} + 147 \beta_{14} + \cdots - 308 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 68 \beta_{18} - 68 \beta_{17} + 21 \beta_{16} + 21 \beta_{15} + 899 \beta_{14} - 899 \beta_{13} + \cdots - 2094 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 50 \beta_{19} - 1567 \beta_{18} + 50 \beta_{17} + 1585 \beta_{16} - 949 \beta_{15} + 1663 \beta_{14} + \cdots - 3557 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1057 \beta_{19} - 7757 \beta_{18} + 8766 \beta_{16} - 764 \beta_{15} + 1898 \beta_{14} - 345 \beta_{13} + \cdots - 2925 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 16514 \beta_{19} - 16514 \beta_{18} - 513 \beta_{17} + 17990 \beta_{16} - 596 \beta_{15} + \cdots + 5311 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 77063 \beta_{19} - 14070 \beta_{18} + 14070 \beta_{17} + 21468 \beta_{16} - 21468 \beta_{15} + \cdots + 15097 \beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 172631 \beta_{19} + 3540 \beta_{18} + 172631 \beta_{17} + 11440 \beta_{16} - 191543 \beta_{15} + \cdots - 43136 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 172989 \beta_{19} + 773659 \beta_{17} + 117234 \beta_{16} - 867100 \beta_{15} - 69075 \beta_{14} + \cdots + 409591 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 6085 \beta_{19} - 6085 \beta_{18} + 1797956 \beta_{17} + 707214 \beta_{16} - 1807851 \beta_{15} + \cdots + 4334562 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 2032620 \beta_{18} + 2032620 \beta_{17} - 891059 \beta_{16} - 891059 \beta_{15} - 8732441 \beta_{14} + \cdots + 19468618 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 330322 \beta_{19} + 18695535 \beta_{18} + 330322 \beta_{17} - 18707565 \beta_{16} + 6436269 \beta_{15} + \cdots + 45263765 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 23228121 \beta_{19} + 79536549 \beta_{18} - 88408294 \beta_{16} + 14388480 \beta_{15} - 30086562 \beta_{14} + \cdots + 53087193 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 194291250 \beta_{19} + 194291250 \beta_{18} - 7913799 \beta_{17} - 220871206 \beta_{16} + 33338924 \beta_{15} + \cdots - 3698703 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(137\)
\(\chi(n)\) \(-\beta_{7}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
2.99334 − 1.23988i
0.953222 − 0.394838i
0.236338 − 0.0978946i
−1.86202 + 0.771273i
−2.32088 + 0.961341i
2.99334 + 1.23988i
0.953222 + 0.394838i
0.236338 + 0.0978946i
−1.86202 − 0.771273i
−2.32088 − 0.961341i
−0.969032 + 2.33945i
−0.826884 + 1.99627i
0.254075 − 0.613391i
0.355063 − 0.857197i
1.18678 − 2.86514i
−0.969032 − 2.33945i
−0.826884 − 1.99627i
0.254075 + 0.613391i
0.355063 + 0.857197i
1.18678 + 2.86514i
−0.707107 − 0.707107i −1.23988 + 2.99334i 1.00000i −2.18859 − 0.458323i 2.99334 − 1.23988i 1.49921 − 0.620992i 0.707107 − 0.707107i −5.30145 − 5.30145i 1.22349 + 1.87165i
9.2 −0.707107 − 0.707107i −0.394838 + 0.953222i 1.00000i 1.21037 − 1.88016i 0.953222 − 0.394838i −0.363965 + 0.150759i 0.707107 − 0.707107i 1.36858 + 1.36858i −2.18534 + 0.473610i
9.3 −0.707107 − 0.707107i −0.0978946 + 0.236338i 1.00000i −0.496623 + 2.18022i 0.236338 − 0.0978946i −3.48449 + 1.44332i 0.707107 − 0.707107i 2.07505 + 2.07505i 1.89281 − 1.19048i
9.4 −0.707107 − 0.707107i 0.771273 − 1.86202i 1.00000i 1.71825 + 1.43095i −1.86202 + 0.771273i 2.47378 − 1.02468i 0.707107 − 0.707107i −0.750924 − 0.750924i −0.203147 − 2.22682i
9.5 −0.707107 − 0.707107i 0.961341 − 2.32088i 1.00000i −1.24340 − 1.85848i −2.32088 + 0.961341i −0.124542 + 0.0515871i 0.707107 − 0.707107i −2.34100 − 2.34100i −0.434925 + 2.19336i
19.1 −0.707107 + 0.707107i −1.23988 − 2.99334i − 1.00000i −2.18859 + 0.458323i 2.99334 + 1.23988i 1.49921 + 0.620992i 0.707107 + 0.707107i −5.30145 + 5.30145i 1.22349 − 1.87165i
19.2 −0.707107 + 0.707107i −0.394838 − 0.953222i − 1.00000i 1.21037 + 1.88016i 0.953222 + 0.394838i −0.363965 − 0.150759i 0.707107 + 0.707107i 1.36858 − 1.36858i −2.18534 − 0.473610i
19.3 −0.707107 + 0.707107i −0.0978946 − 0.236338i − 1.00000i −0.496623 − 2.18022i 0.236338 + 0.0978946i −3.48449 − 1.44332i 0.707107 + 0.707107i 2.07505 − 2.07505i 1.89281 + 1.19048i
19.4 −0.707107 + 0.707107i 0.771273 + 1.86202i − 1.00000i 1.71825 − 1.43095i −1.86202 − 0.771273i 2.47378 + 1.02468i 0.707107 + 0.707107i −0.750924 + 0.750924i −0.203147 + 2.22682i
19.5 −0.707107 + 0.707107i 0.961341 + 2.32088i − 1.00000i −1.24340 + 1.85848i −2.32088 − 0.961341i −0.124542 − 0.0515871i 0.707107 + 0.707107i −2.34100 + 2.34100i −0.434925 − 2.19336i
49.1 0.707107 − 0.707107i −2.33945 + 0.969032i − 1.00000i −0.309710 + 2.21452i −0.969032 + 2.33945i −1.26758 + 3.06021i −0.707107 − 0.707107i 2.41268 − 2.41268i 1.34690 + 1.78490i
49.2 0.707107 − 0.707107i −1.99627 + 0.826884i − 1.00000i −1.27343 − 1.83804i −0.826884 + 1.99627i 1.32795 − 3.20595i −0.707107 − 0.707107i 1.18005 − 1.18005i −2.20014 − 0.399242i
49.3 0.707107 − 0.707107i 0.613391 − 0.254075i − 1.00000i 0.384345 + 2.20279i 0.254075 − 0.613391i 1.97319 − 4.76369i −0.707107 − 0.707107i −1.80963 + 1.80963i 1.82938 + 1.28583i
49.4 0.707107 − 0.707107i 0.857197 − 0.355063i − 1.00000i 2.23422 − 0.0908004i 0.355063 − 0.857197i −0.939960 + 2.26926i −0.707107 − 0.707107i −1.51260 + 1.51260i 1.51563 − 1.64404i
49.5 0.707107 − 0.707107i 2.86514 − 1.18678i − 1.00000i −2.03543 + 0.925748i 1.18678 − 2.86514i −1.09360 + 2.64018i −0.707107 − 0.707107i 4.67924 − 4.67924i −0.784666 + 2.09387i
59.1 0.707107 + 0.707107i −2.33945 − 0.969032i 1.00000i −0.309710 − 2.21452i −0.969032 − 2.33945i −1.26758 − 3.06021i −0.707107 + 0.707107i 2.41268 + 2.41268i 1.34690 − 1.78490i
59.2 0.707107 + 0.707107i −1.99627 − 0.826884i 1.00000i −1.27343 + 1.83804i −0.826884 − 1.99627i 1.32795 + 3.20595i −0.707107 + 0.707107i 1.18005 + 1.18005i −2.20014 + 0.399242i
59.3 0.707107 + 0.707107i 0.613391 + 0.254075i 1.00000i 0.384345 − 2.20279i 0.254075 + 0.613391i 1.97319 + 4.76369i −0.707107 + 0.707107i −1.80963 − 1.80963i 1.82938 − 1.28583i
59.4 0.707107 + 0.707107i 0.857197 + 0.355063i 1.00000i 2.23422 + 0.0908004i 0.355063 + 0.857197i −0.939960 − 2.26926i −0.707107 + 0.707107i −1.51260 − 1.51260i 1.51563 + 1.64404i
59.5 0.707107 + 0.707107i 2.86514 + 1.18678i 1.00000i −2.03543 − 0.925748i 1.18678 + 2.86514i −1.09360 − 2.64018i −0.707107 + 0.707107i 4.67924 + 4.67924i −0.784666 − 2.09387i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.m even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.n.a ✓ 20
5.b even 2 1 170.2.n.b yes 20
5.c odd 4 1 850.2.l.h 20
5.c odd 4 1 850.2.l.i 20
17.d even 8 1 170.2.n.b yes 20
85.k odd 8 1 850.2.l.i 20
85.m even 8 1 inner 170.2.n.a ✓ 20
85.n odd 8 1 850.2.l.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.n.a ✓ 20 1.a even 1 1 trivial
170.2.n.a ✓ 20 85.m even 8 1 inner
170.2.n.b yes 20 5.b even 2 1
170.2.n.b yes 20 17.d even 8 1
850.2.l.h 20 5.c odd 4 1
850.2.l.h 20 85.n odd 8 1
850.2.l.i 20 5.c odd 4 1
850.2.l.i 20 85.k odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 8 T_{3}^{17} + 104 T_{3}^{15} - 20 T_{3}^{14} + 168 T_{3}^{13} + 5349 T_{3}^{12} + \cdots + 2048 \) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} - 8 T^{17} + \cdots + 2048 \) Copy content Toggle raw display
$5$ \( T^{20} + 4 T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} + 28 T^{18} + \cdots + 131072 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 102760448 \) Copy content Toggle raw display
$13$ \( (T^{10} + 12 T^{9} + \cdots - 656)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2015993900449 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 554696704 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 4933025792 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 1394342432 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 104603648 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 22002985088 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 496557944352 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 56865079296 \) Copy content Toggle raw display
$47$ \( (T^{10} - 20 T^{9} + \cdots + 135167744)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 24599695396864 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 44\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 17553552572416 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 215737327296512 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 694577266688 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 740139447894016 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
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