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Properties

Label 279.2.y.c
Level $279$
Weight $2$
Character orbit 279.y
Analytic conductor $2.228$
Analytic rank $0$
Dimension $16$
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] = [279,2,Mod(10,279)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(279, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("279.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 279 = 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 279.y (of order \(15\), degree \(8\), 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: \(2.22782621639\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_{14} + \cdots - \beta_1) q^{2}+ \cdots + (2 \beta_{14} - \beta_{13} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} - \beta_{14} + \cdots - \beta_1) q^{2}+ \cdots + (2 \beta_{15} + 2 \beta_{14} + \cdots - 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{2} - 14 q^{4} + 3 q^{5} + 2 q^{7} - 17 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{2} - 14 q^{4} + 3 q^{5} + 2 q^{7} - 17 q^{8} - 2 q^{10} + 7 q^{11} - 7 q^{13} + 6 q^{14} - 2 q^{16} + 6 q^{17} + 16 q^{19} - 37 q^{20} + 9 q^{22} - q^{23} - 13 q^{25} - 9 q^{26} - 30 q^{28} + 14 q^{29} + 15 q^{31} + 42 q^{32} - 32 q^{34} + 9 q^{35} - 8 q^{37} - 8 q^{38} - q^{40} + 8 q^{41} + 23 q^{43} - 39 q^{44} + 34 q^{46} - 14 q^{47} + 2 q^{49} - 3 q^{50} + 29 q^{52} - 6 q^{53} - 7 q^{55} + 30 q^{56} - 15 q^{58} - 4 q^{59} - 60 q^{61} + 25 q^{62} + 23 q^{64} + 12 q^{65} + 13 q^{67} - 30 q^{68} + 12 q^{70} + 14 q^{71} + 2 q^{73} - 13 q^{74} - 12 q^{76} - 18 q^{77} + 18 q^{79} - 36 q^{80} + 14 q^{82} + 16 q^{83} + 37 q^{85} + 26 q^{86} - 17 q^{88} - q^{89} + 8 q^{91} + 64 q^{92} + 44 q^{94} + 22 q^{95} + 3 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2 \nu^{15} - 4 \nu^{14} + 48 \nu^{13} - 65 \nu^{12} + 458 \nu^{11} - 358 \nu^{10} + 2196 \nu^{9} + \cdots + 255 ) / 186 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2 \nu^{15} + 4 \nu^{14} + 48 \nu^{13} + 65 \nu^{12} + 458 \nu^{11} + 358 \nu^{10} + 2196 \nu^{9} + \cdots - 255 ) / 186 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6 \nu^{15} - 10 \nu^{14} + 144 \nu^{13} - 209 \nu^{12} + 1374 \nu^{11} - 1732 \nu^{10} + 6619 \nu^{9} + \cdots - 463 ) / 186 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17 \nu^{15} + 315 \nu^{13} + 2250 \nu^{11} + 7940 \nu^{9} + 14865 \nu^{7} + 14844 \nu^{5} + 7255 \nu^{3} + \cdots + 93 ) / 186 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6 \nu^{15} + 10 \nu^{14} + 144 \nu^{13} + 209 \nu^{12} + 1374 \nu^{11} + 1732 \nu^{10} + 6619 \nu^{9} + \cdots + 463 ) / 186 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6 \nu^{15} + 28 \nu^{14} - 144 \nu^{13} + 517 \nu^{12} - 1374 \nu^{11} + 3653 \nu^{10} - 6619 \nu^{9} + \cdots + 354 ) / 186 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8 \nu^{15} + 33 \nu^{14} + 130 \nu^{13} + 575 \nu^{12} + 716 \nu^{11} + 3713 \nu^{10} + 1282 \nu^{9} + \cdots - 143 ) / 186 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6 \nu^{15} + 28 \nu^{14} + 144 \nu^{13} + 517 \nu^{12} + 1374 \nu^{11} + 3653 \nu^{10} + 6619 \nu^{9} + \cdots + 354 ) / 186 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36 \nu^{15} + 38 \nu^{14} - 709 \nu^{13} + 695 \nu^{12} - 5485 \nu^{11} + 4827 \nu^{10} + \cdots + 538 ) / 186 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 53 \nu^{15} + 5 \nu^{14} - 962 \nu^{13} + 89 \nu^{12} - 6619 \nu^{11} + 587 \nu^{10} - 21738 \nu^{9} + \cdots + 61 ) / 186 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 50 \nu^{15} - 25 \nu^{14} + 983 \nu^{13} - 445 \nu^{12} + 7575 \nu^{11} - 2966 \nu^{10} + 29263 \nu^{9} + \cdots + 36 ) / 186 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 57 \nu^{15} + 21 \nu^{14} + 1058 \nu^{13} + 380 \nu^{12} + 7535 \nu^{11} + 2608 \nu^{10} + 26161 \nu^{9} + \cdots + 126 ) / 186 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 27 \nu^{15} + 53 \nu^{14} + 524 \nu^{13} + 962 \nu^{12} + 3982 \nu^{11} + 6619 \nu^{10} + 15200 \nu^{9} + \cdots + 597 ) / 186 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 36 \nu^{15} + 68 \nu^{14} + 709 \nu^{13} + 1229 \nu^{12} + 5485 \nu^{11} + 8411 \nu^{10} + 21362 \nu^{9} + \cdots + 470 ) / 186 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 135 \nu^{15} + 34 \nu^{14} + 2558 \nu^{13} + 599 \nu^{12} + 18763 \nu^{11} + 3942 \nu^{10} + \cdots + 359 ) / 186 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{12} + \beta_{11} + \beta_{9} - 2\beta_{8} + \beta_{4} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} - 6 \beta_{8} + 6 \beta_{6} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 8 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} + 2 \beta_{10} - 7 \beta_{9} + 12 \beta_{8} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7 \beta_{15} - 6 \beta_{14} - 4 \beta_{13} - 6 \beta_{12} - 14 \beta_{11} - 7 \beta_{10} + 2 \beta_{9} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 19 \beta_{15} + 55 \beta_{14} - 8 \beta_{12} + 38 \beta_{11} - 19 \beta_{10} + 47 \beta_{9} + \cdots - 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 42 \beta_{15} + 35 \beta_{14} + 44 \beta_{13} + 29 \beta_{12} + 86 \beta_{11} + 42 \beta_{10} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 145 \beta_{15} - 365 \beta_{14} + 14 \beta_{12} - 248 \beta_{11} + 145 \beta_{10} - 309 \beta_{9} + \cdots + 455 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 245 \beta_{15} - 212 \beta_{14} - 356 \beta_{13} - 128 \beta_{12} - 518 \beta_{11} - 245 \beta_{10} + \cdots + 234 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1022 \beta_{15} + 2385 \beta_{14} + 34 \beta_{12} + 1625 \beta_{11} - 1022 \beta_{10} + 2000 \beta_{9} + \cdots - 2780 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1432 \beta_{15} + 1322 \beta_{14} + 2578 \beta_{13} + 518 \beta_{12} + 3129 \beta_{11} + 1432 \beta_{10} + \cdots - 1831 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6922 \beta_{15} - 15445 \beta_{14} - 619 \beta_{12} - 10601 \beta_{11} + 6922 \beta_{10} - 12821 \beta_{9} + \cdots + 17280 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8460 \beta_{15} - 8372 \beta_{14} - 17726 \beta_{13} - 1817 \beta_{12} - 19052 \beta_{11} - 8460 \beta_{10} + \cdots + 13340 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 45841 \beta_{15} + 99462 \beta_{14} + 5217 \beta_{12} + 68772 \beta_{11} - 45841 \beta_{10} + \cdots - 108434 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 50619 \beta_{15} + 53382 \beta_{14} + 118538 \beta_{13} + 4347 \beta_{12} + 116998 \beta_{11} + \cdots - 93153 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(218\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
2.52368i
− 1.03739i
1.83925i
− 1.42343i
− 2.52368i
1.03739i
0.333129i
− 1.14660i
− 0.176392i
2.16544i
0.176392i
− 2.16544i
− 1.83925i
1.42343i
− 0.333129i
1.14660i
0.284315 + 0.206567i 0 −0.579869 − 1.78465i 1.48661 + 2.57489i 0 1.05848 + 0.224987i 0.420982 − 1.29565i 0 −0.109221 + 1.03917i
10.2 1.02470 + 0.744490i 0 −0.122284 − 0.376353i −1.90016 − 3.29117i 0 −2.14115 − 0.455117i 0.937688 − 2.88591i 0 0.503147 − 4.78712i
19.1 −0.557811 − 0.405274i 0 −0.471127 − 1.44998i 1.85376 − 3.21080i 0 0.510810 − 0.567312i −0.750969 + 2.31124i 0 −2.33530 + 1.03974i
19.2 1.86683 + 1.35633i 0 1.02738 + 3.16196i −1.24923 + 2.16373i 0 1.07187 − 1.19043i −0.944583 + 2.90713i 0 −5.26683 + 2.34494i
28.1 0.284315 − 0.206567i 0 −0.579869 + 1.78465i 1.48661 − 2.57489i 0 1.05848 − 0.224987i 0.420982 + 1.29565i 0 −0.109221 − 1.03917i
28.2 1.02470 − 0.744490i 0 −0.122284 + 0.376353i −1.90016 + 3.29117i 0 −2.14115 + 0.455117i 0.937688 + 2.88591i 0 0.503147 + 4.78712i
82.1 −0.640321 − 1.97070i 0 −1.85563 + 1.34820i 1.17396 + 2.03335i 0 0.384094 + 3.65441i 0.492333 + 0.357701i 0 3.25543 − 3.61552i
82.2 0.831304 + 2.55849i 0 −4.23677 + 3.07819i 0.304192 + 0.526876i 0 0.180508 + 1.71742i −7.04481 − 5.11835i 0 −1.09513 + 1.21627i
100.1 −0.380762 − 1.17187i 0 0.389745 − 0.283166i −0.772811 + 1.33855i 0 −3.47491 − 1.54713i −2.47393 − 1.79742i 0 1.86286 + 0.395962i
100.2 0.571745 + 1.75965i 0 −1.15144 + 0.836573i 0.603681 − 1.04561i 0 3.41030 + 1.51837i 0.863288 + 0.627215i 0 2.18505 + 0.464447i
226.1 −0.380762 + 1.17187i 0 0.389745 + 0.283166i −0.772811 − 1.33855i 0 −3.47491 + 1.54713i −2.47393 + 1.79742i 0 1.86286 − 0.395962i
226.2 0.571745 − 1.75965i 0 −1.15144 − 0.836573i 0.603681 + 1.04561i 0 3.41030 − 1.51837i 0.863288 − 0.627215i 0 2.18505 − 0.464447i
235.1 −0.557811 + 0.405274i 0 −0.471127 + 1.44998i 1.85376 + 3.21080i 0 0.510810 + 0.567312i −0.750969 − 2.31124i 0 −2.33530 − 1.03974i
235.2 1.86683 − 1.35633i 0 1.02738 − 3.16196i −1.24923 − 2.16373i 0 1.07187 + 1.19043i −0.944583 − 2.90713i 0 −5.26683 − 2.34494i
262.1 −0.640321 + 1.97070i 0 −1.85563 − 1.34820i 1.17396 − 2.03335i 0 0.384094 − 3.65441i 0.492333 − 0.357701i 0 3.25543 + 3.61552i
262.2 0.831304 − 2.55849i 0 −4.23677 − 3.07819i 0.304192 − 0.526876i 0 0.180508 − 1.71742i −7.04481 + 5.11835i 0 −1.09513 − 1.21627i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 279.2.y.c 16
3.b odd 2 1 31.2.g.a ✓ 16
12.b even 2 1 496.2.bg.c 16
15.d odd 2 1 775.2.bl.a 16
15.e even 4 2 775.2.ck.a 32
31.g even 15 1 inner 279.2.y.c 16
31.g even 15 1 8649.2.a.bf 8
31.h odd 30 1 8649.2.a.be 8
93.c even 2 1 961.2.g.l 16
93.g even 6 1 961.2.d.n 16
93.g even 6 1 961.2.g.j 16
93.h odd 6 1 961.2.d.o 16
93.h odd 6 1 961.2.g.k 16
93.k even 10 1 961.2.c.i 16
93.k even 10 1 961.2.g.j 16
93.k even 10 1 961.2.g.m 16
93.k even 10 1 961.2.g.n 16
93.l odd 10 1 961.2.c.j 16
93.l odd 10 1 961.2.g.k 16
93.l odd 10 1 961.2.g.s 16
93.l odd 10 1 961.2.g.t 16
93.o odd 30 1 31.2.g.a ✓ 16
93.o odd 30 1 961.2.a.i 8
93.o odd 30 1 961.2.c.j 16
93.o odd 30 1 961.2.d.o 16
93.o odd 30 2 961.2.d.p 16
93.o odd 30 1 961.2.g.s 16
93.o odd 30 1 961.2.g.t 16
93.p even 30 1 961.2.a.j 8
93.p even 30 1 961.2.c.i 16
93.p even 30 1 961.2.d.n 16
93.p even 30 2 961.2.d.q 16
93.p even 30 1 961.2.g.l 16
93.p even 30 1 961.2.g.m 16
93.p even 30 1 961.2.g.n 16
372.bd even 30 1 496.2.bg.c 16
465.bl odd 30 1 775.2.bl.a 16
465.bt even 60 2 775.2.ck.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.g.a ✓ 16 3.b odd 2 1
31.2.g.a ✓ 16 93.o odd 30 1
279.2.y.c 16 1.a even 1 1 trivial
279.2.y.c 16 31.g even 15 1 inner
496.2.bg.c 16 12.b even 2 1
496.2.bg.c 16 372.bd even 30 1
775.2.bl.a 16 15.d odd 2 1
775.2.bl.a 16 465.bl odd 30 1
775.2.ck.a 32 15.e even 4 2
775.2.ck.a 32 465.bt even 60 2
961.2.a.i 8 93.o odd 30 1
961.2.a.j 8 93.p even 30 1
961.2.c.i 16 93.k even 10 1
961.2.c.i 16 93.p even 30 1
961.2.c.j 16 93.l odd 10 1
961.2.c.j 16 93.o odd 30 1
961.2.d.n 16 93.g even 6 1
961.2.d.n 16 93.p even 30 1
961.2.d.o 16 93.h odd 6 1
961.2.d.o 16 93.o odd 30 1
961.2.d.p 16 93.o odd 30 2
961.2.d.q 16 93.p even 30 2
961.2.g.j 16 93.g even 6 1
961.2.g.j 16 93.k even 10 1
961.2.g.k 16 93.h odd 6 1
961.2.g.k 16 93.l odd 10 1
961.2.g.l 16 93.c even 2 1
961.2.g.l 16 93.p even 30 1
961.2.g.m 16 93.k even 10 1
961.2.g.m 16 93.p even 30 1
961.2.g.n 16 93.k even 10 1
961.2.g.n 16 93.p even 30 1
961.2.g.s 16 93.l odd 10 1
961.2.g.s 16 93.o odd 30 1
961.2.g.t 16 93.l odd 10 1
961.2.g.t 16 93.o odd 30 1
8649.2.a.be 8 31.h odd 30 1
8649.2.a.bf 8 31.g even 15 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 6 T_{2}^{15} + 29 T_{2}^{14} - 91 T_{2}^{13} + 246 T_{2}^{12} - 523 T_{2}^{11} + 1011 T_{2}^{10} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(279, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 6 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 3 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + \cdots + 68121 \) Copy content Toggle raw display
$11$ \( T^{16} - 7 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$13$ \( T^{16} + 7 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$17$ \( T^{16} - 6 T^{15} + \cdots + 74805201 \) Copy content Toggle raw display
$19$ \( T^{16} - 16 T^{15} + \cdots + 361201 \) Copy content Toggle raw display
$23$ \( T^{16} + T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$29$ \( T^{16} - 14 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 344807761 \) Copy content Toggle raw display
$41$ \( T^{16} - 8 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{16} - 23 T^{15} + \cdots + 7612081 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3306365001 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 366207732801 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 167728401 \) Copy content Toggle raw display
$61$ \( (T^{8} + 30 T^{7} + \cdots + 38161)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 7485883441 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 214944921 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 17441907675201 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 84609661119201 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1446653267361 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 117957215601 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 7131992195241 \) Copy content Toggle raw display
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