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Properties

Label 273.2.l.b
Level $273$
Weight $2$
Character orbit 273.l
Analytic conductor $2.180$
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] = [273,2,Mod(16,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.l (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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 11 x^{14} - 4 x^{13} + 87 x^{12} - 35 x^{11} + 326 x^{10} - 205 x^{9} + 895 x^{8} - 481 x^{7} + 1005 x^{6} - 544 x^{5} + 811 x^{4} - 312 x^{3} + 195 x^{2} + 13 x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 11 x^{14} - 4 x^{13} + 87 x^{12} - 35 x^{11} + 326 x^{10} - 205 x^{9} + 895 x^{8} - 481 x^{7} + 1005 x^{6} - 544 x^{5} + 811 x^{4} - 312 x^{3} + 195 x^{2} + 13 x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 850102445244 \nu^{15} - 602738721141 \nu^{14} - 9256957678228 \nu^{13} - 3074932976676 \nu^{12} - 70844833728257 \nu^{11} + \cdots - 13972229088464 ) / 200652098581830 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 945658852056 \nu^{15} + 5453415441339 \nu^{14} + 13517776750957 \nu^{13} + 52870958416314 \nu^{12} + \cdots - 611267573740024 ) / 200652098581830 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1548397573197 \nu^{15} + 5359246221883 \nu^{14} + 19993119508609 \nu^{13} + 49756879408343 \nu^{12} + \cdots - 10161380439778 ) / 200652098581830 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3810848648686 \nu^{15} + 2398500018441 \nu^{14} - 35957350192522 \nu^{13} + 44493471781581 \nu^{12} + \cdots - 8612982866581 ) / 200652098581830 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22240023182504 \nu^{15} + 33095488971866 \nu^{14} + 219131096234958 \nu^{13} + 229934387339641 \nu^{12} + \cdots - 24\!\cdots\!91 ) / 601956295745490 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23960698558757 \nu^{15} + 80722315132867 \nu^{14} - 248272416035934 \nu^{13} + 945357317741162 \nu^{12} + \cdots - 15\!\cdots\!87 ) / 601956295745490 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1778153235146 \nu^{15} + 3078673249492 \nu^{14} - 23056845757699 \nu^{13} + 36844479871649 \nu^{12} + \cdots + 13468531477239 ) / 40130419716366 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13972229088464 \nu^{15} - 850102445244 \nu^{14} + 153091781251963 \nu^{13} - 65145874032084 \nu^{12} + \cdots + 174261731510394 ) / 200652098581830 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 78334216662179 \nu^{15} + 42305988951004 \nu^{14} - 796493782907553 \nu^{13} + 800506448243009 \nu^{12} + \cdots - 15\!\cdots\!24 ) / 601956295745490 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 40582993882149 \nu^{15} + 45466225490261 \nu^{14} + 457612339093228 \nu^{13} + 320116699841551 \nu^{12} + \cdots + 541126055004554 ) / 200652098581830 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 24469220650027 \nu^{15} + 15238766802188 \nu^{14} - 263471571349674 \nu^{13} + 261485413803346 \nu^{12} + \cdots - 273843217222007 ) / 120391259149098 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 131547920533543 \nu^{15} + 66691456016783 \nu^{14} + \cdots + 20950999529842 ) / 601956295745490 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 46673194766134 \nu^{15} + 504608087166 \nu^{14} + 508750470699368 \nu^{13} - 187060135461519 \nu^{12} + \cdots - 79869396342261 ) / 200652098581830 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 140712016641263 \nu^{15} - 74541657903737 \nu^{14} + \cdots + 431512462062377 ) / 601956295745490 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - 3\beta_{9} + \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} + \beta_{12} - \beta_{6} + \beta_{3} - 4\beta_{2} - 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 7 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} + 15 \beta_{9} - \beta_{8} - 6 \beta_{5} + 6 \beta_{4} + \beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + 8 \beta_{14} + 9 \beta_{13} - \beta_{12} - \beta_{11} - 8 \beta_{9} + \beta_{8} - 8 \beta_{7} + 8 \beta_{6} + \beta_{5} - 8 \beta_{3} + 20 \beta_{2} - 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{15} - \beta_{13} - \beta_{12} + 10 \beta_{11} + 2 \beta_{10} - 9 \beta_{7} + \beta_{6} - 34 \beta_{4} + 45 \beta_{3} - 10 \beta_{2} - 11 \beta _1 + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{15} - 56 \beta_{14} - \beta_{13} - 41 \beta_{12} - 13 \beta_{10} + 56 \beta_{9} - 12 \beta_{8} + 54 \beta_{7} - 10 \beta_{5} + 10 \beta_{4} + 111 \beta _1 - 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 24 \beta_{15} + 286 \beta_{14} + 81 \beta_{13} - 78 \beta_{12} - 78 \beta_{11} + 65 \beta_{10} - 494 \beta_{9} + 78 \beta_{8} + 62 \beta_{7} - 8 \beta_{6} + 198 \beta_{5} - 286 \beta_{3} + 77 \beta_{2} + 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 118 \beta_{15} - 437 \beta_{13} + 335 \beta_{12} + 105 \beta_{11} + 102 \beta_{10} - 16 \beta_{7} - 335 \beta_{6} - 79 \beta_{4} + 382 \beta_{3} - 651 \beta_{2} - 316 \beta _1 + 382 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 437 \beta_{15} - 1818 \beta_{14} - 437 \beta_{13} + 594 \beta_{12} - 644 \beta_{10} + 2986 \beta_{9} - 555 \beta_{8} + 50 \beta_{7} - 1184 \beta_{5} + 1184 \beta_{4} + 547 \beta _1 - 2986 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 762 \beta_{15} + 2583 \beta_{14} + 3017 \beta_{13} - 812 \beta_{12} - 812 \beta_{11} + 168 \beta_{10} - 2596 \beta_{9} + 812 \beta_{8} - 2037 \beta_{7} + 2087 \beta_{6} + 587 \beta_{5} - 2583 \beta_{3} + \cdots - 2087 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4423 \beta_{15} - 1476 \beta_{13} - 98 \beta_{12} + 3779 \beta_{11} + 1574 \beta_{10} - 2849 \beta_{7} + 98 \beta_{6} - 7211 \beta_{4} + 11577 \beta_{3} - 3777 \beta_{2} - 3875 \beta _1 + 18359 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1476 \beta_{15} - 17376 \beta_{14} - 1476 \beta_{13} - 7051 \beta_{12} - 6829 \beta_{10} + 17658 \beta_{9} - 5899 \beta_{8} + 13880 \beta_{7} - 4266 \beta_{5} + 4266 \beta_{4} + 24295 \beta _1 - 17658 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11252 \beta_{15} + 73854 \beta_{14} + 30104 \beta_{13} - 25132 \beta_{12} - 25132 \beta_{11} + 18303 \beta_{10} - 114139 \beta_{9} + 25132 \beta_{8} + 13331 \beta_{7} + 549 \beta_{6} + 44456 \beta_{5} + \cdots - 549 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 48185 \beta_{15} - 116740 \beta_{13} + 80356 \beta_{12} + 41356 \beta_{11} + 36384 \beta_{10} - 11801 \beta_{7} - 80356 \beta_{6} - 30616 \beta_{4} + 116428 \beta_{3} - 151544 \beta_{2} + \cdots + 120172 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(-\beta_{9}\) \(-\beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
1.21707 + 2.10803i
0.857510 + 1.48525i
0.415625 + 0.719884i
0.379240 + 0.656863i
−0.0340180 − 0.0589209i
−0.532778 − 0.922798i
−1.02737 − 1.77946i
−1.27528 − 2.20885i
1.21707 − 2.10803i
0.857510 − 1.48525i
0.415625 − 0.719884i
0.379240 − 0.656863i
−0.0340180 + 0.0589209i
−0.532778 + 0.922798i
−1.02737 + 1.77946i
−1.27528 + 2.20885i
−2.43414 0.500000 − 0.866025i 3.92506 −0.613891 + 1.06329i −1.21707 + 2.10803i 2.20121 + 1.46788i −4.68588 −0.500000 − 0.866025i 1.49430 − 2.58820i
16.2 −1.71502 0.500000 − 0.866025i 0.941295 1.22863 − 2.12806i −0.857510 + 1.48525i −2.38702 − 1.14112i 1.81570 −0.500000 − 0.866025i −2.10713 + 3.64966i
16.3 −0.831251 0.500000 − 0.866025i −1.30902 −1.30847 + 2.26634i −0.415625 + 0.719884i 1.78280 − 1.95490i 2.75063 −0.500000 − 0.866025i 1.08767 − 1.88389i
16.4 −0.758480 0.500000 − 0.866025i −1.42471 −0.357869 + 0.619848i −0.379240 + 0.656863i −1.32400 + 2.29064i 2.59757 −0.500000 − 0.866025i 0.271437 − 0.470142i
16.5 0.0680360 0.500000 − 0.866025i −1.99537 1.52954 − 2.64923i 0.0340180 − 0.0589209i 0.910236 + 2.48424i −0.271829 −0.500000 − 0.866025i 0.104063 − 0.180243i
16.6 1.06556 0.500000 − 0.866025i −0.864591 1.19023 − 2.06154i 0.532778 − 0.922798i −0.813611 − 2.51755i −3.05238 −0.500000 − 0.866025i 1.26826 − 2.19668i
16.7 2.05474 0.500000 − 0.866025i 2.22196 −0.274662 + 0.475728i 1.02737 − 1.77946i 2.59269 − 0.527227i 0.456078 −0.500000 − 0.866025i −0.564359 + 0.977499i
16.8 2.55056 0.500000 − 0.866025i 4.50537 −1.39351 + 2.41363i 1.27528 − 2.20885i −2.46231 − 0.968004i 6.39011 −0.500000 − 0.866025i −3.55423 + 6.15611i
256.1 −2.43414 0.500000 + 0.866025i 3.92506 −0.613891 − 1.06329i −1.21707 − 2.10803i 2.20121 − 1.46788i −4.68588 −0.500000 + 0.866025i 1.49430 + 2.58820i
256.2 −1.71502 0.500000 + 0.866025i 0.941295 1.22863 + 2.12806i −0.857510 − 1.48525i −2.38702 + 1.14112i 1.81570 −0.500000 + 0.866025i −2.10713 − 3.64966i
256.3 −0.831251 0.500000 + 0.866025i −1.30902 −1.30847 − 2.26634i −0.415625 − 0.719884i 1.78280 + 1.95490i 2.75063 −0.500000 + 0.866025i 1.08767 + 1.88389i
256.4 −0.758480 0.500000 + 0.866025i −1.42471 −0.357869 − 0.619848i −0.379240 − 0.656863i −1.32400 − 2.29064i 2.59757 −0.500000 + 0.866025i 0.271437 + 0.470142i
256.5 0.0680360 0.500000 + 0.866025i −1.99537 1.52954 + 2.64923i 0.0340180 + 0.0589209i 0.910236 − 2.48424i −0.271829 −0.500000 + 0.866025i 0.104063 + 0.180243i
256.6 1.06556 0.500000 + 0.866025i −0.864591 1.19023 + 2.06154i 0.532778 + 0.922798i −0.813611 + 2.51755i −3.05238 −0.500000 + 0.866025i 1.26826 + 2.19668i
256.7 2.05474 0.500000 + 0.866025i 2.22196 −0.274662 − 0.475728i 1.02737 + 1.77946i 2.59269 + 0.527227i 0.456078 −0.500000 + 0.866025i −0.564359 − 0.977499i
256.8 2.55056 0.500000 + 0.866025i 4.50537 −1.39351 − 2.41363i 1.27528 + 2.20885i −2.46231 + 0.968004i 6.39011 −0.500000 + 0.866025i −3.55423 − 6.15611i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.l.b yes 16
3.b odd 2 1 819.2.s.e 16
7.c even 3 1 273.2.j.b ✓ 16
13.c even 3 1 273.2.j.b ✓ 16
21.h odd 6 1 819.2.n.e 16
39.i odd 6 1 819.2.n.e 16
91.h even 3 1 inner 273.2.l.b yes 16
273.s odd 6 1 819.2.s.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.j.b ✓ 16 7.c even 3 1
273.2.j.b ✓ 16 13.c even 3 1
273.2.l.b yes 16 1.a even 1 1 trivial
273.2.l.b yes 16 91.h even 3 1 inner
819.2.n.e 16 21.h odd 6 1
819.2.n.e 16 39.i odd 6 1
819.2.s.e 16 3.b odd 2 1
819.2.s.e 16 273.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 11T_{2}^{6} - 2T_{2}^{5} + 34T_{2}^{4} + 13T_{2}^{3} - 26T_{2}^{2} - 13T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 11 T^{6} - 2 T^{5} + 34 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 19 T^{14} + 10 T^{13} + \cdots + 3969 \) Copy content Toggle raw display
$7$ \( T^{16} - T^{15} - 3 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 2 T^{15} + 45 T^{14} + \cdots + 1290496 \) Copy content Toggle raw display
$13$ \( T^{16} - 5 T^{15} + 6 T^{14} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( (T^{8} - 2 T^{7} - 68 T^{6} + 92 T^{5} + \cdots + 6257)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 11 T^{15} + 133 T^{14} + \cdots + 37161216 \) Copy content Toggle raw display
$23$ \( (T^{8} + 4 T^{7} - 50 T^{6} - 158 T^{5} + \cdots - 153)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 15 T^{15} + \cdots + 539772289 \) Copy content Toggle raw display
$31$ \( T^{16} - 3 T^{15} + \cdots + 1771652281 \) Copy content Toggle raw display
$37$ \( (T^{8} + 4 T^{7} - 119 T^{6} - 333 T^{5} + \cdots - 1999)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 19 T^{15} + \cdots + 8083371138129 \) Copy content Toggle raw display
$43$ \( T^{16} - 11 T^{15} + 209 T^{14} + \cdots + 99740169 \) Copy content Toggle raw display
$47$ \( T^{16} - 5 T^{15} + \cdots + 67950412929 \) Copy content Toggle raw display
$53$ \( T^{16} - 36 T^{15} + \cdots + 9921384931329 \) Copy content Toggle raw display
$59$ \( (T^{8} - 17 T^{7} - 66 T^{6} + \cdots + 441561)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 22 T^{15} + \cdots + 124397290000 \) Copy content Toggle raw display
$67$ \( T^{16} - 26 T^{15} + \cdots + 2264374886656 \) Copy content Toggle raw display
$71$ \( T^{16} - 9 T^{15} + \cdots + 83773776969 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 160073179912009 \) Copy content Toggle raw display
$79$ \( T^{16} - 16 T^{15} + \cdots + 131635449856 \) Copy content Toggle raw display
$83$ \( (T^{8} - 18 T^{7} - 196 T^{6} + \cdots - 5580400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 20 T^{7} - 82 T^{6} + \cdots + 683431)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 575206497472369 \) Copy content Toggle raw display
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