Properties

Label 775.2.o.e
Level $775$
Weight $2$
Character orbit 775.o
Analytic conductor $6.188$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(149,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.o (of order \(6\), degree \(2\), not 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 11x^{14} + 88x^{12} - 301x^{10} + 739x^{8} - 825x^{6} + 664x^{4} - 279x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{9} + \beta_{4}) q^{3} + (\beta_{8} - \beta_{7} - 1) q^{4} + (\beta_{7} - 2 \beta_{3} + \beta_{2} + 3) q^{6} + ( - \beta_{11} - \beta_{9} + 2 \beta_{6} + \beta_{4}) q^{7} + ( - \beta_{15} - 2 \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{9} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{5} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{9} + \beta_{4}) q^{3} + (\beta_{8} - \beta_{7} - 1) q^{4} + (\beta_{7} - 2 \beta_{3} + \beta_{2} + 3) q^{6} + ( - \beta_{11} - \beta_{9} + 2 \beta_{6} + \beta_{4}) q^{7} + ( - \beta_{15} - 2 \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{9} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{5} - \beta_{2}) q^{9} + ( - \beta_{10} + 2 \beta_{3} + \beta_{2}) q^{11} + (2 \beta_{9} - \beta_{6} - 2 \beta_{4}) q^{12} + (\beta_{15} - \beta_{13} - \beta_{12} - \beta_{6} + \beta_1) q^{13} + (\beta_{14} - \beta_{10} + 3 \beta_{7} + 3 \beta_{2} + 3) q^{14} + ( - \beta_{14} - 3 \beta_{8} + 1) q^{16} + (2 \beta_{11} + 4 \beta_{9} - \beta_{6} - 2 \beta_{4}) q^{17} + ( - \beta_{15} - 2 \beta_{13} + \beta_{12} + \beta_{6} + \beta_1) q^{18} + ( - 2 \beta_{14} + 2 \beta_{10} - \beta_{8} - \beta_{5} - 2 \beta_{3} + 2) q^{19} + (3 \beta_{10} - \beta_{5} - \beta_{3} - 4 \beta_{2}) q^{21} + (\beta_{15} - \beta_{13} - \beta_{12} - \beta_{6}) q^{22} + (2 \beta_{15} - 2 \beta_{11} - \beta_{4} + \beta_1) q^{23} + (4 \beta_{8} - \beta_{7} + 4 \beta_{5} - \beta_{3} - \beta_{2}) q^{24} + (\beta_{10} - 2 \beta_{5} - 3 \beta_{3}) q^{26} + (2 \beta_{15} - \beta_{13} - \beta_{12} - 2 \beta_{11} - \beta_{9} - \beta_{4} + \beta_1) q^{27} + ( - \beta_{11} - \beta_{9} + \beta_{6} - 4 \beta_{4}) q^{28} + ( - \beta_{8} + 2 \beta_{7} + 4) q^{29} + (\beta_{14} - \beta_{10} - 2 \beta_{8} - 3 \beta_{5} + 2 \beta_{3} + 1) q^{31} + (2 \beta_{15} + \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + \beta_{9} - 2 \beta_{4} + 2 \beta_1) q^{32} + ( - 2 \beta_{15} + 3 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{9}) q^{33} + ( - 2 \beta_{14} + 2 \beta_{10} + 2 \beta_{8} - 3 \beta_{7} + 2 \beta_{5} + 3 \beta_{3} + \cdots - 6) q^{34}+ \cdots + (\beta_{14} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{4} + 20 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{4} + 20 q^{6} + 2 q^{9} + 8 q^{11} + 16 q^{14} + 20 q^{16} + 10 q^{19} + 18 q^{21} - 4 q^{24} - 24 q^{26} + 52 q^{29} + 38 q^{31} - 48 q^{34} - 10 q^{36} + 44 q^{39} - 8 q^{41} + 14 q^{44} - 44 q^{46} + 74 q^{49} - 24 q^{51} - 48 q^{54} - 58 q^{56} - 12 q^{59} - 4 q^{61} - 68 q^{64} + 4 q^{66} + 12 q^{69} + 24 q^{71} - 36 q^{74} - 46 q^{76} + 4 q^{79} + 24 q^{81} - 82 q^{84} - 26 q^{86} - 108 q^{89} - 88 q^{91} + 96 q^{94} + 26 q^{96} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 11x^{14} + 88x^{12} - 301x^{10} + 739x^{8} - 825x^{6} + 664x^{4} - 279x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 917755 \nu^{14} - 18328739 \nu^{12} + 163971211 \nu^{10} - 917953111 \nu^{8} + 2500554337 \nu^{6} - 4578583074 \nu^{4} + 2362112770 \nu^{2} + \cdots - 1001553669 ) / 271834227 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1758328 \nu^{14} + 18577724 \nu^{12} - 147110953 \nu^{10} + 469720072 \nu^{8} - 1130331400 \nu^{6} + 1058930268 \nu^{4} - 1004915092 \nu^{2} + \cdots + 421407288 ) / 271834227 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1758328 \nu^{15} + 18577724 \nu^{13} - 147110953 \nu^{11} + 469720072 \nu^{9} - 1130331400 \nu^{7} + 1058930268 \nu^{5} + \cdots + 421407288 \nu ) / 271834227 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 631 \nu^{14} - 8228 \nu^{12} + 68533 \nu^{10} - 290239 \nu^{8} + 751165 \nu^{6} - 1123230 \nu^{4} + 692959 \nu^{2} - 292581 ) / 46731 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2598901 \nu^{15} - 18826709 \nu^{13} + 130250695 \nu^{11} - 21487033 \nu^{9} - 239891537 \nu^{7} + 2460722538 \nu^{5} - 352282586 \nu^{3} + \cdots + 158739093 \nu ) / 271834227 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1084578 \nu^{14} - 10939177 \nu^{12} + 84531352 \nu^{10} - 240053264 \nu^{8} + 511646207 \nu^{6} - 230883650 \nu^{4} + 98203608 \nu^{2} + \cdots - 23390109 ) / 30203803 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1169454 \nu^{14} - 11786056 \nu^{12} + 91146536 \nu^{10} - 258839152 \nu^{8} + 555167355 \nu^{6} - 248951950 \nu^{4} + 105888744 \nu^{2} + \cdots + 51396348 ) / 30203803 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17132116 \nu^{15} + 220028534 \nu^{13} - 1825849720 \nu^{11} + 7617723388 \nu^{9} - 19649290828 \nu^{7} + 29123514285 \nu^{5} + \cdots + 7638856452 \nu ) / 815502681 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3901087 \nu^{14} - 51196178 \nu^{12} + 426694102 \nu^{10} - 1819894351 \nu^{8} + 4716477541 \nu^{6} - 7176918369 \nu^{4} + 4353856693 \nu^{2} + \cdots - 1838501829 ) / 90611409 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17977667 \nu^{15} + 240737068 \nu^{13} - 2014397198 \nu^{11} + 8761325771 \nu^{9} - 22799007041 \nu^{7} + 35468751036 \nu^{5} + \cdots + 8907660009 \nu ) / 815502681 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7162301 \nu^{15} - 79625884 \nu^{13} + 630531473 \nu^{11} - 2138992343 \nu^{9} + 4844707400 \nu^{7} - 4538675388 \nu^{5} + 1236115058 \nu^{3} + \cdots - 641084301 \nu ) / 271834227 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5100315 \nu^{15} + 51371594 \nu^{13} - 397515460 \nu^{11} + 1128869720 \nu^{9} - 2429841908 \nu^{7} + 1085748875 \nu^{5} + \cdots - 238325848 \nu ) / 90611409 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3930861 \nu^{14} + 39585538 \nu^{12} - 306368924 \nu^{10} + 870030568 \nu^{8} - 1874674553 \nu^{6} + 836796925 \nu^{4} - 355921596 \nu^{2} + \cdots - 156725697 ) / 30203803 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 6692268 \nu^{15} - 67385020 \nu^{13} + 521591312 \nu^{11} - 1481221984 \nu^{9} + 3194181751 \nu^{7} - 1424641900 \nu^{5} + 605954448 \nu^{3} + \cdots + 231851243 \nu ) / 90611409 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{7} - \beta_{5} - 2\beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 2\beta_{13} + \beta_{12} - \beta_{11} + 2\beta_{9} - 5\beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - 9\beta_{5} - 9\beta_{3} + 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{11} + 17\beta_{9} - 6\beta_{6} - 30\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{14} + 67\beta_{8} - 36\beta_{7} - 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -77\beta_{15} - 124\beta_{13} - 36\beta_{12} - 36\beta_{6} - 193\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 77\beta_{14} - 77\beta_{10} + 471\beta_{8} - 229\beta_{7} + 471\beta_{5} + 350\beta_{3} - 229\beta_{2} - 579 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -548\beta_{15} - 865\beta_{13} - 229\beta_{12} + 548\beta_{11} - 865\beta_{9} + 1279\beta_{4} - 1279\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -548\beta_{10} + 3240\beta_{5} + 2329\beta_{3} - 1508\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3788\beta_{11} - 5932\beta_{9} + 1508\beta_{6} + 8585\beta_{4} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -3788\beta_{14} - 22093\beta_{8} + 10093\beta_{7} + 25755 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 25881\beta_{15} + 40398\beta_{13} + 10093\beta_{12} + 10093\beta_{6} + 57941\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 25881 \beta_{14} + 25881 \beta_{10} - 150101 \beta_{8} + 68034 \beta_{7} - 150101 \beta_{5} - 105789 \beta_{3} + 68034 \beta_{2} + 173823 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 175982 \beta_{15} + 274321 \beta_{13} + 68034 \beta_{12} - 175982 \beta_{11} + 274321 \beta_{9} - 391958 \beta_{4} + 391958 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−2.25437 1.30156i
−1.46240 0.844316i
−0.753430 0.434993i
−0.679376 0.392238i
0.679376 + 0.392238i
0.753430 + 0.434993i
1.46240 + 0.844316i
2.25437 + 1.30156i
2.25437 1.30156i
1.46240 0.844316i
0.753430 0.434993i
0.679376 0.392238i
−0.679376 + 0.392238i
−0.753430 + 0.434993i
−1.46240 + 0.844316i
−2.25437 + 1.30156i
2.60312i −1.38834 + 0.801561i −4.77625 0 2.08656 + 3.61403i −2.01898 + 1.16566i 7.22690i −0.215000 + 0.372390i 0
149.2 1.68863i −2.32842 + 1.34432i −0.851477 0 2.27005 + 3.93185i −4.37588 + 2.52642i 1.93943i 2.11437 3.66220i 0
149.3 0.869986i 0.112596 0.0650072i 1.24312 0 −0.0565553 0.0979567i 2.74842 1.58680i 2.82147i −1.49155 + 2.58344i 0
149.4 0.784476i −1.54540 + 0.892238i 1.38460 0 0.699939 + 1.21233i 4.23929 2.44756i 2.65514i 0.0921773 0.159656i 0
149.5 0.784476i 1.54540 0.892238i 1.38460 0 0.699939 + 1.21233i −4.23929 + 2.44756i 2.65514i 0.0921773 0.159656i 0
149.6 0.869986i −0.112596 + 0.0650072i 1.24312 0 −0.0565553 0.0979567i −2.74842 + 1.58680i 2.82147i −1.49155 + 2.58344i 0
149.7 1.68863i 2.32842 1.34432i −0.851477 0 2.27005 + 3.93185i 4.37588 2.52642i 1.93943i 2.11437 3.66220i 0
149.8 2.60312i 1.38834 0.801561i −4.77625 0 2.08656 + 3.61403i 2.01898 1.16566i 7.22690i −0.215000 + 0.372390i 0
749.1 2.60312i 1.38834 + 0.801561i −4.77625 0 2.08656 3.61403i 2.01898 + 1.16566i 7.22690i −0.215000 0.372390i 0
749.2 1.68863i 2.32842 + 1.34432i −0.851477 0 2.27005 3.93185i 4.37588 + 2.52642i 1.93943i 2.11437 + 3.66220i 0
749.3 0.869986i −0.112596 0.0650072i 1.24312 0 −0.0565553 + 0.0979567i −2.74842 1.58680i 2.82147i −1.49155 2.58344i 0
749.4 0.784476i 1.54540 + 0.892238i 1.38460 0 0.699939 1.21233i −4.23929 2.44756i 2.65514i 0.0921773 + 0.159656i 0
749.5 0.784476i −1.54540 0.892238i 1.38460 0 0.699939 1.21233i 4.23929 + 2.44756i 2.65514i 0.0921773 + 0.159656i 0
749.6 0.869986i 0.112596 + 0.0650072i 1.24312 0 −0.0565553 + 0.0979567i 2.74842 + 1.58680i 2.82147i −1.49155 2.58344i 0
749.7 1.68863i −2.32842 1.34432i −0.851477 0 2.27005 3.93185i −4.37588 2.52642i 1.93943i 2.11437 + 3.66220i 0
749.8 2.60312i −1.38834 0.801561i −4.77625 0 2.08656 3.61403i −2.01898 1.16566i 7.22690i −0.215000 0.372390i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.o.e 16
5.b even 2 1 inner 775.2.o.e 16
5.c odd 4 1 155.2.e.c 8
5.c odd 4 1 775.2.e.g 8
31.c even 3 1 inner 775.2.o.e 16
155.j even 6 1 inner 775.2.o.e 16
155.o odd 12 1 155.2.e.c 8
155.o odd 12 1 775.2.e.g 8
155.o odd 12 1 4805.2.a.i 4
155.p even 12 1 4805.2.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.e.c 8 5.c odd 4 1
155.2.e.c 8 155.o odd 12 1
775.2.e.g 8 5.c odd 4 1
775.2.e.g 8 155.o odd 12 1
775.2.o.e 16 1.a even 1 1 trivial
775.2.o.e 16 5.b even 2 1 inner
775.2.o.e 16 31.c even 3 1 inner
775.2.o.e 16 155.j even 6 1 inner
4805.2.a.i 4 155.o odd 12 1
4805.2.a.k 4 155.p even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\):

\( T_{2}^{8} + 11T_{2}^{6} + 33T_{2}^{4} + 31T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{16} - 65 T_{7}^{14} + 2791 T_{7}^{12} - 68818 T_{7}^{10} + 1230127 T_{7}^{8} - 13135494 T_{7}^{6} + 100719190 T_{7}^{4} - 408431844 T_{7}^{2} + 1121513121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 11 T^{6} + 33 T^{4} + 31 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 13 T^{14} + 119 T^{12} - 530 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 65 T^{14} + \cdots + 1121513121 \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + 20 T^{6} + 6 T^{5} + 37 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 33 T^{14} + 799 T^{12} + \cdots + 50625 \) Copy content Toggle raw display
$17$ \( T^{16} - 130 T^{14} + \cdots + 1138678933921 \) Copy content Toggle raw display
$19$ \( (T^{8} - 5 T^{7} + 52 T^{6} - 63 T^{5} + \cdots + 35721)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 75 T^{6} + 1509 T^{4} + \cdots + 24649)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 13 T^{3} + 29 T^{2} + 65 T - 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 19 T^{7} + 244 T^{6} + \cdots + 923521)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 124 T^{14} + 12830 T^{12} + \cdots + 2401 \) Copy content Toggle raw display
$41$ \( (T^{8} + 4 T^{7} + 77 T^{6} + 264 T^{5} + \cdots + 45369)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 117 T^{14} + \cdots + 1073283121 \) Copy content Toggle raw display
$47$ \( (T^{8} + 184 T^{6} + 8066 T^{4} + \cdots + 103041)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} - 179 T^{14} + \cdots + 5393580481 \) Copy content Toggle raw display
$59$ \( (T^{8} + 6 T^{7} + 165 T^{6} + 950 T^{5} + \cdots + 625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + T^{3} - 151 T^{2} - 257 T + 321)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} - 237 T^{14} + \cdots + 13059557667601 \) Copy content Toggle raw display
$71$ \( (T^{8} - 12 T^{7} + 175 T^{6} + \cdots + 164025)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 168 T^{14} + \cdots + 12745506816 \) Copy content Toggle raw display
$79$ \( (T^{8} - 2 T^{7} + 79 T^{6} + 214 T^{5} + \cdots + 80089)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 526 T^{14} + \cdots + 54\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{4} + 27 T^{3} + 206 T^{2} + 274 T - 459)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 471 T^{6} + 60113 T^{4} + \cdots + 4739329)^{2} \) Copy content Toggle raw display
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