Properties

Label 798.2.bo.f
Level $798$
Weight $2$
Character orbit 798.bo
Analytic conductor $6.372$
Analytic rank $0$
Dimension $18$
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] = [798,2,Mod(43,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bo (of order \(9\), degree \(6\), 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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 27 x^{16} - 28 x^{15} + 333 x^{14} - 195 x^{13} + 2778 x^{12} + 519 x^{11} + 13116 x^{10} + 3193 x^{9} + 40191 x^{8} + 15663 x^{7} + 69675 x^{6} + 1134 x^{5} + \cdots + 1369 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} - \beta_{13} q^{3} + ( - \beta_{11} - \beta_{6}) q^{4} + (\beta_{13} + \beta_{8} + \beta_{7} + \beta_{5}) q^{5} + \beta_{6} q^{6} + \beta_{14} q^{7} + ( - \beta_{14} - 1) q^{8} + \beta_{11} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} - \beta_{13} q^{3} + ( - \beta_{11} - \beta_{6}) q^{4} + (\beta_{13} + \beta_{8} + \beta_{7} + \beta_{5}) q^{5} + \beta_{6} q^{6} + \beta_{14} q^{7} + ( - \beta_{14} - 1) q^{8} + \beta_{11} q^{9} + (\beta_{11} - \beta_{10}) q^{10} + ( - \beta_{14} - \beta_{12} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} - 1) q^{11} + \beta_{14} q^{12} + (\beta_{17} - \beta_{15} - \beta_{14} + \beta_{12} + \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 2 \beta_1) q^{13}+ \cdots + ( - \beta_{16} - \beta_{11} - \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 9 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 9 q^{7} - 9 q^{8} - 9 q^{11} - 9 q^{12} + 3 q^{13} - 6 q^{17} + 18 q^{18} + 3 q^{19} + 12 q^{20} - 6 q^{22} - 3 q^{23} - 12 q^{25} - 15 q^{26} - 9 q^{27} - 3 q^{29} - 6 q^{30} - 15 q^{31} + 3 q^{33} - 6 q^{34} - 3 q^{38} + 30 q^{39} - 24 q^{43} - 6 q^{44} - 6 q^{45} - 6 q^{46} - 9 q^{49} - 3 q^{50} + 12 q^{51} + 3 q^{52} - 39 q^{53} - 48 q^{55} + 18 q^{56} + 15 q^{57} + 24 q^{58} + 30 q^{59} + 21 q^{61} - 9 q^{64} - 12 q^{65} + 3 q^{66} + 3 q^{67} - 12 q^{68} - 6 q^{69} + 33 q^{71} + 3 q^{73} - 3 q^{74} + 6 q^{75} + 3 q^{76} + 18 q^{77} - 6 q^{78} - 3 q^{79} + 30 q^{83} - 9 q^{84} - 24 q^{85} - 6 q^{86} - 12 q^{87} - 9 q^{88} + 12 q^{89} + 3 q^{91} - 3 q^{92} + 9 q^{93} + 18 q^{94} - 24 q^{95} + 18 q^{96} + 63 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 27 x^{16} - 28 x^{15} + 333 x^{14} - 195 x^{13} + 2778 x^{12} + 519 x^{11} + 13116 x^{10} + 3193 x^{9} + 40191 x^{8} + 15663 x^{7} + 69675 x^{6} + 1134 x^{5} + \cdots + 1369 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!09 \nu^{17} + \cdots + 11\!\cdots\!24 ) / 44\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26\!\cdots\!99 \nu^{17} + \cdots - 24\!\cdots\!67 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31\!\cdots\!58 \nu^{17} + \cdots + 33\!\cdots\!10 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!14 \nu^{17} + \cdots + 58\!\cdots\!55 ) / 26\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44\!\cdots\!91 \nu^{17} + \cdots - 97\!\cdots\!67 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!31 \nu^{17} + \cdots - 83\!\cdots\!95 ) / 26\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 68\!\cdots\!20 \nu^{17} + \cdots - 60\!\cdots\!16 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!55 \nu^{17} + \cdots + 53\!\cdots\!82 ) / 26\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!18 \nu^{17} + \cdots + 69\!\cdots\!49 ) / 26\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!86 \nu^{17} + \cdots + 12\!\cdots\!41 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!43 \nu^{17} + \cdots - 14\!\cdots\!84 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!15 \nu^{17} + \cdots + 11\!\cdots\!64 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 31\!\cdots\!52 \nu^{17} + \cdots - 28\!\cdots\!48 ) / 16\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 32\!\cdots\!00 \nu^{17} + \cdots - 15\!\cdots\!16 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 58\!\cdots\!83 \nu^{17} + \cdots + 26\!\cdots\!32 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 58\!\cdots\!82 \nu^{17} + \cdots + 61\!\cdots\!45 ) / 97\!\cdots\!63 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} - \beta_{16} - 4 \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{15} + 2 \beta_{13} - 3 \beta_{11} - 3 \beta_{10} + 3 \beta_{8} - \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{3} - 7 \beta_{2} - \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{17} + 10 \beta_{16} - 10 \beta_{15} + 36 \beta_{14} + 6 \beta_{13} + 10 \beta_{12} - \beta_{11} - \beta_{10} - 13 \beta_{9} + 25 \beta_{8} - 14 \beta_{7} + 15 \beta_{6} - 3 \beta_{5} + 10 \beta_{4} - 2 \beta_{3} - 15 \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 21 \beta_{17} + 21 \beta_{16} + 3 \beta_{15} + 71 \beta_{14} + 10 \beta_{13} + 15 \beta_{12} + 10 \beta_{11} + 47 \beta_{10} - 49 \beta_{9} - 20 \beta_{8} - 32 \beta_{7} + 45 \beta_{6} + 17 \beta_{5} + 18 \beta_{3} + 64 \beta_{2} - 61 \beta _1 + 68 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 111 \beta_{17} + 32 \beta_{16} + 143 \beta_{15} + 76 \beta_{13} + 208 \beta_{11} + 185 \beta_{10} - 13 \beta_{9} - 208 \beta_{8} + 122 \beta_{7} + 33 \beta_{6} + 185 \beta_{5} - 109 \beta_{4} + 143 \beta_{3} + 95 \beta_{2} + \cdots + 511 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 63 \beta_{17} - 267 \beta_{16} + 267 \beta_{15} - 1027 \beta_{14} - 394 \beta_{13} - 224 \beta_{12} + 384 \beta_{11} + 30 \beta_{10} + 635 \beta_{9} - 458 \beta_{8} + 665 \beta_{7} - 234 \beta_{6} + 411 \beta_{5} - 224 \beta_{4} + \cdots + 330 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1729 \beta_{17} - 1729 \beta_{16} - 411 \beta_{15} - 4778 \beta_{14} - 2022 \beta_{13} - 1268 \beta_{12} - 2022 \beta_{11} - 2263 \beta_{10} + 2402 \beta_{9} - 675 \beta_{8} + 995 \beta_{7} - 2615 \beta_{6} - 1407 \beta_{5} + \cdots - 4367 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3710 \beta_{17} - 995 \beta_{16} - 4705 \beta_{15} + 1077 \beta_{13} - 8242 \beta_{11} - 8654 \beta_{10} + 337 \beta_{9} + 8242 \beta_{8} - 3547 \beta_{7} - 4287 \beta_{6} - 8654 \beta_{5} + 3210 \beta_{4} + \cdots - 17721 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5107 \beta_{17} + 16193 \beta_{16} - 16193 \beta_{15} + 58885 \beta_{14} + 17482 \beta_{13} + 15361 \beta_{12} - 8861 \beta_{11} - 1349 \beta_{10} - 29623 \beta_{9} + 39343 \beta_{8} - 30972 \beta_{7} + \cdots - 21300 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 64256 \beta_{17} + 64256 \beta_{16} + 14262 \beta_{15} + 186343 \beta_{14} + 59085 \beta_{13} + 44370 \beta_{12} + 59085 \beta_{11} + 108189 \beta_{10} - 111425 \beta_{9} - 4725 \beta_{8} + \cdots + 172081 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 202740 \beta_{17} + 63819 \beta_{16} + 266559 \beta_{15} + 68859 \beta_{13} + 414564 \beta_{11} + 398298 \beta_{10} - 11634 \beta_{9} - 414564 \beta_{8} + 202500 \beta_{7} + 122007 \beta_{6} + \cdots + 940318 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 195798 \beta_{17} - 662406 \beta_{16} + 662406 \beta_{15} - 2444796 \beta_{14} - 807990 \beta_{13} - 598827 \beta_{12} + 574722 \beta_{11} + 25170 \beta_{10} + 1405686 \beta_{9} - 1430922 \beta_{8} + \cdots + 858204 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3374179 \beta_{17} - 3374179 \beta_{16} - 806859 \beta_{15} - 9326644 \beta_{14} - 3649348 \beta_{13} - 2409931 \beta_{12} - 3649348 \beta_{11} - 5039308 \beta_{10} + 5116888 \beta_{9} + \cdots - 8519785 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 8689633 \beta_{17} - 2629377 \beta_{16} - 11319010 \beta_{15} - 1123351 \beta_{13} - 18194130 \beta_{11} - 18364044 \beta_{10} + 99087 \beta_{9} + 18194130 \beta_{8} + \cdots - 40510265 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 10302374 \beta_{17} + 32741764 \beta_{16} - 32741764 \beta_{15} + 118628349 \beta_{14} + 36199212 \beta_{13} + 30740350 \beta_{12} - 20694577 \beta_{11} - 91459 \beta_{10} + \cdots - 43044138 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 148183812 \beta_{17} + 148183812 \beta_{16} + 34878609 \beta_{15} + 412165457 \beta_{14} + 152199271 \beta_{13} + 104912784 \beta_{12} + 152199271 \beta_{11} + \cdots + 377286848 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(1\) \(-\beta_{6} - \beta_{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.973959 + 1.68695i
0.399298 + 0.691605i
−0.873258 1.51253i
−0.848502 + 1.46965i
−0.436629 + 0.756264i
1.78513 3.09194i
−0.848502 1.46965i
−0.436629 0.756264i
1.78513 + 3.09194i
1.79910 3.11613i
0.125533 0.217429i
−1.42463 + 2.46753i
0.973959 1.68695i
0.399298 0.691605i
−0.873258 + 1.51253i
1.79910 + 3.11613i
0.125533 + 0.217429i
−1.42463 2.46753i
−0.939693 + 0.342020i 0.173648 0.984808i 0.766044 0.642788i −0.726148 0.609310i 0.173648 + 0.984808i −0.500000 0.866025i −0.500000 + 0.866025i −0.939693 0.342020i 0.890752 + 0.324207i
43.2 −0.939693 + 0.342020i 0.173648 0.984808i 0.766044 0.642788i 0.154284 + 0.129460i 0.173648 + 0.984808i −0.500000 0.866025i −0.500000 + 0.866025i −0.939693 0.342020i −0.189257 0.0688840i
43.3 −0.939693 + 0.342020i 0.173648 0.984808i 0.766044 0.642788i 2.10395 + 1.76543i 0.173648 + 0.984808i −0.500000 0.866025i −0.500000 + 0.866025i −0.939693 0.342020i −2.58088 0.939364i
85.1 0.173648 + 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i −2.53436 0.922430i 0.766044 + 0.642788i −0.500000 + 0.866025i −0.500000 0.866025i 0.173648 0.984808i 0.468330 2.65603i
85.2 0.173648 + 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i −1.76029 0.640692i 0.766044 + 0.642788i −0.500000 + 0.866025i −0.500000 0.866025i 0.173648 0.984808i 0.325288 1.84480i
85.3 0.173648 + 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 2.41526 + 0.879082i 0.766044 + 0.642788i −0.500000 + 0.866025i −0.500000 0.866025i 0.173648 0.984808i −0.446322 + 2.53122i
169.1 0.173648 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i −2.53436 + 0.922430i 0.766044 0.642788i −0.500000 0.866025i −0.500000 + 0.866025i 0.173648 + 0.984808i 0.468330 + 2.65603i
169.2 0.173648 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i −1.76029 + 0.640692i 0.766044 0.642788i −0.500000 0.866025i −0.500000 + 0.866025i 0.173648 + 0.984808i 0.325288 + 1.84480i
169.3 0.173648 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 2.41526 0.879082i 0.766044 0.642788i −0.500000 0.866025i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.446322 2.53122i
253.1 0.766044 0.642788i −0.939693 0.342020i 0.173648 0.984808i −0.451172 2.55872i −0.939693 + 0.342020i −0.500000 + 0.866025i −0.500000 0.866025i 0.766044 + 0.642788i −1.99033 1.67009i
253.2 0.766044 0.642788i −0.939693 0.342020i 0.173648 0.984808i 0.130051 + 0.737556i −0.939693 + 0.342020i −0.500000 + 0.866025i −0.500000 0.866025i 0.766044 + 0.642788i 0.573717 + 0.481406i
253.3 0.766044 0.642788i −0.939693 0.342020i 0.173648 0.984808i 0.668417 + 3.79078i −0.939693 + 0.342020i −0.500000 + 0.866025i −0.500000 0.866025i 0.766044 + 0.642788i 2.94871 + 2.47426i
631.1 −0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i −0.726148 + 0.609310i 0.173648 0.984808i −0.500000 + 0.866025i −0.500000 0.866025i −0.939693 + 0.342020i 0.890752 0.324207i
631.2 −0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i 0.154284 0.129460i 0.173648 0.984808i −0.500000 + 0.866025i −0.500000 0.866025i −0.939693 + 0.342020i −0.189257 + 0.0688840i
631.3 −0.939693 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i 2.10395 1.76543i 0.173648 0.984808i −0.500000 + 0.866025i −0.500000 0.866025i −0.939693 + 0.342020i −2.58088 + 0.939364i
757.1 0.766044 + 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i −0.451172 + 2.55872i −0.939693 0.342020i −0.500000 0.866025i −0.500000 + 0.866025i 0.766044 0.642788i −1.99033 + 1.67009i
757.2 0.766044 + 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i 0.130051 0.737556i −0.939693 0.342020i −0.500000 0.866025i −0.500000 + 0.866025i 0.766044 0.642788i 0.573717 0.481406i
757.3 0.766044 + 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i 0.668417 3.79078i −0.939693 0.342020i −0.500000 0.866025i −0.500000 + 0.866025i 0.766044 0.642788i 2.94871 2.47426i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.bo.f 18
19.e even 9 1 inner 798.2.bo.f 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bo.f 18 1.a even 1 1 trivial
798.2.bo.f 18 19.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 6 T_{5}^{16} + 23 T_{5}^{15} - 66 T_{5}^{14} - 30 T_{5}^{13} + 581 T_{5}^{12} + 480 T_{5}^{11} + 2838 T_{5}^{10} - 4226 T_{5}^{9} - 23670 T_{5}^{8} + 35892 T_{5}^{7} + 162277 T_{5}^{6} + 147414 T_{5}^{5} + \cdots + 2601 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} + 6 T^{16} + 23 T^{15} + \cdots + 2601 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{18} + 9 T^{17} + 93 T^{16} + \cdots + 1172889 \) Copy content Toggle raw display
$13$ \( T^{18} - 3 T^{17} + 57 T^{16} + \cdots + 87759424 \) Copy content Toggle raw display
$17$ \( T^{18} + 6 T^{17} + 72 T^{16} + \cdots + 633075921 \) Copy content Toggle raw display
$19$ \( T^{18} - 3 T^{17} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + 3 T^{17} - 78 T^{16} + \cdots + 1418481 \) Copy content Toggle raw display
$29$ \( T^{18} + 3 T^{17} + \cdots + 5295181824 \) Copy content Toggle raw display
$31$ \( T^{18} + 15 T^{17} + 240 T^{16} + \cdots + 7789681 \) Copy content Toggle raw display
$37$ \( (T^{9} - 57 T^{7} + 4 T^{6} + 660 T^{5} + \cdots + 37)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + 72 T^{16} + \cdots + 1664742482001 \) Copy content Toggle raw display
$43$ \( T^{18} + 24 T^{17} + 264 T^{16} + \cdots + 64000000 \) Copy content Toggle raw display
$47$ \( T^{18} - 63 T^{16} + \cdots + 38021430487104 \) Copy content Toggle raw display
$53$ \( T^{18} + 39 T^{17} + \cdots + 2997856770624 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 788838225477696 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 826725348020224 \) Copy content Toggle raw display
$67$ \( T^{18} - 3 T^{17} + 108 T^{16} + \cdots + 51955264 \) Copy content Toggle raw display
$71$ \( T^{18} - 33 T^{17} + \cdots + 30582664641 \) Copy content Toggle raw display
$73$ \( T^{18} - 3 T^{17} + \cdots + 51\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{18} + 3 T^{17} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{18} - 30 T^{17} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{18} - 12 T^{17} + \cdots + 466689609 \) Copy content Toggle raw display
$97$ \( T^{18} - 63 T^{17} + \cdots + 86\!\cdots\!36 \) Copy content Toggle raw display
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