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Properties

Label 399.2.bo.b
Level 399399
Weight 22
Character orbit 399.bo
Analytic conductor 3.1863.186
Analytic rank 00
Dimension 1818
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [399,2,Mod(43,399)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(399, 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("399.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 399=3⋅7⋅19 399 = 3 \cdot 7 \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 399.bo (of order 99, degree 66, 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: 3.186031040653.18603104065
Analytic rank: 00
Dimension: 1818
Relative dimension: 33 over Q(ζ9)\Q(\zeta_{9})
Coefficient field: Q[x]/(x18−⋯ )\mathbb{Q}[x]/(x^{18} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x18−3x17+18x16−41x15+177x14−369x13+1063x12−1788x11+⋯+1 x^{18} - 3 x^{17} + 18 x^{16} - 41 x^{15} + 177 x^{14} - 369 x^{13} + 1063 x^{12} - 1788 x^{11} + \cdots + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C9]\mathrm{SU}(2)[C_{9}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,…,β171,\beta_1,\ldots,\beta_{17} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β6q2+β13q3+(−β17+β16+⋯+β1)q4+(β17−β16−β14+⋯+1)q5+β5q6−β11q7+(β17−β14−β13+⋯−1)q8+⋯+(β17−2β16+⋯−β1)q99+O(q100) q + \beta_{6} q^{2} + \beta_{13} q^{3} + ( - \beta_{17} + \beta_{16} + \cdots + \beta_1) q^{4} + (\beta_{17} - \beta_{16} - \beta_{14} + \cdots + 1) q^{5} + \beta_{5} q^{6} - \beta_{11} q^{7} + (\beta_{17} - \beta_{14} - \beta_{13} + \cdots - 1) q^{8}+ \cdots + (\beta_{17} - 2 \beta_{16} + \cdots - \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 18q−3q2+3q4+12q5+3q6+9q7−6q8+6q10+12q11+9q12−6q14−12q15+27q16−3q17−6q18−3q19+3q22−18q23−21q24+⋯+3q99+O(q100) 18 q - 3 q^{2} + 3 q^{4} + 12 q^{5} + 3 q^{6} + 9 q^{7} - 6 q^{8} + 6 q^{10} + 12 q^{11} + 9 q^{12} - 6 q^{14} - 12 q^{15} + 27 q^{16} - 3 q^{17} - 6 q^{18} - 3 q^{19} + 3 q^{22} - 18 q^{23} - 21 q^{24}+ \cdots + 3 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x18−3x17+18x16−41x15+177x14−369x13+1063x12−1788x11+⋯+1 x^{18} - 3 x^{17} + 18 x^{16} - 41 x^{15} + 177 x^{14} - 369 x^{13} + 1063 x^{12} - 1788 x^{11} + \cdots + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (−50 ⁣⋯ ⁣53ν17+⋯+17 ⁣⋯ ⁣19)/27 ⁣⋯ ⁣94 ( - 50\!\cdots\!53 \nu^{17} + \cdots + 17\!\cdots\!19 ) / 27\!\cdots\!94 Copy content Toggle raw display
β3\beta_{3}== (−964152485141125ν17+⋯+23 ⁣⋯ ⁣56)/45 ⁣⋯ ⁣99 ( - 964152485141125 \nu^{17} + \cdots + 23\!\cdots\!56 ) / 45\!\cdots\!99 Copy content Toggle raw display
β4\beta_{4}== (21 ⁣⋯ ⁣03ν17+⋯+65 ⁣⋯ ⁣83)/27 ⁣⋯ ⁣94 ( 21\!\cdots\!03 \nu^{17} + \cdots + 65\!\cdots\!83 ) / 27\!\cdots\!94 Copy content Toggle raw display
β5\beta_{5}== (−13 ⁣⋯ ⁣07ν17+⋯−98 ⁣⋯ ⁣73)/27 ⁣⋯ ⁣94 ( - 13\!\cdots\!07 \nu^{17} + \cdots - 98\!\cdots\!73 ) / 27\!\cdots\!94 Copy content Toggle raw display
β6\beta_{6}== (15 ⁣⋯ ⁣63ν17+⋯−12 ⁣⋯ ⁣12)/27 ⁣⋯ ⁣94 ( 15\!\cdots\!63 \nu^{17} + \cdots - 12\!\cdots\!12 ) / 27\!\cdots\!94 Copy content Toggle raw display
β7\beta_{7}== (51 ⁣⋯ ⁣07ν17+⋯−20 ⁣⋯ ⁣08)/27 ⁣⋯ ⁣94 ( 51\!\cdots\!07 \nu^{17} + \cdots - 20\!\cdots\!08 ) / 27\!\cdots\!94 Copy content Toggle raw display
β8\beta_{8}== (65 ⁣⋯ ⁣83ν17+⋯+31 ⁣⋯ ⁣42)/27 ⁣⋯ ⁣94 ( 65\!\cdots\!83 \nu^{17} + \cdots + 31\!\cdots\!42 ) / 27\!\cdots\!94 Copy content Toggle raw display
β9\beta_{9}== (−69 ⁣⋯ ⁣33ν17+⋯+36 ⁣⋯ ⁣53)/27 ⁣⋯ ⁣94 ( - 69\!\cdots\!33 \nu^{17} + \cdots + 36\!\cdots\!53 ) / 27\!\cdots\!94 Copy content Toggle raw display
β10\beta_{10}== (−38 ⁣⋯ ⁣93ν17+⋯+28 ⁣⋯ ⁣00)/91 ⁣⋯ ⁣98 ( - 38\!\cdots\!93 \nu^{17} + \cdots + 28\!\cdots\!00 ) / 91\!\cdots\!98 Copy content Toggle raw display
β11\beta_{11}== (−23 ⁣⋯ ⁣56ν17+⋯−43 ⁣⋯ ⁣34)/45 ⁣⋯ ⁣99 ( - 23\!\cdots\!56 \nu^{17} + \cdots - 43\!\cdots\!34 ) / 45\!\cdots\!99 Copy content Toggle raw display
β12\beta_{12}== (−16 ⁣⋯ ⁣56ν17+⋯−18 ⁣⋯ ⁣32)/27 ⁣⋯ ⁣94 ( - 16\!\cdots\!56 \nu^{17} + \cdots - 18\!\cdots\!32 ) / 27\!\cdots\!94 Copy content Toggle raw display
β13\beta_{13}== (17 ⁣⋯ ⁣19ν17+⋯+99 ⁣⋯ ⁣80)/27 ⁣⋯ ⁣94 ( 17\!\cdots\!19 \nu^{17} + \cdots + 99\!\cdots\!80 ) / 27\!\cdots\!94 Copy content Toggle raw display
β14\beta_{14}== (−85 ⁣⋯ ⁣37ν17+⋯+71 ⁣⋯ ⁣05)/91 ⁣⋯ ⁣98 ( - 85\!\cdots\!37 \nu^{17} + \cdots + 71\!\cdots\!05 ) / 91\!\cdots\!98 Copy content Toggle raw display
β15\beta_{15}== (47 ⁣⋯ ⁣18ν17+⋯+80 ⁣⋯ ⁣89)/47 ⁣⋯ ⁣42 ( 47\!\cdots\!18 \nu^{17} + \cdots + 80\!\cdots\!89 ) / 47\!\cdots\!42 Copy content Toggle raw display
β16\beta_{16}== (−27 ⁣⋯ ⁣72ν17+⋯−16 ⁣⋯ ⁣02)/27 ⁣⋯ ⁣94 ( - 27\!\cdots\!72 \nu^{17} + \cdots - 16\!\cdots\!02 ) / 27\!\cdots\!94 Copy content Toggle raw display
β17\beta_{17}== (−27 ⁣⋯ ⁣71ν17+⋯−33 ⁣⋯ ⁣00)/27 ⁣⋯ ⁣94 ( - 27\!\cdots\!71 \nu^{17} + \cdots - 33\!\cdots\!00 ) / 27\!\cdots\!94 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β15−β12+2β11−β8−β7−β6+β5−β4−β2 \beta_{15} - \beta_{12} + 2\beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} Copy content Toggle raw display
ν3\nu^{3}== −β17+2β16+β15+β14+β13+β12−β10−2β9+⋯+1 - \beta_{17} + 2 \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - 2 \beta_{9} + \cdots + 1 Copy content Toggle raw display
ν4\nu^{4}== −β17+6β14−5β13−9β11−β9+5β8+5β7+⋯−9 - \beta_{17} + 6 \beta_{14} - 5 \beta_{13} - 9 \beta_{11} - \beta_{9} + 5 \beta_{8} + 5 \beta_{7} + \cdots - 9 Copy content Toggle raw display
ν5\nu^{5}== β17−15β16−6β15−8β12+9β11+8β10+7β9+⋯−36β1 \beta_{17} - 15 \beta_{16} - 6 \beta_{15} - 8 \beta_{12} + 9 \beta_{11} + 8 \beta_{10} + 7 \beta_{9} + \cdots - 36 \beta_1 Copy content Toggle raw display
ν6\nu^{6}== 11β17−20β16−34β15−34β14+25β13+25β12+⋯+46 11 \beta_{17} - 20 \beta_{16} - 34 \beta_{15} - 34 \beta_{14} + 25 \beta_{13} + 25 \beta_{12} + \cdots + 46 Copy content Toggle raw display
ν7\nu^{7}== 45β17−36β14−54β13−61β11−13β10+45β9+54β8+⋯−61 45 \beta_{17} - 36 \beta_{14} - 54 \beta_{13} - 61 \beta_{11} - 13 \beta_{10} + 45 \beta_{9} + 54 \beta_{8} + \cdots - 61 Copy content Toggle raw display
ν8\nu^{8}== −24β17+162β16+199β15+22β13−152β12+248β11+⋯+127β1 - 24 \beta_{17} + 162 \beta_{16} + 199 \beta_{15} + 22 \beta_{13} - 152 \beta_{12} + 248 \beta_{11} + \cdots + 127 \beta_1 Copy content Toggle raw display
ν9\nu^{9}== −403β17+691β16+233β15+233β14+343β13+343β12+⋯+378 - 403 \beta_{17} + 691 \beta_{16} + 233 \beta_{15} + 233 \beta_{14} + 343 \beta_{13} + 343 \beta_{12} + \cdots + 378 Copy content Toggle raw display
ν10\nu^{10}== −504β17+1209β14−873β13+180β12−1381β11+206β10+⋯−1381 - 504 \beta_{17} + 1209 \beta_{14} - 873 \beta_{13} + 180 \beta_{12} - 1381 \beta_{11} + 206 \beta_{10} + \cdots - 1381 Copy content Toggle raw display
ν11\nu^{11}== 882β17−4596β16−1587β15+28β13−2149β12+2247β11+⋯−8711β1 882 \beta_{17} - 4596 \beta_{16} - 1587 \beta_{15} + 28 \beta_{13} - 2149 \beta_{12} + 2247 \beta_{11} + \cdots - 8711 \beta_1 Copy content Toggle raw display
ν12\nu^{12}== 5163β17−8758β16−7559β15−7559β14+3744β13+3744β12+⋯+7859 5163 \beta_{17} - 8758 \beta_{16} - 7559 \beta_{15} - 7559 \beta_{14} + 3744 \beta_{13} + 3744 \beta_{12} + \cdots + 7859 Copy content Toggle raw display
ν13\nu^{13}== 12082β17−11061β14−13201β13+246β12−13048β11+⋯−13048 12082 \beta_{17} - 11061 \beta_{14} - 13201 \beta_{13} + 246 \beta_{12} - 13048 \beta_{11} + \cdots - 13048 Copy content Toggle raw display
ν14\nu^{14}== −11326β17+61866β16+48222β15+9220β13−29552β12+⋯+65974β1 - 11326 \beta_{17} + 61866 \beta_{16} + 48222 \beta_{15} + 9220 \beta_{13} - 29552 \beta_{12} + \cdots + 65974 \beta_1 Copy content Toggle raw display
ν15\nu^{15}== −123308β17+202564β16+77600β15+77600β14+78734β13+⋯+74626 - 123308 \beta_{17} + 202564 \beta_{16} + 77600 \beta_{15} + 77600 \beta_{14} + 78734 \beta_{13} + \cdots + 74626 Copy content Toggle raw display
ν16\nu^{16}== −175850β17+311931β14−172759β13+62490β12−265344β11+⋯−265344 - 175850 \beta_{17} + 311931 \beta_{14} - 172759 \beta_{13} + 62490 \beta_{12} - 265344 \beta_{11} + \cdots - 265344 Copy content Toggle raw display
ν17\nu^{17}== 300884β17−1347894β16−543853β15+9498β13−487563β12+⋯−2318046β1 300884 \beta_{17} - 1347894 \beta_{16} - 543853 \beta_{15} + 9498 \beta_{13} - 487563 \beta_{12} + \cdots - 2318046 \beta_1 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/399Z)×\left(\mathbb{Z}/399\mathbb{Z}\right)^\times.

nn 115115 134134 211211
χ(n)\chi(n) 11 11 β8+β12\beta_{8} + \beta_{12}

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
43.1
−1.15249 + 1.99616i
−0.0269184 + 0.0466240i
0.913360 − 1.58199i
1.30031 + 2.25221i
0.358173 + 0.620373i
−0.218795 − 0.378964i
1.30031 − 2.25221i
0.358173 − 0.620373i
−0.218795 + 0.378964i
−1.15249 − 1.99616i
−0.0269184 − 0.0466240i
0.913360 + 1.58199i
0.952616 + 1.64998i
0.585829 + 1.01469i
−1.21209 − 2.09941i
0.952616 − 1.64998i
0.585829 − 1.01469i
−1.21209 + 2.09941i
−2.16597 + 0.788347i −0.173648 + 0.984808i 2.53783 − 2.12949i −0.626322 − 0.525547i −0.400254 − 2.26995i 0.500000 + 0.866025i −1.51310 + 2.62076i −0.939693 − 0.342020i 1.77091 + 0.644557i
43.2 −0.0505900 + 0.0184132i −0.173648 + 0.984808i −1.52987 + 1.28371i 1.59327 + 1.33691i −0.00934865 − 0.0530188i 0.500000 + 0.866025i 0.107595 − 0.186361i −0.939693 − 0.342020i −0.105220 − 0.0382971i
43.3 1.71656 − 0.624775i −0.173648 + 0.984808i 1.02413 − 0.859347i 1.38035 + 1.15825i 0.317207 + 1.79897i 0.500000 + 0.866025i −0.605643 + 1.04900i −0.939693 − 0.342020i 3.09309 + 1.12579i
85.1 −0.451595 − 2.56112i −0.766044 + 0.642788i −4.47602 + 1.62914i 1.34902 + 0.491004i 1.99220 + 1.67165i 0.500000 − 0.866025i 3.59313 + 6.22348i 0.173648 − 0.984808i 0.648308 − 3.67674i
85.2 −0.124392 − 0.705462i −0.766044 + 0.642788i 1.39718 − 0.508532i −0.780211 − 0.283974i 0.548752 + 0.460458i 0.500000 − 0.866025i −1.24889 − 2.16315i 0.173648 − 0.984808i −0.103281 + 0.585733i
85.3 0.0759867 + 0.430942i −0.766044 + 0.642788i 1.69945 − 0.618549i 2.96328 + 1.07855i −0.335213 − 0.281277i 0.500000 − 0.866025i 0.833284 + 1.44329i 0.173648 − 0.984808i −0.239621 + 1.35896i
169.1 −0.451595 + 2.56112i −0.766044 − 0.642788i −4.47602 − 1.62914i 1.34902 − 0.491004i 1.99220 − 1.67165i 0.500000 + 0.866025i 3.59313 − 6.22348i 0.173648 + 0.984808i 0.648308 + 3.67674i
169.2 −0.124392 + 0.705462i −0.766044 − 0.642788i 1.39718 + 0.508532i −0.780211 + 0.283974i 0.548752 − 0.460458i 0.500000 + 0.866025i −1.24889 + 2.16315i 0.173648 + 0.984808i −0.103281 − 0.585733i
169.3 0.0759867 − 0.430942i −0.766044 − 0.642788i 1.69945 + 0.618549i 2.96328 − 1.07855i −0.335213 + 0.281277i 0.500000 + 0.866025i 0.833284 − 1.44329i 0.173648 + 0.984808i −0.239621 − 1.35896i
232.1 −2.16597 − 0.788347i −0.173648 − 0.984808i 2.53783 + 2.12949i −0.626322 + 0.525547i −0.400254 + 2.26995i 0.500000 − 0.866025i −1.51310 − 2.62076i −0.939693 + 0.342020i 1.77091 − 0.644557i
232.2 −0.0505900 − 0.0184132i −0.173648 − 0.984808i −1.52987 − 1.28371i 1.59327 − 1.33691i −0.00934865 + 0.0530188i 0.500000 − 0.866025i 0.107595 + 0.186361i −0.939693 + 0.342020i −0.105220 + 0.0382971i
232.3 1.71656 + 0.624775i −0.173648 − 0.984808i 1.02413 + 0.859347i 1.38035 − 1.15825i 0.317207 − 1.79897i 0.500000 − 0.866025i −0.605643 − 1.04900i −0.939693 + 0.342020i 3.09309 − 1.12579i
253.1 −1.45949 + 1.22466i 0.939693 + 0.342020i 0.283031 − 1.60515i 0.525672 + 2.98124i −1.79033 + 0.651628i 0.500000 − 0.866025i −0.352551 − 0.610637i 0.766044 + 0.642788i −4.41822 − 3.70732i
253.2 −0.897543 + 0.753128i 0.939693 + 0.342020i −0.108915 + 0.617687i −0.421473 − 2.39029i −1.10100 + 0.400731i 0.500000 − 0.866025i −1.53910 − 2.66580i 0.766044 + 0.642788i 2.17849 + 1.82797i
253.3 1.85704 − 1.55824i 0.939693 + 0.342020i 0.673180 − 3.81779i 0.0164156 + 0.0930973i 2.27799 − 0.829121i 0.500000 − 0.866025i −2.27472 − 3.93994i 0.766044 + 0.642788i 0.175552 + 0.147306i
358.1 −1.45949 − 1.22466i 0.939693 − 0.342020i 0.283031 + 1.60515i 0.525672 − 2.98124i −1.79033 − 0.651628i 0.500000 + 0.866025i −0.352551 + 0.610637i 0.766044 − 0.642788i −4.41822 + 3.70732i
358.2 −0.897543 − 0.753128i 0.939693 − 0.342020i −0.108915 − 0.617687i −0.421473 + 2.39029i −1.10100 − 0.400731i 0.500000 + 0.866025i −1.53910 + 2.66580i 0.766044 − 0.642788i 2.17849 − 1.82797i
358.3 1.85704 + 1.55824i 0.939693 − 0.342020i 0.673180 + 3.81779i 0.0164156 − 0.0930973i 2.27799 + 0.829121i 0.500000 + 0.866025i −2.27472 + 3.93994i 0.766044 − 0.642788i 0.175552 − 0.147306i
nn: 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 399.2.bo.b ✓ 18
19.e even 9 1 inner 399.2.bo.b ✓ 18
19.e even 9 1 7581.2.a.bg 9
19.f odd 18 1 7581.2.a.bh 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.bo.b ✓ 18 1.a even 1 1 trivial
399.2.bo.b ✓ 18 19.e even 9 1 inner
7581.2.a.bg 9 19.e even 9 1
7581.2.a.bh 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T218+3T217+3T216−2T215−6T214+54T213+298T212+⋯+1 T_{2}^{18} + 3 T_{2}^{17} + 3 T_{2}^{16} - 2 T_{2}^{15} - 6 T_{2}^{14} + 54 T_{2}^{13} + 298 T_{2}^{12} + \cdots + 1 acting on S2new(399,[χ])S_{2}^{\mathrm{new}}(399, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T18+3T17+⋯+1 T^{18} + 3 T^{17} + \cdots + 1 Copy content Toggle raw display
33 (T6−T3+1)3 (T^{6} - T^{3} + 1)^{3} Copy content Toggle raw display
55 T18−12T17+⋯+64 T^{18} - 12 T^{17} + \cdots + 64 Copy content Toggle raw display
77 (T2−T+1)9 (T^{2} - T + 1)^{9} Copy content Toggle raw display
1111 T18−12T17+⋯+2809 T^{18} - 12 T^{17} + \cdots + 2809 Copy content Toggle raw display
1313 T18−42T16+⋯+6713281 T^{18} - 42 T^{16} + \cdots + 6713281 Copy content Toggle raw display
1717 T18+⋯+186622921 T^{18} + \cdots + 186622921 Copy content Toggle raw display
1919 T18+⋯+322687697779 T^{18} + \cdots + 322687697779 Copy content Toggle raw display
2323 T18+18T17+⋯+14295961 T^{18} + 18 T^{17} + \cdots + 14295961 Copy content Toggle raw display
2929 T18+⋯+184065882841 T^{18} + \cdots + 184065882841 Copy content Toggle raw display
3131 T18+27T17+⋯+90649441 T^{18} + 27 T^{17} + \cdots + 90649441 Copy content Toggle raw display
3737 (T9−108T7+⋯−937)2 (T^{9} - 108 T^{7} + \cdots - 937)^{2} Copy content Toggle raw display
4141 T18+⋯+8135859601 T^{18} + \cdots + 8135859601 Copy content Toggle raw display
4343 T18+⋯+3524356664929 T^{18} + \cdots + 3524356664929 Copy content Toggle raw display
4747 T18+⋯+39689803729 T^{18} + \cdots + 39689803729 Copy content Toggle raw display
5353 T18+⋯+1696029568489 T^{18} + \cdots + 1696029568489 Copy content Toggle raw display
5959 T18+⋯+14933084401 T^{18} + \cdots + 14933084401 Copy content Toggle raw display
6161 T18+⋯+85 ⁣⋯ ⁣41 T^{18} + \cdots + 85\!\cdots\!41 Copy content Toggle raw display
6767 T18+⋯+10 ⁣⋯ ⁣96 T^{18} + \cdots + 10\!\cdots\!96 Copy content Toggle raw display
7171 T18+⋯+26484846273649 T^{18} + \cdots + 26484846273649 Copy content Toggle raw display
7373 T18+⋯+80577426579049 T^{18} + \cdots + 80577426579049 Copy content Toggle raw display
7979 T18+⋯+29 ⁣⋯ ⁣44 T^{18} + \cdots + 29\!\cdots\!44 Copy content Toggle raw display
8383 T18+⋯+17 ⁣⋯ ⁣69 T^{18} + \cdots + 17\!\cdots\!69 Copy content Toggle raw display
8989 T18+⋯+175843963220689 T^{18} + \cdots + 175843963220689 Copy content Toggle raw display
9797 T18+⋯+40179586755169 T^{18} + \cdots + 40179586755169 Copy content Toggle raw display
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