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Properties

Label 87.2.g.a
Level 8787
Weight 22
Character orbit 87.g
Analytic conductor 0.6950.695
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] = [87,2,Mod(7,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 87=3⋅29 87 = 3 \cdot 29
Weight: k k == 2 2
Character orbit: [χ][\chi] == 87.g (of order 77, 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: 0.6946984975850.694698497585
Analytic rank: 00
Dimension: 1818
Relative dimension: 33 over Q(ζ7)\Q(\zeta_{7})
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: x18−6x17+18x16−37x15+71x14−83x13+225x12−237x11+485x10+⋯+64 x^{18} - 6 x^{17} + 18 x^{16} - 37 x^{15} + 71 x^{14} - 83 x^{13} + 225 x^{12} - 237 x^{11} + 485 x^{10} + \cdots + 64 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)[C7]\mathrm{SU}(2)[C_{7}]

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+(β9−β3)q2−β10q3+(−β16+2β12+⋯+2β1)q4+(−β16+β12+⋯+β2)q5+(β7+β4)q6+⋯+(−2β12+2β10+⋯−2)q99+O(q100) q + (\beta_{9} - \beta_{3}) q^{2} - \beta_{10} q^{3} + ( - \beta_{16} + 2 \beta_{12} + \cdots + 2 \beta_1) q^{4} + ( - \beta_{16} + \beta_{12} + \cdots + \beta_{2}) q^{5} + (\beta_{7} + \beta_{4}) q^{6}+ \cdots + ( - 2 \beta_{12} + 2 \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 18q−4q2−3q3−6q4−q5−4q6−4q7−15q8−3q9−14q10+26q11+22q12+9q13−10q14−q15−14q16+4q17−4q18−10q19−q20+⋯−2q99+O(q100) 18 q - 4 q^{2} - 3 q^{3} - 6 q^{4} - q^{5} - 4 q^{6} - 4 q^{7} - 15 q^{8} - 3 q^{9} - 14 q^{10} + 26 q^{11} + 22 q^{12} + 9 q^{13} - 10 q^{14} - q^{15} - 14 q^{16} + 4 q^{17} - 4 q^{18} - 10 q^{19} - q^{20}+ \cdots - 2 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x18−6x17+18x16−37x15+71x14−83x13+225x12−237x11+485x10+⋯+64 x^{18} - 6 x^{17} + 18 x^{16} - 37 x^{15} + 71 x^{14} - 83 x^{13} + 225 x^{12} - 237 x^{11} + 485 x^{10} + \cdots + 64 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (−27 ⁣⋯ ⁣79ν17+⋯−31 ⁣⋯ ⁣20)/14 ⁣⋯ ⁣56 ( - 27\!\cdots\!79 \nu^{17} + \cdots - 31\!\cdots\!20 ) / 14\!\cdots\!56 Copy content Toggle raw display
β3\beta_{3}== (35 ⁣⋯ ⁣37ν17+⋯+67 ⁣⋯ ⁣32)/14 ⁣⋯ ⁣56 ( 35\!\cdots\!37 \nu^{17} + \cdots + 67\!\cdots\!32 ) / 14\!\cdots\!56 Copy content Toggle raw display
β4\beta_{4}== (62 ⁣⋯ ⁣91ν17+⋯−33 ⁣⋯ ⁣84)/14 ⁣⋯ ⁣56 ( 62\!\cdots\!91 \nu^{17} + \cdots - 33\!\cdots\!84 ) / 14\!\cdots\!56 Copy content Toggle raw display
β5\beta_{5}== (80 ⁣⋯ ⁣03ν17+⋯+26 ⁣⋯ ⁣20)/14 ⁣⋯ ⁣56 ( 80\!\cdots\!03 \nu^{17} + \cdots + 26\!\cdots\!20 ) / 14\!\cdots\!56 Copy content Toggle raw display
β6\beta_{6}== (−85 ⁣⋯ ⁣61ν17+⋯+36 ⁣⋯ ⁣20)/14 ⁣⋯ ⁣56 ( - 85\!\cdots\!61 \nu^{17} + \cdots + 36\!\cdots\!20 ) / 14\!\cdots\!56 Copy content Toggle raw display
β7\beta_{7}== (21 ⁣⋯ ⁣01ν17+⋯−39 ⁣⋯ ⁣56)/28 ⁣⋯ ⁣12 ( 21\!\cdots\!01 \nu^{17} + \cdots - 39\!\cdots\!56 ) / 28\!\cdots\!12 Copy content Toggle raw display
β8\beta_{8}== (−15 ⁣⋯ ⁣25ν17+⋯+20 ⁣⋯ ⁣56)/14 ⁣⋯ ⁣56 ( - 15\!\cdots\!25 \nu^{17} + \cdots + 20\!\cdots\!56 ) / 14\!\cdots\!56 Copy content Toggle raw display
β9\beta_{9}== (−31 ⁣⋯ ⁣54ν17+⋯−69 ⁣⋯ ⁣68)/14 ⁣⋯ ⁣56 ( - 31\!\cdots\!54 \nu^{17} + \cdots - 69\!\cdots\!68 ) / 14\!\cdots\!56 Copy content Toggle raw display
β10\beta_{10}== (81 ⁣⋯ ⁣35ν17+⋯−44 ⁣⋯ ⁣32)/28 ⁣⋯ ⁣12 ( 81\!\cdots\!35 \nu^{17} + \cdots - 44\!\cdots\!32 ) / 28\!\cdots\!12 Copy content Toggle raw display
β11\beta_{11}== (52 ⁣⋯ ⁣81ν17+⋯−17 ⁣⋯ ⁣96)/14 ⁣⋯ ⁣56 ( 52\!\cdots\!81 \nu^{17} + \cdots - 17\!\cdots\!96 ) / 14\!\cdots\!56 Copy content Toggle raw display
β12\beta_{12}== (56 ⁣⋯ ⁣55ν17+⋯−36 ⁣⋯ ⁣08)/14 ⁣⋯ ⁣56 ( 56\!\cdots\!55 \nu^{17} + \cdots - 36\!\cdots\!08 ) / 14\!\cdots\!56 Copy content Toggle raw display
β13\beta_{13}== (86 ⁣⋯ ⁣99ν17+⋯−47 ⁣⋯ ⁣08)/14 ⁣⋯ ⁣56 ( 86\!\cdots\!99 \nu^{17} + \cdots - 47\!\cdots\!08 ) / 14\!\cdots\!56 Copy content Toggle raw display
β14\beta_{14}== (−12 ⁣⋯ ⁣21ν17+⋯+82 ⁣⋯ ⁣88)/14 ⁣⋯ ⁣56 ( - 12\!\cdots\!21 \nu^{17} + \cdots + 82\!\cdots\!88 ) / 14\!\cdots\!56 Copy content Toggle raw display
β15\beta_{15}== (−14 ⁣⋯ ⁣27ν17+⋯+59 ⁣⋯ ⁣32)/14 ⁣⋯ ⁣56 ( - 14\!\cdots\!27 \nu^{17} + \cdots + 59\!\cdots\!32 ) / 14\!\cdots\!56 Copy content Toggle raw display
β16\beta_{16}== (17 ⁣⋯ ⁣45ν17+⋯−11 ⁣⋯ ⁣32)/14 ⁣⋯ ⁣56 ( 17\!\cdots\!45 \nu^{17} + \cdots - 11\!\cdots\!32 ) / 14\!\cdots\!56 Copy content Toggle raw display
β17\beta_{17}== (−39 ⁣⋯ ⁣75ν17+⋯+74 ⁣⋯ ⁣88)/28 ⁣⋯ ⁣12 ( - 39\!\cdots\!75 \nu^{17} + \cdots + 74\!\cdots\!88 ) / 28\!\cdots\!12 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}== β17+β16−β12+2β11−β7+β6−β5−1 \beta_{17} + \beta_{16} - \beta_{12} + 2\beta_{11} - \beta_{7} + \beta_{6} - \beta_{5} - 1 Copy content Toggle raw display
ν3\nu^{3}== 2β17+β16+β15−2β14+2β13−β12+2β10+⋯−1 2 \beta_{17} + \beta_{16} + \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - \beta_{12} + 2 \beta_{10} + \cdots - 1 Copy content Toggle raw display
ν4\nu^{4}== 3β17+β16+8β15−8β14+11β13−β12+8β10+⋯+5 3 \beta_{17} + \beta_{16} + 8 \beta_{15} - 8 \beta_{14} + 11 \beta_{13} - \beta_{12} + 8 \beta_{10} + \cdots + 5 Copy content Toggle raw display
ν5\nu^{5}== 4β17+21β15−13β14+34β13+2β12+2β11+9β10+⋯+19 4 \beta_{17} + 21 \beta_{15} - 13 \beta_{14} + 34 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 9 \beta_{10} + \cdots + 19 Copy content Toggle raw display
ν6\nu^{6}== −18β16+41β15−18β14+87β13+24β12+4β11+⋯+41 - 18 \beta_{16} + 41 \beta_{15} - 18 \beta_{14} + 87 \beta_{13} + 24 \beta_{12} + 4 \beta_{11} + \cdots + 41 Copy content Toggle raw display
ν7\nu^{7}== −63β17−143β16+63β15+143β13+173β12−30β10+⋯+106 - 63 \beta_{17} - 143 \beta_{16} + 63 \beta_{15} + 143 \beta_{13} + 173 \beta_{12} - 30 \beta_{10} + \cdots + 106 Copy content Toggle raw display
ν8\nu^{8}== −457β17−648β16+226β14+788β12−54β11−266β10+⋯+403 - 457 \beta_{17} - 648 \beta_{16} + 226 \beta_{14} + 788 \beta_{12} - 54 \beta_{11} - 266 \beta_{10} + \cdots + 403 Copy content Toggle raw display
ν9\nu^{9}== −2007β17−2007β16−754β15+1504β14−1504β13+2211β12+⋯+1111 - 2007 \beta_{17} - 2007 \beta_{16} - 754 \beta_{15} + 1504 \beta_{14} - 1504 \beta_{13} + 2211 \beta_{12} + \cdots + 1111 Copy content Toggle raw display
ν10\nu^{10}== −6360β17−4816β16−4816β15+6360β14−8884β13+4816β12+⋯+1012 - 6360 \beta_{17} - 4816 \beta_{16} - 4816 \beta_{15} + 6360 \beta_{14} - 8884 \beta_{13} + 4816 \beta_{12} + \cdots + 1012 Copy content Toggle raw display
ν11\nu^{11}== −15550β17−8236β16−20051β15+20051β14−35601β13+⋯−6409 - 15550 \beta_{17} - 8236 \beta_{16} - 20051 \beta_{15} + 20051 \beta_{14} - 35601 \beta_{13} + \cdots - 6409 Copy content Toggle raw display
ν12\nu^{12}== −26842β17−63696β15+49755β14−113451β13−10364β12+⋯−41106 - 26842 \beta_{17} - 63696 \beta_{15} + 49755 \beta_{14} - 113451 \beta_{13} - 10364 \beta_{12} + \cdots - 41106 Copy content Toggle raw display
ν13\nu^{13}== 86625β16−159543β15+86625β14−288853β13−125125β12+⋯−159543 86625 \beta_{16} - 159543 \beta_{15} + 86625 \beta_{14} - 288853 \beta_{13} - 125125 \beta_{12} + \cdots - 159543 Copy content Toggle raw display
ν14\nu^{14}== 279122β17+509987β16−279122β15−509987β13−633169β12+⋯−521858 279122 \beta_{17} + 509987 \beta_{16} - 279122 \beta_{15} - 509987 \beta_{13} - 633169 \beta_{12} + \cdots - 521858 Copy content Toggle raw display
ν15\nu^{15}== 1630935β17+2049943β16−895790β14−2371999β12+182724β11+⋯−1448211 1630935 \beta_{17} + 2049943 \beta_{16} - 895790 \beta_{14} - 2371999 \beta_{12} + 182724 \beta_{11} + \cdots - 1448211 Copy content Toggle raw display
ν16\nu^{16}== 6534172β17+6534172β16+2871755β15−5209837β14+5209837β13+⋯−2972357 6534172 \beta_{17} + 6534172 \beta_{16} + 2871755 \beta_{15} - 5209837 \beta_{14} + 5209837 \beta_{13} + \cdots - 2972357 Copy content Toggle raw display
ν17\nu^{17}== 20832904β17+16642785β16+16642785β15−20832904β14+30023648β13+⋯−1971815 20832904 \beta_{17} + 16642785 \beta_{16} + 16642785 \beta_{15} - 20832904 \beta_{14} + 30023648 \beta_{13} + \cdots - 1971815 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/87Z)×\left(\mathbb{Z}/87\mathbb{Z}\right)^\times.

nn 3131 5959
χ(n)\chi(n) β12\beta_{12} 11

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}
7.1
−1.38228 + 0.665671i
1.03105 − 0.496527i
2.87569 − 1.38486i
1.10857 + 1.39010i
0.102196 + 0.128149i
−1.05678 − 1.32516i
−1.38228 − 0.665671i
1.03105 + 0.496527i
2.87569 + 1.38486i
1.10857 − 1.39010i
0.102196 − 0.128149i
−1.05678 + 1.32516i
0.491931 + 2.15529i
0.183119 + 0.802295i
−0.353498 − 1.54877i
0.491931 − 2.15529i
0.183119 − 0.802295i
−0.353498 + 1.54877i
−0.563916 − 2.47068i −0.900969 − 0.433884i −3.98431 + 1.91874i −0.242440 − 1.06220i −0.563916 + 2.47068i −1.55919 − 0.750867i 3.82730 + 4.79928i 0.623490 + 0.781831i −2.48764 + 1.19798i
7.2 0.0321271 + 0.140758i −0.900969 − 0.433884i 1.78316 − 0.858723i −0.345850 − 1.51527i 0.0321271 − 0.140758i 1.37625 + 0.662766i 0.358196 + 0.449164i 0.623490 + 0.781831i 0.202175 − 0.0973624i
7.3 0.487716 + 2.13682i −0.900969 − 0.433884i −2.52621 + 1.21656i 0.533332 + 2.33668i 0.487716 − 2.13682i −1.21803 − 0.586571i −1.09855 − 1.37753i 0.623490 + 0.781831i −4.73296 + 2.27927i
16.1 −2.50290 − 1.20533i 0.623490 − 0.781831i 3.56470 + 4.46999i 2.28635 + 1.10105i −2.50290 + 1.20533i 0.527912 − 0.661981i −2.29793 − 10.0679i −0.222521 − 0.974928i −4.39538 − 5.51163i
16.2 −1.04865 − 0.505001i 0.623490 − 0.781831i −0.402348 − 0.504528i −2.80239 − 1.34956i −1.04865 + 0.505001i 1.08876 − 1.36527i 0.685121 + 3.00171i −0.222521 − 0.974928i 2.25719 + 2.83042i
16.3 0.626118 + 0.301523i 0.623490 − 0.781831i −0.945872 − 1.18609i 1.81798 + 0.875492i 0.626118 − 0.301523i −1.49319 + 1.87240i −0.543873 − 2.38286i −0.222521 − 0.974928i 0.874288 + 1.09632i
25.1 −0.563916 + 2.47068i −0.900969 + 0.433884i −3.98431 − 1.91874i −0.242440 + 1.06220i −0.563916 − 2.47068i −1.55919 + 0.750867i 3.82730 − 4.79928i 0.623490 − 0.781831i −2.48764 − 1.19798i
25.2 0.0321271 − 0.140758i −0.900969 + 0.433884i 1.78316 + 0.858723i −0.345850 + 1.51527i 0.0321271 + 0.140758i 1.37625 − 0.662766i 0.358196 − 0.449164i 0.623490 − 0.781831i 0.202175 + 0.0973624i
25.3 0.487716 − 2.13682i −0.900969 + 0.433884i −2.52621 − 1.21656i 0.533332 − 2.33668i 0.487716 + 2.13682i −1.21803 + 0.586571i −1.09855 + 1.37753i 0.623490 − 0.781831i −4.73296 − 2.27927i
49.1 −2.50290 + 1.20533i 0.623490 + 0.781831i 3.56470 − 4.46999i 2.28635 − 1.10105i −2.50290 − 1.20533i 0.527912 + 0.661981i −2.29793 + 10.0679i −0.222521 + 0.974928i −4.39538 + 5.51163i
49.2 −1.04865 + 0.505001i 0.623490 + 0.781831i −0.402348 + 0.504528i −2.80239 + 1.34956i −1.04865 − 0.505001i 1.08876 + 1.36527i 0.685121 − 3.00171i −0.222521 + 0.974928i 2.25719 − 2.83042i
49.3 0.626118 − 0.301523i 0.623490 + 0.781831i −0.945872 + 1.18609i 1.81798 − 0.875492i 0.626118 + 0.301523i −1.49319 − 1.87240i −0.543873 + 2.38286i −0.222521 + 0.974928i 0.874288 − 1.09632i
52.1 −0.754870 + 0.946578i −0.222521 + 0.974928i 0.118862 + 0.520769i 1.12131 − 1.40607i −0.754870 − 0.946578i −0.951706 + 4.16970i −2.76431 − 1.33122i −0.900969 − 0.433884i 0.484517 + 2.12281i
52.2 0.110403 − 0.138441i −0.222521 + 0.974928i 0.438065 + 1.91929i −2.15193 + 2.69844i 0.110403 + 0.138441i 0.844514 − 3.70006i 0.633146 + 0.304907i −0.900969 − 0.433884i 0.135995 + 0.595831i
52.3 1.61397 − 2.02385i −0.222521 + 0.974928i −1.04604 − 4.58301i −0.716354 + 0.898279i 1.61397 + 2.02385i −0.615328 + 2.69593i −6.29911 − 3.03349i −0.900969 − 0.433884i 0.661812 + 2.89959i
82.1 −0.754870 − 0.946578i −0.222521 − 0.974928i 0.118862 − 0.520769i 1.12131 + 1.40607i −0.754870 + 0.946578i −0.951706 − 4.16970i −2.76431 + 1.33122i −0.900969 + 0.433884i 0.484517 − 2.12281i
82.2 0.110403 + 0.138441i −0.222521 − 0.974928i 0.438065 − 1.91929i −2.15193 − 2.69844i 0.110403 − 0.138441i 0.844514 + 3.70006i 0.633146 − 0.304907i −0.900969 + 0.433884i 0.135995 − 0.595831i
82.3 1.61397 + 2.02385i −0.222521 − 0.974928i −1.04604 + 4.58301i −0.716354 − 0.898279i 1.61397 − 2.02385i −0.615328 − 2.69593i −6.29911 + 3.03349i −0.900969 + 0.433884i 0.661812 − 2.89959i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.2.g.a ✓ 18
3.b odd 2 1 261.2.k.c 18
29.d even 7 1 inner 87.2.g.a ✓ 18
29.d even 7 1 2523.2.a.r 9
29.e even 14 1 2523.2.a.o 9
87.h odd 14 1 7569.2.a.bm 9
87.j odd 14 1 261.2.k.c 18
87.j odd 14 1 7569.2.a.bj 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.g.a ✓ 18 1.a even 1 1 trivial
87.2.g.a ✓ 18 29.d even 7 1 inner
261.2.k.c 18 3.b odd 2 1
261.2.k.c 18 87.j odd 14 1
2523.2.a.o 9 29.e even 14 1
2523.2.a.r 9 29.d even 7 1
7569.2.a.bj 9 87.j odd 14 1
7569.2.a.bm 9 87.h odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T218+4T217+14T216+45T215+126T214+309T213+828T212+⋯+1 T_{2}^{18} + 4 T_{2}^{17} + 14 T_{2}^{16} + 45 T_{2}^{15} + 126 T_{2}^{14} + 309 T_{2}^{13} + 828 T_{2}^{12} + \cdots + 1 acting on S2new(87,[χ])S_{2}^{\mathrm{new}}(87, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T18+4T17+⋯+1 T^{18} + 4 T^{17} + \cdots + 1 Copy content Toggle raw display
33 (T6+T5+T4+⋯+1)3 (T^{6} + T^{5} + T^{4} + \cdots + 1)^{3} Copy content Toggle raw display
55 T18+T17+⋯+212521 T^{18} + T^{17} + \cdots + 212521 Copy content Toggle raw display
77 T18+4T17+⋯+322624 T^{18} + 4 T^{17} + \cdots + 322624 Copy content Toggle raw display
1111 T18−26T17+⋯+1236544 T^{18} - 26 T^{17} + \cdots + 1236544 Copy content Toggle raw display
1313 T18+⋯+2180609809 T^{18} + \cdots + 2180609809 Copy content Toggle raw display
1717 (T9−2T8+⋯−86143)2 (T^{9} - 2 T^{8} + \cdots - 86143)^{2} Copy content Toggle raw display
1919 T18+⋯+589467701824 T^{18} + \cdots + 589467701824 Copy content Toggle raw display
2323 T18+⋯+880427584 T^{18} + \cdots + 880427584 Copy content Toggle raw display
2929 T18+⋯+14507145975869 T^{18} + \cdots + 14507145975869 Copy content Toggle raw display
3131 T18+⋯+177848896 T^{18} + \cdots + 177848896 Copy content Toggle raw display
3737 T18+⋯+18850466209 T^{18} + \cdots + 18850466209 Copy content Toggle raw display
4141 (T9−12T8+⋯−3560291)2 (T^{9} - 12 T^{8} + \cdots - 3560291)^{2} Copy content Toggle raw display
4343 T18+⋯+271326784 T^{18} + \cdots + 271326784 Copy content Toggle raw display
4747 T18−5T17+⋯+817216 T^{18} - 5 T^{17} + \cdots + 817216 Copy content Toggle raw display
5353 T18+⋯+5317548372361 T^{18} + \cdots + 5317548372361 Copy content Toggle raw display
5959 (T9+16T8+⋯+897224)2 (T^{9} + 16 T^{8} + \cdots + 897224)^{2} Copy content Toggle raw display
6161 T18+⋯+522207124321 T^{18} + \cdots + 522207124321 Copy content Toggle raw display
6767 T18+⋯+2088855616 T^{18} + \cdots + 2088855616 Copy content Toggle raw display
7171 T18+⋯+80 ⁣⋯ ⁣04 T^{18} + \cdots + 80\!\cdots\!04 Copy content Toggle raw display
7373 T18+⋯+61148882089 T^{18} + \cdots + 61148882089 Copy content Toggle raw display
7979 T18+⋯+8744494144 T^{18} + \cdots + 8744494144 Copy content Toggle raw display
8383 T18+⋯+645862966336 T^{18} + \cdots + 645862966336 Copy content Toggle raw display
8989 T18+⋯+21 ⁣⋯ ⁣41 T^{18} + \cdots + 21\!\cdots\!41 Copy content Toggle raw display
9797 T18+⋯+40807578862561 T^{18} + \cdots + 40807578862561 Copy content Toggle raw display
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