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Properties

Label 44.2.g.a
Level 4444
Weight 22
Character orbit 44.g
Analytic conductor 0.3510.351
Analytic rank 00
Dimension 1616
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [44,2,Mod(7,44)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(44, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("44.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 44=22⋅11 44 = 2^{2} \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 44.g (of order 1010, degree 44, 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: 0.3513417688940.351341768894
Analytic rank: 00
Dimension: 1616
Relative dimension: 44 over Q(ζ10)\Q(\zeta_{10})
Coefficient field: Q[x]/(x16−⋯ )\mathbb{Q}[x]/(x^{16} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x16−5x15+13x14−25x13+35x12−30x11−2x10+60x9−116x8+⋯+256 x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 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)[C10]\mathrm{SU}(2)[C_{10}]

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,…,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β2q2+(−β14−β13−β2+⋯−1)q3+(β13+β12+⋯−β2)q4+(β15−β12+β11+⋯−1)q5+(−β15+β14+⋯−β1)q6+⋯+(3β15+8β14+⋯+4)q99+O(q100) q + \beta_{2} q^{2} + ( - \beta_{14} - \beta_{13} - \beta_{2} + \cdots - 1) q^{3} + (\beta_{13} + \beta_{12} + \cdots - \beta_{2}) q^{4} + (\beta_{15} - \beta_{12} + \beta_{11} + \cdots - 1) q^{5} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_1) q^{6}+ \cdots + (3 \beta_{15} + 8 \beta_{14} + \cdots + 4) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q−5q2−q4−6q5−5q6−5q8−10q9−22q12−10q13+8q14+23q16−10q17+20q18+16q20+17q22+25q24+6q25−4q26+20q28+⋯−68q97+O(q100) 16 q - 5 q^{2} - q^{4} - 6 q^{5} - 5 q^{6} - 5 q^{8} - 10 q^{9} - 22 q^{12} - 10 q^{13} + 8 q^{14} + 23 q^{16} - 10 q^{17} + 20 q^{18} + 16 q^{20} + 17 q^{22} + 25 q^{24} + 6 q^{25} - 4 q^{26} + 20 q^{28}+ \cdots - 68 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16−5x15+13x14−25x13+35x12−30x11−2x10+60x9−116x8+⋯+256 x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 : Copy content Toggle raw display

β1\beta_{1}== −ν -\nu Copy content Toggle raw display
β2\beta_{2}== (ν15−5ν14+13ν13−25ν12+35ν11−30ν10−2ν9+60ν8+⋯−640)/128 ( \nu^{15} - 5 \nu^{14} + 13 \nu^{13} - 25 \nu^{12} + 35 \nu^{11} - 30 \nu^{10} - 2 \nu^{9} + 60 \nu^{8} + \cdots - 640 ) / 128 Copy content Toggle raw display
β3\beta_{3}== (−35ν15+94ν14−210ν13+334ν12−328ν11+111ν10+476ν9+⋯+2304)/64 ( - 35 \nu^{15} + 94 \nu^{14} - 210 \nu^{13} + 334 \nu^{12} - 328 \nu^{11} + 111 \nu^{10} + 476 \nu^{9} + \cdots + 2304 ) / 64 Copy content Toggle raw display
β4\beta_{4}== (−30ν15+141ν14−309ν13+573ν12−693ν11+429ν10+398ν9+⋯+9664)/64 ( - 30 \nu^{15} + 141 \nu^{14} - 309 \nu^{13} + 573 \nu^{12} - 693 \nu^{11} + 429 \nu^{10} + 398 \nu^{9} + \cdots + 9664 ) / 64 Copy content Toggle raw display
β5\beta_{5}== (33ν15−147ν14+323ν13−591ν12+705ν11−424ν10−446ν9+⋯−9472)/64 ( 33 \nu^{15} - 147 \nu^{14} + 323 \nu^{13} - 591 \nu^{12} + 705 \nu^{11} - 424 \nu^{10} - 446 \nu^{9} + \cdots - 9472 ) / 64 Copy content Toggle raw display
β6\beta_{6}== (−37ν15+152ν14−334ν13+602ν12−704ν11+405ν10+498ν9+⋯+9088)/64 ( - 37 \nu^{15} + 152 \nu^{14} - 334 \nu^{13} + 602 \nu^{12} - 704 \nu^{11} + 405 \nu^{10} + 498 \nu^{9} + \cdots + 9088 ) / 64 Copy content Toggle raw display
β7\beta_{7}== (35ν15−120ν14+265ν13−455ν12+501ν11−248ν10−475ν9+⋯−5536)/32 ( 35 \nu^{15} - 120 \nu^{14} + 265 \nu^{13} - 455 \nu^{12} + 501 \nu^{11} - 248 \nu^{10} - 475 \nu^{9} + \cdots - 5536 ) / 32 Copy content Toggle raw display
β8\beta_{8}== (−47ν15+153ν14−338ν13+572ν12−618ν11+291ν10+633ν9+⋯+6464)/32 ( - 47 \nu^{15} + 153 \nu^{14} - 338 \nu^{13} + 572 \nu^{12} - 618 \nu^{11} + 291 \nu^{10} + 633 \nu^{9} + \cdots + 6464 ) / 32 Copy content Toggle raw display
β9\beta_{9}== (71ν15−281ν14+619ν13−1107ν12+1281ν11−722ν10−952ν9+⋯−16128)/64 ( 71 \nu^{15} - 281 \nu^{14} + 619 \nu^{13} - 1107 \nu^{12} + 1281 \nu^{11} - 722 \nu^{10} - 952 \nu^{9} + \cdots - 16128 ) / 64 Copy content Toggle raw display
β10\beta_{10}== (−279ν15+987ν14−2179ν13+3775ν12−4213ν11+2162ν10+⋯+48384)/128 ( - 279 \nu^{15} + 987 \nu^{14} - 2179 \nu^{13} + 3775 \nu^{12} - 4213 \nu^{11} + 2162 \nu^{10} + \cdots + 48384 ) / 128 Copy content Toggle raw display
β11\beta_{11}== (307ν15−1097ν14+2421ν13−4209ν12+4711ν11−2444ν10+⋯−54528)/128 ( 307 \nu^{15} - 1097 \nu^{14} + 2421 \nu^{13} - 4209 \nu^{12} + 4711 \nu^{11} - 2444 \nu^{10} + \cdots - 54528 ) / 128 Copy content Toggle raw display
β12\beta_{12}== (−293ν15+1079ν14−2383ν13+4179ν12−4733ν11+2528ν10+⋯+56320)/128 ( - 293 \nu^{15} + 1079 \nu^{14} - 2383 \nu^{13} + 4179 \nu^{12} - 4733 \nu^{11} + 2528 \nu^{10} + \cdots + 56320 ) / 128 Copy content Toggle raw display
β13\beta_{13}== (162ν15−591ν14+1304ν13−2280ν12+2572ν11−1360ν10+⋯−30400)/32 ( 162 \nu^{15} - 591 \nu^{14} + 1304 \nu^{13} - 2280 \nu^{12} + 2572 \nu^{11} - 1360 \nu^{10} + \cdots - 30400 ) / 32 Copy content Toggle raw display
β14\beta_{14}== (−361ν15+1262ν14−2786ν13+4814ν12−5348ν11+2721ν10+⋯+60672)/64 ( - 361 \nu^{15} + 1262 \nu^{14} - 2786 \nu^{13} + 4814 \nu^{12} - 5348 \nu^{11} + 2721 \nu^{10} + \cdots + 60672 ) / 64 Copy content Toggle raw display
β15\beta_{15}== (−785ν15+2815ν14−6215ν13+10819ν12−12133ν11+6324ν10+⋯+141056)/128 ( - 785 \nu^{15} + 2815 \nu^{14} - 6215 \nu^{13} + 10819 \nu^{12} - 12133 \nu^{11} + 6324 \nu^{10} + \cdots + 141056 ) / 128 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}== β14+β13+β11+β10−β9−β8+β1 \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_1 Copy content Toggle raw display
ν3\nu^{3}== β15+β13+β12−β10+2β9+β6+β4+β2+2 \beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + 2\beta_{9} + \beta_{6} + \beta_{4} + \beta_{2} + 2 Copy content Toggle raw display
ν4\nu^{4}== −β15+β13+3β12−β11−β10+2β8−β7+β6+⋯+4 - \beta_{15} + \beta_{13} + 3 \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 4 Copy content Toggle raw display
ν5\nu^{5}== β13−2β11+3β6−2β4−β3−2β2−2β1 \beta_{13} - 2\beta_{11} + 3\beta_{6} - 2\beta_{4} - \beta_{3} - 2\beta_{2} - 2\beta_1 Copy content Toggle raw display
ν6\nu^{6}== β14−2β12+4β11+β10−5β9+β8−β7+⋯+2β1 \beta_{14} - 2 \beta_{12} + 4 \beta_{11} + \beta_{10} - 5 \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 2 \beta_1 Copy content Toggle raw display
ν7\nu^{7}== 6β15−3β14+5β13−β11+β10+5β9+3β8+⋯+10 6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - \beta_{11} + \beta_{10} + 5 \beta_{9} + 3 \beta_{8} + \cdots + 10 Copy content Toggle raw display
ν8\nu^{8}== −5β15−2β14−β13+3β12+2β11−β10−8β9+⋯+6 - 5 \beta_{15} - 2 \beta_{14} - \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - \beta_{10} - 8 \beta_{9} + \cdots + 6 Copy content Toggle raw display
ν9\nu^{9}== −7β15+6β14+3β13+β12+3β11+3β10−6β9+⋯+4 - 7 \beta_{15} + 6 \beta_{14} + 3 \beta_{13} + \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + \cdots + 4 Copy content Toggle raw display
ν10\nu^{10}== −2β15+14β14+9β13−2β12−2β11+2β9+⋯+2β1 - 2 \beta_{15} + 14 \beta_{14} + 9 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{9} + \cdots + 2 \beta_1 Copy content Toggle raw display
ν11\nu^{11}== 26β15+21β14+40β13−24β12−20β11+19β10+⋯+28 26 \beta_{15} + 21 \beta_{14} + 40 \beta_{13} - 24 \beta_{12} - 20 \beta_{11} + 19 \beta_{10} + \cdots + 28 Copy content Toggle raw display
ν12\nu^{12}== 20β15−β14−β13−30β12+β11−23β10−21β9+⋯+10 20 \beta_{15} - \beta_{14} - \beta_{13} - 30 \beta_{12} + \beta_{11} - 23 \beta_{10} - 21 \beta_{9} + \cdots + 10 Copy content Toggle raw display
ν13\nu^{13}== −21β15+8β14+11β13+31β12+24β11−11β10+⋯+50 - 21 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 31 \beta_{12} + 24 \beta_{11} - 11 \beta_{10} + \cdots + 50 Copy content Toggle raw display
ν14\nu^{14}== 7β15+45β13+19β12−33β11−9β10+52β9+62β8+⋯+32 7 \beta_{15} + 45 \beta_{13} + 19 \beta_{12} - 33 \beta_{11} - 9 \beta_{10} + 52 \beta_{9} + 62 \beta_{8} + \cdots + 32 Copy content Toggle raw display
ν15\nu^{15}== 20β15+20β14+61β13−56β12+22β11+40β10+⋯+68 20 \beta_{15} + 20 \beta_{14} + 61 \beta_{13} - 56 \beta_{12} + 22 \beta_{11} + 40 \beta_{10} + \cdots + 68 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/44Z)×\left(\mathbb{Z}/44\mathbb{Z}\right)^\times.

nn 1313 2323
χ(n)\chi(n) −β12-\beta_{12} −1-1

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
1.40958 − 0.114404i
0.0737040 − 1.41229i
−0.544389 + 1.30524i
−1.36594 − 0.366325i
1.40958 + 0.114404i
0.0737040 + 1.41229i
−0.544389 − 1.30524i
−1.36594 + 0.366325i
1.40874 + 0.124276i
1.06665 − 0.928579i
0.656642 + 1.25253i
−0.204982 − 1.39928i
1.40874 − 0.124276i
1.06665 + 0.928579i
0.656642 − 1.25253i
−0.204982 + 1.39928i
−1.40958 − 0.114404i 0.704424 + 0.228881i 1.97382 + 0.322523i 1.09089 − 0.792578i −0.966756 − 0.403215i 0.503194 + 1.54867i −2.74536 − 0.680436i −1.98322 − 1.44090i −1.62837 + 0.992398i
7.2 −0.0737040 − 1.41229i 1.70537 + 0.554109i −1.98914 + 0.208183i −2.39991 + 1.74363i 0.656871 − 2.44932i −0.815620 − 2.51022i 0.440622 + 2.79390i 0.174207 + 0.126569i 2.63940 + 3.26086i
7.3 0.544389 + 1.30524i −0.704424 − 0.228881i −1.40728 + 1.42111i 1.09089 − 0.792578i −0.0847364 − 1.04404i −0.503194 − 1.54867i −2.62099 − 1.06320i −1.98322 − 1.44090i 1.62837 + 0.992398i
7.4 1.36594 − 0.366325i −1.70537 − 0.554109i 1.73161 − 1.00076i −2.39991 + 1.74363i −2.53243 − 0.132161i 0.815620 + 2.51022i 1.99868 − 2.00132i 0.174207 + 0.126569i −2.63940 + 3.26086i
19.1 −1.40958 + 0.114404i 0.704424 − 0.228881i 1.97382 − 0.322523i 1.09089 + 0.792578i −0.966756 + 0.403215i 0.503194 − 1.54867i −2.74536 + 0.680436i −1.98322 + 1.44090i −1.62837 − 0.992398i
19.2 −0.0737040 + 1.41229i 1.70537 − 0.554109i −1.98914 − 0.208183i −2.39991 − 1.74363i 0.656871 + 2.44932i −0.815620 + 2.51022i 0.440622 − 2.79390i 0.174207 − 0.126569i 2.63940 − 3.26086i
19.3 0.544389 − 1.30524i −0.704424 + 0.228881i −1.40728 − 1.42111i 1.09089 + 0.792578i −0.0847364 + 1.04404i −0.503194 + 1.54867i −2.62099 + 1.06320i −1.98322 + 1.44090i 1.62837 − 0.992398i
19.4 1.36594 + 0.366325i −1.70537 + 0.554109i 1.73161 + 1.00076i −2.39991 − 1.74363i −2.53243 + 0.132161i 0.815620 − 2.51022i 1.99868 + 2.00132i 0.174207 − 0.126569i −2.63940 − 3.26086i
35.1 −1.40874 + 0.124276i −1.59814 − 2.19965i 1.96911 − 0.350146i −0.720859 − 2.21858i 2.52473 + 2.90013i 1.04462 + 0.758960i −2.73046 + 0.737979i −1.35736 + 4.17752i 1.29122 + 3.03582i
35.2 −1.06665 − 0.928579i 1.59814 + 2.19965i 0.275480 + 1.98094i −0.720859 − 2.21858i 0.337896 − 3.83025i −1.04462 − 0.758960i 1.54562 − 2.36877i −1.35736 + 4.17752i −1.29122 + 3.03582i
35.3 −0.656642 + 1.25253i 0.539857 + 0.743049i −1.13764 − 1.64492i 0.529876 + 1.63079i −1.28518 + 0.188268i −1.93399 − 1.40513i 2.80733 − 0.344804i 0.666375 − 2.05089i −2.39055 − 0.407162i
35.4 0.204982 − 1.39928i −0.539857 − 0.743049i −1.91596 − 0.573655i 0.529876 + 1.63079i −1.15039 + 0.603098i 1.93399 + 1.40513i −1.19544 + 2.56338i 0.666375 − 2.05089i 2.39055 − 0.407162i
39.1 −1.40874 − 0.124276i −1.59814 + 2.19965i 1.96911 + 0.350146i −0.720859 + 2.21858i 2.52473 − 2.90013i 1.04462 − 0.758960i −2.73046 − 0.737979i −1.35736 − 4.17752i 1.29122 − 3.03582i
39.2 −1.06665 + 0.928579i 1.59814 − 2.19965i 0.275480 − 1.98094i −0.720859 + 2.21858i 0.337896 + 3.83025i −1.04462 + 0.758960i 1.54562 + 2.36877i −1.35736 − 4.17752i −1.29122 − 3.03582i
39.3 −0.656642 − 1.25253i 0.539857 − 0.743049i −1.13764 + 1.64492i 0.529876 − 1.63079i −1.28518 − 0.188268i −1.93399 + 1.40513i 2.80733 + 0.344804i 0.666375 + 2.05089i −2.39055 + 0.407162i
39.4 0.204982 + 1.39928i −0.539857 + 0.743049i −1.91596 + 0.573655i 0.529876 − 1.63079i −1.15039 − 0.603098i 1.93399 − 1.40513i −1.19544 − 2.56338i 0.666375 + 2.05089i 2.39055 + 0.407162i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 44.2.g.a ✓ 16
3.b odd 2 1 396.2.r.a 16
4.b odd 2 1 inner 44.2.g.a ✓ 16
8.b even 2 1 704.2.u.c 16
8.d odd 2 1 704.2.u.c 16
11.b odd 2 1 484.2.g.i 16
11.c even 5 1 484.2.c.d 16
11.c even 5 1 484.2.g.f 16
11.c even 5 1 484.2.g.i 16
11.c even 5 1 484.2.g.j 16
11.d odd 10 1 inner 44.2.g.a ✓ 16
11.d odd 10 1 484.2.c.d 16
11.d odd 10 1 484.2.g.f 16
11.d odd 10 1 484.2.g.j 16
12.b even 2 1 396.2.r.a 16
33.f even 10 1 396.2.r.a 16
44.c even 2 1 484.2.g.i 16
44.g even 10 1 inner 44.2.g.a ✓ 16
44.g even 10 1 484.2.c.d 16
44.g even 10 1 484.2.g.f 16
44.g even 10 1 484.2.g.j 16
44.h odd 10 1 484.2.c.d 16
44.h odd 10 1 484.2.g.f 16
44.h odd 10 1 484.2.g.i 16
44.h odd 10 1 484.2.g.j 16
88.k even 10 1 704.2.u.c 16
88.p odd 10 1 704.2.u.c 16
132.n odd 10 1 396.2.r.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.g.a ✓ 16 1.a even 1 1 trivial
44.2.g.a ✓ 16 4.b odd 2 1 inner
44.2.g.a ✓ 16 11.d odd 10 1 inner
44.2.g.a ✓ 16 44.g even 10 1 inner
396.2.r.a 16 3.b odd 2 1
396.2.r.a 16 12.b even 2 1
396.2.r.a 16 33.f even 10 1
396.2.r.a 16 132.n odd 10 1
484.2.c.d 16 11.c even 5 1
484.2.c.d 16 11.d odd 10 1
484.2.c.d 16 44.g even 10 1
484.2.c.d 16 44.h odd 10 1
484.2.g.f 16 11.c even 5 1
484.2.g.f 16 11.d odd 10 1
484.2.g.f 16 44.g even 10 1
484.2.g.f 16 44.h odd 10 1
484.2.g.i 16 11.b odd 2 1
484.2.g.i 16 11.c even 5 1
484.2.g.i 16 44.c even 2 1
484.2.g.i 16 44.h odd 10 1
484.2.g.j 16 11.c even 5 1
484.2.g.j 16 11.d odd 10 1
484.2.g.j 16 44.g even 10 1
484.2.g.j 16 44.h odd 10 1
704.2.u.c 16 8.b even 2 1
704.2.u.c 16 8.d odd 2 1
704.2.u.c 16 88.k even 10 1
704.2.u.c 16 88.p odd 10 1

Hecke kernels

This newform subspace is the entire newspace S2new(44,[χ])S_{2}^{\mathrm{new}}(44, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16+5T15+⋯+256 T^{16} + 5 T^{15} + \cdots + 256 Copy content Toggle raw display
33 T16−T14+⋯+121 T^{16} - T^{14} + \cdots + 121 Copy content Toggle raw display
55 (T8+3T7+⋯+256)2 (T^{8} + 3 T^{7} + \cdots + 256)^{2} Copy content Toggle raw display
77 T16+11T14+⋯+30976 T^{16} + 11 T^{14} + \cdots + 30976 Copy content Toggle raw display
1111 T16+⋯+214358881 T^{16} + \cdots + 214358881 Copy content Toggle raw display
1313 (T8+5T7+16T6+⋯+16)2 (T^{8} + 5 T^{7} + 16 T^{6} + \cdots + 16)^{2} Copy content Toggle raw display
1717 (T8+5T7−24T6+⋯+1)2 (T^{8} + 5 T^{7} - 24 T^{6} + \cdots + 1)^{2} Copy content Toggle raw display
1919 T16+106T14+⋯+1771561 T^{16} + 106 T^{14} + \cdots + 1771561 Copy content Toggle raw display
2323 (T8+88T6+⋯+2816)2 (T^{8} + 88 T^{6} + \cdots + 2816)^{2} Copy content Toggle raw display
2929 (T8+5T7+⋯+400)2 (T^{8} + 5 T^{7} + \cdots + 400)^{2} Copy content Toggle raw display
3131 T16+⋯+428888770816 T^{16} + \cdots + 428888770816 Copy content Toggle raw display
3737 (T8−9T7+⋯+13456)2 (T^{8} - 9 T^{7} + \cdots + 13456)^{2} Copy content Toggle raw display
4141 (T8−5T7+⋯+2588881)2 (T^{8} - 5 T^{7} + \cdots + 2588881)^{2} Copy content Toggle raw display
4343 (T8−143T6+⋯+148016)2 (T^{8} - 143 T^{6} + \cdots + 148016)^{2} Copy content Toggle raw display
4747 T16+⋯+1206517709056 T^{16} + \cdots + 1206517709056 Copy content Toggle raw display
5353 (T8−19T7+⋯+15376)2 (T^{8} - 19 T^{7} + \cdots + 15376)^{2} Copy content Toggle raw display
5959 T16+⋯+17080137481 T^{16} + \cdots + 17080137481 Copy content Toggle raw display
6161 (T8+5T7+⋯+633616)2 (T^{8} + 5 T^{7} + \cdots + 633616)^{2} Copy content Toggle raw display
6767 (T8+185T6+⋯+1760000)2 (T^{8} + 185 T^{6} + \cdots + 1760000)^{2} Copy content Toggle raw display
7171 T16+⋯+21908736256 T^{16} + \cdots + 21908736256 Copy content Toggle raw display
7373 (T8+15T7+⋯+6561)2 (T^{8} + 15 T^{7} + \cdots + 6561)^{2} Copy content Toggle raw display
7979 T16+⋯+28606986496 T^{16} + \cdots + 28606986496 Copy content Toggle raw display
8383 T16+⋯+435963075625 T^{16} + \cdots + 435963075625 Copy content Toggle raw display
8989 (T4+9T3+⋯+116)4 (T^{4} + 9 T^{3} + \cdots + 116)^{4} Copy content Toggle raw display
9797 (T4+17T3+⋯+3721)4 (T^{4} + 17 T^{3} + \cdots + 3721)^{4} Copy content Toggle raw display
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