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Properties

Label 250.4.d.b
Level 250250
Weight 44
Character orbit 250.d
Analytic conductor 14.75014.750
Analytic rank 00
Dimension 1616
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [250,4,Mod(51,250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(250, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("250.51");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 250=2⋅53 250 = 2 \cdot 5^{3}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 250.d (of order 55, degree 44, 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: 14.750477501414.7504775014
Analytic rank: 00
Dimension: 1616
Relative dimension: 44 over Q(ζ5)\Q(\zeta_{5})
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+135x14+7296x12+200295x10+2912021x8+20937420x6+57578496x4+⋯+952576 x^{16} + 135 x^{14} + 7296 x^{12} + 200295 x^{10} + 2912021 x^{8} + 20937420 x^{6} + 57578496 x^{4} + \cdots + 952576 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 56 5^{6}
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: SU(2)[C5]\mathrm{SU}(2)[C_{5}]

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+2β8q2+(β7+β4−1)q3+(4β9+4β8+4β7−4)q4+(2β7+2β6−2β5+⋯+2)q6+(−β14−β13+β8+⋯+3)q7+⋯+(30β14+30β13+⋯+219)q99+O(q100) q + 2 \beta_{8} q^{2} + (\beta_{7} + \beta_{4} - 1) q^{3} + (4 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} - 4) q^{4} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots + 2) q^{6} + ( - \beta_{14} - \beta_{13} + \beta_{8} + \cdots + 3) q^{7}+ \cdots + (30 \beta_{14} + 30 \beta_{13} + \cdots + 219) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q+8q2−7q3−16q4+14q6+54q7+32q8−83q9+152q11+12q12−77q13+52q14−64q16−76q17−304q18−20q19+257q21+226q22+⋯−1786q99+O(q100) 16 q + 8 q^{2} - 7 q^{3} - 16 q^{4} + 14 q^{6} + 54 q^{7} + 32 q^{8} - 83 q^{9} + 152 q^{11} + 12 q^{12} - 77 q^{13} + 52 q^{14} - 64 q^{16} - 76 q^{17} - 304 q^{18} - 20 q^{19} + 257 q^{21} + 226 q^{22}+ \cdots - 1786 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16+135x14+7296x12+200295x10+2912021x8+20937420x6+57578496x4+⋯+952576 x^{16} + 135 x^{14} + 7296 x^{12} + 200295 x^{10} + 2912021 x^{8} + 20937420 x^{6} + 57578496 x^{4} + \cdots + 952576 : Copy content Toggle raw display

β1\beta_{1}== (−2606260817ν14−299720681931ν12−13455196227932ν10+⋯+704732477177408)/23 ⁣⋯ ⁣80 ( - 2606260817 \nu^{14} - 299720681931 \nu^{12} - 13455196227932 \nu^{10} + \cdots + 704732477177408 ) / 23\!\cdots\!80 Copy content Toggle raw display
β2\beta_{2}== (8091599105ν14+1099677668619ν12+59976834922988ν10+⋯+35 ⁣⋯ ⁣84)/23 ⁣⋯ ⁣80 ( 8091599105 \nu^{14} + 1099677668619 \nu^{12} + 59976834922988 \nu^{10} + \cdots + 35\!\cdots\!84 ) / 23\!\cdots\!80 Copy content Toggle raw display
β3\beta_{3}== (892560297ν14+112376768612ν12+5630820410899ν10+⋯+62 ⁣⋯ ⁣28)/146090958472080 ( 892560297 \nu^{14} + 112376768612 \nu^{12} + 5630820410899 \nu^{10} + \cdots + 62\!\cdots\!28 ) / 146090958472080 Copy content Toggle raw display
β4\beta_{4}== (−630304983ν15−533360406ν14−73305733293ν13−13349395378ν12+⋯+18 ⁣⋯ ⁣04)/46 ⁣⋯ ⁣60 ( - 630304983 \nu^{15} - 533360406 \nu^{14} - 73305733293 \nu^{13} - 13349395378 \nu^{12} + \cdots + 18\!\cdots\!04 ) / 46\!\cdots\!60 Copy content Toggle raw display
β5\beta_{5}== (630304983ν15+721297714ν14+73305733293ν13+106532776630ν12+⋯+24 ⁣⋯ ⁣72)/23 ⁣⋯ ⁣80 ( 630304983 \nu^{15} + 721297714 \nu^{14} + 73305733293 \nu^{13} + 106532776630 \nu^{12} + \cdots + 24\!\cdots\!72 ) / 23\!\cdots\!80 Copy content Toggle raw display
β6\beta_{6}== (630304983ν15−721297714ν14+73305733293ν13−106532776630ν12+⋯−24 ⁣⋯ ⁣72)/23 ⁣⋯ ⁣80 ( 630304983 \nu^{15} - 721297714 \nu^{14} + 73305733293 \nu^{13} - 106532776630 \nu^{12} + \cdots - 24\!\cdots\!72 ) / 23\!\cdots\!80 Copy content Toggle raw display
β7\beta_{7}== (−236761278263ν15−76897207926ν14−32027842535037ν13+⋯+45 ⁣⋯ ⁣56)/57 ⁣⋯ ⁣20 ( - 236761278263 \nu^{15} - 76897207926 \nu^{14} - 32027842535037 \nu^{13} + \cdots + 45\!\cdots\!56 ) / 57\!\cdots\!20 Copy content Toggle raw display
β8\beta_{8}== (236761278263ν15−76897207926ν14+32027842535037ν13+⋯+45 ⁣⋯ ⁣56)/57 ⁣⋯ ⁣20 ( 236761278263 \nu^{15} - 76897207926 \nu^{14} + 32027842535037 \nu^{13} + \cdots + 45\!\cdots\!56 ) / 57\!\cdots\!20 Copy content Toggle raw display
β9\beta_{9}== (388969833455ν15+76897207926ν14+52269860904677ν13+⋯−17 ⁣⋯ ⁣96)/57 ⁣⋯ ⁣20 ( 388969833455 \nu^{15} + 76897207926 \nu^{14} + 52269860904677 \nu^{13} + \cdots - 17\!\cdots\!96 ) / 57\!\cdots\!20 Copy content Toggle raw display
β10\beta_{10}== (−1632378220845ν15+987175090810ν14−211453688342671ν13+⋯+45 ⁣⋯ ⁣08)/57 ⁣⋯ ⁣20 ( - 1632378220845 \nu^{15} + 987175090810 \nu^{14} - 211453688342671 \nu^{13} + \cdots + 45\!\cdots\!08 ) / 57\!\cdots\!20 Copy content Toggle raw display
β11\beta_{11}== (1861751406679ν15+610272512254ν14+258143900192877ν13+⋯+73 ⁣⋯ ⁣80)/57 ⁣⋯ ⁣20 ( 1861751406679 \nu^{15} + 610272512254 \nu^{14} + 258143900192877 \nu^{13} + \cdots + 73\!\cdots\!80 ) / 57\!\cdots\!20 Copy content Toggle raw display
β12\beta_{12}== (−1049256342471ν15+343584860090ν14−145085871363957ν13+⋯+34 ⁣⋯ ⁣12)/28 ⁣⋯ ⁣60 ( - 1049256342471 \nu^{15} + 343584860090 \nu^{14} - 145085871363957 \nu^{13} + \cdots + 34\!\cdots\!12 ) / 28\!\cdots\!60 Copy content Toggle raw display
β13\beta_{13}== (−386689991580ν15+301308049115ν14−52777807690166ν13+⋯+24 ⁣⋯ ⁣40)/71 ⁣⋯ ⁣40 ( - 386689991580 \nu^{15} + 301308049115 \nu^{14} - 52777807690166 \nu^{13} + \cdots + 24\!\cdots\!40 ) / 71\!\cdots\!40 Copy content Toggle raw display
β14\beta_{14}== (3330281210903ν15+2333567184994ν14+454250304056365ν13+⋯+20 ⁣⋯ ⁣76)/57 ⁣⋯ ⁣20 ( 3330281210903 \nu^{15} + 2333567184994 \nu^{14} + 454250304056365 \nu^{13} + \cdots + 20\!\cdots\!76 ) / 57\!\cdots\!20 Copy content Toggle raw display
β15\beta_{15}== (1266846637783ν15−167046673294ν14+170998057181529ν13+⋯−18 ⁣⋯ ⁣16)/14 ⁣⋯ ⁣80 ( 1266846637783 \nu^{15} - 167046673294 \nu^{14} + 170998057181529 \nu^{13} + \cdots - 18\!\cdots\!16 ) / 14\!\cdots\!80 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (2β8+2β7+3β6−β5+4β4−2β1)/5 ( 2\beta_{8} + 2\beta_{7} + 3\beta_{6} - \beta_{5} + 4\beta_{4} - 2\beta_1 ) / 5 Copy content Toggle raw display
ν2\nu^{2}== (2β14+2β13−2β12−2β11+22β8+26β7+3β6+⋯−95)/5 ( 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 22 \beta_{8} + 26 \beta_{7} + 3 \beta_{6} + \cdots - 95 ) / 5 Copy content Toggle raw display
ν3\nu^{3}== (−4β15+5β14−9β13+β12−β11−8β10−50β9+⋯+29)/5 ( - 4 \beta_{15} + 5 \beta_{14} - 9 \beta_{13} + \beta_{12} - \beta_{11} - 8 \beta_{10} - 50 \beta_{9} + \cdots + 29 ) / 5 Copy content Toggle raw display
ν4\nu^{4}== (−54β14−54β13+78β12+78β11−680β8−812β7+⋯+2594)/5 ( - 54 \beta_{14} - 54 \beta_{13} + 78 \beta_{12} + 78 \beta_{11} - 680 \beta_{8} - 812 \beta_{7} + \cdots + 2594 ) / 5 Copy content Toggle raw display
ν5\nu^{5}== (186β15−206β14+392β13−34β12+34β11+414β10+⋯−1495)/5 ( 186 \beta_{15} - 206 \beta_{14} + 392 \beta_{13} - 34 \beta_{12} + 34 \beta_{11} + 414 \beta_{10} + \cdots - 1495 ) / 5 Copy content Toggle raw display
ν6\nu^{6}== (1421β14+1421β13−2795β12−2795β11+22947β8+27163β7+⋯−77349)/5 ( 1421 \beta_{14} + 1421 \beta_{13} - 2795 \beta_{12} - 2795 \beta_{11} + 22947 \beta_{8} + 27163 \beta_{7} + \cdots - 77349 ) / 5 Copy content Toggle raw display
ν7\nu^{7}== (−7036β15+7172β14−14208β13+1348β12−1348β11+⋯+55910)/5 ( - 7036 \beta_{15} + 7172 \beta_{14} - 14208 \beta_{13} + 1348 \beta_{12} - 1348 \beta_{11} + \cdots + 55910 ) / 5 Copy content Toggle raw display
ν8\nu^{8}== (−38778β14−38778β13+96006β12+96006β11−787830β8+⋯+2400613)/5 ( - 38778 \beta_{14} - 38778 \beta_{13} + 96006 \beta_{12} + 96006 \beta_{11} - 787830 \beta_{8} + \cdots + 2400613 ) / 5 Copy content Toggle raw display
ν9\nu^{9}== (252516β15−235731β14+488247β13−54495β12+54495β11+⋯−1852479)/5 ( 252516 \beta_{15} - 235731 \beta_{14} + 488247 \beta_{13} - 54495 \beta_{12} + 54495 \beta_{11} + \cdots - 1852479 ) / 5 Copy content Toggle raw display
ν10\nu^{10}== (1097410β14+1097410β13−3226978β12−3226978β11+27089852β8+⋯−76078198)/5 ( 1097410 \beta_{14} + 1097410 \beta_{13} - 3226978 \beta_{12} - 3226978 \beta_{11} + 27089852 \beta_{8} + \cdots - 76078198 ) / 5 Copy content Toggle raw display
ν11\nu^{11}== (−8848142β15+7559638β14−16407780β13+2145002β12−2145002β11+⋯+57973345)/5 ( - 8848142 \beta_{15} + 7559638 \beta_{14} - 16407780 \beta_{13} + 2145002 \beta_{12} - 2145002 \beta_{11} + \cdots + 57973345 ) / 5 Copy content Toggle raw display
ν12\nu^{12}== (−31856307β14−31856307β13+107278749β12+107278749β11+⋯+2439096967)/5 ( - 31856307 \beta_{14} - 31856307 \beta_{13} + 107278749 \beta_{12} + 107278749 \beta_{11} + \cdots + 2439096967 ) / 5 Copy content Toggle raw display
ν13\nu^{13}== (305600820β15−239777260β14+545378080β13−81805508β12+⋯−1759493642)/5 ( 305600820 \beta_{15} - 239777260 \beta_{14} + 545378080 \beta_{13} - 81805508 \beta_{12} + \cdots - 1759493642 ) / 5 Copy content Toggle raw display
ν14\nu^{14}== (939132274β14+939132274β13−3547108246β12−3547108246β11+⋯−78719495907)/5 ( 939132274 \beta_{14} + 939132274 \beta_{13} - 3547108246 \beta_{12} - 3547108246 \beta_{11} + \cdots - 78719495907 ) / 5 Copy content Toggle raw display
ν15\nu^{15}== (−10452419252β15+7571112493β14−18023531745β13+3037954073β12+⋯+52420041109)/5 ( - 10452419252 \beta_{15} + 7571112493 \beta_{14} - 18023531745 \beta_{13} + 3037954073 \beta_{12} + \cdots + 52420041109 ) / 5 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/250Z)×\left(\mathbb{Z}/250\mathbb{Z}\right)^\times.

nn 127127
χ(n)\chi(n) −1+β7+β8+β9-1 + \beta_{7} + \beta_{8} + \beta_{9}

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}
51.1
4.04577i
0.493198i
− 0.276776i
− 5.43776i
− 5.68871i
− 2.36461i
4.19209i
5.76335i
5.68871i
2.36461i
− 4.19209i
− 5.76335i
− 4.04577i
− 0.493198i
0.276776i
5.43776i
1.61803 − 1.17557i −2.56903 + 7.90665i 1.23607 − 3.80423i 0 5.13805 + 15.8133i 26.4146 −2.47214 − 7.60845i −34.0718 − 24.7546i 0
51.2 1.61803 − 1.17557i −0.480877 + 1.47999i 1.23607 − 3.80423i 0 0.961755 + 2.95998i −19.9781 −2.47214 − 7.60845i 19.8843 + 14.4468i 0
51.3 1.61803 − 1.17557i −0.0282983 + 0.0870932i 1.23607 − 3.80423i 0 0.0565966 + 0.174186i −10.7092 −2.47214 − 7.60845i 21.8367 + 15.8653i 0
51.4 1.61803 − 1.17557i 3.00525 − 9.24922i 1.23607 − 3.80423i 0 −6.01051 − 18.4984i 23.3628 −2.47214 − 7.60845i −54.6730 − 39.7223i 0
101.1 −0.618034 + 1.90211i −6.71931 − 4.88186i −3.23607 − 2.35114i 0 13.4386 − 9.76372i 11.9615 6.47214 − 4.70228i 12.9730 + 39.9269i 0
101.2 −0.618034 + 1.90211i −3.55790 − 2.58496i −3.23607 − 2.35114i 0 7.11579 − 5.16993i −32.4633 6.47214 − 4.70228i −2.36687 − 7.28447i 0
101.3 −0.618034 + 1.90211i 2.67790 + 1.94561i −3.23607 − 2.35114i 0 −5.35580 + 3.89121i 25.0474 6.47214 − 4.70228i −4.95771 − 15.2583i 0
101.4 −0.618034 + 1.90211i 4.17225 + 3.03132i −3.23607 − 2.35114i 0 −8.34450 + 6.06264i 3.36421 6.47214 − 4.70228i −0.124662 − 0.383671i 0
151.1 −0.618034 − 1.90211i −6.71931 + 4.88186i −3.23607 + 2.35114i 0 13.4386 + 9.76372i 11.9615 6.47214 + 4.70228i 12.9730 − 39.9269i 0
151.2 −0.618034 − 1.90211i −3.55790 + 2.58496i −3.23607 + 2.35114i 0 7.11579 + 5.16993i −32.4633 6.47214 + 4.70228i −2.36687 + 7.28447i 0
151.3 −0.618034 − 1.90211i 2.67790 − 1.94561i −3.23607 + 2.35114i 0 −5.35580 − 3.89121i 25.0474 6.47214 + 4.70228i −4.95771 + 15.2583i 0
151.4 −0.618034 − 1.90211i 4.17225 − 3.03132i −3.23607 + 2.35114i 0 −8.34450 − 6.06264i 3.36421 6.47214 + 4.70228i −0.124662 + 0.383671i 0
201.1 1.61803 + 1.17557i −2.56903 − 7.90665i 1.23607 + 3.80423i 0 5.13805 − 15.8133i 26.4146 −2.47214 + 7.60845i −34.0718 + 24.7546i 0
201.2 1.61803 + 1.17557i −0.480877 − 1.47999i 1.23607 + 3.80423i 0 0.961755 − 2.95998i −19.9781 −2.47214 + 7.60845i 19.8843 − 14.4468i 0
201.3 1.61803 + 1.17557i −0.0282983 − 0.0870932i 1.23607 + 3.80423i 0 0.0565966 − 0.174186i −10.7092 −2.47214 + 7.60845i 21.8367 − 15.8653i 0
201.4 1.61803 + 1.17557i 3.00525 + 9.24922i 1.23607 + 3.80423i 0 −6.01051 + 18.4984i 23.3628 −2.47214 + 7.60845i −54.6730 + 39.7223i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 250.4.d.b 16
5.b even 2 1 50.4.d.b ✓ 16
5.c odd 4 2 250.4.e.c 32
25.d even 5 1 inner 250.4.d.b 16
25.d even 5 1 1250.4.a.g 8
25.e even 10 1 50.4.d.b ✓ 16
25.e even 10 1 1250.4.a.j 8
25.f odd 20 2 250.4.e.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.4.d.b ✓ 16 5.b even 2 1
50.4.d.b ✓ 16 25.e even 10 1
250.4.d.b 16 1.a even 1 1 trivial
250.4.d.b 16 25.d even 5 1 inner
250.4.e.c 32 5.c odd 4 2
250.4.e.c 32 25.f odd 20 2
1250.4.a.g 8 25.d even 5 1
1250.4.a.j 8 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T316+7T315+120T314+810T313+7880T312+15911T311+⋯+51609856 T_{3}^{16} + 7 T_{3}^{15} + 120 T_{3}^{14} + 810 T_{3}^{13} + 7880 T_{3}^{12} + 15911 T_{3}^{11} + \cdots + 51609856 acting on S4new(250,[χ])S_{4}^{\mathrm{new}}(250, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4−2T3+4T2+⋯+16)4 (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{4} Copy content Toggle raw display
33 T16+7T15+⋯+51609856 T^{16} + 7 T^{15} + \cdots + 51609856 Copy content Toggle raw display
55 T16 T^{16} Copy content Toggle raw display
77 (T8−27T7+⋯−4320209664)2 (T^{8} - 27 T^{7} + \cdots - 4320209664)^{2} Copy content Toggle raw display
1111 T16+⋯+31 ⁣⋯ ⁣76 T^{16} + \cdots + 31\!\cdots\!76 Copy content Toggle raw display
1313 T16+⋯+26 ⁣⋯ ⁣81 T^{16} + \cdots + 26\!\cdots\!81 Copy content Toggle raw display
1717 T16+⋯+96 ⁣⋯ ⁣56 T^{16} + \cdots + 96\!\cdots\!56 Copy content Toggle raw display
1919 T16+⋯+12 ⁣⋯ ⁣00 T^{16} + \cdots + 12\!\cdots\!00 Copy content Toggle raw display
2323 T16+⋯+43 ⁣⋯ ⁣96 T^{16} + \cdots + 43\!\cdots\!96 Copy content Toggle raw display
2929 T16+⋯+11 ⁣⋯ ⁣25 T^{16} + \cdots + 11\!\cdots\!25 Copy content Toggle raw display
3131 T16+⋯+31 ⁣⋯ ⁣96 T^{16} + \cdots + 31\!\cdots\!96 Copy content Toggle raw display
3737 T16+⋯+23 ⁣⋯ ⁣81 T^{16} + \cdots + 23\!\cdots\!81 Copy content Toggle raw display
4141 T16+⋯+20 ⁣⋯ ⁣96 T^{16} + \cdots + 20\!\cdots\!96 Copy content Toggle raw display
4343 (T8+⋯+15 ⁣⋯ ⁣76)2 (T^{8} + \cdots + 15\!\cdots\!76)^{2} Copy content Toggle raw display
4747 T16+⋯+13 ⁣⋯ ⁣76 T^{16} + \cdots + 13\!\cdots\!76 Copy content Toggle raw display
5353 T16+⋯+10 ⁣⋯ ⁣36 T^{16} + \cdots + 10\!\cdots\!36 Copy content Toggle raw display
5959 T16+⋯+34 ⁣⋯ ⁣00 T^{16} + \cdots + 34\!\cdots\!00 Copy content Toggle raw display
6161 T16+⋯+78 ⁣⋯ ⁣21 T^{16} + \cdots + 78\!\cdots\!21 Copy content Toggle raw display
6767 T16+⋯+42 ⁣⋯ ⁣36 T^{16} + \cdots + 42\!\cdots\!36 Copy content Toggle raw display
7171 T16+⋯+14 ⁣⋯ ⁣76 T^{16} + \cdots + 14\!\cdots\!76 Copy content Toggle raw display
7373 T16+⋯+63 ⁣⋯ ⁣16 T^{16} + \cdots + 63\!\cdots\!16 Copy content Toggle raw display
7979 T16+⋯+92 ⁣⋯ ⁣00 T^{16} + \cdots + 92\!\cdots\!00 Copy content Toggle raw display
8383 T16+⋯+46 ⁣⋯ ⁣96 T^{16} + \cdots + 46\!\cdots\!96 Copy content Toggle raw display
8989 T16+⋯+66 ⁣⋯ ⁣00 T^{16} + \cdots + 66\!\cdots\!00 Copy content Toggle raw display
9797 T16+⋯+12 ⁣⋯ ⁣41 T^{16} + \cdots + 12\!\cdots\!41 Copy content Toggle raw display
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