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Properties

Label 336.4.bc.e
Level 336336
Weight 44
Character orbit 336.bc
Analytic conductor 19.82519.825
Analytic rank 00
Dimension 1616
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,4,Mod(17,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.17"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 336=24⋅3⋅7 336 = 2^{4} \cdot 3 \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 336.bc (of order 66, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 19.824641761919.8246417619
Analytic rank: 00
Dimension: 1616
Relative dimension: 88 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x16−⋯ )\mathbb{Q}[x]/(x^{16} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x16−2x15−x14−2x13+9x12−24x11+714x10−1940x9−2834x8+⋯+43046721 x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} + \cdots + 43046721 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 28⋅311 2^{8}\cdot 3^{11}
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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−β4q3−β9q5+(−β15−β10+β6+⋯−6)q7+(β15+β14−β13+⋯−2)q9+(β15+β14−2β13+⋯−2)q11+⋯+(−36β15−3β14+⋯+243)q99+O(q100) q - \beta_{4} q^{3} - \beta_{9} q^{5} + ( - \beta_{15} - \beta_{10} + \beta_{6} + \cdots - 6) q^{7} + (\beta_{15} + \beta_{14} - \beta_{13} + \cdots - 2) q^{9} + (\beta_{15} + \beta_{14} - 2 \beta_{13} + \cdots - 2) q^{11}+ \cdots + ( - 36 \beta_{15} - 3 \beta_{14} + \cdots + 243) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q−80q7+18q9+342q19−450q21−194q25−804q31+1332q33−962q37−594q39−1732q43−2394q45+820q49−1638q51−2664q57−4620q61+⋯+4284q99+O(q100) 16 q - 80 q^{7} + 18 q^{9} + 342 q^{19} - 450 q^{21} - 194 q^{25} - 804 q^{31} + 1332 q^{33} - 962 q^{37} - 594 q^{39} - 1732 q^{43} - 2394 q^{45} + 820 q^{49} - 1638 q^{51} - 2664 q^{57} - 4620 q^{61}+ \cdots + 4284 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16−2x15−x14−2x13+9x12−24x11+714x10−1940x9−2834x8+⋯+43046721 x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} + \cdots + 43046721 : Copy content Toggle raw display

β1\beta_{1}== (5191ν15−309530ν14−1581934ν13−12539069ν12+1431552ν11+⋯−4281814291149)/980793497376 ( 5191 \nu^{15} - 309530 \nu^{14} - 1581934 \nu^{13} - 12539069 \nu^{12} + 1431552 \nu^{11} + \cdots - 4281814291149 ) / 980793497376 Copy content Toggle raw display
β2\beta_{2}== (−1025ν15+7522ν14+1448ν13−4529ν12−66042ν11+⋯+30596652693)/30024290736 ( - 1025 \nu^{15} + 7522 \nu^{14} + 1448 \nu^{13} - 4529 \nu^{12} - 66042 \nu^{11} + \cdots + 30596652693 ) / 30024290736 Copy content Toggle raw display
β3\beta_{3}== (−98467ν15−3277489ν14−15183785ν13−68558206ν12+⋯−17367563093094)/2942380492128 ( - 98467 \nu^{15} - 3277489 \nu^{14} - 15183785 \nu^{13} - 68558206 \nu^{12} + \cdots - 17367563093094 ) / 2942380492128 Copy content Toggle raw display
β4\beta_{4}== (−10304ν15+5317ν14−182755ν13+524005ν12−438480ν11+⋯+81143069085)/89163045216 ( - 10304 \nu^{15} + 5317 \nu^{14} - 182755 \nu^{13} + 524005 \nu^{12} - 438480 \nu^{11} + \cdots + 81143069085 ) / 89163045216 Copy content Toggle raw display
β5\beta_{5}== (99716ν15+525581ν14+4176229ν13−461611ν12+656424ν11+⋯+74485176237)/735595123032 ( 99716 \nu^{15} + 525581 \nu^{14} + 4176229 \nu^{13} - 461611 \nu^{12} + 656424 \nu^{11} + \cdots + 74485176237 ) / 735595123032 Copy content Toggle raw display
β6\beta_{6}== (−355015ν15−1366828ν14−1922912ν13+5844395ν12+⋯+2323389370347)/980793497376 ( - 355015 \nu^{15} - 1366828 \nu^{14} - 1922912 \nu^{13} + 5844395 \nu^{12} + \cdots + 2323389370347 ) / 980793497376 Copy content Toggle raw display
β7\beta_{7}== (1475767ν15+15672631ν14+32942759ν13+8154460ν12+⋯+67296839372316)/2942380492128 ( 1475767 \nu^{15} + 15672631 \nu^{14} + 32942759 \nu^{13} + 8154460 \nu^{12} + \cdots + 67296839372316 ) / 2942380492128 Copy content Toggle raw display
β8\beta_{8}== (−386047ν15−1698028ν14−7639460ν13−16796293ν12+⋯+470964608523)/735595123032 ( - 386047 \nu^{15} - 1698028 \nu^{14} - 7639460 \nu^{13} - 16796293 \nu^{12} + \cdots + 470964608523 ) / 735595123032 Copy content Toggle raw display
β9\beta_{9}== (−5044301ν15−1066547ν14+74264777ν13+103846498ν12+⋯+184962509874954)/8827141476384 ( - 5044301 \nu^{15} - 1066547 \nu^{14} + 74264777 \nu^{13} + 103846498 \nu^{12} + \cdots + 184962509874954 ) / 8827141476384 Copy content Toggle raw display
β10\beta_{10}== (1999363ν15−277637ν14+2684699ν13+34754776ν12+⋯+22962594135852)/2942380492128 ( 1999363 \nu^{15} - 277637 \nu^{14} + 2684699 \nu^{13} + 34754776 \nu^{12} + \cdots + 22962594135852 ) / 2942380492128 Copy content Toggle raw display
β11\beta_{11}== (−2529436ν15+12956441ν14+26278093ν13+45587477ν12+⋯+52197572223981)/2942380492128 ( - 2529436 \nu^{15} + 12956441 \nu^{14} + 26278093 \nu^{13} + 45587477 \nu^{12} + \cdots + 52197572223981 ) / 2942380492128 Copy content Toggle raw display
β12\beta_{12}== (9038834ν15+13433807ν14−6780077ν13−148095511ν12+⋯+18095200950033)/8827141476384 ( 9038834 \nu^{15} + 13433807 \nu^{14} - 6780077 \nu^{13} - 148095511 \nu^{12} + \cdots + 18095200950033 ) / 8827141476384 Copy content Toggle raw display
β13\beta_{13}== (12998675ν15−19337332ν14−47538200ν13−375390643ν12+⋯−180263553725439)/8827141476384 ( 12998675 \nu^{15} - 19337332 \nu^{14} - 47538200 \nu^{13} - 375390643 \nu^{12} + \cdots - 180263553725439 ) / 8827141476384 Copy content Toggle raw display
β14\beta_{14}== (1736865ν15−3022417ν14−3556645ν13−24737930ν12+⋯−13774105728810)/980793497376 ( 1736865 \nu^{15} - 3022417 \nu^{14} - 3556645 \nu^{13} - 24737930 \nu^{12} + \cdots - 13774105728810 ) / 980793497376 Copy content Toggle raw display
β15\beta_{15}== (6143228ν15−12559ν14−42859679ν13−113257891ν12+⋯−94337802618579)/2942380492128 ( 6143228 \nu^{15} - 12559 \nu^{14} - 42859679 \nu^{13} - 113257891 \nu^{12} + \cdots - 94337802618579 ) / 2942380492128 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (2β14−2β13+2β12+2β11−2β7+β5+2β2+2β1)/12 ( 2\beta_{14} - 2\beta_{13} + 2\beta_{12} + 2\beta_{11} - 2\beta_{7} + \beta_{5} + 2\beta_{2} + 2\beta_1 ) / 12 Copy content Toggle raw display
ν2\nu^{2}== (β15+β14−β13−β12+β11+β9+β3−3β2−2)/3 ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{9} + \beta_{3} - 3\beta_{2} - 2 ) / 3 Copy content Toggle raw display
ν3\nu^{3}== (4β14−4β12+4β11−4β10−4β9+7β8−8β6+⋯+8)/12 ( 4 \beta_{14} - 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 7 \beta_{8} - 8 \beta_{6} + \cdots + 8 ) / 12 Copy content Toggle raw display
ν4\nu^{4}== (2β15−β14−β13+2β12+2β11+3β10−2β9+⋯+5)/3 ( 2 \beta_{15} - \beta_{14} - \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} + \cdots + 5 ) / 3 Copy content Toggle raw display
ν5\nu^{5}== (−8β15+16β14+10β13−74β12−8β11+26β10+⋯+18)/12 ( - 8 \beta_{15} + 16 \beta_{14} + 10 \beta_{13} - 74 \beta_{12} - 8 \beta_{11} + 26 \beta_{10} + \cdots + 18 ) / 12 Copy content Toggle raw display
ν6\nu^{6}== (8β15+66β14+56β12+78β11−58β10+56β9−26β8+⋯−851)/3 ( 8 \beta_{15} + 66 \beta_{14} + 56 \beta_{12} + 78 \beta_{11} - 58 \beta_{10} + 56 \beta_{9} - 26 \beta_{8} + \cdots - 851 ) / 3 Copy content Toggle raw display
ν7\nu^{7}== (904β15−18β14−886β13+150β12−522β11−504β10+⋯+6424)/12 ( 904 \beta_{15} - 18 \beta_{14} - 886 \beta_{13} + 150 \beta_{12} - 522 \beta_{11} - 504 \beta_{10} + \cdots + 6424 ) / 12 Copy content Toggle raw display
ν8\nu^{8}== (−259β15+3β14−685β13+1327β12−259β11−682β10+⋯−426)/3 ( - 259 \beta_{15} + 3 \beta_{14} - 685 \beta_{13} + 1327 \beta_{12} - 259 \beta_{11} - 682 \beta_{10} + \cdots - 426 ) / 3 Copy content Toggle raw display
ν9\nu^{9}== (1520β15+3108β14+740β12+7732β11−1588β10+740β9+⋯+151784)/12 ( 1520 \beta_{15} + 3108 \beta_{14} + 740 \beta_{12} + 7732 \beta_{11} - 1588 \beta_{10} + 740 \beta_{9} + \cdots + 151784 ) / 12 Copy content Toggle raw display
ν10\nu^{10}== (3940β15+7897β14−11837β13+7564β12+8840β11+943β10+⋯−13359)/3 ( 3940 \beta_{15} + 7897 \beta_{14} - 11837 \beta_{13} + 7564 \beta_{12} + 8840 \beta_{11} + 943 \beta_{10} + \cdots - 13359 ) / 3 Copy content Toggle raw display
ν11\nu^{11}== (79680β15+46832β14−67106β13−14686β12+79680β11+⋯−146786)/12 ( 79680 \beta_{15} + 46832 \beta_{14} - 67106 \beta_{13} - 14686 \beta_{12} + 79680 \beta_{11} + \cdots - 146786 ) / 12 Copy content Toggle raw display
ν12\nu^{12}== (10220β15−14848β14−31536β12−980β11+25068β10+⋯+648595)/3 ( 10220 \beta_{15} - 14848 \beta_{14} - 31536 \beta_{12} - 980 \beta_{11} + 25068 \beta_{10} + \cdots + 648595 ) / 3 Copy content Toggle raw display
ν13\nu^{13}== (−241968β15+420802β14−178834β13+96850β12+627458β11+⋯−6565696)/12 ( - 241968 \beta_{15} + 420802 \beta_{14} - 178834 \beta_{13} + 96850 \beta_{12} + 627458 \beta_{11} + \cdots - 6565696 ) / 12 Copy content Toggle raw display
ν14\nu^{14}== (339137β15−43555β14+144427β13−576749β12+339137β11+⋯−194710)/3 ( 339137 \beta_{15} - 43555 \beta_{14} + 144427 \beta_{13} - 576749 \beta_{12} + 339137 \beta_{11} + \cdots - 194710 ) / 3 Copy content Toggle raw display
ν15\nu^{15}== (3314576β15+2956548β14+5874620β12+1803636β11+358028β10+⋯−33554840)/12 ( 3314576 \beta_{15} + 2956548 \beta_{14} + 5874620 \beta_{12} + 1803636 \beta_{11} + 358028 \beta_{10} + \cdots - 33554840 ) / 12 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/336Z)×\left(\mathbb{Z}/336\mathbb{Z}\right)^\times.

nn 8585 113113 127127 241241
χ(n)\chi(n) 11 −1-1 11 −β2-\beta_{2}

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
17.1
2.99617 + 0.151487i
−2.58777 + 1.51770i
−1.62928 − 2.51902i
0.339489 − 2.98073i
−0.0204843 + 2.99993i
2.30541 + 1.91966i
2.41164 − 1.78437i
−2.81518 − 1.03671i
2.99617 − 0.151487i
−2.58777 − 1.51770i
−1.62928 + 2.51902i
0.339489 + 2.98073i
−0.0204843 − 2.99993i
2.30541 − 1.91966i
2.41164 + 1.78437i
−2.81518 + 1.03671i
0 −5.18952 − 0.262384i 0 −2.24534 − 3.88904i 0 9.71288 − 15.7690i 0 26.8623 + 2.72329i 0
17.2 0 −4.48216 + 2.62874i 0 −5.27257 − 9.13236i 0 −17.7029 + 5.44135i 0 13.1794 − 23.5649i 0
17.3 0 −2.82199 − 4.36307i 0 2.24534 + 3.88904i 0 9.71288 − 15.7690i 0 −11.0727 + 24.6251i 0
17.4 0 −0.588012 + 5.16277i 0 4.27911 + 7.41164i 0 6.41772 + 17.3728i 0 −26.3085 − 6.07155i 0
17.5 0 0.0354799 − 5.19603i 0 5.27257 + 9.13236i 0 −17.7029 + 5.44135i 0 −26.9975 − 0.368709i 0
17.6 0 3.99309 + 3.32495i 0 9.90442 + 17.1550i 0 −18.4277 − 1.84901i 0 4.88947 + 26.5536i 0
17.7 0 4.17709 − 3.09062i 0 −4.27911 − 7.41164i 0 6.41772 + 17.3728i 0 7.89612 − 25.8196i 0
17.8 0 4.87603 + 1.79564i 0 −9.90442 − 17.1550i 0 −18.4277 − 1.84901i 0 20.5514 + 17.5112i 0
257.1 0 −5.18952 + 0.262384i 0 −2.24534 + 3.88904i 0 9.71288 + 15.7690i 0 26.8623 − 2.72329i 0
257.2 0 −4.48216 − 2.62874i 0 −5.27257 + 9.13236i 0 −17.7029 − 5.44135i 0 13.1794 + 23.5649i 0
257.3 0 −2.82199 + 4.36307i 0 2.24534 − 3.88904i 0 9.71288 + 15.7690i 0 −11.0727 − 24.6251i 0
257.4 0 −0.588012 − 5.16277i 0 4.27911 − 7.41164i 0 6.41772 − 17.3728i 0 −26.3085 + 6.07155i 0
257.5 0 0.0354799 + 5.19603i 0 5.27257 − 9.13236i 0 −17.7029 − 5.44135i 0 −26.9975 + 0.368709i 0
257.6 0 3.99309 − 3.32495i 0 9.90442 − 17.1550i 0 −18.4277 + 1.84901i 0 4.88947 − 26.5536i 0
257.7 0 4.17709 + 3.09062i 0 −4.27911 + 7.41164i 0 6.41772 − 17.3728i 0 7.89612 + 25.8196i 0
257.8 0 4.87603 − 1.79564i 0 −9.90442 + 17.1550i 0 −18.4277 + 1.84901i 0 20.5514 − 17.5112i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bc.e 16
3.b odd 2 1 inner 336.4.bc.e 16
4.b odd 2 1 42.4.f.a ✓ 16
7.d odd 6 1 inner 336.4.bc.e 16
12.b even 2 1 42.4.f.a ✓ 16
21.g even 6 1 inner 336.4.bc.e 16
28.d even 2 1 294.4.f.a 16
28.f even 6 1 42.4.f.a ✓ 16
28.f even 6 1 294.4.d.a 16
28.g odd 6 1 294.4.d.a 16
28.g odd 6 1 294.4.f.a 16
84.h odd 2 1 294.4.f.a 16
84.j odd 6 1 42.4.f.a ✓ 16
84.j odd 6 1 294.4.d.a 16
84.n even 6 1 294.4.d.a 16
84.n even 6 1 294.4.f.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.f.a ✓ 16 4.b odd 2 1
42.4.f.a ✓ 16 12.b even 2 1
42.4.f.a ✓ 16 28.f even 6 1
42.4.f.a ✓ 16 84.j odd 6 1
294.4.d.a 16 28.f even 6 1
294.4.d.a 16 28.g odd 6 1
294.4.d.a 16 84.j odd 6 1
294.4.d.a 16 84.n even 6 1
294.4.f.a 16 28.d even 2 1
294.4.f.a 16 28.g odd 6 1
294.4.f.a 16 84.h odd 2 1
294.4.f.a 16 84.n even 6 1
336.4.bc.e 16 1.a even 1 1 trivial
336.4.bc.e 16 3.b odd 2 1 inner
336.4.bc.e 16 7.d odd 6 1 inner
336.4.bc.e 16 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(336,[χ])S_{4}^{\mathrm{new}}(336, [\chi]):

T516+597T514+264258T512+45374877T510+5550035922T58+⋯+41 ⁣⋯ ⁣56 T_{5}^{16} + 597 T_{5}^{14} + 264258 T_{5}^{12} + 45374877 T_{5}^{10} + 5550035922 T_{5}^{8} + \cdots + 41\!\cdots\!56 Copy content Toggle raw display
T138+4551T136+2979108T134+570419712T132+16498888704 T_{13}^{8} + 4551T_{13}^{6} + 2979108T_{13}^{4} + 570419712T_{13}^{2} + 16498888704 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T16 T^{16} Copy content Toggle raw display
33 T16+⋯+282429536481 T^{16} + \cdots + 282429536481 Copy content Toggle raw display
55 T16+⋯+41 ⁣⋯ ⁣56 T^{16} + \cdots + 41\!\cdots\!56 Copy content Toggle raw display
77 (T8+40T7+⋯+13841287201)2 (T^{8} + 40 T^{7} + \cdots + 13841287201)^{2} Copy content Toggle raw display
1111 T16+⋯+24 ⁣⋯ ⁣16 T^{16} + \cdots + 24\!\cdots\!16 Copy content Toggle raw display
1313 (T8+4551T6+⋯+16498888704)2 (T^{8} + 4551 T^{6} + \cdots + 16498888704)^{2} Copy content Toggle raw display
1717 T16+⋯+87 ⁣⋯ ⁣00 T^{16} + \cdots + 87\!\cdots\!00 Copy content Toggle raw display
1919 (T8+⋯+69773043356676)2 (T^{8} + \cdots + 69773043356676)^{2} Copy content Toggle raw display
2323 T16+⋯+40 ⁣⋯ ⁣96 T^{16} + \cdots + 40\!\cdots\!96 Copy content Toggle raw display
2929 (T8+⋯+20 ⁣⋯ ⁣96)2 (T^{8} + \cdots + 20\!\cdots\!96)^{2} Copy content Toggle raw display
3131 (T8+⋯+81 ⁣⋯ ⁣01)2 (T^{8} + \cdots + 81\!\cdots\!01)^{2} Copy content Toggle raw display
3737 (T8+⋯+81 ⁣⋯ ⁣56)2 (T^{8} + \cdots + 81\!\cdots\!56)^{2} Copy content Toggle raw display
4141 (T8+⋯+15 ⁣⋯ ⁣76)2 (T^{8} + \cdots + 15\!\cdots\!76)^{2} Copy content Toggle raw display
4343 (T4+433T3+⋯+966156928)4 (T^{4} + 433 T^{3} + \cdots + 966156928)^{4} Copy content Toggle raw display
4747 T16+⋯+20 ⁣⋯ ⁣76 T^{16} + \cdots + 20\!\cdots\!76 Copy content Toggle raw display
5353 T16+⋯+65 ⁣⋯ ⁣00 T^{16} + \cdots + 65\!\cdots\!00 Copy content Toggle raw display
5959 T16+⋯+11 ⁣⋯ ⁣36 T^{16} + \cdots + 11\!\cdots\!36 Copy content Toggle raw display
6161 (T8+⋯+25 ⁣⋯ ⁣44)2 (T^{8} + \cdots + 25\!\cdots\!44)^{2} Copy content Toggle raw display
6767 (T8+⋯+30 ⁣⋯ ⁣04)2 (T^{8} + \cdots + 30\!\cdots\!04)^{2} Copy content Toggle raw display
7171 (T8+⋯+18 ⁣⋯ ⁣16)2 (T^{8} + \cdots + 18\!\cdots\!16)^{2} Copy content Toggle raw display
7373 (T8+⋯+13 ⁣⋯ ⁣64)2 (T^{8} + \cdots + 13\!\cdots\!64)^{2} Copy content Toggle raw display
7979 (T8+⋯+11 ⁣⋯ ⁣61)2 (T^{8} + \cdots + 11\!\cdots\!61)^{2} Copy content Toggle raw display
8383 (T8+⋯+16 ⁣⋯ ⁣56)2 (T^{8} + \cdots + 16\!\cdots\!56)^{2} Copy content Toggle raw display
8989 T16+⋯+18 ⁣⋯ ⁣36 T^{16} + \cdots + 18\!\cdots\!36 Copy content Toggle raw display
9797 (T8+⋯+10 ⁣⋯ ⁣36)2 (T^{8} + \cdots + 10\!\cdots\!36)^{2} Copy content Toggle raw display
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