Properties

Label 448.4.p.h
Level 448448
Weight 44
Character orbit 448.p
Analytic conductor 26.43326.433
Analytic rank 00
Dimension 2020
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,4,Mod(255,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 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("448.255"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 448=267 448 = 2^{6} \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 448.p (of order 66, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0] 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: 26.432855682626.4328556826
Analytic rank: 00
Dimension: 2020
Relative dimension: 1010 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x20)\mathbb{Q}[x]/(x^{20} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x202x1824x17+28x16+56x15192x14+352x13448x12++1073741824 x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + \cdots + 1073741824 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 244 2^{44}
Twist minimal: no (minimal twist has level 28)
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,,β191,\beta_1,\ldots,\beta_{19} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β10q3+(β4+β3)q5+(β13+β2)q7+(β96β16)q9β15q11+(β19β11+β9+4)q13++(β187β17+8β2)q99+O(q100) q + \beta_{10} q^{3} + ( - \beta_{4} + \beta_{3}) q^{5} + (\beta_{13} + \beta_{2}) q^{7} + (\beta_{9} - 6 \beta_1 - 6) q^{9} - \beta_{15} q^{11} + (\beta_{19} - \beta_{11} + \beta_{9} + \cdots - 4) q^{13}+ \cdots + ( - \beta_{18} - 7 \beta_{17} + \cdots - 8 \beta_{2}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 20q+6q556q96q17238q2136q25+352q29+30q33258q37+504q45644q49570q53+1452q57294q61124q65+966q73+378q77++306q93+O(q100) 20 q + 6 q^{5} - 56 q^{9} - 6 q^{17} - 238 q^{21} - 36 q^{25} + 352 q^{29} + 30 q^{33} - 258 q^{37} + 504 q^{45} - 644 q^{49} - 570 q^{53} + 1452 q^{57} - 294 q^{61} - 124 q^{65} + 966 q^{73} + 378 q^{77}+ \cdots + 306 q^{93}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x202x1824x17+28x16+56x15192x14+352x13448x12++1073741824 x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + \cdots + 1073741824 : Copy content Toggle raw display

β1\beta_{1}== (793ν197760ν1870462ν17+82360ν1636188ν15++940060966912)/2280224980992 ( - 793 \nu^{19} - 7760 \nu^{18} - 70462 \nu^{17} + 82360 \nu^{16} - 36188 \nu^{15} + \cdots + 940060966912 ) / 2280224980992 Copy content Toggle raw display
β2\beta_{2}== (253ν19+720ν1811414ν1765832ν16109324ν15++1201651318784)/325746425856 ( - 253 \nu^{19} + 720 \nu^{18} - 11414 \nu^{17} - 65832 \nu^{16} - 109324 \nu^{15} + \cdots + 1201651318784 ) / 325746425856 Copy content Toggle raw display
β3\beta_{3}== (647ν1913656ν18111550ν17+323784ν16943772ν15++3620321886208)/1140112490496 ( 647 \nu^{19} - 13656 \nu^{18} - 111550 \nu^{17} + 323784 \nu^{16} - 943772 \nu^{15} + \cdots + 3620321886208 ) / 1140112490496 Copy content Toggle raw display
β4\beta_{4}== (13969ν19+83816ν18322254ν17+133768ν16++10557834919936)/2280224980992 ( - 13969 \nu^{19} + 83816 \nu^{18} - 322254 \nu^{17} + 133768 \nu^{16} + \cdots + 10557834919936 ) / 2280224980992 Copy content Toggle raw display
β5\beta_{5}== (7197ν19+68984ν18+122498ν17+1668776ν16+9654952263680)/1140112490496 ( - 7197 \nu^{19} + 68984 \nu^{18} + 122498 \nu^{17} + 1668776 \nu^{16} + \cdots - 9654952263680 ) / 1140112490496 Copy content Toggle raw display
β6\beta_{6}== (4809ν19+488ν1847870ν17+103368ν16318364ν15++562506498048)/325746425856 ( - 4809 \nu^{19} + 488 \nu^{18} - 47870 \nu^{17} + 103368 \nu^{16} - 318364 \nu^{15} + \cdots + 562506498048 ) / 325746425856 Copy content Toggle raw display
β7\beta_{7}== (39171ν19+88016ν18+123178ν17159272ν16+596340ν15+10618098679808)/2280224980992 ( 39171 \nu^{19} + 88016 \nu^{18} + 123178 \nu^{17} - 159272 \nu^{16} + 596340 \nu^{15} + \cdots - 10618098679808 ) / 2280224980992 Copy content Toggle raw display
β8\beta_{8}== (1751ν19+1586ν18+12018ν17+98900ν16115692ν15+932611883008)/71257030656 ( 1751 \nu^{19} + 1586 \nu^{18} + 12018 \nu^{17} + 98900 \nu^{16} - 115692 \nu^{15} + \cdots - 932611883008 ) / 71257030656 Copy content Toggle raw display
β9\beta_{9}== (1894ν19+58567ν18+387356ν1754686ν16826352ν15+20322543730688)/285028122624 ( - 1894 \nu^{19} + 58567 \nu^{18} + 387356 \nu^{17} - 54686 \nu^{16} - 826352 \nu^{15} + \cdots - 20322543730688 ) / 285028122624 Copy content Toggle raw display
β10\beta_{10}== (35213ν19+43608ν1869790ν17+86376ν16867004ν15++5266837864448)/1140112490496 ( - 35213 \nu^{19} + 43608 \nu^{18} - 69790 \nu^{17} + 86376 \nu^{16} - 867004 \nu^{15} + \cdots + 5266837864448 ) / 1140112490496 Copy content Toggle raw display
β11\beta_{11}== (7973ν1927196ν18+132966ν1740640ν16+118860ν15+4866399272960)/190018748416 ( 7973 \nu^{19} - 27196 \nu^{18} + 132966 \nu^{17} - 40640 \nu^{16} + 118860 \nu^{15} + \cdots - 4866399272960 ) / 190018748416 Copy content Toggle raw display
β12\beta_{12}== (14861ν19+21795ν18+209574ν17243230ν16861276ν15+10412292571136)/285028122624 ( 14861 \nu^{19} + 21795 \nu^{18} + 209574 \nu^{17} - 243230 \nu^{16} - 861276 \nu^{15} + \cdots - 10412292571136 ) / 285028122624 Copy content Toggle raw display
β13\beta_{13}== (4095ν1910250ν18+27770ν17+32460ν16150956ν15+1487602188288)/81436606464 ( - 4095 \nu^{19} - 10250 \nu^{18} + 27770 \nu^{17} + 32460 \nu^{16} - 150956 \nu^{15} + \cdots - 1487602188288 ) / 81436606464 Copy content Toggle raw display
β14\beta_{14}== (9811ν1911616ν18+66782ν171880ν16190468ν15+2344381054976)/162873212928 ( - 9811 \nu^{19} - 11616 \nu^{18} + 66782 \nu^{17} - 1880 \nu^{16} - 190468 \nu^{15} + \cdots - 2344381054976 ) / 162873212928 Copy content Toggle raw display
β15\beta_{15}== (74377ν1927452ν18793194ν17+594432ν16+1813708ν15++34979287400448)/1140112490496 ( 74377 \nu^{19} - 27452 \nu^{18} - 793194 \nu^{17} + 594432 \nu^{16} + 1813708 \nu^{15} + \cdots + 34979287400448 ) / 1140112490496 Copy content Toggle raw display
β16\beta_{16}== (16233ν19+65704ν1877422ν17+26120ν1686716ν15++2878970265600)/285028122624 ( - 16233 \nu^{19} + 65704 \nu^{18} - 77422 \nu^{17} + 26120 \nu^{16} - 86716 \nu^{15} + \cdots + 2878970265600 ) / 285028122624 Copy content Toggle raw display
β17\beta_{17}== (151077ν19+463976ν18229926ν17371992ν16+7711211126784)/2280224980992 ( - 151077 \nu^{19} + 463976 \nu^{18} - 229926 \nu^{17} - 371992 \nu^{16} + \cdots - 7711211126784 ) / 2280224980992 Copy content Toggle raw display
β18\beta_{18}== (158821ν19488368ν18+901626ν17+2990744ν16+3806280548352)/2280224980992 ( - 158821 \nu^{19} - 488368 \nu^{18} + 901626 \nu^{17} + 2990744 \nu^{16} + \cdots - 3806280548352 ) / 2280224980992 Copy content Toggle raw display
β19\beta_{19}== (25722ν1928761ν18+6684ν17453598ν16+507664ν15++1721912786944)/285028122624 ( 25722 \nu^{19} - 28761 \nu^{18} + 6684 \nu^{17} - 453598 \nu^{16} + 507664 \nu^{15} + \cdots + 1721912786944 ) / 285028122624 Copy content Toggle raw display
ν\nu== (β10β8+β5)/16 ( -\beta_{10} - \beta_{8} + \beta_{5} ) / 16 Copy content Toggle raw display
ν2\nu^{2}== (2β19+β16+β15+β14+β13β12β11β10+1)/8 ( 2 \beta_{19} + \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \cdots - 1 ) / 8 Copy content Toggle raw display
ν3\nu^{3}== (β18β16+4β144β115β10+5β7+β5++28)/8 ( - \beta_{18} - \beta_{16} + 4 \beta_{14} - 4 \beta_{11} - 5 \beta_{10} + 5 \beta_{7} + \beta_{5} + \cdots + 28 ) / 8 Copy content Toggle raw display
ν4\nu^{4}== (β19+β17+8β148β13β12+β1115β10+37)/4 ( - \beta_{19} + \beta_{17} + 8 \beta_{14} - 8 \beta_{13} - \beta_{12} + \beta_{11} - 15 \beta_{10} + \cdots - 37 ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (12β19+β18+11β1610β1510β146β136β12+6)/4 ( 12 \beta_{19} + \beta_{18} + 11 \beta_{16} - 10 \beta_{15} - 10 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + \cdots - 6 ) / 4 Copy content Toggle raw display
ν6\nu^{6}== (3β198β1813β1711β1613β15β14+11β13++274)/2 ( 3 \beta_{19} - 8 \beta_{18} - 13 \beta_{17} - 11 \beta_{16} - 13 \beta_{15} - \beta_{14} + 11 \beta_{13} + \cdots + 274 ) / 2 Copy content Toggle raw display
ν7\nu^{7}== (8β19+24β1744β14+44β138β12+8β11+1436)/2 ( - 8 \beta_{19} + 24 \beta_{17} - 44 \beta_{14} + 44 \beta_{13} - 8 \beta_{12} + 8 \beta_{11} + \cdots - 1436 ) / 2 Copy content Toggle raw display
ν8\nu^{8}== 26β1955β18+46β16+51β15+27β14+59β13++13 - 26 \beta_{19} - 55 \beta_{18} + 46 \beta_{16} + 51 \beta_{15} + 27 \beta_{14} + 59 \beta_{13} + \cdots + 13 Copy content Toggle raw display
ν9\nu^{9}== 114β19+97β1898β17+59β1698β1546β14++1128 - 114 \beta_{19} + 97 \beta_{18} - 98 \beta_{17} + 59 \beta_{16} - 98 \beta_{15} - 46 \beta_{14} + \cdots + 1128 Copy content Toggle raw display
ν10\nu^{10}== 6β19+38β17152β14+152β136β12+6β11++14250 - 6 \beta_{19} + 38 \beta_{17} - 152 \beta_{14} + 152 \beta_{13} - 6 \beta_{12} + 6 \beta_{11} + \cdots + 14250 Copy content Toggle raw display
ν11\nu^{11}== 288β19+1138β18334β16+624β151840β14+232β13+144 288 \beta_{19} + 1138 \beta_{18} - 334 \beta_{16} + 624 \beta_{15} - 1840 \beta_{14} + 232 \beta_{13} + \cdots - 144 Copy content Toggle raw display
ν12\nu^{12}== 2316β19+1892β18+2316β17+1400β16+2316β15+2476β14++75160 2316 \beta_{19} + 1892 \beta_{18} + 2316 \beta_{17} + 1400 \beta_{16} + 2316 \beta_{15} + 2476 \beta_{14} + \cdots + 75160 Copy content Toggle raw display
ν13\nu^{13}== 184β19+13048β17+13392β1413392β13184β12++76120 - 184 \beta_{19} + 13048 \beta_{17} + 13392 \beta_{14} - 13392 \beta_{13} - 184 \beta_{12} + \cdots + 76120 Copy content Toggle raw display
ν14\nu^{14}== 8144β19+960β187976β16+20328β15+21928β1438904β13+4072 8144 \beta_{19} + 960 \beta_{18} - 7976 \beta_{16} + 20328 \beta_{15} + 21928 \beta_{14} - 38904 \beta_{13} + \cdots - 4072 Copy content Toggle raw display
ν15\nu^{15}== 65856β19+30664β1889920β17+90056β1689920β15+388832 65856 \beta_{19} + 30664 \beta_{18} - 89920 \beta_{17} + 90056 \beta_{16} - 89920 \beta_{15} + \cdots - 388832 Copy content Toggle raw display
ν16\nu^{16}== 94256β19102352β17+209920β14209920β1394256β12++3601040 - 94256 \beta_{19} - 102352 \beta_{17} + 209920 \beta_{14} - 209920 \beta_{13} - 94256 \beta_{12} + \cdots + 3601040 Copy content Toggle raw display
ν17\nu^{17}== 209344β19+616944β18+137296β1641440β15717792β14++104672 - 209344 \beta_{19} + 616944 \beta_{18} + 137296 \beta_{16} - 41440 \beta_{15} - 717792 \beta_{14} + \cdots + 104672 Copy content Toggle raw display
ν18\nu^{18}== 789408β192061056β18513632β17+3020128β16513632β15++58623424 789408 \beta_{19} - 2061056 \beta_{18} - 513632 \beta_{17} + 3020128 \beta_{16} - 513632 \beta_{15} + \cdots + 58623424 Copy content Toggle raw display
ν19\nu^{19}== 5736192β191460480β17396928β14+396928β13+5736192β12+6007424 5736192 \beta_{19} - 1460480 \beta_{17} - 396928 \beta_{14} + 396928 \beta_{13} + 5736192 \beta_{12} + \cdots - 6007424 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/448Z)×\left(\mathbb{Z}/448\mathbb{Z}\right)^\times.

nn 127127 129129 197197
χ(n)\chi(n) 1-1 1+β11 + \beta_{1} 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.

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}
255.1
2.82698 0.0903966i
−1.75840 2.21540i
0.448398 + 2.79266i
−2.26510 + 1.69390i
1.31147 2.50600i
−2.82600 0.117237i
2.59951 1.11469i
2.19431 + 1.78465i
−1.03939 2.63053i
−1.49178 + 2.40304i
2.82698 + 0.0903966i
−1.75840 + 2.21540i
0.448398 2.79266i
−2.26510 1.69390i
1.31147 + 2.50600i
−2.82600 + 0.117237i
2.59951 + 1.11469i
2.19431 1.78465i
−1.03939 + 2.63053i
−1.49178 2.40304i
0 −4.65345 + 8.06002i 0 −5.11538 + 2.95337i 0 1.92900 + 18.4195i 0 −29.8092 51.6311i 0
255.2 0 −3.44104 + 5.96006i 0 4.17670 2.41142i 0 5.03893 17.8216i 0 −10.1816 17.6350i 0
255.3 0 −2.11164 + 3.65747i 0 −1.03358 + 0.596737i 0 −16.7651 7.86959i 0 4.58193 + 7.93614i 0
255.4 0 −1.67134 + 2.89484i 0 15.9583 9.21354i 0 15.4841 + 10.1609i 0 7.91327 + 13.7062i 0
255.5 0 −0.0469307 + 0.0812864i 0 −12.4861 + 7.20883i 0 15.0686 10.7674i 0 13.4956 + 23.3751i 0
255.6 0 0.0469307 0.0812864i 0 −12.4861 + 7.20883i 0 −15.0686 + 10.7674i 0 13.4956 + 23.3751i 0
255.7 0 1.67134 2.89484i 0 15.9583 9.21354i 0 −15.4841 10.1609i 0 7.91327 + 13.7062i 0
255.8 0 2.11164 3.65747i 0 −1.03358 + 0.596737i 0 16.7651 + 7.86959i 0 4.58193 + 7.93614i 0
255.9 0 3.44104 5.96006i 0 4.17670 2.41142i 0 −5.03893 + 17.8216i 0 −10.1816 17.6350i 0
255.10 0 4.65345 8.06002i 0 −5.11538 + 2.95337i 0 −1.92900 18.4195i 0 −29.8092 51.6311i 0
383.1 0 −4.65345 8.06002i 0 −5.11538 2.95337i 0 1.92900 18.4195i 0 −29.8092 + 51.6311i 0
383.2 0 −3.44104 5.96006i 0 4.17670 + 2.41142i 0 5.03893 + 17.8216i 0 −10.1816 + 17.6350i 0
383.3 0 −2.11164 3.65747i 0 −1.03358 0.596737i 0 −16.7651 + 7.86959i 0 4.58193 7.93614i 0
383.4 0 −1.67134 2.89484i 0 15.9583 + 9.21354i 0 15.4841 10.1609i 0 7.91327 13.7062i 0
383.5 0 −0.0469307 0.0812864i 0 −12.4861 7.20883i 0 15.0686 + 10.7674i 0 13.4956 23.3751i 0
383.6 0 0.0469307 + 0.0812864i 0 −12.4861 7.20883i 0 −15.0686 10.7674i 0 13.4956 23.3751i 0
383.7 0 1.67134 + 2.89484i 0 15.9583 + 9.21354i 0 −15.4841 + 10.1609i 0 7.91327 13.7062i 0
383.8 0 2.11164 + 3.65747i 0 −1.03358 0.596737i 0 16.7651 7.86959i 0 4.58193 7.93614i 0
383.9 0 3.44104 + 5.96006i 0 4.17670 + 2.41142i 0 −5.03893 17.8216i 0 −10.1816 + 17.6350i 0
383.10 0 4.65345 + 8.06002i 0 −5.11538 2.95337i 0 −1.92900 + 18.4195i 0 −29.8092 + 51.6311i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.p.h 20
4.b odd 2 1 inner 448.4.p.h 20
7.d odd 6 1 inner 448.4.p.h 20
8.b even 2 1 28.4.f.a 20
8.d odd 2 1 28.4.f.a 20
28.f even 6 1 inner 448.4.p.h 20
56.e even 2 1 196.4.f.d 20
56.h odd 2 1 196.4.f.d 20
56.j odd 6 1 28.4.f.a 20
56.j odd 6 1 196.4.d.b 20
56.k odd 6 1 196.4.d.b 20
56.k odd 6 1 196.4.f.d 20
56.m even 6 1 28.4.f.a 20
56.m even 6 1 196.4.d.b 20
56.p even 6 1 196.4.d.b 20
56.p even 6 1 196.4.f.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.f.a 20 8.b even 2 1
28.4.f.a 20 8.d odd 2 1
28.4.f.a 20 56.j odd 6 1
28.4.f.a 20 56.m even 6 1
196.4.d.b 20 56.j odd 6 1
196.4.d.b 20 56.k odd 6 1
196.4.d.b 20 56.m even 6 1
196.4.d.b 20 56.p even 6 1
196.4.f.d 20 56.e even 2 1
196.4.f.d 20 56.h odd 2 1
196.4.f.d 20 56.k odd 6 1
196.4.f.d 20 56.p even 6 1
448.4.p.h 20 1.a even 1 1 trivial
448.4.p.h 20 4.b odd 2 1 inner
448.4.p.h 20 7.d odd 6 1 inner
448.4.p.h 20 28.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T320+163T318+18379T316+1043398T314+42494101T312++51883209 T_{3}^{20} + 163 T_{3}^{18} + 18379 T_{3}^{16} + 1043398 T_{3}^{14} + 42494101 T_{3}^{12} + \cdots + 51883209 acting on S4new(448,[χ])S_{4}^{\mathrm{new}}(448, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T20 T^{20} Copy content Toggle raw display
33 T20+163T18++51883209 T^{20} + 163 T^{18} + \cdots + 51883209 Copy content Toggle raw display
55 (T103T9++81588675)2 (T^{10} - 3 T^{9} + \cdots + 81588675)^{2} Copy content Toggle raw display
77 T20++22 ⁣ ⁣49 T^{20} + \cdots + 22\!\cdots\!49 Copy content Toggle raw display
1111 T20++20 ⁣ ⁣25 T^{20} + \cdots + 20\!\cdots\!25 Copy content Toggle raw display
1313 (T10++10072189747200)2 (T^{10} + \cdots + 10072189747200)^{2} Copy content Toggle raw display
1717 (T10++60 ⁣ ⁣75)2 (T^{10} + \cdots + 60\!\cdots\!75)^{2} Copy content Toggle raw display
1919 T20++12 ⁣ ⁣25 T^{20} + \cdots + 12\!\cdots\!25 Copy content Toggle raw display
2323 T20++17 ⁣ ⁣61 T^{20} + \cdots + 17\!\cdots\!61 Copy content Toggle raw display
2929 (T588T4+5066614016)4 (T^{5} - 88 T^{4} + \cdots - 5066614016)^{4} Copy content Toggle raw display
3131 T20++43 ⁣ ⁣25 T^{20} + \cdots + 43\!\cdots\!25 Copy content Toggle raw display
3737 (T10++48 ⁣ ⁣25)2 (T^{10} + \cdots + 48\!\cdots\!25)^{2} Copy content Toggle raw display
4141 (T10++14 ⁣ ⁣72)2 (T^{10} + \cdots + 14\!\cdots\!72)^{2} Copy content Toggle raw display
4343 (T10++55 ⁣ ⁣56)2 (T^{10} + \cdots + 55\!\cdots\!56)^{2} Copy content Toggle raw display
4747 T20++90 ⁣ ⁣25 T^{20} + \cdots + 90\!\cdots\!25 Copy content Toggle raw display
5353 (T10++23 ⁣ ⁣25)2 (T^{10} + \cdots + 23\!\cdots\!25)^{2} Copy content Toggle raw display
5959 T20++53 ⁣ ⁣25 T^{20} + \cdots + 53\!\cdots\!25 Copy content Toggle raw display
6161 (T10++25 ⁣ ⁣47)2 (T^{10} + \cdots + 25\!\cdots\!47)^{2} Copy content Toggle raw display
6767 T20++16 ⁣ ⁣21 T^{20} + \cdots + 16\!\cdots\!21 Copy content Toggle raw display
7171 (T10++16 ⁣ ⁣00)2 (T^{10} + \cdots + 16\!\cdots\!00)^{2} Copy content Toggle raw display
7373 (T10++39 ⁣ ⁣75)2 (T^{10} + \cdots + 39\!\cdots\!75)^{2} Copy content Toggle raw display
7979 T20++14 ⁣ ⁣25 T^{20} + \cdots + 14\!\cdots\!25 Copy content Toggle raw display
8383 (T10+41 ⁣ ⁣08)2 (T^{10} + \cdots - 41\!\cdots\!08)^{2} Copy content Toggle raw display
8989 (T10++99 ⁣ ⁣27)2 (T^{10} + \cdots + 99\!\cdots\!27)^{2} Copy content Toggle raw display
9797 (T10++12 ⁣ ⁣00)2 (T^{10} + \cdots + 12\!\cdots\!00)^{2} Copy content Toggle raw display
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