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Properties

Label 475.2.l.b
Level 475475
Weight 22
Character orbit 475.l
Analytic conductor 3.7933.793
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] = [475,2,Mod(101,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 475=52⋅19 475 = 5^{2} \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 475.l (of order 99, 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: 3.792894096013.79289409601
Analytic rank: 00
Dimension: 1818
Relative dimension: 33 over Q(ζ9)\Q(\zeta_{9})
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+12x16−8x15+96x14−75x13+448x12−405x11+1521x10+⋯+361 x^{18} + 12 x^{16} - 8 x^{15} + 96 x^{14} - 75 x^{13} + 448 x^{12} - 405 x^{11} + 1521 x^{10} + \cdots + 361 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: no (minimal twist has level 95)
Sato-Tate group: SU(2)[C9]\mathrm{SU}(2)[C_{9}]

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+β14q2+(β15+β7+⋯+β4)q3+(β10−β9+⋯+β2)q4+(β16+β15+⋯+β3)q6+(−β17−β16−β14+⋯+1)q7+⋯+(β15−3β12+β10+⋯−1)q99+O(q100) q + \beta_{14} q^{2} + (\beta_{15} + \beta_{7} + \cdots + \beta_{4}) q^{3} + (\beta_{10} - \beta_{9} + \cdots + \beta_{2}) q^{4} + (\beta_{16} + \beta_{15} + \cdots + \beta_{3}) q^{6} + ( - \beta_{17} - \beta_{16} - \beta_{14} + \cdots + 1) q^{7}+ \cdots + (\beta_{15} - 3 \beta_{12} + \beta_{10} + \cdots - 1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 18q−3q2+3q3−3q4−6q6+12q8−21q9−6q12+3q13+24q14+21q16+24q17+12q18−12q19+3q21−15q22−21q23+21q24−21q26+⋯+3q99+O(q100) 18 q - 3 q^{2} + 3 q^{3} - 3 q^{4} - 6 q^{6} + 12 q^{8} - 21 q^{9} - 6 q^{12} + 3 q^{13} + 24 q^{14} + 21 q^{16} + 24 q^{17} + 12 q^{18} - 12 q^{19} + 3 q^{21} - 15 q^{22} - 21 q^{23} + 21 q^{24} - 21 q^{26}+ \cdots + 3 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x18+12x16−8x15+96x14−75x13+448x12−405x11+1521x10+⋯+361 x^{18} + 12 x^{16} - 8 x^{15} + 96 x^{14} - 75 x^{13} + 448 x^{12} - 405 x^{11} + 1521 x^{10} + \cdots + 361 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (−136164871502623ν17+⋯−12 ⁣⋯ ⁣58)/23 ⁣⋯ ⁣13 ( - 136164871502623 \nu^{17} + \cdots - 12\!\cdots\!58 ) / 23\!\cdots\!13 Copy content Toggle raw display
β3\beta_{3}== (−305669388244104ν17+493873766181026ν16+⋯+47 ⁣⋯ ⁣16)/78 ⁣⋯ ⁣71 ( - 305669388244104 \nu^{17} + 493873766181026 \nu^{16} + \cdots + 47\!\cdots\!16 ) / 78\!\cdots\!71 Copy content Toggle raw display
β4\beta_{4}== (−13 ⁣⋯ ⁣19ν17+⋯+40 ⁣⋯ ⁣81)/23 ⁣⋯ ⁣13 ( - 13\!\cdots\!19 \nu^{17} + \cdots + 40\!\cdots\!81 ) / 23\!\cdots\!13 Copy content Toggle raw display
β5\beta_{5}== (−13 ⁣⋯ ⁣95ν17+⋯+49 ⁣⋯ ⁣59)/23 ⁣⋯ ⁣13 ( - 13\!\cdots\!95 \nu^{17} + \cdots + 49\!\cdots\!59 ) / 23\!\cdots\!13 Copy content Toggle raw display
β6\beta_{6}== (−38 ⁣⋯ ⁣82ν17+⋯+73 ⁣⋯ ⁣88)/23 ⁣⋯ ⁣13 ( - 38\!\cdots\!82 \nu^{17} + \cdots + 73\!\cdots\!88 ) / 23\!\cdots\!13 Copy content Toggle raw display
β7\beta_{7}== (13 ⁣⋯ ⁣56ν17+305669388244104ν16+⋯−15 ⁣⋯ ⁣37)/78 ⁣⋯ ⁣71 ( 13\!\cdots\!56 \nu^{17} + 305669388244104 \nu^{16} + \cdots - 15\!\cdots\!37 ) / 78\!\cdots\!71 Copy content Toggle raw display
β8\beta_{8}== (42 ⁣⋯ ⁣70ν17+⋯−79 ⁣⋯ ⁣20)/23 ⁣⋯ ⁣13 ( 42\!\cdots\!70 \nu^{17} + \cdots - 79\!\cdots\!20 ) / 23\!\cdots\!13 Copy content Toggle raw display
β9\beta_{9}== (−116934812256397ν17−400329673137974ν16+⋯−19 ⁣⋯ ⁣38)/63 ⁣⋯ ⁣49 ( - 116934812256397 \nu^{17} - 400329673137974 \nu^{16} + \cdots - 19\!\cdots\!38 ) / 63\!\cdots\!49 Copy content Toggle raw display
β10\beta_{10}== (47 ⁣⋯ ⁣03ν17+⋯+52 ⁣⋯ ⁣54)/23 ⁣⋯ ⁣13 ( 47\!\cdots\!03 \nu^{17} + \cdots + 52\!\cdots\!54 ) / 23\!\cdots\!13 Copy content Toggle raw display
β11\beta_{11}== (−61 ⁣⋯ ⁣77ν17+828266836359098ν16+⋯+60 ⁣⋯ ⁣85)/23 ⁣⋯ ⁣13 ( - 61\!\cdots\!77 \nu^{17} + 828266836359098 \nu^{16} + \cdots + 60\!\cdots\!85 ) / 23\!\cdots\!13 Copy content Toggle raw display
β12\beta_{12}== (84 ⁣⋯ ⁣25ν17+⋯−58 ⁣⋯ ⁣05)/23 ⁣⋯ ⁣13 ( 84\!\cdots\!25 \nu^{17} + \cdots - 58\!\cdots\!05 ) / 23\!\cdots\!13 Copy content Toggle raw display
β13\beta_{13}== (84 ⁣⋯ ⁣44ν17+⋯−15 ⁣⋯ ⁣70)/23 ⁣⋯ ⁣13 ( 84\!\cdots\!44 \nu^{17} + \cdots - 15\!\cdots\!70 ) / 23\!\cdots\!13 Copy content Toggle raw display
β14\beta_{14}== (−87 ⁣⋯ ⁣64ν17+⋯+47 ⁣⋯ ⁣08)/23 ⁣⋯ ⁣13 ( - 87\!\cdots\!64 \nu^{17} + \cdots + 47\!\cdots\!08 ) / 23\!\cdots\!13 Copy content Toggle raw display
β15\beta_{15}== (92 ⁣⋯ ⁣56ν17+⋯+30 ⁣⋯ ⁣25)/23 ⁣⋯ ⁣13 ( 92\!\cdots\!56 \nu^{17} + \cdots + 30\!\cdots\!25 ) / 23\!\cdots\!13 Copy content Toggle raw display
β16\beta_{16}== (316478299376213ν17−588887179277519ν16+⋯−38 ⁣⋯ ⁣43)/63 ⁣⋯ ⁣49 ( 316478299376213 \nu^{17} - 588887179277519 \nu^{16} + \cdots - 38\!\cdots\!43 ) / 63\!\cdots\!49 Copy content Toggle raw display
β17\beta_{17}== (17 ⁣⋯ ⁣19ν17+⋯−11 ⁣⋯ ⁣95)/23 ⁣⋯ ⁣13 ( 17\!\cdots\!19 \nu^{17} + \cdots - 11\!\cdots\!95 ) / 23\!\cdots\!13 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}== −β16+3β7+β2 -\beta_{16} + 3\beta_{7} + \beta_{2} Copy content Toggle raw display
ν3\nu^{3}== β17−β12−2β10+β9−β4−3β3−2β2−3β1+1 \beta_{17} - \beta_{12} - 2\beta_{10} + \beta_{9} - \beta_{4} - 3\beta_{3} - 2\beta_{2} - 3\beta _1 + 1 Copy content Toggle raw display
ν4\nu^{4}== −β17+6β16+β14−β13+2β12+6β11+6β10+⋯−12 - \beta_{17} + 6 \beta_{16} + \beta_{14} - \beta_{13} + 2 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} + \cdots - 12 Copy content Toggle raw display
ν5\nu^{5}== −9β16−β15−β14−β13−β12−7β11+9β10+⋯+7 - 9 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - 7 \beta_{11} + 9 \beta_{10} + \cdots + 7 Copy content Toggle raw display
ν6\nu^{6}== 3β17−8β16+10β15−9β14+10β13−12β12−33β11+⋯+57 3 \beta_{17} - 8 \beta_{16} + 10 \beta_{15} - 9 \beta_{14} + 10 \beta_{13} - 12 \beta_{12} - 33 \beta_{11} + \cdots + 57 Copy content Toggle raw display
ν7\nu^{7}== −46β17+69β16−10β15+22β14−12β13+77β12+⋯−112 - 46 \beta_{17} + 69 \beta_{16} - 10 \beta_{15} + 22 \beta_{14} - 12 \beta_{13} + 77 \beta_{12} + \cdots - 112 Copy content Toggle raw display
ν8\nu^{8}== 52β17−176β16−63β15−14β14−14β13−76β12+⋯+39 52 \beta_{17} - 176 \beta_{16} - 63 \beta_{15} - 14 \beta_{14} - 14 \beta_{13} - 76 \beta_{12} + \cdots + 39 Copy content Toggle raw display
ν9\nu^{9}== 252β17−53β16+181β15−105β14+181β13−417β12+⋯+518 252 \beta_{17} - 53 \beta_{16} + 181 \beta_{15} - 105 \beta_{14} + 181 \beta_{13} - 417 \beta_{12} + \cdots + 518 Copy content Toggle raw display
ν10\nu^{10}== −682β17+1497β16−133β15+550β14−417β13+1349β12+⋯−2209 - 682 \beta_{17} + 1497 \beta_{16} - 133 \beta_{15} + 550 \beta_{14} - 417 \beta_{13} + 1349 \beta_{12} + \cdots - 2209 Copy content Toggle raw display
ν11\nu^{11}== 487β17−3061β16−816β15−533β14−533β13−1081β12+⋯+1560 487 \beta_{17} - 3061 \beta_{16} - 816 \beta_{15} - 533 \beta_{14} - 533 \beta_{13} - 1081 \beta_{12} + \cdots + 1560 Copy content Toggle raw display
ν12\nu^{12}== 2804β17−2093β16+3829β15−2748β14+3829β13−5996β12+⋯+11782 2804 \beta_{17} - 2093 \beta_{16} + 3829 \beta_{15} - 2748 \beta_{14} + 3829 \beta_{13} - 5996 \beta_{12} + \cdots + 11782 Copy content Toggle raw display
ν13\nu^{13}== −13860β17+24807β16−3646β15+9642β14−5996β13+⋯−36433 - 13860 \beta_{17} + 24807 \beta_{16} - 3646 \beta_{15} + 9642 \beta_{14} - 5996 \beta_{13} + \cdots - 36433 Copy content Toggle raw display
ν14\nu^{14}== 13621β17−54352β16−18275β15−8140β14−8140β13+⋯+20828 13621 \beta_{17} - 54352 \beta_{16} - 18275 \beta_{15} - 8140 \beta_{14} - 8140 \beta_{13} + \cdots + 20828 Copy content Toggle raw display
ν15\nu^{15}== 65420β17−28968β16+67593β15−42795β14+67593β13+⋯+185488 65420 \beta_{17} - 28968 \beta_{16} + 67593 \beta_{15} - 42795 \beta_{14} + 67593 \beta_{13} + \cdots + 185488 Copy content Toggle raw display
ν16\nu^{16}== −239831β17+463488β16−58869β15+181667β14−122798β13+⋯−669036 - 239831 \beta_{17} + 463488 \beta_{16} - 58869 \beta_{15} + 181667 \beta_{14} - 122798 \beta_{13} + \cdots - 669036 Copy content Toggle raw display
ν17\nu^{17}== 208482β17−982637β16−300623β15−168782β14−168782β13+⋯+436673 208482 \beta_{17} - 982637 \beta_{16} - 300623 \beta_{15} - 168782 \beta_{14} - 168782 \beta_{13} + \cdots + 436673 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/475Z)×\left(\mathbb{Z}/475\mathbb{Z}\right)^\times.

nn 7777 401401
χ(n)\chi(n) 11 β4+β8\beta_{4} + \beta_{8}

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}
101.1
−0.908512 − 1.57359i
0.296923 + 0.514286i
0.785237 + 1.36007i
1.01081 − 1.75077i
0.597039 − 1.03410i
−0.841804 + 1.45805i
−1.31112 + 2.27092i
−0.359728 + 0.623068i
0.731154 − 1.26640i
1.01081 + 1.75077i
0.597039 + 1.03410i
−0.841804 − 1.45805i
−0.908512 + 1.57359i
0.296923 − 0.514286i
0.785237 − 1.36007i
−1.31112 − 2.27092i
−0.359728 − 0.623068i
0.731154 + 1.26640i
−1.39192 + 1.16796i 0.166424 + 0.0605732i 0.226016 − 1.28180i 0 −0.302396 + 0.110063i 0.536732 − 0.929646i −0.634528 − 1.09903i −2.27411 − 1.90820i 0
101.2 0.454913 − 0.381717i −1.81364 − 0.660111i −0.286059 + 1.62232i 0 −1.07702 + 0.392004i 0.530259 − 0.918436i 1.08298 + 1.87578i 0.555408 + 0.466042i 0
101.3 1.20305 − 1.00948i 2.14722 + 0.781523i 0.0809872 − 0.459301i 0 3.37214 − 1.22736i −2.00668 + 3.47568i 1.20425 + 2.08582i 1.70162 + 1.42783i 0
176.1 −1.89970 + 0.691434i 0.330026 − 1.87167i 1.59869 − 1.34146i 0 0.667187 + 3.78381i 1.54609 + 2.67790i −0.0878797 + 0.152212i −0.575162 − 0.209342i 0
176.2 −1.12207 + 0.408399i −0.394733 + 2.23864i −0.439845 + 0.369074i 0 −0.471342 − 2.67312i −1.09825 − 1.90222i 1.53688 − 2.66196i −2.03663 − 0.741274i 0
176.3 1.58207 − 0.575828i 0.564707 − 3.20261i 0.639290 − 0.536428i 0 −0.950745 − 5.39194i −0.274194 − 0.474919i −0.981094 + 1.69930i −7.11876 − 2.59101i 0
226.1 −0.455347 + 2.58240i 0.711484 + 0.597006i −4.58206 − 1.66773i 0 −1.86568 + 1.56549i −1.91579 − 3.31824i 3.77094 − 6.53146i −0.371151 − 2.10490i 0
226.2 −0.124932 + 0.708527i 0.945515 + 0.793382i 1.39298 + 0.507004i 0 −0.680257 + 0.570804i 0.645970 + 1.11885i −1.25271 + 2.16976i −0.256400 − 1.45411i 0
226.3 0.253927 − 1.44009i −1.15700 − 0.970838i −0.130002 − 0.0473169i 0 −1.69189 + 1.41966i 2.03586 + 3.52622i 1.36116 − 2.35759i −0.124823 − 0.707907i 0
251.1 −1.89970 − 0.691434i 0.330026 + 1.87167i 1.59869 + 1.34146i 0 0.667187 − 3.78381i 1.54609 − 2.67790i −0.0878797 − 0.152212i −0.575162 + 0.209342i 0
251.2 −1.12207 − 0.408399i −0.394733 − 2.23864i −0.439845 − 0.369074i 0 −0.471342 + 2.67312i −1.09825 + 1.90222i 1.53688 + 2.66196i −2.03663 + 0.741274i 0
251.3 1.58207 + 0.575828i 0.564707 + 3.20261i 0.639290 + 0.536428i 0 −0.950745 + 5.39194i −0.274194 + 0.474919i −0.981094 − 1.69930i −7.11876 + 2.59101i 0
301.1 −1.39192 − 1.16796i 0.166424 − 0.0605732i 0.226016 + 1.28180i 0 −0.302396 − 0.110063i 0.536732 + 0.929646i −0.634528 + 1.09903i −2.27411 + 1.90820i 0
301.2 0.454913 + 0.381717i −1.81364 + 0.660111i −0.286059 − 1.62232i 0 −1.07702 − 0.392004i 0.530259 + 0.918436i 1.08298 − 1.87578i 0.555408 − 0.466042i 0
301.3 1.20305 + 1.00948i 2.14722 − 0.781523i 0.0809872 + 0.459301i 0 3.37214 + 1.22736i −2.00668 − 3.47568i 1.20425 − 2.08582i 1.70162 − 1.42783i 0
351.1 −0.455347 − 2.58240i 0.711484 − 0.597006i −4.58206 + 1.66773i 0 −1.86568 − 1.56549i −1.91579 + 3.31824i 3.77094 + 6.53146i −0.371151 + 2.10490i 0
351.2 −0.124932 − 0.708527i 0.945515 − 0.793382i 1.39298 − 0.507004i 0 −0.680257 − 0.570804i 0.645970 − 1.11885i −1.25271 − 2.16976i −0.256400 + 1.45411i 0
351.3 0.253927 + 1.44009i −1.15700 + 0.970838i −0.130002 + 0.0473169i 0 −1.69189 − 1.41966i 2.03586 − 3.52622i 1.36116 + 2.35759i −0.124823 + 0.707907i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.l.b 18
5.b even 2 1 95.2.k.b ✓ 18
5.c odd 4 2 475.2.u.c 36
15.d odd 2 1 855.2.bs.b 18
19.e even 9 1 inner 475.2.l.b 18
19.e even 9 1 9025.2.a.ce 9
19.f odd 18 1 9025.2.a.cd 9
95.o odd 18 1 1805.2.a.u 9
95.p even 18 1 95.2.k.b ✓ 18
95.p even 18 1 1805.2.a.t 9
95.q odd 36 2 475.2.u.c 36
285.bd odd 18 1 855.2.bs.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.b ✓ 18 5.b even 2 1
95.2.k.b ✓ 18 95.p even 18 1
475.2.l.b 18 1.a even 1 1 trivial
475.2.l.b 18 19.e even 9 1 inner
475.2.u.c 36 5.c odd 4 2
475.2.u.c 36 95.q odd 36 2
855.2.bs.b 18 15.d odd 2 1
855.2.bs.b 18 285.bd odd 18 1
1805.2.a.t 9 95.p even 18 1
1805.2.a.u 9 95.o odd 18 1
9025.2.a.cd 9 19.f odd 18 1
9025.2.a.ce 9 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T218+3T217+6T216+5T215−15T214−24T213+70T212+⋯+361 T_{2}^{18} + 3 T_{2}^{17} + 6 T_{2}^{16} + 5 T_{2}^{15} - 15 T_{2}^{14} - 24 T_{2}^{13} + 70 T_{2}^{12} + \cdots + 361 acting on S2new(475,[χ])S_{2}^{\mathrm{new}}(475, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T18+3T17+⋯+361 T^{18} + 3 T^{17} + \cdots + 361 Copy content Toggle raw display
33 T18−3T17+⋯+361 T^{18} - 3 T^{17} + \cdots + 361 Copy content Toggle raw display
55 T18 T^{18} Copy content Toggle raw display
77 T18+33T16+⋯+117649 T^{18} + 33 T^{16} + \cdots + 117649 Copy content Toggle raw display
1111 T18+36T16+⋯+361 T^{18} + 36 T^{16} + \cdots + 361 Copy content Toggle raw display
1313 T18−3T17+⋯+98743969 T^{18} - 3 T^{17} + \cdots + 98743969 Copy content Toggle raw display
1717 T18−24T17+⋯+1 T^{18} - 24 T^{17} + \cdots + 1 Copy content Toggle raw display
1919 T18+⋯+322687697779 T^{18} + \cdots + 322687697779 Copy content Toggle raw display
2323 T18+⋯+3993860809 T^{18} + \cdots + 3993860809 Copy content Toggle raw display
2929 T18+⋯+492805404001 T^{18} + \cdots + 492805404001 Copy content Toggle raw display
3131 T18+⋯+993257404129 T^{18} + \cdots + 993257404129 Copy content Toggle raw display
3737 (T9−30T8+⋯−27721)2 (T^{9} - 30 T^{8} + \cdots - 27721)^{2} Copy content Toggle raw display
4141 T18+⋯+132479256529 T^{18} + \cdots + 132479256529 Copy content Toggle raw display
4343 T18+⋯+101989801 T^{18} + \cdots + 101989801 Copy content Toggle raw display
4747 T18+⋯+32737816559809 T^{18} + \cdots + 32737816559809 Copy content Toggle raw display
5353 T18+⋯+19250147475001 T^{18} + \cdots + 19250147475001 Copy content Toggle raw display
5959 T18+⋯+333749999521 T^{18} + \cdots + 333749999521 Copy content Toggle raw display
6161 T18+⋯+3468418292161 T^{18} + \cdots + 3468418292161 Copy content Toggle raw display
6767 T18+⋯+36 ⁣⋯ ⁣81 T^{18} + \cdots + 36\!\cdots\!81 Copy content Toggle raw display
7171 T18+⋯+8590138489 T^{18} + \cdots + 8590138489 Copy content Toggle raw display
7373 T18+⋯+1047660743809 T^{18} + \cdots + 1047660743809 Copy content Toggle raw display
7979 T18+⋯+316957384070209 T^{18} + \cdots + 316957384070209 Copy content Toggle raw display
8383 T18+171T16+⋯+96609241 T^{18} + 171 T^{16} + \cdots + 96609241 Copy content Toggle raw display
8989 T18+⋯+143304841 T^{18} + \cdots + 143304841 Copy content Toggle raw display
9797 T18+⋯+36 ⁣⋯ ⁣09 T^{18} + \cdots + 36\!\cdots\!09 Copy content Toggle raw display
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