↑

Properties

Label 18.11.d.a
Level 1818
Weight 1111
Character orbit 18.d
Analytic conductor 11.43611.436
Analytic rank 00
Dimension 2020
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,11,Mod(5,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.5");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: N N == 18=2⋅32 18 = 2 \cdot 3^{2}
Weight: k k == 11 11
Character orbit: [χ][\chi] == 18.d (of order 66, degree 22, 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: 11.436430548111.4364305481
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)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x20−10x19+63819x18−574086x17+1685636151x16−13472077884x15+⋯+42 ⁣⋯ ⁣17 x^{20} - 10 x^{19} + 63819 x^{18} - 574086 x^{17} + 1685636151 x^{16} - 13472077884 x^{15} + \cdots + 42\!\cdots\!17 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 246⋅337 2^{46}\cdot 3^{37}
Twist minimal: yes
Sato-Tate group: SU(2)[C6]\mathrm{SU}(2)[C_{6}]

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,…,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+β2q2+(−β4+β3−4β1−2)q3+512β1q4+(β9+4β4+β3+⋯−662)q5+(−β14−β11+⋯+608)q6+⋯+(6642β19+59025β18+⋯+2182827372)q99+O(q100) q + \beta_{2} q^{2} + ( - \beta_{4} + \beta_{3} - 4 \beta_1 - 2) q^{3} + 512 \beta_1 q^{4} + (\beta_{9} + 4 \beta_{4} + \beta_{3} + \cdots - 662) q^{5} + ( - \beta_{14} - \beta_{11} + \cdots + 608) q^{6}+ \cdots + (6642 \beta_{19} + 59025 \beta_{18} + \cdots + 2182827372) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 20q−84q3+5120q4−9918q5+12864q6+12238q7+79248q9−327582q11+9216q12−280550q13+175680q14−2685042q15−2621440q16+3925632q18−2966240q19+⋯+41160676842q99+O(q100) 20 q - 84 q^{3} + 5120 q^{4} - 9918 q^{5} + 12864 q^{6} + 12238 q^{7} + 79248 q^{9} - 327582 q^{11} + 9216 q^{12} - 280550 q^{13} + 175680 q^{14} - 2685042 q^{15} - 2621440 q^{16} + 3925632 q^{18} - 2966240 q^{19}+ \cdots + 41160676842 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x20−10x19+63819x18−574086x17+1685636151x16−13472077884x15+⋯+42 ⁣⋯ ⁣17 x^{20} - 10 x^{19} + 63819 x^{18} - 574086 x^{17} + 1685636151 x^{16} - 13472077884 x^{15} + \cdots + 42\!\cdots\!17 : Copy content Toggle raw display

β1\beta_{1}== (53 ⁣⋯ ⁣04ν19+⋯+96 ⁣⋯ ⁣63)/21 ⁣⋯ ⁣00 ( 53\!\cdots\!04 \nu^{19} + \cdots + 96\!\cdots\!63 ) / 21\!\cdots\!00 Copy content Toggle raw display
β2\beta_{2}== (−42 ⁣⋯ ⁣57ν19+⋯−33 ⁣⋯ ⁣02)/66 ⁣⋯ ⁣00 ( - 42\!\cdots\!57 \nu^{19} + \cdots - 33\!\cdots\!02 ) / 66\!\cdots\!00 Copy content Toggle raw display
β3\beta_{3}== (42 ⁣⋯ ⁣57ν19+⋯−35 ⁣⋯ ⁣49)/66 ⁣⋯ ⁣00 ( 42\!\cdots\!57 \nu^{19} + \cdots - 35\!\cdots\!49 ) / 66\!\cdots\!00 Copy content Toggle raw display
β4\beta_{4}== (11 ⁣⋯ ⁣72ν19+⋯−64 ⁣⋯ ⁣73)/13 ⁣⋯ ⁣00 ( 11\!\cdots\!72 \nu^{19} + \cdots - 64\!\cdots\!73 ) / 13\!\cdots\!00 Copy content Toggle raw display
β5\beta_{5}== (19 ⁣⋯ ⁣03ν19+⋯+78 ⁣⋯ ⁣76)/13 ⁣⋯ ⁣00 ( 19\!\cdots\!03 \nu^{19} + \cdots + 78\!\cdots\!76 ) / 13\!\cdots\!00 Copy content Toggle raw display
β6\beta_{6}== (−25 ⁣⋯ ⁣50ν19+⋯+89 ⁣⋯ ⁣70)/41 ⁣⋯ ⁣00 ( - 25\!\cdots\!50 \nu^{19} + \cdots + 89\!\cdots\!70 ) / 41\!\cdots\!00 Copy content Toggle raw display
β7\beta_{7}== (30 ⁣⋯ ⁣53ν19+⋯−36 ⁣⋯ ⁣26)/41 ⁣⋯ ⁣00 ( 30\!\cdots\!53 \nu^{19} + \cdots - 36\!\cdots\!26 ) / 41\!\cdots\!00 Copy content Toggle raw display
β8\beta_{8}== (30 ⁣⋯ ⁣54ν19+⋯+18 ⁣⋯ ⁣29)/41 ⁣⋯ ⁣00 ( 30\!\cdots\!54 \nu^{19} + \cdots + 18\!\cdots\!29 ) / 41\!\cdots\!00 Copy content Toggle raw display
β9\beta_{9}== (40 ⁣⋯ ⁣62ν19+⋯−20 ⁣⋯ ⁣14)/41 ⁣⋯ ⁣00 ( 40\!\cdots\!62 \nu^{19} + \cdots - 20\!\cdots\!14 ) / 41\!\cdots\!00 Copy content Toggle raw display
β10\beta_{10}== (−47 ⁣⋯ ⁣51ν19+⋯+98 ⁣⋯ ⁣78)/41 ⁣⋯ ⁣00 ( - 47\!\cdots\!51 \nu^{19} + \cdots + 98\!\cdots\!78 ) / 41\!\cdots\!00 Copy content Toggle raw display
β11\beta_{11}== (−19 ⁣⋯ ⁣99ν19+⋯−13 ⁣⋯ ⁣28)/17 ⁣⋯ ⁣00 ( - 19\!\cdots\!99 \nu^{19} + \cdots - 13\!\cdots\!28 ) / 17\!\cdots\!00 Copy content Toggle raw display
β12\beta_{12}== (−25 ⁣⋯ ⁣54ν19+⋯−30 ⁣⋯ ⁣25)/20 ⁣⋯ ⁣00 ( - 25\!\cdots\!54 \nu^{19} + \cdots - 30\!\cdots\!25 ) / 20\!\cdots\!00 Copy content Toggle raw display
β13\beta_{13}== (95 ⁣⋯ ⁣07ν19+⋯+36 ⁣⋯ ⁣31)/41 ⁣⋯ ⁣00 ( 95\!\cdots\!07 \nu^{19} + \cdots + 36\!\cdots\!31 ) / 41\!\cdots\!00 Copy content Toggle raw display
β14\beta_{14}== (−55 ⁣⋯ ⁣53ν19+⋯+18 ⁣⋯ ⁣64)/17 ⁣⋯ ⁣00 ( - 55\!\cdots\!53 \nu^{19} + \cdots + 18\!\cdots\!64 ) / 17\!\cdots\!00 Copy content Toggle raw display
β15\beta_{15}== (−14 ⁣⋯ ⁣25ν19+⋯+68 ⁣⋯ ⁣34)/41 ⁣⋯ ⁣00 ( - 14\!\cdots\!25 \nu^{19} + \cdots + 68\!\cdots\!34 ) / 41\!\cdots\!00 Copy content Toggle raw display
β16\beta_{16}== (20 ⁣⋯ ⁣83ν19+⋯+38 ⁣⋯ ⁣87)/41 ⁣⋯ ⁣00 ( 20\!\cdots\!83 \nu^{19} + \cdots + 38\!\cdots\!87 ) / 41\!\cdots\!00 Copy content Toggle raw display
β17\beta_{17}== (−26 ⁣⋯ ⁣05ν19+⋯+46 ⁣⋯ ⁣86)/41 ⁣⋯ ⁣00 ( - 26\!\cdots\!05 \nu^{19} + \cdots + 46\!\cdots\!86 ) / 41\!\cdots\!00 Copy content Toggle raw display
β18\beta_{18}== (54 ⁣⋯ ⁣34ν19+⋯−50 ⁣⋯ ⁣91)/41 ⁣⋯ ⁣00 ( 54\!\cdots\!34 \nu^{19} + \cdots - 50\!\cdots\!91 ) / 41\!\cdots\!00 Copy content Toggle raw display
β19\beta_{19}== (80 ⁣⋯ ⁣05ν19+⋯+40 ⁣⋯ ⁣69)/41 ⁣⋯ ⁣00 ( 80\!\cdots\!05 \nu^{19} + \cdots + 40\!\cdots\!69 ) / 41\!\cdots\!00 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (10β17−30β14+8β13−10β12+8β11+2β10+⋯+1492)/3888 ( 10 \beta_{17} - 30 \beta_{14} + 8 \beta_{13} - 10 \beta_{12} + 8 \beta_{11} + 2 \beta_{10} + \cdots + 1492 ) / 3888 Copy content Toggle raw display
ν2\nu^{2}== (316β17+648β15−516β14+170β13−46β12−262β11+⋯−12396926)/1944 ( 316 \beta_{17} + 648 \beta_{15} - 516 \beta_{14} + 170 \beta_{13} - 46 \beta_{12} - 262 \beta_{11} + \cdots - 12396926 ) / 1944 Copy content Toggle raw display
ν3\nu^{3}== (−14256β19−14256β18−104222β17+6480β16−1296β15+⋯−253049042)/3888 ( - 14256 \beta_{19} - 14256 \beta_{18} - 104222 \beta_{17} + 6480 \beta_{16} - 1296 \beta_{15} + \cdots - 253049042 ) / 3888 Copy content Toggle raw display
ν4\nu^{4}== (−115020β19+100764β18−2498350β17+3240β16−4524012β15+⋯+67970418806)/972 ( - 115020 \beta_{19} + 100764 \beta_{18} - 2498350 \beta_{17} + 3240 \beta_{16} - 4524012 \beta_{15} + \cdots + 67970418806 ) / 972 Copy content Toggle raw display
ν5\nu^{5}== (270059184β19+272217024β18+1229184508β17−456229584β16+⋯+5748673158226)/3888 ( 270059184 \beta_{19} + 272217024 \beta_{18} + 1229184508 \beta_{17} - 456229584 \beta_{16} + \cdots + 5748673158226 ) / 3888 Copy content Toggle raw display
ν6\nu^{6}== (5926077288β19−5112591696β18+71672748418β17−684360576β16+⋯−16 ⁣⋯ ⁣38)/1944 ( 5926077288 \beta_{19} - 5112591696 \beta_{18} + 71672748418 \beta_{17} - 684360576 \beta_{16} + \cdots - 16\!\cdots\!38 ) / 1944 Copy content Toggle raw display
ν7\nu^{7}== (−4534948582416β19−4612226817744β18−15407050889594β17+⋯−10 ⁣⋯ ⁣20)/3888 ( - 4534948582416 \beta_{19} - 4612226817744 \beta_{18} - 15407050889594 \beta_{17} + \cdots - 10\!\cdots\!20 ) / 3888 Copy content Toggle raw display
ν8\nu^{8}== (−29605866477210β19+25031329693974β18−250500745288382β17+⋯+55 ⁣⋯ ⁣23)/486 ( - 29605866477210 \beta_{19} + 25031329693974 \beta_{18} - 250500745288382 \beta_{17} + \cdots + 55\!\cdots\!23 ) / 486 Copy content Toggle raw display
ν9\nu^{9}== (73 ⁣⋯ ⁣36β19+⋯+18 ⁣⋯ ⁣78)/3888 ( 73\!\cdots\!36 \beta_{19} + \cdots + 18\!\cdots\!78 ) / 3888 Copy content Toggle raw display
ν10\nu^{10}== (21 ⁣⋯ ⁣48β19+⋯−30 ⁣⋯ ⁣44)/1944 ( 21\!\cdots\!48 \beta_{19} + \cdots - 30\!\cdots\!44 ) / 1944 Copy content Toggle raw display
ν11\nu^{11}== (−11 ⁣⋯ ⁣20β19+⋯−31 ⁣⋯ ⁣06)/3888 ( - 11\!\cdots\!20 \beta_{19} + \cdots - 31\!\cdots\!06 ) / 3888 Copy content Toggle raw display
ν12\nu^{12}== (−19 ⁣⋯ ⁣16β19+⋯+21 ⁣⋯ ⁣34)/972 ( - 19\!\cdots\!16 \beta_{19} + \cdots + 21\!\cdots\!34 ) / 972 Copy content Toggle raw display
ν13\nu^{13}== (18 ⁣⋯ ⁣40β19+⋯+50 ⁣⋯ ⁣56)/3888 ( 18\!\cdots\!40 \beta_{19} + \cdots + 50\!\cdots\!56 ) / 3888 Copy content Toggle raw display
ν14\nu^{14}== (65 ⁣⋯ ⁣56β19+⋯−61 ⁣⋯ ⁣02)/1944 ( 65\!\cdots\!56 \beta_{19} + \cdots - 61\!\cdots\!02 ) / 1944 Copy content Toggle raw display
ν15\nu^{15}== (−29 ⁣⋯ ⁣64β19+⋯−80 ⁣⋯ ⁣42)/3888 ( - 29\!\cdots\!64 \beta_{19} + \cdots - 80\!\cdots\!42 ) / 3888 Copy content Toggle raw display
ν16\nu^{16}== (−27 ⁣⋯ ⁣90β19+⋯+22 ⁣⋯ ⁣81)/486 ( - 27\!\cdots\!90 \beta_{19} + \cdots + 22\!\cdots\!81 ) / 486 Copy content Toggle raw display
ν17\nu^{17}== (47 ⁣⋯ ⁣52β19+⋯+12 ⁣⋯ ⁣50)/3888 ( 47\!\cdots\!52 \beta_{19} + \cdots + 12\!\cdots\!50 ) / 3888 Copy content Toggle raw display
ν18\nu^{18}== (18 ⁣⋯ ⁣16β19+⋯−13 ⁣⋯ ⁣90)/1944 ( 18\!\cdots\!16 \beta_{19} + \cdots - 13\!\cdots\!90 ) / 1944 Copy content Toggle raw display
ν19\nu^{19}== (−74 ⁣⋯ ⁣32β19+⋯−20 ⁣⋯ ⁣92)/3888 ( - 74\!\cdots\!32 \beta_{19} + \cdots - 20\!\cdots\!92 ) / 3888 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/18Z)×\left(\mathbb{Z}/18\mathbb{Z}\right)^\times.

nn 1111
χ(n)\chi(n) β1\beta_{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}
5.1
0.500000 − 79.2536i
0.500000 − 100.759i
0.500000 + 66.6144i
0.500000 − 10.2011i
0.500000 + 125.561i
0.500000 − 18.4488i
0.500000 + 111.664i
0.500000 − 106.731i
0.500000 + 42.2052i
0.500000 − 32.3837i
0.500000 + 79.2536i
0.500000 + 100.759i
0.500000 − 66.6144i
0.500000 + 10.2011i
0.500000 − 125.561i
0.500000 + 18.4488i
0.500000 − 111.664i
0.500000 + 106.731i
0.500000 − 42.2052i
0.500000 + 32.3837i
−19.5959 + 11.3137i −237.011 − 53.6159i 256.000 − 443.405i −1517.40 − 876.072i 5251.05 − 1630.82i −13322.5 − 23075.2i 11585.2i 53299.7 + 25415.2i 39646.5
5.2 −19.5959 + 11.3137i −196.987 + 142.285i 256.000 − 443.405i 3091.89 + 1785.10i 2250.37 − 5016.87i 12604.4 + 21831.4i 11585.2i 18558.8 − 56056.7i −80784.5
5.3 −19.5959 + 11.3137i 16.6955 − 242.426i 256.000 − 443.405i 2723.51 + 1572.42i 2415.57 + 4939.44i −1768.70 − 3063.49i 11585.2i −58491.5 − 8094.85i −71159.5
5.4 −19.5959 + 11.3137i 57.7167 + 236.046i 256.000 − 443.405i −2305.51 − 1331.09i −3801.57 − 3972.55i −3513.89 − 6086.24i 11585.2i −52386.6 + 27247.6i 60238.1
5.5 −19.5959 + 11.3137i 208.763 − 124.366i 256.000 − 443.405i −4471.98 − 2581.90i −2683.86 + 4798.96i 7939.56 + 13751.7i 11585.2i 28115.1 − 51926.2i 116843.
5.6 19.5959 − 11.3137i −230.699 + 76.3332i 256.000 − 443.405i 1617.35 + 933.780i −3657.16 + 4105.89i −504.464 − 873.757i − 11585.2i 47395.5 − 35220.0i 42258.0
5.7 19.5959 − 11.3137i −120.535 − 210.998i 256.000 − 443.405i −1022.89 − 590.567i −4749.17 − 2771.00i −2684.05 − 4648.92i − 11585.2i −29991.5 + 50865.5i −26726.0
5.8 19.5959 − 11.3137i 1.63905 + 242.994i 256.000 − 443.405i −1947.00 − 1124.10i 2781.29 + 4743.16i 14864.4 + 25746.0i − 11585.2i −59043.6 + 796.560i −50871.1
5.9 19.5959 − 11.3137i 223.765 − 94.7529i 256.000 − 443.405i 3162.64 + 1825.95i 3312.88 − 4388.38i 5852.43 + 10136.7i − 11585.2i 41092.8 − 42404.8i 82633.0
5.10 19.5959 − 11.3137i 234.653 + 63.1409i 256.000 − 443.405i −4289.59 − 2476.60i 5312.61 − 1417.50i −13348.2 − 23119.8i − 11585.2i 51075.4 + 29632.5i −112078.
11.1 −19.5959 − 11.3137i −237.011 + 53.6159i 256.000 + 443.405i −1517.40 + 876.072i 5251.05 + 1630.82i −13322.5 + 23075.2i − 11585.2i 53299.7 − 25415.2i 39646.5
11.2 −19.5959 − 11.3137i −196.987 − 142.285i 256.000 + 443.405i 3091.89 − 1785.10i 2250.37 + 5016.87i 12604.4 − 21831.4i − 11585.2i 18558.8 + 56056.7i −80784.5
11.3 −19.5959 − 11.3137i 16.6955 + 242.426i 256.000 + 443.405i 2723.51 − 1572.42i 2415.57 − 4939.44i −1768.70 + 3063.49i − 11585.2i −58491.5 + 8094.85i −71159.5
11.4 −19.5959 − 11.3137i 57.7167 − 236.046i 256.000 + 443.405i −2305.51 + 1331.09i −3801.57 + 3972.55i −3513.89 + 6086.24i − 11585.2i −52386.6 − 27247.6i 60238.1
11.5 −19.5959 − 11.3137i 208.763 + 124.366i 256.000 + 443.405i −4471.98 + 2581.90i −2683.86 − 4798.96i 7939.56 − 13751.7i − 11585.2i 28115.1 + 51926.2i 116843.
11.6 19.5959 + 11.3137i −230.699 − 76.3332i 256.000 + 443.405i 1617.35 − 933.780i −3657.16 − 4105.89i −504.464 + 873.757i 11585.2i 47395.5 + 35220.0i 42258.0
11.7 19.5959 + 11.3137i −120.535 + 210.998i 256.000 + 443.405i −1022.89 + 590.567i −4749.17 + 2771.00i −2684.05 + 4648.92i 11585.2i −29991.5 − 50865.5i −26726.0
11.8 19.5959 + 11.3137i 1.63905 − 242.994i 256.000 + 443.405i −1947.00 + 1124.10i 2781.29 − 4743.16i 14864.4 − 25746.0i 11585.2i −59043.6 − 796.560i −50871.1
11.9 19.5959 + 11.3137i 223.765 + 94.7529i 256.000 + 443.405i 3162.64 − 1825.95i 3312.88 + 4388.38i 5852.43 − 10136.7i 11585.2i 41092.8 + 42404.8i 82633.0
11.10 19.5959 + 11.3137i 234.653 − 63.1409i 256.000 + 443.405i −4289.59 + 2476.60i 5312.61 + 1417.50i −13348.2 + 23119.8i 11585.2i 51075.4 − 29632.5i −112078.
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.11.d.a ✓ 20
3.b odd 2 1 54.11.d.a 20
9.c even 3 1 54.11.d.a 20
9.c even 3 1 162.11.b.c 20
9.d odd 6 1 inner 18.11.d.a ✓ 20
9.d odd 6 1 162.11.b.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.11.d.a ✓ 20 1.a even 1 1 trivial
18.11.d.a ✓ 20 9.d odd 6 1 inner
54.11.d.a 20 3.b odd 2 1
54.11.d.a 20 9.c even 3 1
162.11.b.c 20 9.c even 3 1
162.11.b.c 20 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace S11new(18,[χ])S_{11}^{\mathrm{new}}(18, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4−512T2+262144)5 (T^{4} - 512 T^{2} + 262144)^{5} Copy content Toggle raw display
33 T20+⋯+51 ⁣⋯ ⁣01 T^{20} + \cdots + 51\!\cdots\!01 Copy content Toggle raw display
55 T20+⋯+58 ⁣⋯ ⁣00 T^{20} + \cdots + 58\!\cdots\!00 Copy content Toggle raw display
77 T20+⋯+17 ⁣⋯ ⁣00 T^{20} + \cdots + 17\!\cdots\!00 Copy content Toggle raw display
1111 T20+⋯+18 ⁣⋯ ⁣81 T^{20} + \cdots + 18\!\cdots\!81 Copy content Toggle raw display
1313 T20+⋯+53 ⁣⋯ ⁣56 T^{20} + \cdots + 53\!\cdots\!56 Copy content Toggle raw display
1717 T20+⋯+75 ⁣⋯ ⁣00 T^{20} + \cdots + 75\!\cdots\!00 Copy content Toggle raw display
1919 (T10+⋯+33 ⁣⋯ ⁣40)2 (T^{10} + \cdots + 33\!\cdots\!40)^{2} Copy content Toggle raw display
2323 T20+⋯+48 ⁣⋯ ⁣56 T^{20} + \cdots + 48\!\cdots\!56 Copy content Toggle raw display
2929 T20+⋯+39 ⁣⋯ ⁣00 T^{20} + \cdots + 39\!\cdots\!00 Copy content Toggle raw display
3131 T20+⋯+34 ⁣⋯ ⁣00 T^{20} + \cdots + 34\!\cdots\!00 Copy content Toggle raw display
3737 (T10+⋯+64 ⁣⋯ ⁣68)2 (T^{10} + \cdots + 64\!\cdots\!68)^{2} Copy content Toggle raw display
4141 T20+⋯+75 ⁣⋯ ⁣25 T^{20} + \cdots + 75\!\cdots\!25 Copy content Toggle raw display
4343 T20+⋯+12 ⁣⋯ ⁣25 T^{20} + \cdots + 12\!\cdots\!25 Copy content Toggle raw display
4747 T20+⋯+90 ⁣⋯ ⁣16 T^{20} + \cdots + 90\!\cdots\!16 Copy content Toggle raw display
5353 T20+⋯+10 ⁣⋯ ⁣00 T^{20} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
5959 T20+⋯+60 ⁣⋯ ⁣25 T^{20} + \cdots + 60\!\cdots\!25 Copy content Toggle raw display
6161 T20+⋯+35 ⁣⋯ ⁣04 T^{20} + \cdots + 35\!\cdots\!04 Copy content Toggle raw display
6767 T20+⋯+41 ⁣⋯ ⁣25 T^{20} + \cdots + 41\!\cdots\!25 Copy content Toggle raw display
7171 T20+⋯+47 ⁣⋯ ⁣64 T^{20} + \cdots + 47\!\cdots\!64 Copy content Toggle raw display
7373 (T10+⋯+20 ⁣⋯ ⁣20)2 (T^{10} + \cdots + 20\!\cdots\!20)^{2} Copy content Toggle raw display
7979 T20+⋯+39 ⁣⋯ ⁣00 T^{20} + \cdots + 39\!\cdots\!00 Copy content Toggle raw display
8383 T20+⋯+22 ⁣⋯ ⁣84 T^{20} + \cdots + 22\!\cdots\!84 Copy content Toggle raw display
8989 T20+⋯+20 ⁣⋯ ⁣00 T^{20} + \cdots + 20\!\cdots\!00 Copy content Toggle raw display
9797 T20+⋯+45 ⁣⋯ ⁣25 T^{20} + \cdots + 45\!\cdots\!25 Copy content Toggle raw display
show more
show less