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Properties

Label 33.6.e.b
Level 3333
Weight 66
Character orbit 33.e
Analytic conductor 5.2935.293
Analytic rank 00
Dimension 2020
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [33,6,Mod(4,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 33=3⋅11 33 = 3 \cdot 11
Weight: k k == 6 6
Character orbit: [χ][\chi] == 33.e (of order 55, degree 44, 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: 5.292666053835.29266605383
Analytic rank: 00
Dimension: 2020
Relative dimension: 55 over Q(ζ5)\Q(\zeta_{5})
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−x19+78x18+79x17+10573x16−33409x15+1262953x14+⋯+25599187870096 x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} + \cdots + 25599187870096 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 24⋅112 2^{4}\cdot 11^{2}
Twist minimal: yes
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,…,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+(β6−β5)q2+9β7q3+(β16+β12+18β7+⋯−12)q4+(β17−β14+β11+⋯−3)q5+(9β2−9β1)q6+⋯+(−81β19+324β17+⋯−4212)q99+O(q100) q + (\beta_{6} - \beta_{5}) q^{2} + 9 \beta_{7} q^{3} + (\beta_{16} + \beta_{12} + 18 \beta_{7} + \cdots - 12) q^{4} + (\beta_{17} - \beta_{14} + \beta_{11} + \cdots - 3) q^{5} + (9 \beta_{2} - 9 \beta_1) q^{6}+ \cdots + ( - 81 \beta_{19} + 324 \beta_{17} + \cdots - 4212) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 20q+6q2+45q3+8q4−11q5−54q6−139q7−76q8−405q9−424q10+2289q11−2682q12−847q13+2022q14−396q15−7148q16+2482q17+⋯−115506q99+O(q100) 20 q + 6 q^{2} + 45 q^{3} + 8 q^{4} - 11 q^{5} - 54 q^{6} - 139 q^{7} - 76 q^{8} - 405 q^{9} - 424 q^{10} + 2289 q^{11} - 2682 q^{12} - 847 q^{13} + 2022 q^{14} - 396 q^{15} - 7148 q^{16} + 2482 q^{17}+ \cdots - 115506 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x20−x19+78x18+79x17+10573x16−33409x15+1262953x14+⋯+25599187870096 x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} + \cdots + 25599187870096 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (24 ⁣⋯ ⁣63ν19+⋯−65 ⁣⋯ ⁣64)/34 ⁣⋯ ⁣02 ( 24\!\cdots\!63 \nu^{19} + \cdots - 65\!\cdots\!64 ) / 34\!\cdots\!02 Copy content Toggle raw display
β3\beta_{3}== (−98 ⁣⋯ ⁣84ν19+⋯+78 ⁣⋯ ⁣30)/13 ⁣⋯ ⁣11 ( - 98\!\cdots\!84 \nu^{19} + \cdots + 78\!\cdots\!30 ) / 13\!\cdots\!11 Copy content Toggle raw display
β4\beta_{4}== (−32 ⁣⋯ ⁣82ν19+⋯−50 ⁣⋯ ⁣28)/27 ⁣⋯ ⁣22 ( - 32\!\cdots\!82 \nu^{19} + \cdots - 50\!\cdots\!28 ) / 27\!\cdots\!22 Copy content Toggle raw display
β5\beta_{5}== (68 ⁣⋯ ⁣67ν19+⋯+56 ⁣⋯ ⁣00)/54 ⁣⋯ ⁣44 ( 68\!\cdots\!67 \nu^{19} + \cdots + 56\!\cdots\!00 ) / 54\!\cdots\!44 Copy content Toggle raw display
β6\beta_{6}== (30 ⁣⋯ ⁣65ν19+⋯+20 ⁣⋯ ⁣20)/13 ⁣⋯ ⁣08 ( 30\!\cdots\!65 \nu^{19} + \cdots + 20\!\cdots\!20 ) / 13\!\cdots\!08 Copy content Toggle raw display
β7\beta_{7}== (17 ⁣⋯ ⁣06ν19+⋯+32 ⁣⋯ ⁣16)/69 ⁣⋯ ⁣04 ( 17\!\cdots\!06 \nu^{19} + \cdots + 32\!\cdots\!16 ) / 69\!\cdots\!04 Copy content Toggle raw display
β8\beta_{8}== (−56 ⁣⋯ ⁣13ν19+⋯−29 ⁣⋯ ⁣88)/54 ⁣⋯ ⁣44 ( - 56\!\cdots\!13 \nu^{19} + \cdots - 29\!\cdots\!88 ) / 54\!\cdots\!44 Copy content Toggle raw display
β9\beta_{9}== (16 ⁣⋯ ⁣73ν19+⋯−39 ⁣⋯ ⁣64)/34 ⁣⋯ ⁣02 ( 16\!\cdots\!73 \nu^{19} + \cdots - 39\!\cdots\!64 ) / 34\!\cdots\!02 Copy content Toggle raw display
β10\beta_{10}== (81 ⁣⋯ ⁣62ν19+⋯+18 ⁣⋯ ⁣84)/15 ⁣⋯ ⁣76 ( 81\!\cdots\!62 \nu^{19} + \cdots + 18\!\cdots\!84 ) / 15\!\cdots\!76 Copy content Toggle raw display
β11\beta_{11}== (24 ⁣⋯ ⁣41ν19+⋯+16 ⁣⋯ ⁣84)/31 ⁣⋯ ⁣52 ( 24\!\cdots\!41 \nu^{19} + \cdots + 16\!\cdots\!84 ) / 31\!\cdots\!52 Copy content Toggle raw display
β12\beta_{12}== (−56 ⁣⋯ ⁣84ν19+⋯+10 ⁣⋯ ⁣64)/69 ⁣⋯ ⁣04 ( - 56\!\cdots\!84 \nu^{19} + \cdots + 10\!\cdots\!64 ) / 69\!\cdots\!04 Copy content Toggle raw display
β13\beta_{13}== (−40 ⁣⋯ ⁣71ν19+⋯−24 ⁣⋯ ⁣12)/40 ⁣⋯ ⁣32 ( - 40\!\cdots\!71 \nu^{19} + \cdots - 24\!\cdots\!12 ) / 40\!\cdots\!32 Copy content Toggle raw display
β14\beta_{14}== (−21 ⁣⋯ ⁣25ν19+⋯+40 ⁣⋯ ⁣80)/20 ⁣⋯ ⁣16 ( - 21\!\cdots\!25 \nu^{19} + \cdots + 40\!\cdots\!80 ) / 20\!\cdots\!16 Copy content Toggle raw display
β15\beta_{15}== (−23 ⁣⋯ ⁣85ν19+⋯+58 ⁣⋯ ⁣24)/20 ⁣⋯ ⁣16 ( - 23\!\cdots\!85 \nu^{19} + \cdots + 58\!\cdots\!24 ) / 20\!\cdots\!16 Copy content Toggle raw display
β16\beta_{16}== (−16 ⁣⋯ ⁣35ν19+⋯−10 ⁣⋯ ⁣84)/13 ⁣⋯ ⁣08 ( - 16\!\cdots\!35 \nu^{19} + \cdots - 10\!\cdots\!84 ) / 13\!\cdots\!08 Copy content Toggle raw display
β17\beta_{17}== (−61 ⁣⋯ ⁣69ν19+⋯−13 ⁣⋯ ⁣12)/40 ⁣⋯ ⁣32 ( - 61\!\cdots\!69 \nu^{19} + \cdots - 13\!\cdots\!12 ) / 40\!\cdots\!32 Copy content Toggle raw display
β18\beta_{18}== (−62 ⁣⋯ ⁣69ν19+⋯−11 ⁣⋯ ⁣44)/20 ⁣⋯ ⁣16 ( - 62\!\cdots\!69 \nu^{19} + \cdots - 11\!\cdots\!44 ) / 20\!\cdots\!16 Copy content Toggle raw display
β19\beta_{19}== (−80 ⁣⋯ ⁣34ν19+⋯−26 ⁣⋯ ⁣28)/20 ⁣⋯ ⁣16 ( - 80\!\cdots\!34 \nu^{19} + \cdots - 26\!\cdots\!28 ) / 20\!\cdots\!16 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}== −β12+β9−β8−49β7−6β6+β4+6 -\beta_{12} + \beta_{9} - \beta_{8} - 49\beta_{7} - 6\beta_{6} + \beta_{4} + 6 Copy content Toggle raw display
ν3\nu^{3}== −β19+β16+2β14+3β13+β12−2β11+2β9+⋯−35 - \beta_{19} + \beta_{16} + 2 \beta_{14} + 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{9} + \cdots - 35 Copy content Toggle raw display
ν4\nu^{4}== 6β19−3β18+5β17−109β16+3β15−5β14+⋯+2β1 6 \beta_{19} - 3 \beta_{18} + 5 \beta_{17} - 109 \beta_{16} + 3 \beta_{15} - 5 \beta_{14} + \cdots + 2 \beta_1 Copy content Toggle raw display
ν5\nu^{5}== −17β19+17β18−249β17−219β16+249β13−408β11+⋯+7332 - 17 \beta_{19} + 17 \beta_{18} - 249 \beta_{17} - 219 \beta_{16} + 249 \beta_{13} - 408 \beta_{11} + \cdots + 7332 Copy content Toggle raw display
ν6\nu^{6}== −391β18−943β17−470β15+472β14+2129β12−943β11+⋯−129800 - 391 \beta_{18} - 943 \beta_{17} - 470 \beta_{15} + 472 \beta_{14} + 2129 \beta_{12} - 943 \beta_{11} + \cdots - 129800 Copy content Toggle raw display
ν7\nu^{7}== 3378β19+10216β18−18712β17+18726β16+3378β15+⋯+101768 3378 \beta_{19} + 10216 \beta_{18} - 18712 \beta_{17} + 18726 \beta_{16} + 3378 \beta_{15} + \cdots + 101768 Copy content Toggle raw display
ν8\nu^{8}== −103772β19+1140633β16−45416β15−74628β14−33804β13+⋯−24013831 - 103772 \beta_{19} + 1140633 \beta_{16} - 45416 \beta_{15} - 74628 \beta_{14} - 33804 \beta_{13} + \cdots - 24013831 Copy content Toggle raw display
ν9\nu^{9}== 1450621β19−960777β18+2035773β17−2777337β16+960777β15+⋯+26404325β1 1450621 \beta_{19} - 960777 \beta_{18} + 2035773 \beta_{17} - 2777337 \beta_{16} + 960777 \beta_{15} + \cdots + 26404325 \beta_1 Copy content Toggle raw display
ν10\nu^{10}== −5314715β19+5314715β18+4888469β17+37443407β16+⋯+2321917415 - 5314715 \beta_{19} + 5314715 \beta_{18} + 4888469 \beta_{17} + 37443407 \beta_{16} + \cdots + 2321917415 Copy content Toggle raw display
ν11\nu^{11}== −63162745β18+291186469β17−93206829β15+215238939β14+⋯−5229729401 - 63162745 \beta_{18} + 291186469 \beta_{17} - 93206829 \beta_{15} + 215238939 \beta_{14} + \cdots - 5229729401 Copy content Toggle raw display
ν12\nu^{12}== 628941271β19+732584482β18−444888904β17−4464778965β16+⋯+206326475808 628941271 \beta_{19} + 732584482 \beta_{18} - 444888904 \beta_{17} - 4464778965 \beta_{16} + \cdots + 206326475808 Copy content Toggle raw display
ν13\nu^{13}== −17032821338β19+43657317140β16−7707637614β15+30587060194β14+⋯−1324846024046 - 17032821338 \beta_{19} + 43657317140 \beta_{16} - 7707637614 \beta_{15} + 30587060194 \beta_{14} + \cdots - 1324846024046 Copy content Toggle raw display
ν14\nu^{14}== 153953121956β19−79410379408β18+53176977104β17−1277679603561β16+⋯+2160057360596β1 153953121956 \beta_{19} - 79410379408 \beta_{18} + 53176977104 \beta_{17} - 1277679603561 \beta_{16} + \cdots + 2160057360596 \beta_1 Copy content Toggle raw display
ν15\nu^{15}== −913604260816β19+913604260816β18−3235932238730β17+863704227014β16+⋯+144241411965627 - 913604260816 \beta_{19} + 913604260816 \beta_{18} - 3235932238730 \beta_{17} + 863704227014 \beta_{16} + \cdots + 144241411965627 Copy content Toggle raw display
ν16\nu^{16}== −8796134072883β18+3823389968623β17−8576511792099β15+⋯−26 ⁣⋯ ⁣65 - 8796134072883 \beta_{18} + 3823389968623 \beta_{17} - 8576511792099 \beta_{15} + \cdots - 26\!\cdots\!65 Copy content Toggle raw display
ν17\nu^{17}== 106507268471761β19+99981415905261β18−246103968451527β17+⋯+14 ⁣⋯ ⁣77 106507268471761 \beta_{19} + 99981415905261 \beta_{18} - 246103968451527 \beta_{17} + \cdots + 14\!\cdots\!77 Copy content Toggle raw display
ν18\nu^{18}== −19 ⁣⋯ ⁣09β19+⋯−25 ⁣⋯ ⁣20 - 19\!\cdots\!09 \beta_{19} + \cdots - 25\!\cdots\!20 Copy content Toggle raw display
ν19\nu^{19}== 22 ⁣⋯ ⁣26β19+⋯+55 ⁣⋯ ⁣22β1 22\!\cdots\!26 \beta_{19} + \cdots + 55\!\cdots\!22 \beta_1 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/33Z)×\left(\mathbb{Z}/33\mathbb{Z}\right)^\times.

nn 1313 2323
χ(n)\chi(n) −1−β2+β6+β7-1 - \beta_{2} + \beta_{6} + \beta_{7} 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.

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}
4.1
−8.59311 + 6.24326i
−3.89647 + 2.83095i
−0.490010 + 0.356013i
4.70404 − 3.41769i
7.96653 − 5.78803i
−2.34585 + 7.21978i
−1.64776 + 5.07128i
0.527694 − 1.62407i
1.34436 − 4.13753i
2.93057 − 9.01936i
−8.59311 − 6.24326i
−3.89647 − 2.83095i
−0.490010 − 0.356013i
4.70404 + 3.41769i
7.96653 + 5.78803i
−2.34585 − 7.21978i
−1.64776 − 5.07128i
0.527694 + 1.62407i
1.34436 + 4.13753i
2.93057 + 9.01936i
−7.78409 − 5.65548i −2.78115 + 8.55951i 18.7192 + 57.6117i 48.1408 − 34.9763i 70.0568 − 50.8993i 9.43666 + 29.0431i 84.9654 − 261.497i −65.5304 − 47.6106i −572.540
4.2 −3.08746 − 2.24317i −2.78115 + 8.55951i −5.38796 − 16.5824i 3.77724 − 2.74432i 27.7871 − 20.1885i 49.4196 + 152.098i −58.2998 + 179.428i −65.5304 − 47.6106i −17.8180
4.3 0.319007 + 0.231772i −2.78115 + 8.55951i −9.84050 − 30.2859i −17.3054 + 12.5731i −2.87106 + 2.08595i −52.3175 − 161.017i 7.77944 − 23.9427i −65.5304 − 47.6106i −8.43465
4.4 5.51306 + 4.00547i −2.78115 + 8.55951i 4.46148 + 13.7310i −62.5986 + 45.4806i −49.6176 + 36.0493i 59.3828 + 182.761i 36.9828 − 113.821i −65.5304 − 47.6106i −527.281
4.5 8.77555 + 6.37581i −2.78115 + 8.55951i 26.4708 + 81.4687i 31.3852 − 22.8027i −78.9800 + 57.3823i −67.6896 − 208.327i −179.871 + 553.584i −65.5304 − 47.6106i 420.808
16.1 −2.65487 − 8.17084i 7.28115 + 5.29007i −33.8257 + 24.5758i −21.1802 + 65.1858i 23.8938 − 73.5376i −196.384 + 142.681i 68.1911 + 49.5438i 25.0304 + 77.0356i 588.853
16.2 −1.95678 − 6.02234i 7.28115 + 5.29007i −6.55106 + 4.75963i 8.65323 − 26.6319i 17.6110 − 54.2011i 121.539 − 88.3034i −122.450 − 88.9651i 25.0304 + 77.0356i −177.319
16.3 0.218677 + 0.673018i 7.28115 + 5.29007i 25.4834 − 18.5148i −21.0138 + 64.6739i −1.96809 + 6.05716i 98.3565 − 71.4602i 36.3535 + 26.4124i 25.0304 + 77.0356i −48.1219
16.4 1.03535 + 3.18647i 7.28115 + 5.29007i 16.8069 − 12.2109i 32.9083 − 101.281i −9.31813 + 28.6783i −32.7853 + 23.8199i 143.049 + 103.931i 25.0304 + 77.0356i 356.801
16.5 2.62155 + 8.06830i 7.28115 + 5.29007i −32.3365 + 23.4938i −8.26670 + 25.4423i −23.5940 + 72.6147i −58.4588 + 42.4728i −54.7010 − 39.7426i 25.0304 + 77.0356i −226.948
25.1 −7.78409 + 5.65548i −2.78115 − 8.55951i 18.7192 − 57.6117i 48.1408 + 34.9763i 70.0568 + 50.8993i 9.43666 − 29.0431i 84.9654 + 261.497i −65.5304 + 47.6106i −572.540
25.2 −3.08746 + 2.24317i −2.78115 − 8.55951i −5.38796 + 16.5824i 3.77724 + 2.74432i 27.7871 + 20.1885i 49.4196 − 152.098i −58.2998 − 179.428i −65.5304 + 47.6106i −17.8180
25.3 0.319007 − 0.231772i −2.78115 − 8.55951i −9.84050 + 30.2859i −17.3054 − 12.5731i −2.87106 − 2.08595i −52.3175 + 161.017i 7.77944 + 23.9427i −65.5304 + 47.6106i −8.43465
25.4 5.51306 − 4.00547i −2.78115 − 8.55951i 4.46148 − 13.7310i −62.5986 − 45.4806i −49.6176 − 36.0493i 59.3828 − 182.761i 36.9828 + 113.821i −65.5304 + 47.6106i −527.281
25.5 8.77555 − 6.37581i −2.78115 − 8.55951i 26.4708 − 81.4687i 31.3852 + 22.8027i −78.9800 − 57.3823i −67.6896 + 208.327i −179.871 − 553.584i −65.5304 + 47.6106i 420.808
31.1 −2.65487 + 8.17084i 7.28115 − 5.29007i −33.8257 − 24.5758i −21.1802 − 65.1858i 23.8938 + 73.5376i −196.384 − 142.681i 68.1911 − 49.5438i 25.0304 − 77.0356i 588.853
31.2 −1.95678 + 6.02234i 7.28115 − 5.29007i −6.55106 − 4.75963i 8.65323 + 26.6319i 17.6110 + 54.2011i 121.539 + 88.3034i −122.450 + 88.9651i 25.0304 − 77.0356i −177.319
31.3 0.218677 − 0.673018i 7.28115 − 5.29007i 25.4834 + 18.5148i −21.0138 − 64.6739i −1.96809 − 6.05716i 98.3565 + 71.4602i 36.3535 − 26.4124i 25.0304 − 77.0356i −48.1219
31.4 1.03535 − 3.18647i 7.28115 − 5.29007i 16.8069 + 12.2109i 32.9083 + 101.281i −9.31813 − 28.6783i −32.7853 − 23.8199i 143.049 − 103.931i 25.0304 − 77.0356i 356.801
31.5 2.62155 − 8.06830i 7.28115 − 5.29007i −32.3365 − 23.4938i −8.26670 − 25.4423i −23.5940 − 72.6147i −58.4588 − 42.4728i −54.7010 + 39.7426i 25.0304 − 77.0356i −226.948
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.6.e.b ✓ 20
3.b odd 2 1 99.6.f.b 20
11.c even 5 1 inner 33.6.e.b ✓ 20
11.c even 5 1 363.6.a.r 10
11.d odd 10 1 363.6.a.t 10
33.f even 10 1 1089.6.a.bi 10
33.h odd 10 1 99.6.f.b 20
33.h odd 10 1 1089.6.a.bk 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.b ✓ 20 1.a even 1 1 trivial
33.6.e.b ✓ 20 11.c even 5 1 inner
99.6.f.b 20 3.b odd 2 1
99.6.f.b 20 33.h odd 10 1
363.6.a.r 10 11.c even 5 1
363.6.a.t 10 11.d odd 10 1
1089.6.a.bi 10 33.f even 10 1
1089.6.a.bk 10 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T220−6T219+94T218−488T217+12401T216−30756T215+⋯+1371559530496 T_{2}^{20} - 6 T_{2}^{19} + 94 T_{2}^{18} - 488 T_{2}^{17} + 12401 T_{2}^{16} - 30756 T_{2}^{15} + \cdots + 1371559530496 acting on S6new(33,[χ])S_{6}^{\mathrm{new}}(33, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T20+⋯+1371559530496 T^{20} + \cdots + 1371559530496 Copy content Toggle raw display
33 (T4−9T3+⋯+6561)5 (T^{4} - 9 T^{3} + \cdots + 6561)^{5} Copy content Toggle raw display
55 T20+⋯+43 ⁣⋯ ⁣61 T^{20} + \cdots + 43\!\cdots\!61 Copy content Toggle raw display
77 T20+⋯+20 ⁣⋯ ⁣61 T^{20} + \cdots + 20\!\cdots\!61 Copy content Toggle raw display
1111 T20+⋯+11 ⁣⋯ ⁣01 T^{20} + \cdots + 11\!\cdots\!01 Copy content Toggle raw display
1313 T20+⋯+14 ⁣⋯ ⁣76 T^{20} + \cdots + 14\!\cdots\!76 Copy content Toggle raw display
1717 T20+⋯+96 ⁣⋯ ⁣00 T^{20} + \cdots + 96\!\cdots\!00 Copy content Toggle raw display
1919 T20+⋯+20 ⁣⋯ ⁣00 T^{20} + \cdots + 20\!\cdots\!00 Copy content Toggle raw display
2323 (T10+⋯+12 ⁣⋯ ⁣56)2 (T^{10} + \cdots + 12\!\cdots\!56)^{2} Copy content Toggle raw display
2929 T20+⋯+13 ⁣⋯ ⁣00 T^{20} + \cdots + 13\!\cdots\!00 Copy content Toggle raw display
3131 T20+⋯+91 ⁣⋯ ⁣01 T^{20} + \cdots + 91\!\cdots\!01 Copy content Toggle raw display
3737 T20+⋯+15 ⁣⋯ ⁣16 T^{20} + \cdots + 15\!\cdots\!16 Copy content Toggle raw display
4141 T20+⋯+54 ⁣⋯ ⁣36 T^{20} + \cdots + 54\!\cdots\!36 Copy content Toggle raw display
4343 (T10+⋯+51 ⁣⋯ ⁣00)2 (T^{10} + \cdots + 51\!\cdots\!00)^{2} Copy content Toggle raw display
4747 T20+⋯+39 ⁣⋯ ⁣16 T^{20} + \cdots + 39\!\cdots\!16 Copy content Toggle raw display
5353 T20+⋯+16 ⁣⋯ ⁣61 T^{20} + \cdots + 16\!\cdots\!61 Copy content Toggle raw display
5959 T20+⋯+17 ⁣⋯ ⁣25 T^{20} + \cdots + 17\!\cdots\!25 Copy content Toggle raw display
6161 T20+⋯+12 ⁣⋯ ⁣16 T^{20} + \cdots + 12\!\cdots\!16 Copy content Toggle raw display
6767 (T10+⋯+33 ⁣⋯ ⁣36)2 (T^{10} + \cdots + 33\!\cdots\!36)^{2} Copy content Toggle raw display
7171 T20+⋯+31 ⁣⋯ ⁣36 T^{20} + \cdots + 31\!\cdots\!36 Copy content Toggle raw display
7373 T20+⋯+77 ⁣⋯ ⁣36 T^{20} + \cdots + 77\!\cdots\!36 Copy content Toggle raw display
7979 T20+⋯+53 ⁣⋯ ⁣25 T^{20} + \cdots + 53\!\cdots\!25 Copy content Toggle raw display
8383 T20+⋯+75 ⁣⋯ ⁣61 T^{20} + \cdots + 75\!\cdots\!61 Copy content Toggle raw display
8989 (T10+⋯−11 ⁣⋯ ⁣20)2 (T^{10} + \cdots - 11\!\cdots\!20)^{2} Copy content Toggle raw display
9797 T20+⋯+48 ⁣⋯ ⁣25 T^{20} + \cdots + 48\!\cdots\!25 Copy content Toggle raw display
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