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Properties

Label 4.45.b.b
Level 44
Weight 4545
Character orbit 4.b
Analytic conductor 49.04849.048
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] = [4,45,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 45, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 45);
 
N := Newforms(S);
 
Level: N N == 4=22 4 = 2^{2}
Weight: k k == 45 45
Character orbit: [χ][\chi] == 4.b (of order 22, degree 11, 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: 49.047893991749.0478939917
Analytic rank: 00
Dimension: 2020
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+⋯+47 ⁣⋯ ⁣00 x^{20} + \cdots + 47\!\cdots\!00 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: multiple of 2380⋅344⋅512⋅72⋅114 2^{380}\cdot 3^{44}\cdot 5^{12}\cdot 7^{2}\cdot 11^{4}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

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+(β1+239996)q2+(β2+431β1+86)q3+(−β3+37β2+⋯−1531938272161)q4+(−β4+16β3+⋯−64326974711593)q5+(−β5+β4+⋯−75 ⁣⋯ ⁣08)q6+⋯+(−23 ⁣⋯ ⁣96β19+⋯+18 ⁣⋯ ⁣78)q99+O(q100) q + (\beta_1 + 239996) q^{2} + (\beta_{2} + 431 \beta_1 + 86) q^{3} + ( - \beta_{3} + 37 \beta_{2} + \cdots - 1531938272161) q^{4} + ( - \beta_{4} + 16 \beta_{3} + \cdots - 64326974711593) q^{5} + ( - \beta_{5} + \beta_{4} + \cdots - 75\!\cdots\!08) q^{6}+ \cdots + ( - 23\!\cdots\!96 \beta_{19} + \cdots + 18\!\cdots\!78) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 20q+4799916q2−30638766490096q4−12 ⁣⋯ ⁣00q5−15 ⁣⋯ ⁣24q6+17 ⁣⋯ ⁣96q8−67 ⁣⋯ ⁣92q9−12 ⁣⋯ ⁣00q10−10 ⁣⋯ ⁣80q12−50 ⁣⋯ ⁣88q13+⋯−38 ⁣⋯ ⁣44q98+O(q100) 20 q + 4799916 q^{2} - 30638766490096 q^{4} - 12\!\cdots\!00 q^{5} - 15\!\cdots\!24 q^{6} + 17\!\cdots\!96 q^{8} - 67\!\cdots\!92 q^{9} - 12\!\cdots\!00 q^{10} - 10\!\cdots\!80 q^{12} - 50\!\cdots\!88 q^{13}+ \cdots - 38\!\cdots\!44 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x20+⋯+47 ⁣⋯ ⁣00 x^{20} + \cdots + 47\!\cdots\!00 : Copy content Toggle raw display

β1\beta_{1}== (−21 ⁣⋯ ⁣75ν19+⋯−13 ⁣⋯ ⁣00)/58 ⁣⋯ ⁣00 ( - 21\!\cdots\!75 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 58\!\cdots\!00 Copy content Toggle raw display
β2\beta_{2}== (92 ⁣⋯ ⁣25ν19+⋯+56 ⁣⋯ ⁣00)/58 ⁣⋯ ⁣00 ( 92\!\cdots\!25 \nu^{19} + \cdots + 56\!\cdots\!00 ) / 58\!\cdots\!00 Copy content Toggle raw display
β3\beta_{3}== (−86 ⁣⋯ ⁣65ν19+⋯−12 ⁣⋯ ⁣00)/58 ⁣⋯ ⁣00 ( - 86\!\cdots\!65 \nu^{19} + \cdots - 12\!\cdots\!00 ) / 58\!\cdots\!00 Copy content Toggle raw display
β4\beta_{4}== (−65 ⁣⋯ ⁣15ν19+⋯−81 ⁣⋯ ⁣00)/58 ⁣⋯ ⁣00 ( - 65\!\cdots\!15 \nu^{19} + \cdots - 81\!\cdots\!00 ) / 58\!\cdots\!00 Copy content Toggle raw display
β5\beta_{5}== (97 ⁣⋯ ⁣49ν19+⋯−16 ⁣⋯ ⁣00)/58 ⁣⋯ ⁣00 ( 97\!\cdots\!49 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 58\!\cdots\!00 Copy content Toggle raw display
β6\beta_{6}== (−10 ⁣⋯ ⁣15ν19+⋯+85 ⁣⋯ ⁣00)/10 ⁣⋯ ⁣00 ( - 10\!\cdots\!15 \nu^{19} + \cdots + 85\!\cdots\!00 ) / 10\!\cdots\!00 Copy content Toggle raw display
β7\beta_{7}== (77 ⁣⋯ ⁣43ν19+⋯+30 ⁣⋯ ⁣00)/20 ⁣⋯ ⁣00 ( 77\!\cdots\!43 \nu^{19} + \cdots + 30\!\cdots\!00 ) / 20\!\cdots\!00 Copy content Toggle raw display
β8\beta_{8}== (15 ⁣⋯ ⁣05ν19+⋯−89 ⁣⋯ ⁣00)/29 ⁣⋯ ⁣00 ( 15\!\cdots\!05 \nu^{19} + \cdots - 89\!\cdots\!00 ) / 29\!\cdots\!00 Copy content Toggle raw display
β9\beta_{9}== (−22 ⁣⋯ ⁣95ν19+⋯+94 ⁣⋯ ⁣00)/29 ⁣⋯ ⁣00 ( - 22\!\cdots\!95 \nu^{19} + \cdots + 94\!\cdots\!00 ) / 29\!\cdots\!00 Copy content Toggle raw display
β10\beta_{10}== (47 ⁣⋯ ⁣39ν19+⋯−32 ⁣⋯ ⁣00)/29 ⁣⋯ ⁣00 ( 47\!\cdots\!39 \nu^{19} + \cdots - 32\!\cdots\!00 ) / 29\!\cdots\!00 Copy content Toggle raw display
β11\beta_{11}== (−25 ⁣⋯ ⁣65ν19+⋯−16 ⁣⋯ ⁣00)/58 ⁣⋯ ⁣00 ( - 25\!\cdots\!65 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 58\!\cdots\!00 Copy content Toggle raw display
β12\beta_{12}== (15 ⁣⋯ ⁣89ν19+⋯−36 ⁣⋯ ⁣00)/82 ⁣⋯ ⁣00 ( 15\!\cdots\!89 \nu^{19} + \cdots - 36\!\cdots\!00 ) / 82\!\cdots\!00 Copy content Toggle raw display
β13\beta_{13}== (−18 ⁣⋯ ⁣83ν19+⋯+68 ⁣⋯ ⁣00)/82 ⁣⋯ ⁣00 ( - 18\!\cdots\!83 \nu^{19} + \cdots + 68\!\cdots\!00 ) / 82\!\cdots\!00 Copy content Toggle raw display
β14\beta_{14}== (15 ⁣⋯ ⁣53ν19+⋯−16 ⁣⋯ ⁣00)/58 ⁣⋯ ⁣00 ( 15\!\cdots\!53 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 58\!\cdots\!00 Copy content Toggle raw display
β15\beta_{15}== (−10 ⁣⋯ ⁣77ν19+⋯+89 ⁣⋯ ⁣00)/27 ⁣⋯ ⁣00 ( - 10\!\cdots\!77 \nu^{19} + \cdots + 89\!\cdots\!00 ) / 27\!\cdots\!00 Copy content Toggle raw display
β16\beta_{16}== (−81 ⁣⋯ ⁣55ν19+⋯−18 ⁣⋯ ⁣00)/13 ⁣⋯ ⁣00 ( - 81\!\cdots\!55 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 13\!\cdots\!00 Copy content Toggle raw display
β17\beta_{17}== (−47 ⁣⋯ ⁣13ν19+⋯−13 ⁣⋯ ⁣00)/63 ⁣⋯ ⁣00 ( - 47\!\cdots\!13 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 63\!\cdots\!00 Copy content Toggle raw display
β18\beta_{18}== (51 ⁣⋯ ⁣03ν19+⋯+81 ⁣⋯ ⁣00)/58 ⁣⋯ ⁣00 ( 51\!\cdots\!03 \nu^{19} + \cdots + 81\!\cdots\!00 ) / 58\!\cdots\!00 Copy content Toggle raw display
β19\beta_{19}== (−19 ⁣⋯ ⁣99ν19+⋯−26 ⁣⋯ ⁣00)/18 ⁣⋯ ⁣00 ( - 19\!\cdots\!99 \nu^{19} + \cdots - 26\!\cdots\!00 ) / 18\!\cdots\!00 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β2+431β1+86)/16 ( \beta_{2} + 431\beta _1 + 86 ) / 16 Copy content Toggle raw display
ν2\nu^{2}== (β10−3β8−2β7−465β6−485β5−4808β4+⋯−13 ⁣⋯ ⁣44)/256 ( \beta_{10} - 3 \beta_{8} - 2 \beta_{7} - 465 \beta_{6} - 485 \beta_{5} - 4808 \beta_{4} + \cdots - 13\!\cdots\!44 ) / 256 Copy content Toggle raw display
ν3\nu^{3}== (46576β19−5961920β18+10435980β17+699168β16+⋯−21 ⁣⋯ ⁣80)/4096 ( 46576 \beta_{19} - 5961920 \beta_{18} + 10435980 \beta_{17} + 699168 \beta_{16} + \cdots - 21\!\cdots\!80 ) / 4096 Copy content Toggle raw display
ν4\nu^{4}== (−72 ⁣⋯ ⁣82β17+⋯+13 ⁣⋯ ⁣34)/32768 ( - 72\!\cdots\!82 \beta_{17} + \cdots + 13\!\cdots\!34 ) / 32768 Copy content Toggle raw display
ν5\nu^{5}== (−50 ⁣⋯ ⁣60β19+⋯+35 ⁣⋯ ⁣36)/524288 ( - 50\!\cdots\!60 \beta_{19} + \cdots + 35\!\cdots\!36 ) / 524288 Copy content Toggle raw display
ν6\nu^{6}== (27 ⁣⋯ ⁣17β17+⋯−41 ⁣⋯ ⁣76)/1048576 ( 27\!\cdots\!17 \beta_{17} + \cdots - 41\!\cdots\!76 ) / 1048576 Copy content Toggle raw display
ν7\nu^{7}== (30 ⁣⋯ ⁣36β19+⋯−13 ⁣⋯ ⁣51)/16777216 ( 30\!\cdots\!36 \beta_{19} + \cdots - 13\!\cdots\!51 ) / 16777216 Copy content Toggle raw display
ν8\nu^{8}== (−98 ⁣⋯ ⁣32β17+⋯+12 ⁣⋯ ⁣54)/33554432 ( - 98\!\cdots\!32 \beta_{17} + \cdots + 12\!\cdots\!54 ) / 33554432 Copy content Toggle raw display
ν9\nu^{9}== (−14 ⁣⋯ ⁣20β19+⋯+51 ⁣⋯ ⁣34)/536870912 ( - 14\!\cdots\!20 \beta_{19} + \cdots + 51\!\cdots\!34 ) / 536870912 Copy content Toggle raw display
ν10\nu^{10}== (34 ⁣⋯ ⁣64β17+⋯−40 ⁣⋯ ⁣94)/1073741824 ( 34\!\cdots\!64 \beta_{17} + \cdots - 40\!\cdots\!94 ) / 1073741824 Copy content Toggle raw display
ν11\nu^{11}== (17 ⁣⋯ ⁣12β19+⋯−58 ⁣⋯ ⁣76)/536870912 ( 17\!\cdots\!12 \beta_{19} + \cdots - 58\!\cdots\!76 ) / 536870912 Copy content Toggle raw display
ν12\nu^{12}== (−39 ⁣⋯ ⁣08β17+⋯+41 ⁣⋯ ⁣74)/1073741824 ( - 39\!\cdots\!08 \beta_{17} + \cdots + 41\!\cdots\!74 ) / 1073741824 Copy content Toggle raw display
ν13\nu^{13}== (−21 ⁣⋯ ⁣56β19+⋯+67 ⁣⋯ ⁣94)/536870912 ( - 21\!\cdots\!56 \beta_{19} + \cdots + 67\!\cdots\!94 ) / 536870912 Copy content Toggle raw display
ν14\nu^{14}== (22 ⁣⋯ ⁣00β17+⋯−21 ⁣⋯ ⁣94)/536870912 ( 22\!\cdots\!00 \beta_{17} + \cdots - 21\!\cdots\!94 ) / 536870912 Copy content Toggle raw display
ν15\nu^{15}== (13 ⁣⋯ ⁣68β19+⋯−38 ⁣⋯ ⁣68)/268435456 ( 13\!\cdots\!68 \beta_{19} + \cdots - 38\!\cdots\!68 ) / 268435456 Copy content Toggle raw display
ν16\nu^{16}== (−25 ⁣⋯ ⁣56β17+⋯+21 ⁣⋯ ⁣90)/536870912 ( - 25\!\cdots\!56 \beta_{17} + \cdots + 21\!\cdots\!90 ) / 536870912 Copy content Toggle raw display
ν17\nu^{17}== (−30 ⁣⋯ ⁣72β19+⋯+86 ⁣⋯ ⁣74)/536870912 ( - 30\!\cdots\!72 \beta_{19} + \cdots + 86\!\cdots\!74 ) / 536870912 Copy content Toggle raw display
ν18\nu^{18}== (58 ⁣⋯ ⁣56β17+⋯−45 ⁣⋯ ⁣22)/1073741824 ( 58\!\cdots\!56 \beta_{17} + \cdots - 45\!\cdots\!22 ) / 1073741824 Copy content Toggle raw display
ν19\nu^{19}== (35 ⁣⋯ ⁣20β19+⋯−98 ⁣⋯ ⁣76)/536870912 ( 35\!\cdots\!20 \beta_{19} + \cdots - 98\!\cdots\!76 ) / 536870912 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/4Z)×\left(\mathbb{Z}/4\mathbb{Z}\right)^\times.

nn 33
χ(n)\chi(n) −1-1

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}
3.1
− 3.35197e9i
3.35197e9i
2.38483e9i
− 2.38483e9i
1.23618e9i
− 1.23618e9i
− 1.22466e9i
1.22466e9i
− 1.35492e9i
1.35492e9i
3.08594e9i
− 3.08594e9i
− 3.00773e9i
3.00773e9i
3.19621e8i
− 3.19621e8i
1.49929e9i
− 1.49929e9i
− 2.98896e9i
2.98896e9i
−3.68465e6 − 2.00389e6i − 5.36316e10i 9.56104e12 + 1.47672e13i 7.16734e14 −1.07472e17 + 1.97613e17i 1.83098e18i −5.63715e18 − 7.35713e19i −1.89157e21 −2.64091e21 − 1.43626e21i
3.2 −3.68465e6 + 2.00389e6i 5.36316e10i 9.56104e12 − 1.47672e13i 7.16734e14 −1.07472e17 − 1.97613e17i − 1.83098e18i −5.63715e18 + 7.35713e19i −1.89157e21 −2.64091e21 + 1.43626e21i
3.3 −3.36765e6 − 2.50023e6i 3.81573e10i 5.08992e12 + 1.68398e13i −2.81252e15 9.54019e16 − 1.28500e17i 3.86238e18i 2.49622e19 − 6.94364e19i −4.71210e20 9.47159e21 + 7.03195e21i
3.4 −3.36765e6 + 2.50023e6i − 3.81573e10i 5.08992e12 − 1.68398e13i −2.81252e15 9.54019e16 + 1.28500e17i − 3.86238e18i 2.49622e19 + 6.94364e19i −4.71210e20 9.47159e21 − 7.03195e21i
3.5 −2.67504e6 − 3.23054e6i 1.97789e10i −3.28055e12 + 1.72836e13i 4.32114e15 6.38964e16 − 5.29092e16i − 4.05908e18i 6.46109e19 − 3.56363e19i 5.93567e20 −1.15592e22 − 1.39596e22i
3.6 −2.67504e6 + 3.23054e6i − 1.97789e10i −3.28055e12 − 1.72836e13i 4.32114e15 6.38964e16 + 5.29092e16i 4.05908e18i 6.46109e19 + 3.56363e19i 5.93567e20 −1.15592e22 + 1.39596e22i
3.7 −1.53066e6 − 3.90503e6i − 1.95946e10i −1.29064e13 + 1.19545e13i −3.24586e15 −7.65175e16 + 2.99926e16i − 5.50179e18i 6.64381e19 + 3.21014e19i 6.00823e20 4.96831e21 + 1.26752e22i
3.8 −1.53066e6 + 3.90503e6i 1.95946e10i −1.29064e13 − 1.19545e13i −3.24586e15 −7.65175e16 − 2.99926e16i 5.50179e18i 6.64381e19 − 3.21014e19i 6.00823e20 4.96831e21 − 1.26752e22i
3.9 −174802. − 4.19066e6i − 2.16788e10i −1.75311e13 + 1.46508e12i 1.03983e15 −9.08484e16 + 3.78950e15i 7.00373e18i 9.20411e18 + 7.32107e19i 5.14802e20 −1.81765e20 − 4.35758e21i
3.10 −174802. + 4.19066e6i 2.16788e10i −1.75311e13 − 1.46508e12i 1.03983e15 −9.08484e16 − 3.78950e15i − 7.00373e18i 9.20411e18 − 7.32107e19i 5.14802e20 −1.81765e20 + 4.35758e21i
3.11 692907. − 4.13667e6i 4.93751e10i −1.66319e13 − 5.73266e12i 3.31017e14 2.04249e17 + 3.42123e16i − 3.30487e17i −3.52385e19 + 6.48287e19i −1.45313e21 2.29364e20 − 1.36931e21i
3.12 692907. + 4.13667e6i − 4.93751e10i −1.66319e13 + 5.73266e12i 3.31017e14 2.04249e17 − 3.42123e16i 3.30487e17i −3.52385e19 − 6.48287e19i −1.45313e21 2.29364e20 + 1.36931e21i
3.13 2.45341e6 − 3.40190e6i − 4.81236e10i −5.55370e12 − 1.66926e13i 2.94946e15 −1.63712e17 − 1.18067e17i − 5.91201e18i −7.04120e19 − 2.20606e19i −1.33111e21 7.23624e21 − 1.00338e22i
3.14 2.45341e6 + 3.40190e6i 4.81236e10i −5.55370e12 + 1.66926e13i 2.94946e15 −1.63712e17 + 1.18067e17i 5.91201e18i −7.04120e19 + 2.20606e19i −1.33111e21 7.23624e21 + 1.00338e22i
3.15 2.63672e6 − 3.26188e6i 5.11393e9i −3.68760e12 − 1.72014e13i −1.83599e15 1.66811e16 + 1.34840e16i − 1.24564e18i −6.58320e19 − 3.33266e19i 9.58619e20 −4.84100e21 + 5.98879e21i
3.16 2.63672e6 + 3.26188e6i − 5.11393e9i −3.68760e12 + 1.72014e13i −1.83599e15 1.66811e16 − 1.34840e16i 1.24564e18i −6.58320e19 + 3.33266e19i 9.58619e20 −4.84100e21 − 5.98879e21i
3.17 3.98169e6 − 1.31845e6i 2.39886e10i 1.41156e13 − 1.04993e13i 1.91424e15 3.16279e16 + 9.55154e16i 2.07168e18i 4.23609e19 − 6.04158e19i 4.09316e20 7.62190e21 − 2.52383e21i
3.18 3.98169e6 + 1.31845e6i − 2.39886e10i 1.41156e13 + 1.04993e13i 1.91424e15 3.16279e16 − 9.55154e16i − 2.07168e18i 4.23609e19 + 6.04158e19i 4.09316e20 7.62190e21 + 2.52383e21i
3.19 4.06802e6 − 1.02149e6i − 4.78233e10i 1.55053e13 − 8.31084e12i −4.02131e15 −4.88508e16 − 1.94546e17i 5.21410e18i 5.45865e19 − 4.96471e19i −1.30230e21 −1.63587e22 + 4.10771e21i
3.20 4.06802e6 + 1.02149e6i 4.78233e10i 1.55053e13 + 8.31084e12i −4.02131e15 −4.88508e16 + 1.94546e17i − 5.21410e18i 5.45865e19 + 4.96471e19i −1.30230e21 −1.63587e22 − 4.10771e21i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.45.b.b ✓ 20
4.b odd 2 1 inner 4.45.b.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.45.b.b ✓ 20 1.a even 1 1 trivial
4.45.b.b ✓ 20 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T320+⋯+57 ⁣⋯ ⁣00 T_{3}^{20} + \cdots + 57\!\cdots\!00 acting on S45new(4,[χ])S_{45}^{\mathrm{new}}(4, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T20+⋯+28 ⁣⋯ ⁣76 T^{20} + \cdots + 28\!\cdots\!76 Copy content Toggle raw display
33 T20+⋯+57 ⁣⋯ ⁣00 T^{20} + \cdots + 57\!\cdots\!00 Copy content Toggle raw display
55 (T10+⋯+40 ⁣⋯ ⁣00)2 (T^{10} + \cdots + 40\!\cdots\!00)^{2} Copy content Toggle raw display
77 T20+⋯+84 ⁣⋯ ⁣00 T^{20} + \cdots + 84\!\cdots\!00 Copy content Toggle raw display
1111 T20+⋯+25 ⁣⋯ ⁣00 T^{20} + \cdots + 25\!\cdots\!00 Copy content Toggle raw display
1313 (T10+⋯−42 ⁣⋯ ⁣00)2 (T^{10} + \cdots - 42\!\cdots\!00)^{2} Copy content Toggle raw display
1717 (T10+⋯+27 ⁣⋯ ⁣00)2 (T^{10} + \cdots + 27\!\cdots\!00)^{2} Copy content Toggle raw display
1919 T20+⋯+22 ⁣⋯ ⁣00 T^{20} + \cdots + 22\!\cdots\!00 Copy content Toggle raw display
2323 T20+⋯+41 ⁣⋯ ⁣00 T^{20} + \cdots + 41\!\cdots\!00 Copy content Toggle raw display
2929 (T10+⋯−60 ⁣⋯ ⁣56)2 (T^{10} + \cdots - 60\!\cdots\!56)^{2} Copy content Toggle raw display
3131 T20+⋯+65 ⁣⋯ ⁣00 T^{20} + \cdots + 65\!\cdots\!00 Copy content Toggle raw display
3737 (T10+⋯−13 ⁣⋯ ⁣00)2 (T^{10} + \cdots - 13\!\cdots\!00)^{2} Copy content Toggle raw display
4141 (T10+⋯−17 ⁣⋯ ⁣16)2 (T^{10} + \cdots - 17\!\cdots\!16)^{2} Copy content Toggle raw display
4343 T20+⋯+21 ⁣⋯ ⁣00 T^{20} + \cdots + 21\!\cdots\!00 Copy content Toggle raw display
4747 T20+⋯+12 ⁣⋯ ⁣00 T^{20} + \cdots + 12\!\cdots\!00 Copy content Toggle raw display
5353 (T10+⋯+56 ⁣⋯ ⁣00)2 (T^{10} + \cdots + 56\!\cdots\!00)^{2} Copy content Toggle raw display
5959 T20+⋯+78 ⁣⋯ ⁣00 T^{20} + \cdots + 78\!\cdots\!00 Copy content Toggle raw display
6161 (T10+⋯−24 ⁣⋯ ⁣56)2 (T^{10} + \cdots - 24\!\cdots\!56)^{2} Copy content Toggle raw display
6767 T20+⋯+57 ⁣⋯ ⁣00 T^{20} + \cdots + 57\!\cdots\!00 Copy content Toggle raw display
7171 T20+⋯+39 ⁣⋯ ⁣00 T^{20} + \cdots + 39\!\cdots\!00 Copy content Toggle raw display
7373 (T10+⋯−10 ⁣⋯ ⁣00)2 (T^{10} + \cdots - 10\!\cdots\!00)^{2} Copy content Toggle raw display
7979 T20+⋯+57 ⁣⋯ ⁣00 T^{20} + \cdots + 57\!\cdots\!00 Copy content Toggle raw display
8383 T20+⋯+89 ⁣⋯ ⁣00 T^{20} + \cdots + 89\!\cdots\!00 Copy content Toggle raw display
8989 (T10+⋯+14 ⁣⋯ ⁣24)2 (T^{10} + \cdots + 14\!\cdots\!24)^{2} Copy content Toggle raw display
9797 (T10+⋯−10 ⁣⋯ ⁣00)2 (T^{10} + \cdots - 10\!\cdots\!00)^{2} Copy content Toggle raw display
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