Properties

Label 8.22.b.a
Level 88
Weight 2222
Character orbit 8.b
Analytic conductor 22.35822.358
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] = [8,22,Mod(5,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.5");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: N N == 8=23 8 = 2^{3}
Weight: k k == 22 22
Character orbit: [χ][\chi] == 8.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: 22.358187543022.3581875430
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: x2010x19+8499037025x1876491332940x17++35 ⁣ ⁣00 x^{20} - 10 x^{19} + 8499037025 x^{18} - 76491332940 x^{17} + \cdots + 35\!\cdots\!00 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: multiple of 21903145477 2^{190}\cdot 3^{14}\cdot 5^{4}\cdot 7^{7}
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+14)q2+(β2+3β1+1)q3+(β314β1+20490)q4+(β5β39β2+365)q5+(β4+3β3++5911873)q6++(18680248β19+11 ⁣ ⁣99)q99+O(q100) q + ( - \beta_1 + 14) q^{2} + ( - \beta_{2} + 3 \beta_1 + 1) q^{3} + ( - \beta_{3} - 14 \beta_1 + 20490) q^{4} + (\beta_{5} - \beta_{3} - 9 \beta_{2} + \cdots - 365) q^{5} + (\beta_{4} + 3 \beta_{3} + \cdots + 5911873) q^{6}+ \cdots + (18680248 \beta_{19} + \cdots - 11\!\cdots\!99) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 20q+286q2+409876q4+118236748q6+564950496q7+3649699336q862762119220q951269339528q10+65316900136q121924309104464q142285680856096q15+7793508571920q16++13 ⁣ ⁣30q98+O(q100) 20 q + 286 q^{2} + 409876 q^{4} + 118236748 q^{6} + 564950496 q^{7} + 3649699336 q^{8} - 62762119220 q^{9} - 51269339528 q^{10} + 65316900136 q^{12} - 1924309104464 q^{14} - 2285680856096 q^{15} + 7793508571920 q^{16}+ \cdots + 13\!\cdots\!30 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x2010x19+8499037025x1876491332940x17++35 ⁣ ⁣00 x^{20} - 10 x^{19} + 8499037025 x^{18} - 76491332940 x^{17} + \cdots + 35\!\cdots\!00 : Copy content Toggle raw display

β1\beta_{1}== (29 ⁣ ⁣75ν19++36 ⁣ ⁣00)/16 ⁣ ⁣00 ( 29\!\cdots\!75 \nu^{19} + \cdots + 36\!\cdots\!00 ) / 16\!\cdots\!00 Copy content Toggle raw display
β2\beta_{2}== (29 ⁣ ⁣75ν19++36 ⁣ ⁣00)/53 ⁣ ⁣00 ( 29\!\cdots\!75 \nu^{19} + \cdots + 36\!\cdots\!00 ) / 53\!\cdots\!00 Copy content Toggle raw display
β3\beta_{3}== (14 ⁣ ⁣18ν19++80 ⁣ ⁣00)/12 ⁣ ⁣00 ( 14\!\cdots\!18 \nu^{19} + \cdots + 80\!\cdots\!00 ) / 12\!\cdots\!00 Copy content Toggle raw display
β4\beta_{4}== (62 ⁣ ⁣41ν19+68 ⁣ ⁣00)/16 ⁣ ⁣00 ( 62\!\cdots\!41 \nu^{19} + \cdots - 68\!\cdots\!00 ) / 16\!\cdots\!00 Copy content Toggle raw display
β5\beta_{5}== (26 ⁣ ⁣78ν19++42 ⁣ ⁣00)/12 ⁣ ⁣00 ( 26\!\cdots\!78 \nu^{19} + \cdots + 42\!\cdots\!00 ) / 12\!\cdots\!00 Copy content Toggle raw display
β6\beta_{6}== (19 ⁣ ⁣79ν19+28 ⁣ ⁣00)/48 ⁣ ⁣00 ( 19\!\cdots\!79 \nu^{19} + \cdots - 28\!\cdots\!00 ) / 48\!\cdots\!00 Copy content Toggle raw display
β7\beta_{7}== (99 ⁣ ⁣17ν19+14 ⁣ ⁣00)/12 ⁣ ⁣00 ( 99\!\cdots\!17 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 12\!\cdots\!00 Copy content Toggle raw display
β8\beta_{8}== (31 ⁣ ⁣63ν19+64 ⁣ ⁣00)/48 ⁣ ⁣00 ( 31\!\cdots\!63 \nu^{19} + \cdots - 64\!\cdots\!00 ) / 48\!\cdots\!00 Copy content Toggle raw display
β9\beta_{9}== (10 ⁣ ⁣53ν19+26 ⁣ ⁣00)/16 ⁣ ⁣00 ( 10\!\cdots\!53 \nu^{19} + \cdots - 26\!\cdots\!00 ) / 16\!\cdots\!00 Copy content Toggle raw display
β10\beta_{10}== (40 ⁣ ⁣99ν19++77 ⁣ ⁣00)/48 ⁣ ⁣00 ( 40\!\cdots\!99 \nu^{19} + \cdots + 77\!\cdots\!00 ) / 48\!\cdots\!00 Copy content Toggle raw display
β11\beta_{11}== (24 ⁣ ⁣83ν19+17 ⁣ ⁣00)/13 ⁣ ⁣00 ( 24\!\cdots\!83 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 13\!\cdots\!00 Copy content Toggle raw display
β12\beta_{12}== (16 ⁣ ⁣83ν19++45 ⁣ ⁣00)/40 ⁣ ⁣00 ( 16\!\cdots\!83 \nu^{19} + \cdots + 45\!\cdots\!00 ) / 40\!\cdots\!00 Copy content Toggle raw display
β13\beta_{13}== (17 ⁣ ⁣97ν19++34 ⁣ ⁣00)/40 ⁣ ⁣00 ( 17\!\cdots\!97 \nu^{19} + \cdots + 34\!\cdots\!00 ) / 40\!\cdots\!00 Copy content Toggle raw display
β14\beta_{14}== (54 ⁣ ⁣50ν19+59 ⁣ ⁣00)/12 ⁣ ⁣00 ( 54\!\cdots\!50 \nu^{19} + \cdots - 59\!\cdots\!00 ) / 12\!\cdots\!00 Copy content Toggle raw display
β15\beta_{15}== (56 ⁣ ⁣29ν19+34 ⁣ ⁣00)/12 ⁣ ⁣00 ( - 56\!\cdots\!29 \nu^{19} + \cdots - 34\!\cdots\!00 ) / 12\!\cdots\!00 Copy content Toggle raw display
β16\beta_{16}== (12 ⁣ ⁣95ν19++35 ⁣ ⁣00)/48 ⁣ ⁣00 ( 12\!\cdots\!95 \nu^{19} + \cdots + 35\!\cdots\!00 ) / 48\!\cdots\!00 Copy content Toggle raw display
β17\beta_{17}== (87 ⁣ ⁣49ν19++16 ⁣ ⁣00)/20 ⁣ ⁣00 ( - 87\!\cdots\!49 \nu^{19} + \cdots + 16\!\cdots\!00 ) / 20\!\cdots\!00 Copy content Toggle raw display
β18\beta_{18}== (83 ⁣ ⁣59ν19++14 ⁣ ⁣00)/16 ⁣ ⁣00 ( - 83\!\cdots\!59 \nu^{19} + \cdots + 14\!\cdots\!00 ) / 16\!\cdots\!00 Copy content Toggle raw display
β19\beta_{19}== (32 ⁣ ⁣15ν19++53 ⁣ ⁣00)/48 ⁣ ⁣00 ( - 32\!\cdots\!15 \nu^{19} + \cdots + 53\!\cdots\!00 ) / 48\!\cdots\!00 Copy content Toggle raw display
ν\nu== (β23β1+1)/4 ( \beta_{2} - 3\beta _1 + 1 ) / 4 Copy content Toggle raw display
ν2\nu^{2}== (β9+β74β6β5467β3+315β2+963976β113598169814)/16 ( \beta_{9} + \beta_{7} - 4\beta_{6} - \beta_{5} - 467\beta_{3} + 315\beta_{2} + 963976\beta _1 - 13598169814 ) / 16 Copy content Toggle raw display
ν3\nu^{3}== (216β19+128β18148β17+330β16+1961β15+65924238814)/64 ( 216 \beta_{19} + 128 \beta_{18} - 148 \beta_{17} + 330 \beta_{16} + 1961 \beta_{15} + \cdots - 65924238814 ) / 64 Copy content Toggle raw display
ν4\nu^{4}== (6479142β196477766β18+5751370β176368267β16++15 ⁣ ⁣16)/128 ( 6479142 \beta_{19} - 6477766 \beta_{18} + 5751370 \beta_{17} - 6368267 \beta_{16} + \cdots + 15\!\cdots\!16 ) / 128 Copy content Toggle raw display
ν5\nu^{5}== (4492587005148β19+1507086129732β18+3284020257464β17++16 ⁣ ⁣30)/512 ( - 4492587005148 \beta_{19} + 1507086129732 \beta_{18} + 3284020257464 \beta_{17} + \cdots + 16\!\cdots\!30 ) / 512 Copy content Toggle raw display
ν6\nu^{6}== (95 ⁣ ⁣96β19+10 ⁣ ⁣90)/512 ( - 95\!\cdots\!96 \beta_{19} + \cdots - 10\!\cdots\!90 ) / 512 Copy content Toggle raw display
ν7\nu^{7}== (24 ⁣ ⁣23β19+11 ⁣ ⁣98)/128 ( 24\!\cdots\!23 \beta_{19} + \cdots - 11\!\cdots\!98 ) / 128 Copy content Toggle raw display
ν8\nu^{8}== (12 ⁣ ⁣16β19++10 ⁣ ⁣62)/256 ( 12\!\cdots\!16 \beta_{19} + \cdots + 10\!\cdots\!62 ) / 256 Copy content Toggle raw display
ν9\nu^{9}== (20 ⁣ ⁣12β19++11 ⁣ ⁣94)/512 ( - 20\!\cdots\!12 \beta_{19} + \cdots + 11\!\cdots\!94 ) / 512 Copy content Toggle raw display
ν10\nu^{10}== (55 ⁣ ⁣64β19+38 ⁣ ⁣78)/512 ( - 55\!\cdots\!64 \beta_{19} + \cdots - 38\!\cdots\!78 ) / 512 Copy content Toggle raw display
ν11\nu^{11}== (20 ⁣ ⁣58β19+13 ⁣ ⁣78)/256 ( 20\!\cdots\!58 \beta_{19} + \cdots - 13\!\cdots\!78 ) / 256 Copy content Toggle raw display
ν12\nu^{12}== (29 ⁣ ⁣82β19++18 ⁣ ⁣64)/128 ( 29\!\cdots\!82 \beta_{19} + \cdots + 18\!\cdots\!64 ) / 128 Copy content Toggle raw display
ν13\nu^{13}== (83 ⁣ ⁣96β19++64 ⁣ ⁣30)/512 ( - 83\!\cdots\!96 \beta_{19} + \cdots + 64\!\cdots\!30 ) / 512 Copy content Toggle raw display
ν14\nu^{14}== (23 ⁣ ⁣64β19+15 ⁣ ⁣62)/512 ( - 23\!\cdots\!64 \beta_{19} + \cdots - 15\!\cdots\!62 ) / 512 Copy content Toggle raw display
ν15\nu^{15}== (42 ⁣ ⁣79β19+36 ⁣ ⁣92)/128 ( 42\!\cdots\!79 \beta_{19} + \cdots - 36\!\cdots\!92 ) / 128 Copy content Toggle raw display
ν16\nu^{16}== (23 ⁣ ⁣64β19++15 ⁣ ⁣18)/256 ( 23\!\cdots\!64 \beta_{19} + \cdots + 15\!\cdots\!18 ) / 256 Copy content Toggle raw display
ν17\nu^{17}== (35 ⁣ ⁣92β19++33 ⁣ ⁣50)/512 ( - 35\!\cdots\!92 \beta_{19} + \cdots + 33\!\cdots\!50 ) / 512 Copy content Toggle raw display
ν18\nu^{18}== (94 ⁣ ⁣56β19+60 ⁣ ⁣58)/512 ( - 94\!\cdots\!56 \beta_{19} + \cdots - 60\!\cdots\!58 ) / 512 Copy content Toggle raw display
ν19\nu^{19}== (36 ⁣ ⁣66β19+37 ⁣ ⁣90)/256 ( 36\!\cdots\!66 \beta_{19} + \cdots - 37\!\cdots\!90 ) / 256 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/8Z)×\left(\mathbb{Z}/8\mathbb{Z}\right)^\times.

nn 55 77
χ(n)\chi(n) 1-1 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.

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 + 19382.6i
0.500000 19382.6i
0.500000 32835.5i
0.500000 + 32835.5i
0.500000 + 21396.5i
0.500000 21396.5i
0.500000 6359.74i
0.500000 + 6359.74i
0.500000 37074.8i
0.500000 + 37074.8i
0.500000 + 45387.1i
0.500000 45387.1i
0.500000 6610.70i
0.500000 + 6610.70i
0.500000 44665.4i
0.500000 + 44665.4i
0.500000 + 15078.6i
0.500000 15078.6i
0.500000 + 29091.7i
0.500000 29091.7i
−1394.30 391.258i 77530.5i 1.79099e6 + 1.09106e6i 2.27953e7i −3.03344e7 + 1.08101e8i 1.10984e9 −2.07029e9 2.22200e9i 4.44938e9 −8.91883e9 + 3.17834e10i
5.2 −1394.30 + 391.258i 77530.5i 1.79099e6 1.09106e6i 2.27953e7i −3.03344e7 1.08101e8i 1.10984e9 −2.07029e9 + 2.22200e9i 4.44938e9 −8.91883e9 3.17834e10i
5.3 −1358.45 501.755i 131342.i 1.59363e6 + 1.36322e6i 4.61121e6i 6.59016e7 1.78422e8i −9.67293e8 −1.48087e9 2.65149e9i −6.79041e9 −2.31370e9 + 6.26411e9i
5.4 −1358.45 + 501.755i 131342.i 1.59363e6 1.36322e6i 4.61121e6i 6.59016e7 + 1.78422e8i −9.67293e8 −1.48087e9 + 2.65149e9i −6.79041e9 −2.31370e9 6.26411e9i
5.5 −962.766 1081.77i 85585.8i −243314. + 2.08299e6i 2.31314e7i −9.25844e7 + 8.23991e7i −3.19239e8 2.48758e9 1.74222e9i 3.13542e9 2.50229e10 2.22701e10i
5.6 −962.766 + 1081.77i 85585.8i −243314. 2.08299e6i 2.31314e7i −9.25844e7 8.23991e7i −3.19239e8 2.48758e9 + 1.74222e9i 3.13542e9 2.50229e10 + 2.22701e10i
5.7 −476.143 1367.64i 25439.0i −1.64373e6 + 1.30238e6i 3.81460e7i 3.47914e7 1.21126e7i 7.30446e7 2.56384e9 + 1.62791e9i 9.81321e9 −5.21701e10 + 1.81630e10i
5.8 −476.143 + 1367.64i 25439.0i −1.64373e6 1.30238e6i 3.81460e7i 3.47914e7 + 1.21126e7i 7.30446e7 2.56384e9 1.62791e9i 9.81321e9 −5.21701e10 1.81630e10i
5.9 −377.837 1398.00i 148299.i −1.81163e6 + 1.05643e6i 2.51716e7i 2.07321e8 5.60329e7i 8.96078e8 2.16139e9 + 2.13349e9i −1.15323e10 3.51898e10 9.51079e9i
5.10 −377.837 + 1398.00i 148299.i −1.81163e6 1.05643e6i 2.51716e7i 2.07321e8 + 5.60329e7i 8.96078e8 2.16139e9 2.13349e9i −1.15323e10 3.51898e10 + 9.51079e9i
5.11 228.196 1430.06i 181549.i −1.99301e6 652669.i 6.89272e6i −2.59626e8 4.14287e7i 1.45891e8 −1.38815e9 + 2.70119e9i −2.24995e10 −9.85702e9 1.57289e9i
5.12 228.196 + 1430.06i 181549.i −1.99301e6 + 652669.i 6.89272e6i −2.59626e8 + 4.14287e7i 1.45891e8 −1.38815e9 2.70119e9i −2.24995e10 −9.85702e9 + 1.57289e9i
5.13 582.905 1325.66i 26442.8i −1.41760e6 1.54547e6i 7.17621e6i 3.50542e7 + 1.54136e7i −5.46394e8 −2.87509e9 + 9.78391e8i 9.76113e9 9.51321e9 + 4.18305e9i
5.14 582.905 + 1325.66i 26442.8i −1.41760e6 + 1.54547e6i 7.17621e6i 3.50542e7 1.54136e7i −5.46394e8 −2.87509e9 9.78391e8i 9.76113e9 9.51321e9 4.18305e9i
5.15 1174.53 847.130i 178662.i 661894. 1.98996e6i 3.12530e7i 1.51350e8 + 2.09844e8i 2.98846e8 −9.08340e8 2.89798e9i −2.14596e10 −2.64753e10 3.67076e10i
5.16 1174.53 + 847.130i 178662.i 661894. + 1.98996e6i 3.12530e7i 1.51350e8 2.09844e8i 2.98846e8 −9.08340e8 + 2.89798e9i −2.14596e10 −2.64753e10 + 3.67076e10i
5.17 1282.51 672.555i 60314.6i 1.19249e6 1.72511e6i 1.19399e7i −4.05649e7 7.73538e7i 8.57498e8 3.69144e8 3.01448e9i 6.82251e9 8.03026e9 + 1.53130e10i
5.18 1282.51 + 672.555i 60314.6i 1.19249e6 + 1.72511e6i 1.19399e7i −4.05649e7 + 7.73538e7i 8.57498e8 3.69144e8 + 3.01448e9i 6.82251e9 8.03026e9 1.53130e10i
5.19 1444.36 104.758i 116367.i 2.07520e6 302616.i 3.48989e7i −1.21903e7 1.68076e8i −1.26580e9 2.96564e9 6.54481e8i −3.08088e9 −3.65593e9 5.04066e10i
5.20 1444.36 + 104.758i 116367.i 2.07520e6 + 302616.i 3.48989e7i −1.21903e7 + 1.68076e8i −1.26580e9 2.96564e9 + 6.54481e8i −3.08088e9 −3.65593e9 + 5.04066e10i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.22.b.a 20
4.b odd 2 1 32.22.b.a 20
8.b even 2 1 inner 8.22.b.a 20
8.d odd 2 1 32.22.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.22.b.a 20 1.a even 1 1 trivial
8.22.b.a 20 8.b even 2 1 inner
32.22.b.a 20 4.b odd 2 1
32.22.b.a 20 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace S22new(8,[χ])S_{22}^{\mathrm{new}}(8, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T20++16 ⁣ ⁣24 T^{20} + \cdots + 16\!\cdots\!24 Copy content Toggle raw display
33 T20++39 ⁣ ⁣48 T^{20} + \cdots + 39\!\cdots\!48 Copy content Toggle raw display
55 T20++22 ⁣ ⁣00 T^{20} + \cdots + 22\!\cdots\!00 Copy content Toggle raw display
77 (T10++58 ⁣ ⁣52)2 (T^{10} + \cdots + 58\!\cdots\!52)^{2} Copy content Toggle raw display
1111 T20++29 ⁣ ⁣00 T^{20} + \cdots + 29\!\cdots\!00 Copy content Toggle raw display
1313 T20++23 ⁣ ⁣00 T^{20} + \cdots + 23\!\cdots\!00 Copy content Toggle raw display
1717 (T10++48 ⁣ ⁣24)2 (T^{10} + \cdots + 48\!\cdots\!24)^{2} Copy content Toggle raw display
1919 T20++21 ⁣ ⁣28 T^{20} + \cdots + 21\!\cdots\!28 Copy content Toggle raw display
2323 (T10+42 ⁣ ⁣24)2 (T^{10} + \cdots - 42\!\cdots\!24)^{2} Copy content Toggle raw display
2929 T20++50 ⁣ ⁣00 T^{20} + \cdots + 50\!\cdots\!00 Copy content Toggle raw display
3131 (T10+19 ⁣ ⁣72)2 (T^{10} + \cdots - 19\!\cdots\!72)^{2} Copy content Toggle raw display
3737 T20++33 ⁣ ⁣12 T^{20} + \cdots + 33\!\cdots\!12 Copy content Toggle raw display
4141 (T10++21 ⁣ ⁣00)2 (T^{10} + \cdots + 21\!\cdots\!00)^{2} Copy content Toggle raw display
4343 T20++63 ⁣ ⁣92 T^{20} + \cdots + 63\!\cdots\!92 Copy content Toggle raw display
4747 (T10++61 ⁣ ⁣64)2 (T^{10} + \cdots + 61\!\cdots\!64)^{2} Copy content Toggle raw display
5353 T20++48 ⁣ ⁣28 T^{20} + \cdots + 48\!\cdots\!28 Copy content Toggle raw display
5959 T20++11 ⁣ ⁣92 T^{20} + \cdots + 11\!\cdots\!92 Copy content Toggle raw display
6161 T20++54 ⁣ ⁣00 T^{20} + \cdots + 54\!\cdots\!00 Copy content Toggle raw display
6767 T20++43 ⁣ ⁣08 T^{20} + \cdots + 43\!\cdots\!08 Copy content Toggle raw display
7171 (T10+18 ⁣ ⁣16)2 (T^{10} + \cdots - 18\!\cdots\!16)^{2} Copy content Toggle raw display
7373 (T10+39 ⁣ ⁣52)2 (T^{10} + \cdots - 39\!\cdots\!52)^{2} Copy content Toggle raw display
7979 (T10++55 ⁣ ⁣00)2 (T^{10} + \cdots + 55\!\cdots\!00)^{2} Copy content Toggle raw display
8383 T20++55 ⁣ ⁣32 T^{20} + \cdots + 55\!\cdots\!32 Copy content Toggle raw display
8989 (T10++18 ⁣ ⁣00)2 (T^{10} + \cdots + 18\!\cdots\!00)^{2} Copy content Toggle raw display
9797 (T10++11 ⁣ ⁣96)2 (T^{10} + \cdots + 11\!\cdots\!96)^{2} Copy content Toggle raw display
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