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Properties

Label 460.2.e.a
Level 460460
Weight 22
Character orbit 460.e
Analytic conductor 3.6733.673
Analytic rank 00
Dimension 1616
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [460,2,Mod(91,460)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(460, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("460.91"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 460=22⋅5⋅23 460 = 2^{2} \cdot 5 \cdot 23
Weight: k k == 2 2
Character orbit: [χ][\chi] == 460.e (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 3.673118492983.67311849298
Analytic rank: 00
Dimension: 1616
Coefficient field: 16.0.7465802011608416256.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x16−x14+x12+8x10−20x8+32x6+16x4−64x2+256 x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 28 2^{8}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,…,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−β4q2+β8q3+(β8−β2)q4+β3q5+(β10−β8+β6+⋯+1)q6+(−2β15−β13+2β12)q7+⋯+(5β15−2β13+⋯−9β5)q99+O(q100) q - \beta_{4} q^{2} + \beta_{8} q^{3} + (\beta_{8} - \beta_{2}) q^{4} + \beta_{3} q^{5} + (\beta_{10} - \beta_{8} + \beta_{6} + \cdots + 1) q^{6} + ( - 2 \beta_{15} - \beta_{13} + 2 \beta_{12}) q^{7}+ \cdots + (5 \beta_{15} - 2 \beta_{13} + \cdots - 9 \beta_{5}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 16q+4q4+14q6+4q9−30q12+4q13+4q16+30q18+2q24−16q25−54q26−48q29+34q36−36q41−40q46+18q48+68q49+34q52−40q54+⋯−18q98+O(q100) 16 q + 4 q^{4} + 14 q^{6} + 4 q^{9} - 30 q^{12} + 4 q^{13} + 4 q^{16} + 30 q^{18} + 2 q^{24} - 16 q^{25} - 54 q^{26} - 48 q^{29} + 34 q^{36} - 36 q^{41} - 40 q^{46} + 18 q^{48} + 68 q^{49} + 34 q^{52} - 40 q^{54}+ \cdots - 18 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16−x14+x12+8x10−20x8+32x6+16x4−64x2+256 x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 : Copy content Toggle raw display

β1\beta_{1}== (−ν15+5ν13+3ν11+4ν9+172ν7−96ν5−112ν3−256ν)/1152 ( -\nu^{15} + 5\nu^{13} + 3\nu^{11} + 4\nu^{9} + 172\nu^{7} - 96\nu^{5} - 112\nu^{3} - 256\nu ) / 1152 Copy content Toggle raw display
β2\beta_{2}== (−ν14−ν12−15ν10−2ν8+28ν6−96ν4+80ν2−160)/288 ( -\nu^{14} - \nu^{12} - 15\nu^{10} - 2\nu^{8} + 28\nu^{6} - 96\nu^{4} + 80\nu^{2} - 160 ) / 288 Copy content Toggle raw display
β3\beta_{3}== (ν15+7ν13+9ν11−16ν9+44ν7+144ν5−80ν3+640ν)/1152 ( \nu^{15} + 7\nu^{13} + 9\nu^{11} - 16\nu^{9} + 44\nu^{7} + 144\nu^{5} - 80\nu^{3} + 640\nu ) / 1152 Copy content Toggle raw display
β4\beta_{4}== (−2ν14−3ν13+4ν12+15ν11−12ν10−27ν9−22ν8+⋯+64)/576 ( - 2 \nu^{14} - 3 \nu^{13} + 4 \nu^{12} + 15 \nu^{11} - 12 \nu^{10} - 27 \nu^{9} - 22 \nu^{8} + \cdots + 64 ) / 576 Copy content Toggle raw display
β5\beta_{5}== (−ν15+6ν14+11ν13−6ν12−15ν11+30ν10+22ν9+⋯+768)/1152 ( - \nu^{15} + 6 \nu^{14} + 11 \nu^{13} - 6 \nu^{12} - 15 \nu^{11} + 30 \nu^{10} + 22 \nu^{9} + \cdots + 768 ) / 1152 Copy content Toggle raw display
β6\beta_{6}== (2ν14−3ν13−4ν12+15ν11+12ν10−27ν9+22ν8+24ν7+⋯−64)/576 ( 2 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 15 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 24 \nu^{7} + \cdots - 64 ) / 576 Copy content Toggle raw display
β7\beta_{7}== (ν15−9ν13+9ν11−16ν9−36ν7+144ν5−80ν3)/384 ( \nu^{15} - 9\nu^{13} + 9\nu^{11} - 16\nu^{9} - 36\nu^{7} + 144\nu^{5} - 80\nu^{3} ) / 384 Copy content Toggle raw display
β8\beta_{8}== (−3ν15−4ν14−3ν13+8ν12+3ν11−24ν10−54ν9+⋯+128)/1152 ( - 3 \nu^{15} - 4 \nu^{14} - 3 \nu^{13} + 8 \nu^{12} + 3 \nu^{11} - 24 \nu^{10} - 54 \nu^{9} + \cdots + 128 ) / 1152 Copy content Toggle raw display
β9\beta_{9}== (2ν14+9ν13−4ν12−9ν11+12ν10−27ν9+22ν8+36ν7+⋯−64)/576 ( 2 \nu^{14} + 9 \nu^{13} - 4 \nu^{12} - 9 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 36 \nu^{7} + \cdots - 64 ) / 576 Copy content Toggle raw display
β10\beta_{10}== (−3ν15+8ν14+3ν13−4ν12−39ν11+36ν10−12ν9+⋯+896)/1152 ( - 3 \nu^{15} + 8 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} - 39 \nu^{11} + 36 \nu^{10} - 12 \nu^{9} + \cdots + 896 ) / 1152 Copy content Toggle raw display
β11\beta_{11}== (−ν15−6ν14+11ν13+30ν12−15ν11−6ν10+22ν9+⋯+1152)/1152 ( - \nu^{15} - 6 \nu^{14} + 11 \nu^{13} + 30 \nu^{12} - 15 \nu^{11} - 6 \nu^{10} + 22 \nu^{9} + \cdots + 1152 ) / 1152 Copy content Toggle raw display
β12\beta_{12}== (ν15−4ν12+8ν10+5ν9−8ν8−12ν6+32ν4−8ν3−128ν2+64)/192 ( \nu^{15} - 4\nu^{12} + 8\nu^{10} + 5\nu^{9} - 8\nu^{8} - 12\nu^{6} + 32\nu^{4} - 8\nu^{3} - 128\nu^{2} + 64 ) / 192 Copy content Toggle raw display
β13\beta_{13}== (ν15+4ν12+5ν9+20ν6−96ν4−8ν3+96ν2+64)/192 ( \nu^{15} + 4\nu^{12} + 5\nu^{9} + 20\nu^{6} - 96\nu^{4} - 8\nu^{3} + 96\nu^{2} + 64 ) / 192 Copy content Toggle raw display
β14\beta_{14}== (3ν15+8ν14−3ν13−4ν12+39ν11+36ν10+12ν9+⋯+896)/1152 ( 3 \nu^{15} + 8 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 39 \nu^{11} + 36 \nu^{10} + 12 \nu^{9} + \cdots + 896 ) / 1152 Copy content Toggle raw display
β15\beta_{15}== (−ν15−4ν12+8ν10−5ν9−8ν8−12ν6+32ν4+8ν3−128ν2+64)/192 ( -\nu^{15} - 4\nu^{12} + 8\nu^{10} - 5\nu^{9} - 8\nu^{8} - 12\nu^{6} + 32\nu^{4} + 8\nu^{3} - 128\nu^{2} + 64 ) / 192 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (−β14+β10−β9−2β8−β7+β6+2β4+β3−2β1)/4 ( -\beta_{14} + \beta_{10} - \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + 2\beta_{4} + \beta_{3} - 2\beta_1 ) / 4 Copy content Toggle raw display
ν2\nu^{2}== (β15+β14+β13+2β11+β10−β7+4β6−4β5+⋯+1)/4 ( \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{10} - \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + \cdots + 1 ) / 4 Copy content Toggle raw display
ν3\nu^{3}== (β15−β12+β9−3β8+β7−2β6+2β4)/2 ( \beta_{15} - \beta_{12} + \beta_{9} - 3\beta_{8} + \beta_{7} - 2\beta_{6} + 2\beta_{4} ) / 2 Copy content Toggle raw display
ν4\nu^{4}== (−5β15+β14−7β13+2β12+4β11+β10+β7+⋯−1)/4 ( - 5 \beta_{15} + \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \beta_{10} + \beta_{7} + \cdots - 1 ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (2β15−β14−2β12+β10−3β9+5β7−3β6+⋯+2β1)/4 ( 2 \beta_{15} - \beta_{14} - 2 \beta_{12} + \beta_{10} - 3 \beta_{9} + 5 \beta_{7} - 3 \beta_{6} + \cdots + 2 \beta_1 ) / 4 Copy content Toggle raw display
ν6\nu^{6}== (β15+β12+5β11+5β7+3β6+5β5−3β4+6β2+5β1−5)/2 ( \beta_{15} + \beta_{12} + 5\beta_{11} + 5\beta_{7} + 3\beta_{6} + 5\beta_{5} - 3\beta_{4} + 6\beta_{2} + 5\beta _1 - 5 ) / 2 Copy content Toggle raw display
ν7\nu^{7}== (−5β14+5β10−β9−10β8+7β7−3β6+6β4+5β3+26β1)/4 ( -5\beta_{14} + 5\beta_{10} - \beta_{9} - 10\beta_{8} + 7\beta_{7} - 3\beta_{6} + 6\beta_{4} + 5\beta_{3} + 26\beta_1 ) / 4 Copy content Toggle raw display
ν8\nu^{8}== (−9β15−25β14−β13−8β12−18β11−25β10+9β7+⋯+23)/4 ( - 9 \beta_{15} - 25 \beta_{14} - \beta_{13} - 8 \beta_{12} - 18 \beta_{11} - 25 \beta_{10} + 9 \beta_{7} + \cdots + 23 ) / 4 Copy content Toggle raw display
ν9\nu^{9}== (−β15+β12−21β9−9β8−13β7+6β6−6β4)/2 ( -\beta_{15} + \beta_{12} - 21\beta_{9} - 9\beta_{8} - 13\beta_{7} + 6\beta_{6} - 6\beta_{4} ) / 2 Copy content Toggle raw display
ν10\nu^{10}== (49β15−13β14+43β13+6β12+12β11−13β10+3β7+⋯−35)/4 ( 49 \beta_{15} - 13 \beta_{14} + 43 \beta_{13} + 6 \beta_{12} + 12 \beta_{11} - 13 \beta_{10} + 3 \beta_{7} + \cdots - 35 ) / 4 Copy content Toggle raw display
ν11\nu^{11}== (22β15+61β14−22β12−61β10+7β9−17β7+⋯+22β1)/4 ( 22 \beta_{15} + 61 \beta_{14} - 22 \beta_{12} - 61 \beta_{10} + 7 \beta_{9} - 17 \beta_{7} + \cdots + 22 \beta_1 ) / 4 Copy content Toggle raw display
ν12\nu^{12}== (−29β15−29β12−β11−β7−39β6−β5+39β4+⋯−31)/2 ( - 29 \beta_{15} - 29 \beta_{12} - \beta_{11} - \beta_{7} - 39 \beta_{6} - \beta_{5} + 39 \beta_{4} + \cdots - 31 ) / 2 Copy content Toggle raw display
ν13\nu^{13}== (65β14−65β10+45β9+130β8−91β7−25β6+⋯−50β1)/4 ( 65 \beta_{14} - 65 \beta_{10} + 45 \beta_{9} + 130 \beta_{8} - 91 \beta_{7} - 25 \beta_{6} + \cdots - 50 \beta_1 ) / 4 Copy content Toggle raw display
ν14\nu^{14}== (−43β15+229β14+109β13−152β12−86β11+229β10+⋯−203)/4 ( - 43 \beta_{15} + 229 \beta_{14} + 109 \beta_{13} - 152 \beta_{12} - 86 \beta_{11} + 229 \beta_{10} + \cdots - 203 ) / 4 Copy content Toggle raw display
ν15\nu^{15}== (−179β15+179β12+113β9+21β8+73β7−46β6+46β4)/2 ( -179\beta_{15} + 179\beta_{12} + 113\beta_{9} + 21\beta_{8} + 73\beta_{7} - 46\beta_{6} + 46\beta_{4} ) / 2 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/460Z)×\left(\mathbb{Z}/460\mathbb{Z}\right)^\times.

nn 231231 277277 281281
χ(n)\chi(n) −1-1 11 −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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
91.1
1.18353 − 0.774115i
−0.0786378 + 1.41203i
−0.0786378 − 1.41203i
1.18353 + 0.774115i
1.37379 − 0.335728i
0.977642 − 1.02187i
0.977642 + 1.02187i
1.37379 + 0.335728i
−0.977642 + 1.02187i
−1.37379 + 0.335728i
−1.37379 − 0.335728i
−0.977642 − 1.02187i
0.0786378 − 1.41203i
−1.18353 + 0.774115i
−1.18353 − 0.774115i
0.0786378 + 1.41203i
−1.35760 − 0.396143i 1.47175i 1.68614 + 1.07561i − 1.00000i 0.583024 − 1.99804i −3.53986 −1.86301 − 2.12819i 0.833952 −0.396143 + 1.35760i
91.2 −1.35760 − 0.396143i 1.47175i 1.68614 + 1.07561i 1.00000i 0.583024 − 1.99804i 3.53986 −1.86301 − 2.12819i 0.833952 0.396143 − 1.35760i
91.3 −1.35760 + 0.396143i − 1.47175i 1.68614 − 1.07561i − 1.00000i 0.583024 + 1.99804i 3.53986 −1.86301 + 2.12819i 0.833952 0.396143 + 1.35760i
91.4 −1.35760 + 0.396143i − 1.47175i 1.68614 − 1.07561i 1.00000i 0.583024 + 1.99804i −3.53986 −1.86301 + 2.12819i 0.833952 −0.396143 − 1.35760i
91.5 −0.637910 − 1.26217i 2.87247i −1.18614 + 1.61030i − 1.00000i 3.62554 − 1.83238i −2.68161 2.78912 + 0.469882i −5.25109 −1.26217 + 0.637910i
91.6 −0.637910 − 1.26217i 2.87247i −1.18614 + 1.61030i 1.00000i 3.62554 − 1.83238i 2.68161 2.78912 + 0.469882i −5.25109 1.26217 − 0.637910i
91.7 −0.637910 + 1.26217i − 2.87247i −1.18614 − 1.61030i − 1.00000i 3.62554 + 1.83238i 2.68161 2.78912 − 0.469882i −5.25109 1.26217 + 0.637910i
91.8 −0.637910 + 1.26217i − 2.87247i −1.18614 − 1.61030i 1.00000i 3.62554 + 1.83238i −2.68161 2.78912 − 0.469882i −5.25109 −1.26217 − 0.637910i
91.9 0.637910 − 1.26217i − 0.348132i −1.18614 − 1.61030i − 1.00000i −0.439402 − 0.222077i −4.89140 −2.78912 + 0.469882i 2.87880 −1.26217 − 0.637910i
91.10 0.637910 − 1.26217i − 0.348132i −1.18614 − 1.61030i 1.00000i −0.439402 − 0.222077i 4.89140 −2.78912 + 0.469882i 2.87880 1.26217 + 0.637910i
91.11 0.637910 + 1.26217i 0.348132i −1.18614 + 1.61030i − 1.00000i −0.439402 + 0.222077i 4.89140 −2.78912 − 0.469882i 2.87880 1.26217 − 0.637910i
91.12 0.637910 + 1.26217i 0.348132i −1.18614 + 1.61030i 1.00000i −0.439402 + 0.222077i −4.89140 −2.78912 − 0.469882i 2.87880 −1.26217 + 0.637910i
91.13 1.35760 − 0.396143i − 0.679463i 1.68614 − 1.07561i − 1.00000i −0.269165 − 0.922437i 1.16300 1.86301 − 2.12819i 2.53833 −0.396143 − 1.35760i
91.14 1.35760 − 0.396143i − 0.679463i 1.68614 − 1.07561i 1.00000i −0.269165 − 0.922437i −1.16300 1.86301 − 2.12819i 2.53833 0.396143 + 1.35760i
91.15 1.35760 + 0.396143i 0.679463i 1.68614 + 1.07561i − 1.00000i −0.269165 + 0.922437i −1.16300 1.86301 + 2.12819i 2.53833 0.396143 − 1.35760i
91.16 1.35760 + 0.396143i 0.679463i 1.68614 + 1.07561i 1.00000i −0.269165 + 0.922437i 1.16300 1.86301 + 2.12819i 2.53833 −0.396143 + 1.35760i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.e.a ✓ 16
4.b odd 2 1 inner 460.2.e.a ✓ 16
23.b odd 2 1 inner 460.2.e.a ✓ 16
92.b even 2 1 inner 460.2.e.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.e.a ✓ 16 1.a even 1 1 trivial
460.2.e.a ✓ 16 4.b odd 2 1 inner
460.2.e.a ✓ 16 23.b odd 2 1 inner
460.2.e.a ✓ 16 92.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T38+11T36+24T34+11T32+1 T_{3}^{8} + 11T_{3}^{6} + 24T_{3}^{4} + 11T_{3}^{2} + 1 acting on S2new(460,[χ])S_{2}^{\mathrm{new}}(460, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T8−T6−4T2+16)2 (T^{8} - T^{6} - 4 T^{2} + 16)^{2} Copy content Toggle raw display
33 (T8+11T6+24T4+⋯+1)2 (T^{8} + 11 T^{6} + 24 T^{4} + \cdots + 1)^{2} Copy content Toggle raw display
55 (T2+1)8 (T^{2} + 1)^{8} Copy content Toggle raw display
77 (T8−45T6+⋯+2916)2 (T^{8} - 45 T^{6} + \cdots + 2916)^{2} Copy content Toggle raw display
1111 (T8−45T6+⋯+2916)2 (T^{8} - 45 T^{6} + \cdots + 2916)^{2} Copy content Toggle raw display
1313 (T4−T3−24T2+⋯+31)4 (T^{4} - T^{3} - 24 T^{2} + \cdots + 31)^{4} Copy content Toggle raw display
1717 (T8+99T6+⋯+11664)2 (T^{8} + 99 T^{6} + \cdots + 11664)^{2} Copy content Toggle raw display
1919 (T8−45T6+⋯+2916)2 (T^{8} - 45 T^{6} + \cdots + 2916)^{2} Copy content Toggle raw display
2323 (T8−16T6+⋯+279841)2 (T^{8} - 16 T^{6} + \cdots + 279841)^{2} Copy content Toggle raw display
2929 (T4+12T3+⋯+12)4 (T^{4} + 12 T^{3} + \cdots + 12)^{4} Copy content Toggle raw display
3131 (T8+207T6+⋯+4124961)2 (T^{8} + 207 T^{6} + \cdots + 4124961)^{2} Copy content Toggle raw display
3737 (T8+108T6+⋯+46656)2 (T^{8} + 108 T^{6} + \cdots + 46656)^{2} Copy content Toggle raw display
4141 (T4+9T3+⋯−681)4 (T^{4} + 9 T^{3} + \cdots - 681)^{4} Copy content Toggle raw display
4343 (T8−180T6+⋯+46656)2 (T^{8} - 180 T^{6} + \cdots + 46656)^{2} Copy content Toggle raw display
4747 (T8+362T6+⋯+163216)2 (T^{8} + 362 T^{6} + \cdots + 163216)^{2} Copy content Toggle raw display
5353 (T4+180T2+6912)4 (T^{4} + 180 T^{2} + 6912)^{4} Copy content Toggle raw display
5959 (T4+76T2+256)4 (T^{4} + 76 T^{2} + 256)^{4} Copy content Toggle raw display
6161 (T8+243T6+⋯+13089924)2 (T^{8} + 243 T^{6} + \cdots + 13089924)^{2} Copy content Toggle raw display
6767 (T8−288T6+⋯+6718464)2 (T^{8} - 288 T^{6} + \cdots + 6718464)^{2} Copy content Toggle raw display
7171 (T8+215T6+⋯+6889)2 (T^{8} + 215 T^{6} + \cdots + 6889)^{2} Copy content Toggle raw display
7373 (T4−2T3+⋯−908)4 (T^{4} - 2 T^{3} + \cdots - 908)^{4} Copy content Toggle raw display
7979 (T8−324T6+⋯+3779136)2 (T^{8} - 324 T^{6} + \cdots + 3779136)^{2} Copy content Toggle raw display
8383 (T4−252T2+5184)4 (T^{4} - 252 T^{2} + 5184)^{4} Copy content Toggle raw display
8989 (T8+396T6+⋯+13483584)2 (T^{8} + 396 T^{6} + \cdots + 13483584)^{2} Copy content Toggle raw display
9797 (T8+531T6+⋯+29746116)2 (T^{8} + 531 T^{6} + \cdots + 29746116)^{2} Copy content Toggle raw display
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