Properties

Label 28.6.d.b
Level $28$
Weight $6$
Character orbit 28.d
Analytic conductor $4.491$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [28,6,Mod(27,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.27");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 28.d (of order \(2\), degree \(1\), 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: \(4.49074695476\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 674 x^{14} + 3404 x^{13} + 173721 x^{12} - 919512 x^{11} - 21981508 x^{10} + 94817024 x^{9} + 1652635052 x^{8} - 4373672096 x^{7} + \cdots + 224266997486896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{46} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{2} - \beta_{2} q^{3} + (\beta_{5} - \beta_{3} - 3) q^{4} + \beta_{9} q^{5} + ( - \beta_{7} + \beta_{2}) q^{6} + ( - \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{3} - 2) q^{7} + ( - 2 \beta_{10} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_1 + 34) q^{8} + ( - \beta_{12} - \beta_{10} - \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 13 \beta_{3} - 4 \beta_1 + 91) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{2} - \beta_{2} q^{3} + (\beta_{5} - \beta_{3} - 3) q^{4} + \beta_{9} q^{5} + ( - \beta_{7} + \beta_{2}) q^{6} + ( - \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{3} - 2) q^{7} + ( - 2 \beta_{10} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_1 + 34) q^{8} + ( - \beta_{12} - \beta_{10} - \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 13 \beta_{3} - 4 \beta_1 + 91) q^{9} + ( - \beta_{11} - \beta_{8} - 2 \beta_{2} - 1) q^{10} + ( - \beta_{12} + \beta_{10} + 4 \beta_{6} + 2 \beta_{5} + 3 \beta_{3} + \beta_1 - 1) q^{11} + (\beta_{15} + \beta_{12} + 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 3 \beta_{2}) q^{12} + ( - \beta_{14} + \beta_{13} + \beta_{11} - \beta_{9} - 2 \beta_{7} + \beta_{5}) q^{13} + (\beta_{14} + \beta_{12} - 6 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - 5 \beta_{4} - 2 \beta_{3} + \cdots + 67) q^{14}+ \cdots + ( - 145 \beta_{12} - 253 \beta_{10} - 108 \beta_{6} - 796 \beta_{5} + \cdots + 2871) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 48 q^{4} + 608 q^{8} + 1616 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 48 q^{4} + 608 q^{8} + 1616 q^{9} + 1176 q^{14} - 3008 q^{16} - 7640 q^{18} - 1792 q^{21} + 1552 q^{22} - 9776 q^{25} - 14672 q^{28} + 26592 q^{29} + 11200 q^{30} + 11648 q^{32} + 31760 q^{36} - 26272 q^{37} + 7616 q^{42} - 12256 q^{44} + 20208 q^{46} + 8848 q^{49} + 5992 q^{50} - 41888 q^{53} + 38304 q^{56} - 60288 q^{57} - 144400 q^{58} - 132992 q^{60} + 45312 q^{64} + 66688 q^{65} + 79296 q^{70} + 44512 q^{72} + 348464 q^{74} + 320992 q^{77} - 344256 q^{78} + 56336 q^{81} - 194432 q^{84} + 78080 q^{85} - 78448 q^{86} - 66112 q^{88} + 446944 q^{92} - 335616 q^{93} + 224840 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 674 x^{14} + 3404 x^{13} + 173721 x^{12} - 919512 x^{11} - 21981508 x^{10} + 94817024 x^{9} + 1652635052 x^{8} - 4373672096 x^{7} + \cdots + 224266997486896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\!\cdots\!41 \nu^{15} + \cdots + 46\!\cdots\!92 ) / 71\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 93\!\cdots\!21 \nu^{15} + \cdots - 62\!\cdots\!76 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 93\!\cdots\!21 \nu^{15} + \cdots - 58\!\cdots\!96 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!19 \nu^{15} + \cdots - 18\!\cdots\!04 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!87 \nu^{15} + \cdots + 85\!\cdots\!28 ) / 71\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!27 \nu^{15} + \cdots - 55\!\cdots\!52 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!19 \nu^{15} + \cdots - 95\!\cdots\!04 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!11 \nu^{15} + \cdots - 25\!\cdots\!96 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20\!\cdots\!64 \nu^{15} + \cdots - 11\!\cdots\!68 ) / 37\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 28\!\cdots\!37 \nu^{15} + \cdots - 82\!\cdots\!52 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 33\!\cdots\!53 \nu^{15} + \cdots + 35\!\cdots\!68 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 61\!\cdots\!75 \nu^{15} + \cdots + 86\!\cdots\!52 ) / 35\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 63\!\cdots\!42 \nu^{15} + \cdots - 17\!\cdots\!28 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 67\!\cdots\!07 \nu^{15} + \cdots - 51\!\cdots\!72 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 92\!\cdots\!09 \nu^{15} + \cdots + 63\!\cdots\!04 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{12} - \beta_{10} + 2 \beta_{7} - \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} - 2 \beta_{2} - 4 \beta _1 + 331 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{15} + 2 \beta_{14} - 8 \beta_{13} + 4 \beta_{12} + 4 \beta_{11} - 14 \beta_{10} + 4 \beta_{9} + 7 \beta_{8} - 35 \beta_{7} + 48 \beta_{6} - 46 \beta_{5} + 26 \beta_{4} - 1008 \beta_{3} + 575 \beta_{2} + 56 \beta _1 - 473 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 64 \beta_{15} - 16 \beta_{14} + 24 \beta_{13} - 389 \beta_{12} + 20 \beta_{11} - 197 \beta_{10} + 440 \beta_{9} - 60 \beta_{8} + 1060 \beta_{7} - 597 \beta_{6} + 2329 \beta_{5} - 944 \beta_{4} + 9383 \beta_{3} - 3100 \beta_{2} + \cdots + 94421 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3276 \beta_{15} + 746 \beta_{14} - 3564 \beta_{13} + 461 \beta_{12} + 732 \beta_{11} - 16051 \beta_{10} - 6750 \beta_{9} + 478 \beta_{8} - 24426 \beta_{7} + 26825 \beta_{6} - 49645 \beta_{5} + \cdots - 1040817 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 27408 \beta_{15} - 1286 \beta_{14} + 3516 \beta_{13} - 58644 \beta_{12} + 9842 \beta_{11} + 53762 \beta_{10} + 173322 \beta_{9} - 16200 \beta_{8} + 272964 \beta_{7} - 175832 \beta_{6} + \cdots + 16536506 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 994386 \beta_{15} + 97504 \beta_{14} - 466370 \beta_{13} - 173379 \beta_{12} - 143694 \beta_{11} - 6313423 \beta_{10} - 4132090 \beta_{9} + 48896 \beta_{8} - 7326348 \beta_{7} + \cdots - 325364495 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4458492 \beta_{15} + 125016 \beta_{14} - 387120 \beta_{13} - 2508560 \beta_{12} + 1968504 \beta_{11} + 20295682 \beta_{10} + 28558992 \beta_{9} - 1708260 \beta_{8} + \cdots + 1522190365 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 146781779 \beta_{15} + 47246 \beta_{14} + 2493160 \beta_{13} - 81824902 \beta_{12} - 58256618 \beta_{11} - 1086989293 \beta_{10} - 823255652 \beta_{9} + \cdots - 36505630337 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2607280116 \beta_{15} + 201519766 \beta_{14} - 826376492 \beta_{13} + 1073823469 \beta_{12} + 1422038670 \beta_{11} + 16817530523 \beta_{10} + 17618120222 \beta_{9} + \cdots + 518623160875 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 83490538352 \beta_{15} - 6391578346 \beta_{14} + 32231694568 \beta_{13} - 86420623275 \beta_{12} - 48369260454 \beta_{11} - 698198497995 \beta_{10} + \cdots - 12370219291491 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 358178757498 \beta_{15} + 44814331584 \beta_{14} - 202874410600 \beta_{13} + 414987622535 \beta_{12} + 237626189708 \beta_{11} + 2877019856637 \beta_{10} + \cdots + 31729488258250 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 11339647088759 \beta_{15} - 1612797452186 \beta_{14} + 7911692216368 \beta_{13} - 17904084161658 \beta_{12} - 8276600843780 \beta_{11} + \cdots - 494319932192243 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 187342485511976 \beta_{15} + 33290093313390 \beta_{14} - 157186917116908 \beta_{13} + 343507344114547 \beta_{12} + 148796550614158 \beta_{11} + \cdots - 31\!\cdots\!19 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 58\!\cdots\!12 \beta_{15} + \cdots + 39\!\cdots\!47 ) / 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
11.9248 1.80141i
−16.2859 1.80141i
11.9248 + 1.80141i
−16.2859 + 1.80141i
2.37022 2.69664i
−4.07684 2.69664i
2.37022 + 2.69664i
−4.07684 + 2.69664i
9.86747 2.48366i
−7.16085 2.48366i
9.86747 + 2.48366i
−7.16085 + 2.48366i
10.5645 0.902498i
−5.20336 0.902498i
10.5645 + 0.902498i
−5.20336 + 0.902498i
−4.36116 3.60282i −28.2107 6.03935 + 31.4249i 57.4402i 123.031 + 101.638i 15.0189 + 128.769i 86.8799 158.808i 552.846 −206.947 + 250.506i
27.2 −4.36116 3.60282i 28.2107 6.03935 + 31.4249i 57.4402i −123.031 101.638i −15.0189 + 128.769i 86.8799 158.808i 552.846 206.947 250.506i
27.3 −4.36116 + 3.60282i −28.2107 6.03935 31.4249i 57.4402i 123.031 101.638i 15.0189 128.769i 86.8799 + 158.808i 552.846 −206.947 250.506i
27.4 −4.36116 + 3.60282i 28.2107 6.03935 31.4249i 57.4402i −123.031 + 101.638i −15.0189 128.769i 86.8799 + 158.808i 552.846 206.947 + 250.506i
27.5 −1.70662 5.39328i −6.44707 −26.1749 + 18.4085i 76.3023i 11.0027 + 34.7708i 120.438 47.9752i 143.953 + 109.752i −201.435 411.519 130.219i
27.6 −1.70662 5.39328i 6.44707 −26.1749 + 18.4085i 76.3023i −11.0027 34.7708i −120.438 47.9752i 143.953 + 109.752i −201.435 −411.519 + 130.219i
27.7 −1.70662 + 5.39328i −6.44707 −26.1749 18.4085i 76.3023i 11.0027 34.7708i 120.438 + 47.9752i 143.953 109.752i −201.435 411.519 + 130.219i
27.8 −1.70662 + 5.39328i 6.44707 −26.1749 18.4085i 76.3023i −11.0027 + 34.7708i −120.438 + 47.9752i 143.953 109.752i −201.435 −411.519 130.219i
27.9 2.70662 4.96731i −17.0283 −17.3484 26.8893i 28.3847i −46.0892 + 84.5850i −117.282 + 55.2436i −180.523 + 13.3961i 46.9637 140.996 + 76.8266i
27.10 2.70662 4.96731i 17.0283 −17.3484 26.8893i 28.3847i 46.0892 84.5850i 117.282 + 55.2436i −180.523 + 13.3961i 46.9637 −140.996 76.8266i
27.11 2.70662 + 4.96731i −17.0283 −17.3484 + 26.8893i 28.3847i −46.0892 84.5850i −117.282 55.2436i −180.523 13.3961i 46.9637 140.996 76.8266i
27.12 2.70662 + 4.96731i 17.0283 −17.3484 + 26.8893i 28.3847i 46.0892 + 84.5850i 117.282 55.2436i −180.523 13.3961i 46.9637 −140.996 + 76.8266i
27.13 5.36116 1.80500i −15.7679 25.4840 19.3537i 70.8301i −84.5340 + 28.4610i 78.9551 102.826i 101.690 149.757i 5.62572 −127.848 379.731i
27.14 5.36116 1.80500i 15.7679 25.4840 19.3537i 70.8301i 84.5340 28.4610i −78.9551 102.826i 101.690 149.757i 5.62572 127.848 + 379.731i
27.15 5.36116 + 1.80500i −15.7679 25.4840 + 19.3537i 70.8301i −84.5340 28.4610i 78.9551 + 102.826i 101.690 + 149.757i 5.62572 −127.848 + 379.731i
27.16 5.36116 + 1.80500i 15.7679 25.4840 + 19.3537i 70.8301i 84.5340 + 28.4610i −78.9551 + 102.826i 101.690 + 149.757i 5.62572 127.848 379.731i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.6.d.b 16
3.b odd 2 1 252.6.b.d 16
4.b odd 2 1 inner 28.6.d.b 16
7.b odd 2 1 inner 28.6.d.b 16
8.b even 2 1 448.6.f.d 16
8.d odd 2 1 448.6.f.d 16
12.b even 2 1 252.6.b.d 16
21.c even 2 1 252.6.b.d 16
28.d even 2 1 inner 28.6.d.b 16
56.e even 2 1 448.6.f.d 16
56.h odd 2 1 448.6.f.d 16
84.h odd 2 1 252.6.b.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.d.b 16 1.a even 1 1 trivial
28.6.d.b 16 4.b odd 2 1 inner
28.6.d.b 16 7.b odd 2 1 inner
28.6.d.b 16 28.d even 2 1 inner
252.6.b.d 16 3.b odd 2 1
252.6.b.d 16 12.b even 2 1
252.6.b.d 16 21.c even 2 1
252.6.b.d 16 84.h odd 2 1
448.6.f.d 16 8.b even 2 1
448.6.f.d 16 8.d odd 2 1
448.6.f.d 16 56.e even 2 1
448.6.f.d 16 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 1376T_{3}^{6} + 556192T_{3}^{4} - 78187008T_{3}^{2} + 2384750592 \) acting on \(S_{6}^{\mathrm{new}}(28, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 4 T^{7} + 20 T^{6} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - 1376 T^{6} + \cdots + 2384750592)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 14944 T^{6} + \cdots + 77644413167616)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 4424 T^{14} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{8} + 289072 T^{6} + \cdots + 16\!\cdots\!48)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 1669600 T^{6} + \cdots + 34\!\cdots\!24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 7386624 T^{6} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 14532320 T^{6} + \cdots + 43\!\cdots\!08)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 9935536 T^{6} + \cdots + 27\!\cdots\!08)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 6648 T^{3} + \cdots - 47450817355248)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 101160448 T^{6} + \cdots + 19\!\cdots\!72)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 6568 T^{3} + \cdots - 39\!\cdots\!84)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 721188736 T^{6} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 313622576 T^{6} + \cdots + 22\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 633042432 T^{6} + \cdots + 51\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 10472 T^{3} + \cdots + 85\!\cdots\!76)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 534584288 T^{6} + \cdots + 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 2196041824 T^{6} + \cdots + 75\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 9454705072 T^{6} + \cdots + 51\!\cdots\!92)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 6241223408 T^{6} + \cdots + 50\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 7874786432 T^{6} + \cdots + 22\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 12613036912 T^{6} + \cdots + 63\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 11901939552 T^{6} + \cdots + 62\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 18068143744 T^{6} + \cdots + 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 35259705856 T^{6} + \cdots + 31\!\cdots\!84)^{2} \) Copy content Toggle raw display
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