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Properties

Label 36.3.f.c
Level $36$
Weight $3$
Character orbit 36.f
Analytic conductor $0.981$
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] = [36,3,Mod(7,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.f (of order \(6\), degree \(2\), 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: \(0.980928951697\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{14} q^{3} + (\beta_{3} - \beta_{2}) q^{4} + (\beta_{13} - \beta_{9} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - \beta_{14} - \beta_{13} + \beta_{12} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{14} q^{3} + (\beta_{3} - \beta_{2}) q^{4} + (\beta_{13} - \beta_{9} + \cdots + \beta_{2}) q^{5}+ \cdots + (4 \beta_{15} - 12 \beta_{14} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} - 5 q^{4} + 6 q^{5} + 9 q^{6} - 54 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{2} - 5 q^{4} + 6 q^{5} + 9 q^{6} - 54 q^{8} + 18 q^{9} + 20 q^{10} - 36 q^{12} - 46 q^{13} - 12 q^{14} - 17 q^{16} + 12 q^{17} + 48 q^{18} + 36 q^{20} - 66 q^{21} + 33 q^{22} + 129 q^{24} - 30 q^{25} + 72 q^{26} + 12 q^{28} + 42 q^{29} + 84 q^{30} + 87 q^{32} - 168 q^{33} + 11 q^{34} - 81 q^{36} + 56 q^{37} - 99 q^{38} + 68 q^{40} + 84 q^{41} - 354 q^{42} - 222 q^{44} + 174 q^{45} - 264 q^{46} - 189 q^{48} + 58 q^{49} - 219 q^{50} + 110 q^{52} - 72 q^{53} - 105 q^{54} + 270 q^{56} + 366 q^{57} - 16 q^{58} + 432 q^{60} - 34 q^{61} + 516 q^{62} - 254 q^{64} - 30 q^{65} + 510 q^{66} + 375 q^{68} - 54 q^{69} + 150 q^{70} - 45 q^{72} + 116 q^{73} - 372 q^{74} - 15 q^{76} - 330 q^{77} - 294 q^{78} - 720 q^{80} - 102 q^{81} + 254 q^{82} - 714 q^{84} - 140 q^{85} - 273 q^{86} + 75 q^{88} - 384 q^{89} + 108 q^{90} + 258 q^{92} - 486 q^{93} + 36 q^{94} + 900 q^{96} - 148 q^{97} + 1170 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + 17 \nu^{14} + 83 \nu^{13} - 394 \nu^{12} + 204 \nu^{11} - 2224 \nu^{10} + 6280 \nu^{9} + \cdots - 1228800 ) / 540672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 17 \nu^{14} + 83 \nu^{13} - 394 \nu^{12} + 204 \nu^{11} - 2224 \nu^{10} + 6280 \nu^{9} + \cdots - 1228800 ) / 540672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{15} + 19 \nu^{14} - 91 \nu^{13} + 32 \nu^{12} - 520 \nu^{11} + 1464 \nu^{10} - 1688 \nu^{9} + \cdots - 16384 ) / 135168 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 3 \nu^{14} - 5 \nu^{13} + 22 \nu^{12} + 8 \nu^{11} + 72 \nu^{10} - 328 \nu^{9} + \cdots + 122880 ) / 24576 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} + 5 \nu^{14} + 15 \nu^{13} - 8 \nu^{12} - 36 \nu^{11} - 128 \nu^{10} + 136 \nu^{9} + \cdots - 40960 ) / 24576 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7 \nu^{15} - 97 \nu^{14} + 211 \nu^{13} - 366 \nu^{12} + 1454 \nu^{11} - 3572 \nu^{10} + \cdots - 679936 ) / 135168 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25 \nu^{15} - 7 \nu^{14} + 15 \nu^{13} - 72 \nu^{12} + 136 \nu^{11} - 280 \nu^{10} + \cdots - 819200 ) / 270336 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 61 \nu^{15} + 371 \nu^{14} - 839 \nu^{13} + 1506 \nu^{12} - 5404 \nu^{11} + 13168 \nu^{10} + \cdots - 16384 ) / 540672 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 63 \nu^{15} + 73 \nu^{14} + 51 \nu^{13} + 358 \nu^{12} - 972 \nu^{11} - 1216 \nu^{10} + \cdots - 1523712 ) / 540672 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{15} - 23 \nu^{14} + 11 \nu^{13} + 118 \nu^{12} + 156 \nu^{11} - 176 \nu^{10} + \cdots + 393216 ) / 49152 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25 \nu^{15} - 37 \nu^{14} + 95 \nu^{13} - 610 \nu^{12} + 942 \nu^{11} - 1612 \nu^{10} + \cdots - 712704 ) / 135168 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26 \nu^{15} - 119 \nu^{14} + 365 \nu^{13} - 1015 \nu^{12} + 2642 \nu^{11} - 6300 \nu^{10} + \cdots - 1220608 ) / 135168 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 113 \nu^{15} - 147 \nu^{14} + 535 \nu^{13} - 2590 \nu^{12} + 3692 \nu^{11} - 7728 \nu^{10} + \cdots - 3866624 ) / 540672 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 137 \nu^{15} - 487 \nu^{14} + 811 \nu^{13} - 3994 \nu^{12} + 8940 \nu^{11} - 17456 \nu^{10} + \cdots - 3080192 ) / 540672 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{14} - \beta_{13} + 2\beta_{10} - \beta_{6} + \beta_{5} - 2\beta_{4} - 2\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{15} + 3\beta_{12} + 2\beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} + 4\beta_{2} + 3\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5 \beta_{14} + \beta_{13} + 5 \beta_{12} + 6 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} + \cdots - 12 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{15} + 5 \beta_{14} - 11 \beta_{13} - 2 \beta_{11} + 5 \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5 \beta_{15} + 2 \beta_{14} + 11 \beta_{12} + 22 \beta_{11} - 17 \beta_{10} - 11 \beta_{7} + \cdots - 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 23 \beta_{14} + 35 \beta_{13} - 13 \beta_{12} + 30 \beta_{10} - 22 \beta_{9} + 13 \beta_{8} + \cdots - 64 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5 \beta_{15} - 97 \beta_{14} - 5 \beta_{13} - 70 \beta_{11} - 97 \beta_{10} - 70 \beta_{9} + 85 \beta_{8} + \cdots - 80 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 83 \beta_{15} - 18 \beta_{14} + 61 \beta_{12} + 10 \beta_{11} - 247 \beta_{10} + 67 \beta_{7} + \cdots + 176 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 95 \beta_{14} + 245 \beta_{13} - 171 \beta_{12} + 98 \beta_{10} + 134 \beta_{9} + 171 \beta_{8} + \cdots - 624 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 637 \beta_{15} - 359 \beta_{14} + 365 \beta_{13} - 490 \beta_{11} - 359 \beta_{10} - 490 \beta_{9} + \cdots - 944 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 949 \beta_{15} + 706 \beta_{14} + 187 \beta_{12} - 730 \beta_{11} - 337 \beta_{10} + 1253 \beta_{7} + \cdots + 3408 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1111 \beta_{14} + 739 \beta_{13} + 2179 \beta_{12} + 654 \beta_{10} + 2506 \beta_{9} + \cdots - 208 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2459 \beta_{15} + 1439 \beta_{14} + 1131 \beta_{13} - 1478 \beta_{11} + 1439 \beta_{10} + \cdots - 13136 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.93353 − 0.511345i
1.84233 + 0.778342i
1.63139 − 1.15696i
0.186266 − 1.99131i
−0.523926 − 1.93016i
−0.710719 + 1.86946i
−1.26364 + 1.55023i
−1.59523 − 1.20633i
1.93353 + 0.511345i
1.84233 − 0.778342i
1.63139 + 1.15696i
0.186266 + 1.99131i
−0.523926 + 1.93016i
−0.710719 − 1.86946i
−1.26364 − 1.55023i
−1.59523 + 1.20633i
−1.93353 + 0.511345i −2.76570 − 1.16229i 3.47705 − 1.97740i −4.03104 − 6.98197i 5.94188 + 0.833101i −3.90254 − 2.25313i −5.71184 + 5.60133i 6.29815 + 6.42910i 11.3643 + 11.4386i
7.2 −1.84233 − 0.778342i 0.262217 + 2.98852i 2.78837 + 2.86793i 1.10093 + 1.90686i 1.84300 − 5.70994i 7.23844 + 4.17912i −2.90487 − 7.45397i −8.86248 + 1.56728i −0.544081 − 4.36996i
7.3 −1.63139 + 1.15696i 2.67178 − 1.36441i 1.32286 − 3.77492i 3.07403 + 5.32438i −2.78013 + 5.31703i −0.511543 − 0.295340i 2.20934 + 7.68888i 5.27677 − 7.29079i −11.1751 − 5.12959i
7.4 −0.186266 + 1.99131i −2.67178 + 1.36441i −3.93061 − 0.741826i 3.07403 + 5.32438i −2.21930 − 5.57447i 0.511543 + 0.295340i 2.20934 − 7.68888i 5.27677 − 7.29079i −11.1751 + 5.12959i
7.5 0.523926 + 1.93016i 2.76570 + 1.16229i −3.45100 + 2.02252i −4.03104 − 6.98197i −0.794388 + 5.94718i 3.90254 + 2.25313i −5.71184 − 5.60133i 6.29815 + 6.42910i 11.3643 − 11.4386i
7.6 0.710719 − 1.86946i 2.32245 + 1.89900i −2.98976 − 2.65732i 1.35609 + 2.34881i 5.20072 − 2.99207i −10.0431 − 5.79837i −7.09263 + 3.70062i 1.78756 + 8.82069i 5.35481 − 0.865806i
7.7 1.26364 − 1.55023i −2.32245 − 1.89900i −0.806428 − 3.91787i 1.35609 + 2.34881i −5.87864 + 1.20068i 10.0431 + 5.79837i −7.09263 − 3.70062i 1.78756 + 8.82069i 5.35481 + 0.865806i
7.8 1.59523 + 1.20633i −0.262217 − 2.98852i 1.08951 + 3.84876i 1.10093 + 1.90686i 3.18686 − 5.08369i −7.23844 − 4.17912i −2.90487 + 7.45397i −8.86248 + 1.56728i −0.544081 + 4.36996i
31.1 −1.93353 − 0.511345i −2.76570 + 1.16229i 3.47705 + 1.97740i −4.03104 + 6.98197i 5.94188 − 0.833101i −3.90254 + 2.25313i −5.71184 − 5.60133i 6.29815 − 6.42910i 11.3643 − 11.4386i
31.2 −1.84233 + 0.778342i 0.262217 − 2.98852i 2.78837 − 2.86793i 1.10093 − 1.90686i 1.84300 + 5.70994i 7.23844 − 4.17912i −2.90487 + 7.45397i −8.86248 − 1.56728i −0.544081 + 4.36996i
31.3 −1.63139 − 1.15696i 2.67178 + 1.36441i 1.32286 + 3.77492i 3.07403 − 5.32438i −2.78013 − 5.31703i −0.511543 + 0.295340i 2.20934 − 7.68888i 5.27677 + 7.29079i −11.1751 + 5.12959i
31.4 −0.186266 − 1.99131i −2.67178 − 1.36441i −3.93061 + 0.741826i 3.07403 − 5.32438i −2.21930 + 5.57447i 0.511543 − 0.295340i 2.20934 + 7.68888i 5.27677 + 7.29079i −11.1751 − 5.12959i
31.5 0.523926 − 1.93016i 2.76570 − 1.16229i −3.45100 − 2.02252i −4.03104 + 6.98197i −0.794388 − 5.94718i 3.90254 − 2.25313i −5.71184 + 5.60133i 6.29815 − 6.42910i 11.3643 + 11.4386i
31.6 0.710719 + 1.86946i 2.32245 − 1.89900i −2.98976 + 2.65732i 1.35609 − 2.34881i 5.20072 + 2.99207i −10.0431 + 5.79837i −7.09263 − 3.70062i 1.78756 − 8.82069i 5.35481 + 0.865806i
31.7 1.26364 + 1.55023i −2.32245 + 1.89900i −0.806428 + 3.91787i 1.35609 − 2.34881i −5.87864 − 1.20068i 10.0431 − 5.79837i −7.09263 + 3.70062i 1.78756 − 8.82069i 5.35481 − 0.865806i
31.8 1.59523 − 1.20633i −0.262217 + 2.98852i 1.08951 − 3.84876i 1.10093 − 1.90686i 3.18686 + 5.08369i −7.23844 + 4.17912i −2.90487 − 7.45397i −8.86248 − 1.56728i −0.544081 − 4.36996i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.3.f.c ✓ 16
3.b odd 2 1 108.3.f.c 16
4.b odd 2 1 inner 36.3.f.c ✓ 16
8.b even 2 1 576.3.o.g 16
8.d odd 2 1 576.3.o.g 16
9.c even 3 1 inner 36.3.f.c ✓ 16
9.c even 3 1 324.3.d.i 8
9.d odd 6 1 108.3.f.c 16
9.d odd 6 1 324.3.d.g 8
12.b even 2 1 108.3.f.c 16
24.f even 2 1 1728.3.o.g 16
24.h odd 2 1 1728.3.o.g 16
36.f odd 6 1 inner 36.3.f.c ✓ 16
36.f odd 6 1 324.3.d.i 8
36.h even 6 1 108.3.f.c 16
36.h even 6 1 324.3.d.g 8
72.j odd 6 1 1728.3.o.g 16
72.l even 6 1 1728.3.o.g 16
72.n even 6 1 576.3.o.g 16
72.p odd 6 1 576.3.o.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.c ✓ 16 1.a even 1 1 trivial
36.3.f.c ✓ 16 4.b odd 2 1 inner
36.3.f.c ✓ 16 9.c even 3 1 inner
36.3.f.c ✓ 16 36.f odd 6 1 inner
108.3.f.c 16 3.b odd 2 1
108.3.f.c 16 9.d odd 6 1
108.3.f.c 16 12.b even 2 1
108.3.f.c 16 36.h even 6 1
324.3.d.g 8 9.d odd 6 1
324.3.d.g 8 36.h even 6 1
324.3.d.i 8 9.c even 3 1
324.3.d.i 8 36.f odd 6 1
576.3.o.g 16 8.b even 2 1
576.3.o.g 16 8.d odd 2 1
576.3.o.g 16 72.n even 6 1
576.3.o.g 16 72.p odd 6 1
1728.3.o.g 16 24.f even 2 1
1728.3.o.g 16 24.h odd 2 1
1728.3.o.g 16 72.j odd 6 1
1728.3.o.g 16 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(36, [\chi])\):

\( T_{5}^{8} - 3T_{5}^{7} + 62T_{5}^{6} - 351T_{5}^{5} + 3870T_{5}^{4} - 15291T_{5}^{3} + 49337T_{5}^{2} - 75480T_{5} + 87616 \) Copy content Toggle raw display
\( T_{7}^{16} - 225 T_{7}^{14} + 37002 T_{7}^{12} - 2674161 T_{7}^{10} + 141530490 T_{7}^{8} + \cdots + 4430766096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 3 T^{15} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( T^{16} - 9 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{8} - 3 T^{7} + \cdots + 87616)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 4430766096 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 134421415700625 \) Copy content Toggle raw display
$13$ \( (T^{8} + 23 T^{7} + \cdots + 12100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 3 T^{3} + \cdots + 2200)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1215 T^{6} + \cdots + 7464960000)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{8} - 21 T^{7} + \cdots + 3658072324)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} - 14 T^{3} + \cdots + 5920)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 42 T^{7} + \cdots + 12391919761)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{4} + 18 T^{3} + \cdots - 16160)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 933144135518464)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 29 T^{3} + \cdots + 20112040)^{4} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + 96 T^{3} + \cdots - 4957424)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 74 T^{7} + \cdots + 4182855625)^{2} \) Copy content Toggle raw display
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