↑

Properties

Label 72.2.l.b
Level $72$
Weight $2$
Character orbit 72.l
Analytic conductor $0.575$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,2,Mod(11,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.l (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: \(0.574922894553\)
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: \( x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + (\beta_{6} - \beta_{3}) q^{3} + (\beta_{15} + 2 \beta_{11} - \beta_{9} + \cdots - 1) q^{4}+ \cdots + (2 \beta_{15} - \beta_{13} + 2 \beta_{11} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + (\beta_{6} - \beta_{3}) q^{3} + (\beta_{15} + 2 \beta_{11} - \beta_{9} + \cdots - 1) q^{4}+ \cdots + (2 \beta_{15} - \beta_{13} + 2 \beta_{11} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} - 6 q^{3} - 5 q^{4} - 3 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{2} - 6 q^{3} - 5 q^{4} - 3 q^{6} - 6 q^{9} + 12 q^{11} - 6 q^{12} - 18 q^{14} + 7 q^{16} - 4 q^{19} + 18 q^{20} - q^{22} + 21 q^{24} - 14 q^{25} - 36 q^{27} - 12 q^{28} + 12 q^{30} + 27 q^{32} + 12 q^{33} - 13 q^{34} + 27 q^{36} - 15 q^{38} - 12 q^{40} - 36 q^{41} + 42 q^{42} + 8 q^{43} + 12 q^{46} - 27 q^{48} + 10 q^{49} + 51 q^{50} + 18 q^{51} - 18 q^{52} + 39 q^{54} - 66 q^{56} + 18 q^{57} + 12 q^{58} + 12 q^{59} - 72 q^{60} + 34 q^{64} - 6 q^{65} - 24 q^{66} - 16 q^{67} - 9 q^{68} + 18 q^{70} - 21 q^{72} - 4 q^{73} - 60 q^{74} + 78 q^{75} - 7 q^{76} - 72 q^{78} - 6 q^{81} - 22 q^{82} + 54 q^{83} + 12 q^{84} - 51 q^{86} - 13 q^{88} - 66 q^{90} - 36 q^{91} + 84 q^{92} + 24 q^{94} + 42 q^{96} + 8 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + \cdots + 1280 ) / 896 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + \nu^{14} + 25 \nu^{13} - 38 \nu^{12} + 46 \nu^{11} - 24 \nu^{10} + 8 \nu^{9} + 68 \nu^{8} + \cdots - 512 ) / 896 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{15} + 6 \nu^{14} - 11 \nu^{13} + 24 \nu^{12} - 39 \nu^{11} + 10 \nu^{10} + 20 \nu^{9} + \cdots + 512 ) / 448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{15} - 11 \nu^{14} + 47 \nu^{13} - 58 \nu^{12} + 40 \nu^{11} + 40 \nu^{10} - 144 \nu^{9} + \cdots + 1152 ) / 896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{15} + 23 \nu^{14} - 13 \nu^{13} - 6 \nu^{12} + 50 \nu^{11} - 132 \nu^{10} + 240 \nu^{9} + \cdots - 1920 ) / 896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3 \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 2 \nu^{12} + 23 \nu^{11} - 40 \nu^{10} + 32 \nu^{9} + \cdots + 192 ) / 448 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{15} + 3 \nu^{14} - 9 \nu^{13} + 26 \nu^{12} - 30 \nu^{11} + 12 \nu^{10} + 52 \nu^{9} + \cdots - 192 ) / 448 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4 \nu^{15} + 17 \nu^{14} - 37 \nu^{13} + 19 \nu^{12} + 12 \nu^{11} - 86 \nu^{10} + 192 \nu^{9} + \cdots - 1088 ) / 448 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5 \nu^{15} - 9 \nu^{14} + 27 \nu^{13} - 36 \nu^{12} + 34 \nu^{11} - 8 \nu^{10} - 72 \nu^{9} + \cdots - 320 ) / 448 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5 \nu^{15} - 2 \nu^{14} - 8 \nu^{13} + 27 \nu^{12} - 36 \nu^{11} + 48 \nu^{10} - 16 \nu^{9} + \cdots + 128 ) / 448 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 6 \nu^{15} + 15 \nu^{14} - 24 \nu^{13} + 18 \nu^{12} - 3 \nu^{11} - 66 \nu^{10} + 120 \nu^{9} + \cdots + 384 ) / 448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5 \nu^{15} - 16 \nu^{14} + 48 \nu^{13} - 71 \nu^{12} + 76 \nu^{11} - 22 \nu^{10} - 100 \nu^{9} + \cdots - 768 ) / 448 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3 \nu^{14} - 7 \nu^{13} + 15 \nu^{12} - 22 \nu^{11} + 16 \nu^{10} + 4 \nu^{9} - 56 \nu^{8} + 92 \nu^{7} + \cdots + 384 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3 \nu^{15} - 18 \nu^{14} + 47 \nu^{13} - 100 \nu^{12} + 117 \nu^{11} - 100 \nu^{10} - 60 \nu^{9} + \cdots - 1536 ) / 448 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{8} - \beta_{6} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{12} - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + \beta_{7} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} - \beta_{14} - 6 \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{6} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{15} - \beta_{14} - 2 \beta_{13} - 3 \beta_{12} + 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 4 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2 \beta_{15} - 3 \beta_{14} - 4 \beta_{11} - 3 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 3 \beta_{6} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - \beta_{15} - 3 \beta_{14} - 6 \beta_{13} - 7 \beta_{12} - 7 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 17 \beta_{15} - 4 \beta_{14} + 18 \beta_{13} + 7 \beta_{12} - 8 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8 \beta_{15} - 11 \beta_{14} - 24 \beta_{13} + 18 \beta_{12} + 40 \beta_{11} - 7 \beta_{10} - 34 \beta_{9} + \cdots - 22 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5 \beta_{15} - 13 \beta_{14} - 6 \beta_{13} + \beta_{12} - \beta_{10} - 18 \beta_{9} + 18 \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 17 \beta_{15} + 18 \beta_{14} - 34 \beta_{13} - 43 \beta_{12} + 16 \beta_{11} + 18 \beta_{10} + \cdots - 25 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 50 \beta_{15} + 45 \beta_{14} - 20 \beta_{13} - 20 \beta_{12} + 48 \beta_{11} + 25 \beta_{10} + 6 \beta_{9} + \cdots + 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.608741 − 1.27649i
−0.186766 − 1.40183i
−0.533474 − 1.30973i
1.40985 + 0.111062i
1.12063 + 0.862658i
−1.37702 − 0.322193i
0.867527 + 1.11687i
−0.409484 + 1.35363i
0.608741 + 1.27649i
−0.186766 + 1.40183i
−0.533474 + 1.30973i
1.40985 − 0.111062i
1.12063 − 0.862658i
−1.37702 + 0.322193i
0.867527 − 1.11687i
−0.409484 − 1.35363i
−1.40985 + 0.111062i −1.71646 − 0.231865i 1.97533 − 0.313160i 1.74322 + 3.01934i 2.44570 + 0.136260i 1.80802 + 1.04386i −2.75013 + 0.660890i 2.89248 + 0.795973i −2.79300 − 4.06320i
11.2 −1.12063 + 0.862658i 0.418594 − 1.68071i 0.511643 − 1.93345i −1.60936 − 2.78750i 0.980785 + 2.24456i 1.82223 + 1.05206i 1.09454 + 2.60806i −2.64956 − 1.40707i 4.20817 + 1.73544i
11.3 −0.867527 + 1.11687i 0.925606 + 1.46399i −0.494795 − 1.93783i 0.895377 + 1.55084i −2.43807 − 0.236266i −2.08793 − 1.20546i 2.59355 + 1.12850i −1.28651 + 2.71015i −2.50885 − 0.345375i
11.4 −0.608741 − 1.27649i −1.71646 − 0.231865i −1.25887 + 1.55411i −1.74322 − 3.01934i 0.748906 + 2.33220i −1.80802 − 1.04386i 2.75013 + 0.660890i 2.89248 + 0.795973i −2.79300 + 4.06320i
11.5 0.186766 − 1.40183i 0.418594 − 1.68071i −1.93024 − 0.523628i 1.60936 + 2.78750i −2.27788 − 0.900696i −1.82223 − 1.05206i −1.09454 + 2.60806i −2.64956 − 1.40707i 4.20817 − 1.73544i
11.6 0.409484 + 1.35363i −1.12774 + 1.31461i −1.66465 + 1.10858i −0.565188 − 0.978934i −2.24129 − 0.988231i 3.71499 + 2.14485i −2.18226 − 1.79937i −0.456412 − 2.96508i 1.09368 − 1.16591i
11.7 0.533474 − 1.30973i 0.925606 + 1.46399i −1.43081 − 1.39742i −0.895377 − 1.55084i 2.41122 − 0.431300i 2.08793 + 1.20546i −2.59355 + 1.12850i −1.28651 + 2.71015i −2.50885 + 0.345375i
11.8 1.37702 − 0.322193i −1.12774 + 1.31461i 1.79238 − 0.887333i 0.565188 + 0.978934i −1.12936 + 2.17360i −3.71499 − 2.14485i 2.18226 − 1.79937i −0.456412 − 2.96508i 1.09368 + 1.16591i
59.1 −1.40985 − 0.111062i −1.71646 + 0.231865i 1.97533 + 0.313160i 1.74322 − 3.01934i 2.44570 − 0.136260i 1.80802 − 1.04386i −2.75013 − 0.660890i 2.89248 − 0.795973i −2.79300 + 4.06320i
59.2 −1.12063 − 0.862658i 0.418594 + 1.68071i 0.511643 + 1.93345i −1.60936 + 2.78750i 0.980785 − 2.24456i 1.82223 − 1.05206i 1.09454 − 2.60806i −2.64956 + 1.40707i 4.20817 − 1.73544i
59.3 −0.867527 − 1.11687i 0.925606 − 1.46399i −0.494795 + 1.93783i 0.895377 − 1.55084i −2.43807 + 0.236266i −2.08793 + 1.20546i 2.59355 − 1.12850i −1.28651 − 2.71015i −2.50885 + 0.345375i
59.4 −0.608741 + 1.27649i −1.71646 + 0.231865i −1.25887 − 1.55411i −1.74322 + 3.01934i 0.748906 − 2.33220i −1.80802 + 1.04386i 2.75013 − 0.660890i 2.89248 − 0.795973i −2.79300 − 4.06320i
59.5 0.186766 + 1.40183i 0.418594 + 1.68071i −1.93024 + 0.523628i 1.60936 − 2.78750i −2.27788 + 0.900696i −1.82223 + 1.05206i −1.09454 − 2.60806i −2.64956 + 1.40707i 4.20817 + 1.73544i
59.6 0.409484 − 1.35363i −1.12774 − 1.31461i −1.66465 − 1.10858i −0.565188 + 0.978934i −2.24129 + 0.988231i 3.71499 − 2.14485i −2.18226 + 1.79937i −0.456412 + 2.96508i 1.09368 + 1.16591i
59.7 0.533474 + 1.30973i 0.925606 − 1.46399i −1.43081 + 1.39742i −0.895377 + 1.55084i 2.41122 + 0.431300i 2.08793 − 1.20546i −2.59355 − 1.12850i −1.28651 − 2.71015i −2.50885 − 0.345375i
59.8 1.37702 + 0.322193i −1.12774 − 1.31461i 1.79238 + 0.887333i 0.565188 − 0.978934i −1.12936 − 2.17360i −3.71499 + 2.14485i 2.18226 + 1.79937i −0.456412 + 2.96508i 1.09368 − 1.16591i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.l.b ✓ 16
3.b odd 2 1 216.2.l.b 16
4.b odd 2 1 288.2.p.b 16
8.b even 2 1 288.2.p.b 16
8.d odd 2 1 inner 72.2.l.b ✓ 16
9.c even 3 1 216.2.l.b 16
9.c even 3 1 648.2.f.b 16
9.d odd 6 1 inner 72.2.l.b ✓ 16
9.d odd 6 1 648.2.f.b 16
12.b even 2 1 864.2.p.b 16
24.f even 2 1 216.2.l.b 16
24.h odd 2 1 864.2.p.b 16
36.f odd 6 1 864.2.p.b 16
36.f odd 6 1 2592.2.f.b 16
36.h even 6 1 288.2.p.b 16
36.h even 6 1 2592.2.f.b 16
72.j odd 6 1 288.2.p.b 16
72.j odd 6 1 2592.2.f.b 16
72.l even 6 1 inner 72.2.l.b ✓ 16
72.l even 6 1 648.2.f.b 16
72.n even 6 1 864.2.p.b 16
72.n even 6 1 2592.2.f.b 16
72.p odd 6 1 216.2.l.b 16
72.p odd 6 1 648.2.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.b ✓ 16 1.a even 1 1 trivial
72.2.l.b ✓ 16 8.d odd 2 1 inner
72.2.l.b ✓ 16 9.d odd 6 1 inner
72.2.l.b ✓ 16 72.l even 6 1 inner
216.2.l.b 16 3.b odd 2 1
216.2.l.b 16 9.c even 3 1
216.2.l.b 16 24.f even 2 1
216.2.l.b 16 72.p odd 6 1
288.2.p.b 16 4.b odd 2 1
288.2.p.b 16 8.b even 2 1
288.2.p.b 16 36.h even 6 1
288.2.p.b 16 72.j odd 6 1
648.2.f.b 16 9.c even 3 1
648.2.f.b 16 9.d odd 6 1
648.2.f.b 16 72.l even 6 1
648.2.f.b 16 72.p odd 6 1
864.2.p.b 16 12.b even 2 1
864.2.p.b 16 24.h odd 2 1
864.2.p.b 16 36.f odd 6 1
864.2.p.b 16 72.n even 6 1
2592.2.f.b 16 36.f odd 6 1
2592.2.f.b 16 36.h even 6 1
2592.2.f.b 16 72.j odd 6 1
2592.2.f.b 16 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 27 T_{5}^{14} + 498 T_{5}^{12} + 4923 T_{5}^{10} + 35106 T_{5}^{8} + 123903 T_{5}^{6} + \cdots + 266256 \) acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 3 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{8} + 3 T^{7} + 6 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 27 T^{14} + \cdots + 266256 \) Copy content Toggle raw display
$7$ \( T^{16} - 33 T^{14} + \cdots + 4260096 \) Copy content Toggle raw display
$11$ \( (T^{8} - 6 T^{7} + 8 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 57 T^{14} + \cdots + 266256 \) Copy content Toggle raw display
$17$ \( (T^{8} + 35 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} - 12 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 639280656 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 17449353216 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 279189651456 \) Copy content Toggle raw display
$37$ \( (T^{8} + 156 T^{6} + \cdots + 74304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 18 T^{7} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 4 T^{7} + \cdots + 6889)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 111 T^{14} + \cdots + 266256 \) Copy content Toggle raw display
$53$ \( (T^{8} - 228 T^{6} + \cdots + 297216)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 6 T^{7} + \cdots + 528529)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1192149524736 \) Copy content Toggle raw display
$67$ \( (T^{8} + 8 T^{7} + \cdots + 582169)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 168 T^{6} + \cdots + 74304)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + T^{3} - 78 T^{2} + \cdots + 172)^{4} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 74509345296 \) Copy content Toggle raw display
$83$ \( (T^{8} - 27 T^{7} + \cdots + 432964)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 272 T^{6} + \cdots + 891136)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 4 T^{7} + \cdots + 1018081)^{2} \) Copy content Toggle raw display
show more
show less