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Properties

Label 44.2.g.a
Level $44$
Weight $2$
Character orbit 44.g
Analytic conductor $0.351$
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] = [44,2,Mod(7,44)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(44, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("44.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 44 = 2^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 44.g (of order \(10\), degree \(4\), 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: \(0.351341768894\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{14} - \beta_{13} - \beta_{2} + \cdots - 1) q^{3}+ \cdots + ( - \beta_{12} - \beta_{11} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{14} - \beta_{13} - \beta_{2} + \cdots - 1) q^{3}+ \cdots + (3 \beta_{15} + 8 \beta_{14} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 5 q^{2} - q^{4} - 6 q^{5} - 5 q^{6} - 5 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 5 q^{2} - q^{4} - 6 q^{5} - 5 q^{6} - 5 q^{8} - 10 q^{9} - 22 q^{12} - 10 q^{13} + 8 q^{14} + 23 q^{16} - 10 q^{17} + 20 q^{18} + 16 q^{20} + 17 q^{22} + 25 q^{24} + 6 q^{25} - 4 q^{26} + 20 q^{28} - 10 q^{29} - 12 q^{33} - 6 q^{34} - 30 q^{36} + 18 q^{37} - 38 q^{38} - 40 q^{40} + 10 q^{41} - 26 q^{42} - 28 q^{44} + 40 q^{45} - 30 q^{46} - 36 q^{48} + 6 q^{49} - 15 q^{50} - 10 q^{52} + 38 q^{53} - 12 q^{56} + 30 q^{58} + 52 q^{60} - 10 q^{61} + 70 q^{62} + 23 q^{64} + 36 q^{66} + 60 q^{68} - 16 q^{69} + 12 q^{70} + 45 q^{72} - 30 q^{73} + 40 q^{74} + 2 q^{77} + 4 q^{78} - 28 q^{80} - 4 q^{81} - 59 q^{82} - 10 q^{84} - 50 q^{85} - 39 q^{86} - 53 q^{88} - 36 q^{89} - 50 q^{90} + 36 q^{92} - 38 q^{93} - 30 q^{94} - 68 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 5 \nu^{14} + 13 \nu^{13} - 25 \nu^{12} + 35 \nu^{11} - 30 \nu^{10} - 2 \nu^{9} + 60 \nu^{8} + \cdots - 640 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35 \nu^{15} + 94 \nu^{14} - 210 \nu^{13} + 334 \nu^{12} - 328 \nu^{11} + 111 \nu^{10} + 476 \nu^{9} + \cdots + 2304 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30 \nu^{15} + 141 \nu^{14} - 309 \nu^{13} + 573 \nu^{12} - 693 \nu^{11} + 429 \nu^{10} + 398 \nu^{9} + \cdots + 9664 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33 \nu^{15} - 147 \nu^{14} + 323 \nu^{13} - 591 \nu^{12} + 705 \nu^{11} - 424 \nu^{10} - 446 \nu^{9} + \cdots - 9472 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 37 \nu^{15} + 152 \nu^{14} - 334 \nu^{13} + 602 \nu^{12} - 704 \nu^{11} + 405 \nu^{10} + 498 \nu^{9} + \cdots + 9088 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35 \nu^{15} - 120 \nu^{14} + 265 \nu^{13} - 455 \nu^{12} + 501 \nu^{11} - 248 \nu^{10} - 475 \nu^{9} + \cdots - 5536 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 47 \nu^{15} + 153 \nu^{14} - 338 \nu^{13} + 572 \nu^{12} - 618 \nu^{11} + 291 \nu^{10} + 633 \nu^{9} + \cdots + 6464 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 71 \nu^{15} - 281 \nu^{14} + 619 \nu^{13} - 1107 \nu^{12} + 1281 \nu^{11} - 722 \nu^{10} - 952 \nu^{9} + \cdots - 16128 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 279 \nu^{15} + 987 \nu^{14} - 2179 \nu^{13} + 3775 \nu^{12} - 4213 \nu^{11} + 2162 \nu^{10} + \cdots + 48384 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 307 \nu^{15} - 1097 \nu^{14} + 2421 \nu^{13} - 4209 \nu^{12} + 4711 \nu^{11} - 2444 \nu^{10} + \cdots - 54528 ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 293 \nu^{15} + 1079 \nu^{14} - 2383 \nu^{13} + 4179 \nu^{12} - 4733 \nu^{11} + 2528 \nu^{10} + \cdots + 56320 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 162 \nu^{15} - 591 \nu^{14} + 1304 \nu^{13} - 2280 \nu^{12} + 2572 \nu^{11} - 1360 \nu^{10} + \cdots - 30400 ) / 32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 361 \nu^{15} + 1262 \nu^{14} - 2786 \nu^{13} + 4814 \nu^{12} - 5348 \nu^{11} + 2721 \nu^{10} + \cdots + 60672 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 785 \nu^{15} + 2815 \nu^{14} - 6215 \nu^{13} + 10819 \nu^{12} - 12133 \nu^{11} + 6324 \nu^{10} + \cdots + 141056 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + 2\beta_{9} + \beta_{6} + \beta_{4} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + \beta_{13} + 3 \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 2\beta_{11} + 3\beta_{6} - 2\beta_{4} - \beta_{3} - 2\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - 2 \beta_{12} + 4 \beta_{11} + \beta_{10} - 5 \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - \beta_{11} + \beta_{10} + 5 \beta_{9} + 3 \beta_{8} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5 \beta_{15} - 2 \beta_{14} - \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - \beta_{10} - 8 \beta_{9} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7 \beta_{15} + 6 \beta_{14} + 3 \beta_{13} + \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2 \beta_{15} + 14 \beta_{14} + 9 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{9} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 26 \beta_{15} + 21 \beta_{14} + 40 \beta_{13} - 24 \beta_{12} - 20 \beta_{11} + 19 \beta_{10} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 20 \beta_{15} - \beta_{14} - \beta_{13} - 30 \beta_{12} + \beta_{11} - 23 \beta_{10} - 21 \beta_{9} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 21 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 31 \beta_{12} + 24 \beta_{11} - 11 \beta_{10} + \cdots + 50 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 7 \beta_{15} + 45 \beta_{13} + 19 \beta_{12} - 33 \beta_{11} - 9 \beta_{10} + 52 \beta_{9} + 62 \beta_{8} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 20 \beta_{15} + 20 \beta_{14} + 61 \beta_{13} - 56 \beta_{12} + 22 \beta_{11} + 40 \beta_{10} + \cdots + 68 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/44\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-\beta_{12}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.40958 − 0.114404i
0.0737040 − 1.41229i
−0.544389 + 1.30524i
−1.36594 − 0.366325i
1.40958 + 0.114404i
0.0737040 + 1.41229i
−0.544389 − 1.30524i
−1.36594 + 0.366325i
1.40874 + 0.124276i
1.06665 − 0.928579i
0.656642 + 1.25253i
−0.204982 − 1.39928i
1.40874 − 0.124276i
1.06665 + 0.928579i
0.656642 − 1.25253i
−0.204982 + 1.39928i
−1.40958 − 0.114404i 0.704424 + 0.228881i 1.97382 + 0.322523i 1.09089 − 0.792578i −0.966756 − 0.403215i 0.503194 + 1.54867i −2.74536 − 0.680436i −1.98322 − 1.44090i −1.62837 + 0.992398i
7.2 −0.0737040 − 1.41229i 1.70537 + 0.554109i −1.98914 + 0.208183i −2.39991 + 1.74363i 0.656871 − 2.44932i −0.815620 − 2.51022i 0.440622 + 2.79390i 0.174207 + 0.126569i 2.63940 + 3.26086i
7.3 0.544389 + 1.30524i −0.704424 − 0.228881i −1.40728 + 1.42111i 1.09089 − 0.792578i −0.0847364 − 1.04404i −0.503194 − 1.54867i −2.62099 − 1.06320i −1.98322 − 1.44090i 1.62837 + 0.992398i
7.4 1.36594 − 0.366325i −1.70537 − 0.554109i 1.73161 − 1.00076i −2.39991 + 1.74363i −2.53243 − 0.132161i 0.815620 + 2.51022i 1.99868 − 2.00132i 0.174207 + 0.126569i −2.63940 + 3.26086i
19.1 −1.40958 + 0.114404i 0.704424 − 0.228881i 1.97382 − 0.322523i 1.09089 + 0.792578i −0.966756 + 0.403215i 0.503194 − 1.54867i −2.74536 + 0.680436i −1.98322 + 1.44090i −1.62837 − 0.992398i
19.2 −0.0737040 + 1.41229i 1.70537 − 0.554109i −1.98914 − 0.208183i −2.39991 − 1.74363i 0.656871 + 2.44932i −0.815620 + 2.51022i 0.440622 − 2.79390i 0.174207 − 0.126569i 2.63940 − 3.26086i
19.3 0.544389 − 1.30524i −0.704424 + 0.228881i −1.40728 − 1.42111i 1.09089 + 0.792578i −0.0847364 + 1.04404i −0.503194 + 1.54867i −2.62099 + 1.06320i −1.98322 + 1.44090i 1.62837 − 0.992398i
19.4 1.36594 + 0.366325i −1.70537 + 0.554109i 1.73161 + 1.00076i −2.39991 − 1.74363i −2.53243 + 0.132161i 0.815620 − 2.51022i 1.99868 + 2.00132i 0.174207 − 0.126569i −2.63940 − 3.26086i
35.1 −1.40874 + 0.124276i −1.59814 − 2.19965i 1.96911 − 0.350146i −0.720859 − 2.21858i 2.52473 + 2.90013i 1.04462 + 0.758960i −2.73046 + 0.737979i −1.35736 + 4.17752i 1.29122 + 3.03582i
35.2 −1.06665 − 0.928579i 1.59814 + 2.19965i 0.275480 + 1.98094i −0.720859 − 2.21858i 0.337896 − 3.83025i −1.04462 − 0.758960i 1.54562 − 2.36877i −1.35736 + 4.17752i −1.29122 + 3.03582i
35.3 −0.656642 + 1.25253i 0.539857 + 0.743049i −1.13764 − 1.64492i 0.529876 + 1.63079i −1.28518 + 0.188268i −1.93399 − 1.40513i 2.80733 − 0.344804i 0.666375 − 2.05089i −2.39055 − 0.407162i
35.4 0.204982 − 1.39928i −0.539857 − 0.743049i −1.91596 − 0.573655i 0.529876 + 1.63079i −1.15039 + 0.603098i 1.93399 + 1.40513i −1.19544 + 2.56338i 0.666375 − 2.05089i 2.39055 − 0.407162i
39.1 −1.40874 − 0.124276i −1.59814 + 2.19965i 1.96911 + 0.350146i −0.720859 + 2.21858i 2.52473 − 2.90013i 1.04462 − 0.758960i −2.73046 − 0.737979i −1.35736 − 4.17752i 1.29122 − 3.03582i
39.2 −1.06665 + 0.928579i 1.59814 − 2.19965i 0.275480 − 1.98094i −0.720859 + 2.21858i 0.337896 + 3.83025i −1.04462 + 0.758960i 1.54562 + 2.36877i −1.35736 − 4.17752i −1.29122 − 3.03582i
39.3 −0.656642 − 1.25253i 0.539857 − 0.743049i −1.13764 + 1.64492i 0.529876 − 1.63079i −1.28518 − 0.188268i −1.93399 + 1.40513i 2.80733 + 0.344804i 0.666375 + 2.05089i −2.39055 + 0.407162i
39.4 0.204982 + 1.39928i −0.539857 + 0.743049i −1.91596 + 0.573655i 0.529876 − 1.63079i −1.15039 − 0.603098i 1.93399 − 1.40513i −1.19544 − 2.56338i 0.666375 + 2.05089i 2.39055 + 0.407162i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 44.2.g.a ✓ 16
3.b odd 2 1 396.2.r.a 16
4.b odd 2 1 inner 44.2.g.a ✓ 16
8.b even 2 1 704.2.u.c 16
8.d odd 2 1 704.2.u.c 16
11.b odd 2 1 484.2.g.i 16
11.c even 5 1 484.2.c.d 16
11.c even 5 1 484.2.g.f 16
11.c even 5 1 484.2.g.i 16
11.c even 5 1 484.2.g.j 16
11.d odd 10 1 inner 44.2.g.a ✓ 16
11.d odd 10 1 484.2.c.d 16
11.d odd 10 1 484.2.g.f 16
11.d odd 10 1 484.2.g.j 16
12.b even 2 1 396.2.r.a 16
33.f even 10 1 396.2.r.a 16
44.c even 2 1 484.2.g.i 16
44.g even 10 1 inner 44.2.g.a ✓ 16
44.g even 10 1 484.2.c.d 16
44.g even 10 1 484.2.g.f 16
44.g even 10 1 484.2.g.j 16
44.h odd 10 1 484.2.c.d 16
44.h odd 10 1 484.2.g.f 16
44.h odd 10 1 484.2.g.i 16
44.h odd 10 1 484.2.g.j 16
88.k even 10 1 704.2.u.c 16
88.p odd 10 1 704.2.u.c 16
132.n odd 10 1 396.2.r.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.g.a ✓ 16 1.a even 1 1 trivial
44.2.g.a ✓ 16 4.b odd 2 1 inner
44.2.g.a ✓ 16 11.d odd 10 1 inner
44.2.g.a ✓ 16 44.g even 10 1 inner
396.2.r.a 16 3.b odd 2 1
396.2.r.a 16 12.b even 2 1
396.2.r.a 16 33.f even 10 1
396.2.r.a 16 132.n odd 10 1
484.2.c.d 16 11.c even 5 1
484.2.c.d 16 11.d odd 10 1
484.2.c.d 16 44.g even 10 1
484.2.c.d 16 44.h odd 10 1
484.2.g.f 16 11.c even 5 1
484.2.g.f 16 11.d odd 10 1
484.2.g.f 16 44.g even 10 1
484.2.g.f 16 44.h odd 10 1
484.2.g.i 16 11.b odd 2 1
484.2.g.i 16 11.c even 5 1
484.2.g.i 16 44.c even 2 1
484.2.g.i 16 44.h odd 10 1
484.2.g.j 16 11.c even 5 1
484.2.g.j 16 11.d odd 10 1
484.2.g.j 16 44.g even 10 1
484.2.g.j 16 44.h odd 10 1
704.2.u.c 16 8.b even 2 1
704.2.u.c 16 8.d odd 2 1
704.2.u.c 16 88.k even 10 1
704.2.u.c 16 88.p odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(44, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 5 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} - T^{14} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( (T^{8} + 3 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 11 T^{14} + \cdots + 30976 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} + 5 T^{7} + 16 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 5 T^{7} - 24 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 106 T^{14} + \cdots + 1771561 \) Copy content Toggle raw display
$23$ \( (T^{8} + 88 T^{6} + \cdots + 2816)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 5 T^{7} + \cdots + 400)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 428888770816 \) Copy content Toggle raw display
$37$ \( (T^{8} - 9 T^{7} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 5 T^{7} + \cdots + 2588881)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 143 T^{6} + \cdots + 148016)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1206517709056 \) Copy content Toggle raw display
$53$ \( (T^{8} - 19 T^{7} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 17080137481 \) Copy content Toggle raw display
$61$ \( (T^{8} + 5 T^{7} + \cdots + 633616)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 185 T^{6} + \cdots + 1760000)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 21908736256 \) Copy content Toggle raw display
$73$ \( (T^{8} + 15 T^{7} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 28606986496 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 435963075625 \) Copy content Toggle raw display
$89$ \( (T^{4} + 9 T^{3} + \cdots + 116)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 17 T^{3} + \cdots + 3721)^{4} \) Copy content Toggle raw display
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