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Properties

Label 52.2.l.b
Level $52$
Weight $2$
Character orbit 52.l
Analytic conductor $0.415$
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] = [52,2,Mod(7,52)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(52, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("52.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 52.l (of order \(12\), 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: \(0.415222090511\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.102930383934669717504.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{12} q^{2} + (\beta_{14} - \beta_{13} + \cdots + \beta_{3}) q^{3}+ \cdots + (\beta_{12} - \beta_{10} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{2} + (\beta_{14} - \beta_{13} + \cdots + \beta_{3}) q^{3}+ \cdots + ( - 2 \beta_{14} + 2 \beta_{13} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 6 q^{4} - 12 q^{5} - 14 q^{6} + 10 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} - 6 q^{4} - 12 q^{5} - 14 q^{6} + 10 q^{8} + 4 q^{9} - 12 q^{13} + 8 q^{14} - 2 q^{16} + 12 q^{17} - 6 q^{18} + 2 q^{20} - 28 q^{21} + 10 q^{24} + 16 q^{26} + 12 q^{28} - 8 q^{29} + 42 q^{30} + 28 q^{32} - 20 q^{33} + 14 q^{34} - 6 q^{36} - 16 q^{37} - 40 q^{40} + 48 q^{41} - 28 q^{42} - 8 q^{44} + 20 q^{45} - 46 q^{46} - 10 q^{48} + 60 q^{49} + 10 q^{50} - 32 q^{52} - 32 q^{53} - 16 q^{54} - 60 q^{56} + 12 q^{57} - 48 q^{58} - 24 q^{60} + 4 q^{61} - 18 q^{62} - 8 q^{65} + 56 q^{66} + 16 q^{68} - 12 q^{69} + 28 q^{70} + 56 q^{72} + 20 q^{73} + 4 q^{74} + 22 q^{76} + 68 q^{78} + 44 q^{80} + 48 q^{81} + 84 q^{84} + 20 q^{85} + 16 q^{86} + 36 q^{88} - 52 q^{89} - 12 q^{92} - 92 q^{93} - 38 q^{94} - 72 q^{96} - 28 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19 \nu^{15} + 26 \nu^{14} + 49 \nu^{13} + 12 \nu^{12} - 163 \nu^{11} - 146 \nu^{10} + 172 \nu^{9} + \cdots - 1152 ) / 1408 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - 4 \nu^{14} + 5 \nu^{13} - 2 \nu^{12} + 5 \nu^{11} - 8 \nu^{10} - 12 \nu^{9} + 32 \nu^{8} + \cdots - 512 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25 \nu^{15} + 6 \nu^{14} + 31 \nu^{13} + 76 \nu^{12} - 61 \nu^{11} - 182 \nu^{10} + 40 \nu^{8} + \cdots - 384 ) / 1408 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{15} - 62 \nu^{14} + 27 \nu^{13} + 56 \nu^{12} + 167 \nu^{11} - 146 \nu^{10} - 400 \nu^{9} + \cdots - 2560 ) / 1408 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10 \nu^{15} - 37 \nu^{14} - 12 \nu^{13} + 7 \nu^{12} + 106 \nu^{11} + 87 \nu^{10} - 266 \nu^{9} + \cdots + 1216 ) / 704 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25 \nu^{15} + 72 \nu^{14} - 13 \nu^{13} - 34 \nu^{12} - 149 \nu^{11} - 28 \nu^{10} + 396 \nu^{9} + \cdots + 2432 ) / 1408 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7 \nu^{15} + 17 \nu^{14} + \nu^{13} + 5 \nu^{12} - 57 \nu^{11} - 9 \nu^{10} + 100 \nu^{9} + \cdots + 544 ) / 352 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{15} + 28 \nu^{14} - 21 \nu^{13} - 6 \nu^{12} - 57 \nu^{11} + 24 \nu^{10} + 166 \nu^{9} + \cdots + 1600 ) / 352 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 47 \nu^{15} + 78 \nu^{14} + 13 \nu^{13} + 32 \nu^{12} - 191 \nu^{11} - 150 \nu^{10} + 420 \nu^{9} + \cdots + 3200 ) / 704 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 53 \nu^{15} - 139 \nu^{14} + 61 \nu^{13} + 3 \nu^{12} + 271 \nu^{11} - 59 \nu^{10} - 748 \nu^{9} + \cdots - 8704 ) / 704 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 34 \nu^{15} - 83 \nu^{14} + 31 \nu^{13} - 7 \nu^{12} + 173 \nu^{11} - \nu^{10} - 467 \nu^{9} + \cdots - 4608 ) / 352 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 73 \nu^{15} - 204 \nu^{14} + 109 \nu^{13} + 6 \nu^{12} + 365 \nu^{11} - 112 \nu^{10} - 1068 \nu^{9} + \cdots - 13568 ) / 704 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 82 \nu^{15} + 201 \nu^{14} - 52 \nu^{13} + 13 \nu^{12} - 434 \nu^{11} - 51 \nu^{10} + 1138 \nu^{9} + \cdots + 9728 ) / 704 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 46 \nu^{15} + 129 \nu^{14} - 72 \nu^{13} + 3 \nu^{12} - 230 \nu^{11} + 87 \nu^{10} + 654 \nu^{9} + \cdots + 9760 ) / 352 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{5} - \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{8} + 2 \beta_{5} + \beta_{4} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + 2\beta_{12} + \beta_{9} + 2\beta_{8} + \beta_{7} + 2\beta_{4} + 3\beta_{3} - 2\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{15} + 4 \beta_{14} + 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + \beta_{9} + 3 \beta_{8} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{15} + 3 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} - 3 \beta_{11} + 4 \beta_{9} + \beta_{8} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{15} + 4 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} + 8 \beta_{11} + 4 \beta_{10} + \cdots - 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{15} + 7 \beta_{14} + 5 \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + 4 \beta_{9} - 5 \beta_{8} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3 \beta_{15} + 2 \beta_{14} + 10 \beta_{13} - 10 \beta_{12} - 6 \beta_{11} - 11 \beta_{9} + 5 \beta_{7} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 13 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} - 22 \beta_{11} - 16 \beta_{9} - 2 \beta_{8} - 17 \beta_{7} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 7 \beta_{15} - 20 \beta_{13} - 4 \beta_{12} - 20 \beta_{10} - 27 \beta_{9} + 7 \beta_{8} - 23 \beta_{7} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13 \beta_{15} - 13 \beta_{14} + 13 \beta_{13} + 14 \beta_{12} - 35 \beta_{11} + 16 \beta_{10} - 30 \beta_{9} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 30 \beta_{14} - 78 \beta_{13} + 54 \beta_{12} + 30 \beta_{11} - 60 \beta_{10} - 78 \beta_{9} + 45 \beta_{8} + \cdots - 60 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 78 \beta_{15} + 15 \beta_{14} + 45 \beta_{13} + 90 \beta_{12} - 27 \beta_{11} - 72 \beta_{10} + \cdots + 61 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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
1.31256 + 0.526485i
−0.873468 + 1.11223i
1.08916 − 0.902074i
−1.39427 − 0.236640i
−0.00757716 − 1.41419i
1.41121 − 0.0921725i
1.17605 + 0.785427i
−0.713659 + 1.22094i
1.31256 − 0.526485i
−0.873468 − 1.11223i
1.08916 + 0.902074i
−1.39427 + 0.236640i
−0.00757716 + 1.41419i
1.41121 + 0.0921725i
1.17605 − 0.785427i
−0.713659 − 1.22094i
−1.11223 − 0.873468i −2.16981 − 1.25274i 0.474107 + 1.94299i −2.19962 − 2.19962i 1.31910 + 3.28859i 0.152604 − 0.569525i 1.16983 − 2.57517i 1.63871 + 2.83834i 0.525184 + 4.36778i
7.2 −0.526485 + 1.31256i 2.16981 + 1.25274i −1.44563 − 1.38209i −2.19962 − 2.19962i −2.78667 + 2.18846i −0.152604 + 0.569525i 2.57517 − 1.16983i 1.63871 + 2.83834i 4.04520 − 1.72907i
7.3 0.236640 − 1.39427i 0.736159 + 0.425021i −1.88800 − 0.659882i −0.166404 − 0.166404i 0.766801 − 0.925830i −0.684384 + 2.55416i −1.36683 + 2.47624i −1.13871 − 1.97231i −0.271390 + 0.192635i
7.4 0.902074 + 1.08916i −0.736159 − 0.425021i −0.372527 + 1.96500i −0.166404 − 0.166404i −0.201154 − 1.18519i 0.684384 − 2.55416i −2.47624 + 1.36683i −1.13871 − 1.97231i 0.0311314 − 0.331348i
11.1 −1.22094 − 0.713659i 1.40004 − 0.808315i 0.981383 + 1.74267i −1.52798 − 1.52798i −2.28623 − 0.0122495i 1.97429 − 0.529008i 0.0454612 − 2.82806i −0.193255 + 0.334727i 0.775114 + 2.95603i
11.2 −0.785427 + 1.17605i 1.81380 − 1.04720i −0.766209 − 1.84741i 0.894007 + 0.894007i −0.193046 + 2.95563i −4.37156 + 1.17136i 2.77446 + 0.549903i 0.693255 − 1.20075i −1.75358 + 0.349224i
11.3 0.0921725 + 1.41121i −1.81380 + 1.04720i −1.98301 + 0.260149i 0.894007 + 0.894007i −1.64500 − 2.46313i 4.37156 − 1.17136i −0.549903 − 2.77446i 0.693255 − 1.20075i −1.17923 + 1.34403i
11.4 1.41419 − 0.00757716i −1.40004 + 0.808315i 1.99989 − 0.0214311i −1.52798 − 1.52798i −1.97381 + 1.15372i −1.97429 + 0.529008i 2.82806 − 0.0454612i −0.193255 + 0.334727i −2.17244 − 2.14928i
15.1 −1.11223 + 0.873468i −2.16981 + 1.25274i 0.474107 − 1.94299i −2.19962 + 2.19962i 1.31910 − 3.28859i 0.152604 + 0.569525i 1.16983 + 2.57517i 1.63871 − 2.83834i 0.525184 − 4.36778i
15.2 −0.526485 − 1.31256i 2.16981 − 1.25274i −1.44563 + 1.38209i −2.19962 + 2.19962i −2.78667 − 2.18846i −0.152604 − 0.569525i 2.57517 + 1.16983i 1.63871 − 2.83834i 4.04520 + 1.72907i
15.3 0.236640 + 1.39427i 0.736159 − 0.425021i −1.88800 + 0.659882i −0.166404 + 0.166404i 0.766801 + 0.925830i −0.684384 − 2.55416i −1.36683 − 2.47624i −1.13871 + 1.97231i −0.271390 − 0.192635i
15.4 0.902074 − 1.08916i −0.736159 + 0.425021i −0.372527 − 1.96500i −0.166404 + 0.166404i −0.201154 + 1.18519i 0.684384 + 2.55416i −2.47624 − 1.36683i −1.13871 + 1.97231i 0.0311314 + 0.331348i
19.1 −1.22094 + 0.713659i 1.40004 + 0.808315i 0.981383 − 1.74267i −1.52798 + 1.52798i −2.28623 + 0.0122495i 1.97429 + 0.529008i 0.0454612 + 2.82806i −0.193255 − 0.334727i 0.775114 − 2.95603i
19.2 −0.785427 − 1.17605i 1.81380 + 1.04720i −0.766209 + 1.84741i 0.894007 − 0.894007i −0.193046 − 2.95563i −4.37156 − 1.17136i 2.77446 − 0.549903i 0.693255 + 1.20075i −1.75358 − 0.349224i
19.3 0.0921725 − 1.41121i −1.81380 − 1.04720i −1.98301 − 0.260149i 0.894007 − 0.894007i −1.64500 + 2.46313i 4.37156 + 1.17136i −0.549903 + 2.77446i 0.693255 + 1.20075i −1.17923 − 1.34403i
19.4 1.41419 + 0.00757716i −1.40004 − 0.808315i 1.99989 + 0.0214311i −1.52798 + 1.52798i −1.97381 − 1.15372i −1.97429 − 0.529008i 2.82806 + 0.0454612i −0.193255 − 0.334727i −2.17244 + 2.14928i
\(n\): 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
13.f odd 12 1 inner
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.2.l.b ✓ 16
3.b odd 2 1 468.2.cb.f 16
4.b odd 2 1 inner 52.2.l.b ✓ 16
8.b even 2 1 832.2.bu.n 16
8.d odd 2 1 832.2.bu.n 16
12.b even 2 1 468.2.cb.f 16
13.b even 2 1 676.2.l.k 16
13.c even 3 1 676.2.f.h 16
13.c even 3 1 676.2.l.m 16
13.d odd 4 1 676.2.l.i 16
13.d odd 4 1 676.2.l.m 16
13.e even 6 1 676.2.f.i 16
13.e even 6 1 676.2.l.i 16
13.f odd 12 1 inner 52.2.l.b ✓ 16
13.f odd 12 1 676.2.f.h 16
13.f odd 12 1 676.2.f.i 16
13.f odd 12 1 676.2.l.k 16
39.k even 12 1 468.2.cb.f 16
52.b odd 2 1 676.2.l.k 16
52.f even 4 1 676.2.l.i 16
52.f even 4 1 676.2.l.m 16
52.i odd 6 1 676.2.f.i 16
52.i odd 6 1 676.2.l.i 16
52.j odd 6 1 676.2.f.h 16
52.j odd 6 1 676.2.l.m 16
52.l even 12 1 inner 52.2.l.b ✓ 16
52.l even 12 1 676.2.f.h 16
52.l even 12 1 676.2.f.i 16
52.l even 12 1 676.2.l.k 16
104.u even 12 1 832.2.bu.n 16
104.x odd 12 1 832.2.bu.n 16
156.v odd 12 1 468.2.cb.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.l.b ✓ 16 1.a even 1 1 trivial
52.2.l.b ✓ 16 4.b odd 2 1 inner
52.2.l.b ✓ 16 13.f odd 12 1 inner
52.2.l.b ✓ 16 52.l even 12 1 inner
468.2.cb.f 16 3.b odd 2 1
468.2.cb.f 16 12.b even 2 1
468.2.cb.f 16 39.k even 12 1
468.2.cb.f 16 156.v odd 12 1
676.2.f.h 16 13.c even 3 1
676.2.f.h 16 13.f odd 12 1
676.2.f.h 16 52.j odd 6 1
676.2.f.h 16 52.l even 12 1
676.2.f.i 16 13.e even 6 1
676.2.f.i 16 13.f odd 12 1
676.2.f.i 16 52.i odd 6 1
676.2.f.i 16 52.l even 12 1
676.2.l.i 16 13.d odd 4 1
676.2.l.i 16 13.e even 6 1
676.2.l.i 16 52.f even 4 1
676.2.l.i 16 52.i odd 6 1
676.2.l.k 16 13.b even 2 1
676.2.l.k 16 13.f odd 12 1
676.2.l.k 16 52.b odd 2 1
676.2.l.k 16 52.l even 12 1
676.2.l.m 16 13.c even 3 1
676.2.l.m 16 13.d odd 4 1
676.2.l.m 16 52.f even 4 1
676.2.l.m 16 52.j odd 6 1
832.2.bu.n 16 8.b even 2 1
832.2.bu.n 16 8.d odd 2 1
832.2.bu.n 16 104.u even 12 1
832.2.bu.n 16 104.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 14T_{3}^{14} + 131T_{3}^{12} - 686T_{3}^{10} + 2605T_{3}^{8} - 5824T_{3}^{6} + 9164T_{3}^{4} - 5824T_{3}^{2} + 2704 \) acting on \(S_{2}^{\mathrm{new}}(52, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} - 14 T^{14} + \cdots + 2704 \) Copy content Toggle raw display
$5$ \( (T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 30 T^{14} + \cdots + 43264 \) Copy content Toggle raw display
$11$ \( T^{16} + 18 T^{14} + \cdots + 692224 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 6 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} - 54 T^{14} + \cdots + 77228944 \) Copy content Toggle raw display
$23$ \( T^{16} + 106 T^{14} + \cdots + 77228944 \) Copy content Toggle raw display
$29$ \( (T^{8} + 4 T^{7} + \cdots + 51529)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1235663104 \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + 5 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + \cdots + 81)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 5671027857664 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 5671027857664 \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} + \cdots - 128)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 171720267307264 \) Copy content Toggle raw display
$61$ \( (T^{8} - 2 T^{7} + \cdots + 6889)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2205735869584 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 5067731344 \) Copy content Toggle raw display
$73$ \( (T^{8} - 10 T^{7} + \cdots + 676)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 160 T^{6} + \cdots + 53248)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 538409280507904 \) Copy content Toggle raw display
$89$ \( (T^{8} + 26 T^{7} + \cdots + 2896804)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 14 T^{7} + \cdots + 2116)^{2} \) Copy content Toggle raw display
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