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Properties

Label 784.2.x.k
Level $784$
Weight $2$
Character orbit 784.x
Analytic conductor $6.260$
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] = [784,2,Mod(165,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.x (of order \(12\), degree \(4\), not 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 4 x^{14} + 4 x^{13} - 13 x^{12} + 32 x^{11} - 4 x^{10} - 34 x^{9} + 121 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 112)
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.

\(f(q)\) \(=\) \( q + (\beta_{12} + \beta_{9}) q^{2} + (\beta_{13} - \beta_{8}) q^{3} + (\beta_{11} - \beta_{10} + \beta_{8} + \cdots - 1) q^{4}+ \cdots + (\beta_{15} - \beta_{14} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} + \beta_{9}) q^{2} + (\beta_{13} - \beta_{8}) q^{3} + (\beta_{11} - \beta_{10} + \beta_{8} + \cdots - 1) q^{4}+ \cdots + (2 \beta_{14} + 4 \beta_{13} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 4 q^{4} + 4 q^{5} - 32 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} - 4 q^{4} + 4 q^{5} - 32 q^{6} - 8 q^{8} - 12 q^{10} + 12 q^{12} - 16 q^{15} - 8 q^{16} - 24 q^{17} - 6 q^{18} + 12 q^{19} - 16 q^{20} - 8 q^{22} - 8 q^{26} + 24 q^{27} - 32 q^{29} - 20 q^{30} - 16 q^{31} + 28 q^{32} + 24 q^{33} - 24 q^{34} + 48 q^{36} - 16 q^{37} + 16 q^{38} - 28 q^{40} - 64 q^{43} + 32 q^{44} + 8 q^{45} - 20 q^{46} - 24 q^{47} + 40 q^{48} - 28 q^{50} - 8 q^{51} - 32 q^{52} + 8 q^{53} + 16 q^{54} - 12 q^{58} + 28 q^{59} + 28 q^{60} - 28 q^{61} + 40 q^{62} - 64 q^{64} + 48 q^{65} + 16 q^{66} - 28 q^{68} + 88 q^{69} - 44 q^{72} + 4 q^{74} - 28 q^{75} - 48 q^{76} + 24 q^{78} + 24 q^{79} + 12 q^{80} - 40 q^{81} - 4 q^{82} - 80 q^{85} + 40 q^{88} - 32 q^{90} + 72 q^{92} - 16 q^{93} - 28 q^{94} - 16 q^{95} - 8 q^{96} - 64 q^{97} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 4 x^{14} + 4 x^{13} - 13 x^{12} + 32 x^{11} - 4 x^{10} - 34 x^{9} + 121 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 46 \nu^{15} + 465 \nu^{14} - 180 \nu^{13} + 992 \nu^{12} + 3150 \nu^{11} - 1905 \nu^{10} + \cdots + 70080 ) / 3008 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 99 \nu^{15} + 1526 \nu^{14} - 900 \nu^{13} + 1012 \nu^{12} + 10439 \nu^{11} - 9008 \nu^{10} + \cdots + 166912 ) / 6016 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 117 \nu^{15} - 26 \nu^{14} + 4 \nu^{13} + 684 \nu^{12} - 1057 \nu^{11} + 528 \nu^{10} + 1724 \nu^{9} + \cdots - 20608 ) / 6016 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 44 \nu^{15} - 79 \nu^{14} + 23 \nu^{13} - 438 \nu^{12} - 520 \nu^{11} + 263 \nu^{10} + \cdots - 16224 ) / 1504 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 243 \nu^{15} - 336 \nu^{14} + 688 \nu^{13} + 1276 \nu^{12} - 2687 \nu^{11} + 4994 \nu^{10} + \cdots - 31232 ) / 6016 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 70 \nu^{15} + 26 \nu^{14} - 51 \nu^{13} - 402 \nu^{12} + 258 \nu^{11} - 434 \nu^{10} - 1301 \nu^{9} + \cdots - 1952 ) / 1504 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 429 \nu^{15} - 1474 \nu^{14} + 1080 \nu^{13} + 1380 \nu^{12} - 11145 \nu^{11} + 10584 \nu^{10} + \cdots - 167808 ) / 6016 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 119 \nu^{15} - 392 \nu^{14} + 317 \nu^{13} + 392 \nu^{12} - 2939 \nu^{11} + 3022 \nu^{10} + \cdots - 44960 ) / 1504 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 262 \nu^{15} + 53 \nu^{14} - 254 \nu^{13} - 1980 \nu^{12} + 826 \nu^{11} - 2085 \nu^{10} + \cdots - 17920 ) / 3008 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 293 \nu^{15} + 274 \nu^{14} - 476 \nu^{13} - 1872 \nu^{12} + 2361 \nu^{11} - 3800 \nu^{10} + \cdots + 15872 ) / 3008 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 607 \nu^{15} + 2182 \nu^{14} - 1536 \nu^{13} - 1900 \nu^{12} + 16211 \nu^{11} - 15128 \nu^{10} + \cdots + 243072 ) / 6016 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 619 \nu^{15} + 806 \nu^{14} - 1252 \nu^{13} - 3532 \nu^{12} + 6447 \nu^{11} - 10352 \nu^{10} + \cdots + 79360 ) / 6016 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 767 \nu^{15} + 630 \nu^{14} - 1384 \nu^{13} - 4860 \nu^{12} + 5843 \nu^{11} - 10856 \nu^{10} + \cdots + 49536 ) / 6016 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 108 \nu^{15} - 164 \nu^{14} - 29 \nu^{13} - 1058 \nu^{12} - 926 \nu^{11} - 68 \nu^{10} + \cdots - 26560 ) / 752 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1311 \nu^{15} - 1764 \nu^{14} + 2296 \nu^{13} + 7404 \nu^{12} - 14283 \nu^{11} + 19662 \nu^{10} + \cdots - 145920 ) / 6016 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \cdots - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{3} - \beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} + \beta_{14} - 3\beta_{13} - \beta_{9} - \beta_{6} - 3\beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{15} + \beta_{14} + \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \cdots + 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{13} + 3\beta_{12} + 3\beta_{11} - 3\beta_{10} + 7\beta_{8} - \beta_{7} - \beta_{3} + \beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{15} - 3\beta_{14} + 9\beta_{13} - 3\beta_{9} - \beta_{6} + 9\beta_{5} + 5\beta_{4} + \beta_{3} - 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9 \beta_{15} + 5 \beta_{14} - 13 \beta_{12} - \beta_{11} + 3 \beta_{10} - 13 \beta_{9} + \beta_{8} + \cdots + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -5\beta_{13} - 3\beta_{12} - 15\beta_{11} + 7\beta_{10} - 9\beta_{8} - 17\beta_{7} + 5\beta_{3} - 3\beta_{2} + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11 \beta_{15} + 7 \beta_{14} + \beta_{13} + 21 \beta_{9} + 19 \beta_{6} + \beta_{5} - 33 \beta_{4} + \cdots - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21 \beta_{15} - 19 \beta_{14} + 21 \beta_{12} + 11 \beta_{11} + 37 \beta_{10} + 21 \beta_{9} + \cdots - 11 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 11 \beta_{13} - 25 \beta_{12} - \beta_{11} + 19 \beta_{10} - 15 \beta_{8} + 39 \beta_{7} - 31 \beta_{3} + \cdots + 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 43 \beta_{15} - 17 \beta_{14} - 53 \beta_{13} - 49 \beta_{9} - 25 \beta_{6} - 53 \beta_{5} + \cdots + 71 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 17 \beta_{15} + 23 \beta_{14} + 37 \beta_{12} + 13 \beta_{11} - 157 \beta_{10} + 37 \beta_{9} + \cdots - 13 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 57 \beta_{13} + 69 \beta_{12} + 123 \beta_{11} - 15 \beta_{10} + 33 \beta_{8} + 67 \beta_{7} + \cdots - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 49 \beta_{15} + 11 \beta_{14} + 79 \beta_{13} + 109 \beta_{9} - 167 \beta_{6} + 79 \beta_{5} + 35 \beta_{4} + \cdots - 85 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−1.33068 − 0.478848i
0.640069 + 1.26108i
0.840224 − 1.13755i
1.21641 + 0.721349i
−1.40526 − 0.158880i
0.772089 − 1.18485i
0.0165007 − 1.41412i
0.250645 + 1.39183i
−1.40526 + 0.158880i
0.772089 + 1.18485i
0.0165007 + 1.41412i
0.250645 − 1.39183i
−1.33068 + 0.478848i
0.640069 − 1.26108i
0.840224 + 1.13755i
1.21641 − 0.721349i
−1.19412 + 0.757684i 0.538823 + 2.01092i 0.851831 − 1.80953i 0.129523 − 0.483385i −2.16706 − 1.99301i 0 0.353863 + 2.80620i −1.15537 + 0.667056i 0.211588 + 0.675356i
165.2 −1.14477 − 0.830359i −0.261809 − 0.977085i 0.621007 + 1.90114i 0.317608 − 1.18533i −0.511620 + 1.33594i 0 0.867721 − 2.69204i 1.71192 − 0.988380i −1.34784 + 1.09320i
165.3 0.105414 − 1.41028i −0.719263 − 2.68432i −1.97778 − 0.297327i −0.229791 + 0.857592i −3.86147 + 0.731395i 0 −0.627801 + 2.75787i −4.09018 + 2.36147i 1.18522 + 0.414472i
165.4 0.867450 + 1.11693i 0.442248 + 1.65049i −0.495063 + 1.93776i −0.949390 + 3.54317i −1.45986 + 1.92568i 0 −2.59378 + 1.12796i 0.0695300 − 0.0401432i −4.78102 + 2.01312i
373.1 −1.27404 + 0.613848i 2.68432 + 0.719263i 1.24638 − 1.56414i 0.857592 − 0.229791i −3.86147 + 0.731395i 0 −0.627801 + 2.75787i 4.09018 + 2.36147i −0.951553 + 0.819195i
373.2 −0.146726 + 1.40658i 0.977085 + 0.261809i −1.95694 − 0.412764i −1.18533 + 0.317608i −0.511620 + 1.33594i 0 0.867721 − 2.69204i −1.71192 − 0.988380i −0.272823 − 1.71386i
373.3 0.533564 − 1.30970i −1.65049 − 0.442248i −1.43062 − 1.39762i 3.54317 − 0.949390i −1.45986 + 1.92568i 0 −2.59378 + 1.12796i −0.0695300 − 0.0401432i 0.647096 − 5.14705i
373.4 1.25323 + 0.655294i −2.01092 − 0.538823i 1.14118 + 1.64247i −0.483385 + 0.129523i −2.16706 − 1.99301i 0 0.353863 + 2.80620i 1.15537 + 0.667056i −0.690669 − 0.154437i
557.1 −1.27404 − 0.613848i 2.68432 − 0.719263i 1.24638 + 1.56414i 0.857592 + 0.229791i −3.86147 − 0.731395i 0 −0.627801 − 2.75787i 4.09018 − 2.36147i −0.951553 − 0.819195i
557.2 −0.146726 − 1.40658i 0.977085 − 0.261809i −1.95694 + 0.412764i −1.18533 − 0.317608i −0.511620 − 1.33594i 0 0.867721 + 2.69204i −1.71192 + 0.988380i −0.272823 + 1.71386i
557.3 0.533564 + 1.30970i −1.65049 + 0.442248i −1.43062 + 1.39762i 3.54317 + 0.949390i −1.45986 − 1.92568i 0 −2.59378 − 1.12796i −0.0695300 + 0.0401432i 0.647096 + 5.14705i
557.4 1.25323 − 0.655294i −2.01092 + 0.538823i 1.14118 − 1.64247i −0.483385 − 0.129523i −2.16706 + 1.99301i 0 0.353863 − 2.80620i 1.15537 − 0.667056i −0.690669 + 0.154437i
765.1 −1.19412 − 0.757684i 0.538823 − 2.01092i 0.851831 + 1.80953i 0.129523 + 0.483385i −2.16706 + 1.99301i 0 0.353863 − 2.80620i −1.15537 − 0.667056i 0.211588 − 0.675356i
765.2 −1.14477 + 0.830359i −0.261809 + 0.977085i 0.621007 − 1.90114i 0.317608 + 1.18533i −0.511620 − 1.33594i 0 0.867721 + 2.69204i 1.71192 + 0.988380i −1.34784 − 1.09320i
765.3 0.105414 + 1.41028i −0.719263 + 2.68432i −1.97778 + 0.297327i −0.229791 − 0.857592i −3.86147 − 0.731395i 0 −0.627801 − 2.75787i −4.09018 − 2.36147i 1.18522 − 0.414472i
765.4 0.867450 − 1.11693i 0.442248 − 1.65049i −0.495063 − 1.93776i −0.949390 − 3.54317i −1.45986 − 1.92568i 0 −2.59378 − 1.12796i 0.0695300 + 0.0401432i −4.78102 − 2.01312i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.k 16
7.b odd 2 1 784.2.x.j 16
7.c even 3 1 112.2.m.c ✓ 8
7.c even 3 1 inner 784.2.x.k 16
7.d odd 6 1 784.2.m.g 8
7.d odd 6 1 784.2.x.j 16
16.e even 4 1 inner 784.2.x.k 16
28.g odd 6 1 448.2.m.c 8
56.k odd 6 1 896.2.m.f 8
56.p even 6 1 896.2.m.e 8
112.l odd 4 1 784.2.x.j 16
112.u odd 12 1 448.2.m.c 8
112.u odd 12 1 896.2.m.f 8
112.w even 12 1 112.2.m.c ✓ 8
112.w even 12 1 inner 784.2.x.k 16
112.w even 12 1 896.2.m.e 8
112.x odd 12 1 784.2.m.g 8
112.x odd 12 1 784.2.x.j 16
224.bd even 24 2 7168.2.a.bc 8
224.bf odd 24 2 7168.2.a.bd 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.c ✓ 8 7.c even 3 1
112.2.m.c ✓ 8 112.w even 12 1
448.2.m.c 8 28.g odd 6 1
448.2.m.c 8 112.u odd 12 1
784.2.m.g 8 7.d odd 6 1
784.2.m.g 8 112.x odd 12 1
784.2.x.j 16 7.b odd 2 1
784.2.x.j 16 7.d odd 6 1
784.2.x.j 16 112.l odd 4 1
784.2.x.j 16 112.x odd 12 1
784.2.x.k 16 1.a even 1 1 trivial
784.2.x.k 16 7.c even 3 1 inner
784.2.x.k 16 16.e even 4 1 inner
784.2.x.k 16 112.w even 12 1 inner
896.2.m.e 8 56.p even 6 1
896.2.m.e 8 112.w even 12 1
896.2.m.f 8 56.k odd 6 1
896.2.m.f 8 112.u odd 12 1
7168.2.a.bc 8 224.bd even 24 2
7168.2.a.bd 8 224.bf odd 24 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{16} - 8 T_{3}^{13} - 44 T_{3}^{12} + 32 T_{3}^{11} + 32 T_{3}^{10} + 136 T_{3}^{9} + 1708 T_{3}^{8} + \cdots + 10000 \) Copy content Toggle raw display
\( T_{5}^{16} - 4 T_{5}^{15} + 8 T_{5}^{14} - 56 T_{5}^{13} + 100 T_{5}^{12} + 184 T_{5}^{11} + 32 T_{5}^{10} + \cdots + 16 \) 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
$3$ \( T^{16} - 8 T^{13} + \cdots + 10000 \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 64 T^{13} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{8} - 4 T^{5} + \cdots + 28900)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 12 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 1536953616 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 136048896 \) Copy content Toggle raw display
$29$ \( (T^{8} + 16 T^{7} + \cdots + 144)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 8 T^{7} + \cdots + 238144)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 189747360000 \) Copy content Toggle raw display
$41$ \( (T^{8} + 288 T^{6} + \cdots + 5198400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 32 T^{7} + \cdots + 10863616)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 12 T^{7} + \cdots + 141376)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} - 8 T^{15} + \cdots + 21381376 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 95489560559376 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 26115852816 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 245635219456 \) Copy content Toggle raw display
$71$ \( (T^{8} + 496 T^{6} + \cdots + 145926400)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 959512576 \) Copy content Toggle raw display
$79$ \( (T^{8} - 12 T^{7} + \cdots + 4665600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 4 T^{5} + \cdots + 100)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1665379926016 \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots + 712)^{4} \) Copy content Toggle raw display
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