Properties

Label 816.2.bq.f
Level $816$
Weight $2$
Character orbit 816.bq
Analytic conductor $6.516$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(49,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.bq (of order \(8\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 36x^{14} + 466x^{12} + 2956x^{10} + 10049x^{8} + 18032x^{6} + 14800x^{4} + 3200x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 408)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{12} q^{3} + ( - \beta_{11} - \beta_{9} - \beta_{7} - \beta_{5}) q^{5} + (\beta_{12} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{7} - \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{3} + ( - \beta_{11} - \beta_{9} - \beta_{7} - \beta_{5}) q^{5} + (\beta_{12} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{7} - \beta_{8} q^{9} + (\beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2}) q^{11} + (\beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{6} - \beta_{5}) q^{13} + ( - \beta_{14} - \beta_{9} - \beta_{7} - \beta_1 - 1) q^{15} + ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} + \cdots - 1) q^{17}+ \cdots + ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{11} - 8 q^{15} + 8 q^{19} + 8 q^{23} - 8 q^{25} + 32 q^{29} - 8 q^{31} - 8 q^{33} - 16 q^{35} + 8 q^{37} + 8 q^{39} + 40 q^{41} + 24 q^{43} + 8 q^{45} + 24 q^{49} - 8 q^{51} - 24 q^{53} - 8 q^{57} + 16 q^{59} - 64 q^{65} - 8 q^{69} + 16 q^{71} + 24 q^{73} + 96 q^{79} - 32 q^{85} - 16 q^{87} + 24 q^{93} + 8 q^{95} + 56 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 36x^{14} + 466x^{12} + 2956x^{10} + 10049x^{8} + 18032x^{6} + 14800x^{4} + 3200x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2 \nu^{15} + 12 \nu^{14} + 51 \nu^{13} + 393 \nu^{12} + 264 \nu^{11} + 4324 \nu^{10} - 1058 \nu^{9} + 21690 \nu^{8} - 12266 \nu^{7} + 52488 \nu^{6} - 34685 \nu^{5} + 54073 \nu^{4} + \cdots - 2168 ) / 1088 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2 \nu^{15} - 12 \nu^{14} + 51 \nu^{13} - 393 \nu^{12} + 264 \nu^{11} - 4324 \nu^{10} - 1058 \nu^{9} - 21690 \nu^{8} - 12266 \nu^{7} - 52488 \nu^{6} - 34685 \nu^{5} - 54073 \nu^{4} + \cdots + 1080 ) / 1088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39 \nu^{14} + 1288 \nu^{12} + 14390 \nu^{10} + 73964 \nu^{8} + 187047 \nu^{6} + 214076 \nu^{4} + 75928 \nu^{2} + 7456 ) / 2176 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6 \nu^{15} - 31 \nu^{14} + 141 \nu^{13} - 1006 \nu^{12} + 400 \nu^{11} - 10854 \nu^{10} - 7454 \nu^{9} - 52416 \nu^{8} - 57814 \nu^{7} - 118207 \nu^{6} - 152083 \nu^{5} - 105370 \nu^{4} + \cdots + 3952 ) / 2176 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47 \nu^{15} + 1492 \nu^{13} + 15446 \nu^{11} + 69732 \nu^{9} + 137983 \nu^{7} + 75336 \nu^{5} - 67080 \nu^{3} - 30720 \nu ) / 4352 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23 \nu^{15} + 73 \nu^{14} + 776 \nu^{13} + 2448 \nu^{12} + 9062 \nu^{11} + 27962 \nu^{10} + 50684 \nu^{9} + 146292 \nu^{8} + 150727 \nu^{7} + 368665 \nu^{6} + 238604 \nu^{5} + \cdots + 416 ) / 4352 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23 \nu^{15} - 73 \nu^{14} + 776 \nu^{13} - 2448 \nu^{12} + 9062 \nu^{11} - 27962 \nu^{10} + 50684 \nu^{9} - 146292 \nu^{8} + 150727 \nu^{7} - 368665 \nu^{6} + 238604 \nu^{5} + \cdots - 416 ) / 4352 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15 \nu^{15} - 124 \nu^{14} - 454 \nu^{13} - 4032 \nu^{12} - 4174 \nu^{11} - 43768 \nu^{10} - 13464 \nu^{9} - 215056 \nu^{8} + 1993 \nu^{7} - 507420 \nu^{6} + 87270 \nu^{5} + \cdots - 6016 ) / 4352 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15 \nu^{15} - 124 \nu^{14} + 454 \nu^{13} - 4032 \nu^{12} + 4174 \nu^{11} - 43768 \nu^{10} + 13464 \nu^{9} - 215056 \nu^{8} - 1993 \nu^{7} - 507420 \nu^{6} - 87270 \nu^{5} + \cdots - 6016 ) / 4352 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{15} + 172 \nu^{14} - 6 \nu^{13} + 5656 \nu^{12} + 590 \nu^{11} + 62536 \nu^{10} + 10000 \nu^{9} + 313744 \nu^{8} + 63047 \nu^{7} + 750460 \nu^{6} + 179198 \nu^{5} + 746264 \nu^{4} + \cdots - 64 ) / 4352 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{15} - 172 \nu^{14} - 6 \nu^{13} - 5656 \nu^{12} + 590 \nu^{11} - 62536 \nu^{10} + 10000 \nu^{9} - 313744 \nu^{8} + 63047 \nu^{7} - 750460 \nu^{6} + 179198 \nu^{5} - 746264 \nu^{4} + \cdots + 64 ) / 4352 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 93 \nu^{15} - 61 \nu^{14} - 3052 \nu^{13} - 1940 \nu^{12} - 33650 \nu^{11} - 20210 \nu^{10} - 168604 \nu^{9} - 93196 \nu^{8} - 405757 \nu^{7} - 199037 \nu^{6} - 419096 \nu^{5} + \cdots + 3776 ) / 4352 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3 \nu^{15} - 186 \nu^{14} + 172 \nu^{13} - 6044 \nu^{12} + 3374 \nu^{11} - 65476 \nu^{10} + 28372 \nu^{9} - 319888 \nu^{8} + 113635 \nu^{7} - 743834 \nu^{6} + 216896 \nu^{5} + \cdots + 1888 ) / 4352 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 93 \nu^{15} - 61 \nu^{14} + 3052 \nu^{13} - 1940 \nu^{12} + 33650 \nu^{11} - 20210 \nu^{10} + 168604 \nu^{9} - 93196 \nu^{8} + 405757 \nu^{7} - 199037 \nu^{6} + 419096 \nu^{5} + \cdots + 3776 ) / 4352 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 8 \nu^{15} + 197 \nu^{14} - 322 \nu^{13} + 6480 \nu^{12} - 4888 \nu^{11} + 71730 \nu^{10} - 37220 \nu^{9} + 361348 \nu^{8} - 152720 \nu^{7} + 876085 \nu^{6} - 325874 \nu^{5} + \cdots + 6432 ) / 4352 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{15} + \beta_{8} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{14} + \beta_{13} - 2\beta_{12} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 7 \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - 11 \beta_{8} - 10 \beta_{7} - 3 \beta_{6} + 6 \beta_{5} - \beta_{4} - 3 \beta_{2} - 3 \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 42 \beta_{14} - 16 \beta_{13} + 42 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} - 21 \beta_{8} + 25 \beta_{7} - 25 \beta_{6} - 16 \beta_{4} - 19 \beta_{3} - 14 \beta_{2} + 14 \beta _1 + 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 85 \beta_{15} + 20 \beta_{14} - 25 \beta_{13} - 20 \beta_{12} - 63 \beta_{11} - 63 \beta_{10} - 59 \beta_{9} + 169 \beta_{8} + 149 \beta_{7} + 64 \beta_{6} - 122 \beta_{5} + 25 \beta_{4} + 54 \beta_{2} + 54 \beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 746 \beta_{14} + 275 \beta_{13} - 746 \beta_{12} + 108 \beta_{11} - 108 \beta_{10} + 90 \beta_{9} + 365 \beta_{8} - 463 \beta_{7} + 463 \beta_{6} + 275 \beta_{4} + 315 \beta_{3} + 223 \beta_{2} - 223 \beta _1 - 532 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1305 \beta_{15} - 341 \beta_{14} + 463 \beta_{13} + 341 \beta_{12} + 1129 \beta_{11} + 1129 \beta_{10} + 1025 \beta_{9} - 2793 \beta_{8} - 2460 \beta_{7} - 1155 \beta_{6} + 2122 \beta_{5} - 463 \beta_{4} + \cdots - 903 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12794 \beta_{14} - 4718 \beta_{13} + 12794 \beta_{12} - 1933 \beta_{11} + 1933 \beta_{10} - 1481 \beta_{9} - 6199 \beta_{8} + 8035 \beta_{7} - 8035 \beta_{6} - 4718 \beta_{4} - 5233 \beta_{3} - 3696 \beta_{2} + \cdots + 8355 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 21487 \beta_{15} + 5726 \beta_{14} - 8035 \beta_{13} - 5726 \beta_{12} - 19445 \beta_{11} - 19445 \beta_{10} - 17401 \beta_{9} + 46923 \beta_{8} + 41461 \beta_{7} + 19974 \beta_{6} - 36054 \beta_{5} + \cdots + 15128 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 217346 \beta_{14} + 80351 \beta_{13} - 217346 \beta_{12} + 33206 \beta_{11} - 33206 \beta_{10} + 24572 \beta_{9} + 104923 \beta_{8} - 136985 \beta_{7} + 136985 \beta_{6} + 80351 \beta_{4} + \cdots - 137862 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 360615 \beta_{15} - 96401 \beta_{14} + 136985 \beta_{13} + 96401 \beta_{12} + 330903 \beta_{11} + 330903 \beta_{10} + 294303 \beta_{9} - 791903 \beta_{8} - 701174 \beta_{7} - 340559 \beta_{6} + \cdots - 254771 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3681746 \beta_{14} - 1362980 \beta_{13} + 3681746 \beta_{12} - 564289 \beta_{11} + 564289 \beta_{10} - 412025 \beta_{9} - 1775005 \beta_{8} + 2323009 \beta_{7} - 2323009 \beta_{6} + \cdots + 2312571 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6085205 \beta_{15} + 1627248 \beta_{14} - 2323009 \beta_{13} - 1627248 \beta_{12} - 5609015 \beta_{11} - 5609015 \beta_{10} - 4976403 \beta_{9} + 13384617 \beta_{8} + 11862505 \beta_{7} + \cdots + 4302122 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 62309682 \beta_{14} + 23080535 \beta_{13} - 62309682 \beta_{12} + 9559272 \beta_{11} - 9559272 \beta_{10} + 6945486 \beta_{9} + 30026021 \beta_{8} - 39328087 \beta_{7} + \cdots - 39006928 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 102854737 \beta_{15} - 27503217 \beta_{14} + 39328087 \beta_{13} + 27503217 \beta_{12} + 94949489 \beta_{11} + 94949489 \beta_{10} + 84161897 \beta_{9} - 226344721 \beta_{8} + \cdots - 72726587 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(545\) \(613\)
\(\chi(n)\) \(-\beta_{8}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.94001i
1.58886i
1.71472i
0.535137i
0.149074i
4.11290i
2.44964i
1.88319i
1.94001i
1.58886i
1.71472i
0.535137i
0.149074i
4.11290i
2.44964i
1.88319i
0 −0.382683 0.923880i 0 −0.868455 + 0.359726i 0 4.01891 + 1.66469i 0 −0.707107 + 0.707107i 0
49.2 0 −0.382683 0.923880i 0 2.39179 0.990712i 0 −2.00524 0.830597i 0 −0.707107 + 0.707107i 0
49.3 0 0.382683 + 0.923880i 0 −2.50807 + 1.03888i 0 −2.22010 0.919595i 0 −0.707107 + 0.707107i 0
49.4 0 0.382683 + 0.923880i 0 −0.429478 + 0.177896i 0 1.62064 + 0.671292i 0 −0.707107 + 0.707107i 0
145.1 0 −0.923880 0.382683i 0 −0.325635 + 0.786153i 0 −0.663444 1.60170i 0 0.707107 + 0.707107i 0
145.2 0 −0.923880 0.382683i 0 1.19125 2.87594i 0 0.497533 + 1.20115i 0 0.707107 + 0.707107i 0
145.3 0 0.923880 + 0.382683i 0 −0.554753 + 1.33929i 0 0.0103761 + 0.0250502i 0 0.707107 + 0.707107i 0
145.4 0 0.923880 + 0.382683i 0 1.10335 2.66372i 0 −1.25868 3.03872i 0 0.707107 + 0.707107i 0
433.1 0 −0.382683 + 0.923880i 0 −0.868455 0.359726i 0 4.01891 1.66469i 0 −0.707107 0.707107i 0
433.2 0 −0.382683 + 0.923880i 0 2.39179 + 0.990712i 0 −2.00524 + 0.830597i 0 −0.707107 0.707107i 0
433.3 0 0.382683 0.923880i 0 −2.50807 1.03888i 0 −2.22010 + 0.919595i 0 −0.707107 0.707107i 0
433.4 0 0.382683 0.923880i 0 −0.429478 0.177896i 0 1.62064 0.671292i 0 −0.707107 0.707107i 0
529.1 0 −0.923880 + 0.382683i 0 −0.325635 0.786153i 0 −0.663444 + 1.60170i 0 0.707107 0.707107i 0
529.2 0 −0.923880 + 0.382683i 0 1.19125 + 2.87594i 0 0.497533 1.20115i 0 0.707107 0.707107i 0
529.3 0 0.923880 0.382683i 0 −0.554753 1.33929i 0 0.0103761 0.0250502i 0 0.707107 0.707107i 0
529.4 0 0.923880 0.382683i 0 1.10335 + 2.66372i 0 −1.25868 + 3.03872i 0 0.707107 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.bq.f 16
4.b odd 2 1 408.2.ba.a 16
12.b even 2 1 1224.2.bq.e 16
17.d even 8 1 inner 816.2.bq.f 16
68.g odd 8 1 408.2.ba.a 16
68.i even 16 1 6936.2.a.bl 8
68.i even 16 1 6936.2.a.bo 8
204.p even 8 1 1224.2.bq.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.2.ba.a 16 4.b odd 2 1
408.2.ba.a 16 68.g odd 8 1
816.2.bq.f 16 1.a even 1 1 trivial
816.2.bq.f 16 17.d even 8 1 inner
1224.2.bq.e 16 12.b even 2 1
1224.2.bq.e 16 204.p even 8 1
6936.2.a.bl 8 68.i even 16 1
6936.2.a.bo 8 68.i even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 4 T_{5}^{14} + 32 T_{5}^{13} + 8 T_{5}^{12} - 56 T_{5}^{11} + 336 T_{5}^{10} - 72 T_{5}^{9} + 69 T_{5}^{8} + 2920 T_{5}^{7} + 14908 T_{5}^{6} + 34536 T_{5}^{5} + 46856 T_{5}^{4} + 41056 T_{5}^{3} + 23672 T_{5}^{2} + \cdots + 1156 \) acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{14} + 32 T^{13} + \cdots + 1156 \) Copy content Toggle raw display
$7$ \( T^{16} - 12 T^{14} - 40 T^{13} + 72 T^{12} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{15} + 64 T^{14} + \cdots + 73984 \) Copy content Toggle raw display
$13$ \( T^{16} + 100 T^{14} + 3470 T^{12} + \cdots + 150544 \) Copy content Toggle raw display
$17$ \( T^{16} + 52 T^{14} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} - 8 T^{15} + 32 T^{14} + 88 T^{13} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{16} - 8 T^{15} + 56 T^{14} + \cdots + 4624 \) Copy content Toggle raw display
$29$ \( T^{16} - 32 T^{15} + \cdots + 277155904 \) Copy content Toggle raw display
$31$ \( T^{16} + 8 T^{15} + 136 T^{14} + \cdots + 75203584 \) Copy content Toggle raw display
$37$ \( T^{16} - 8 T^{15} + \cdots + 8599223824 \) Copy content Toggle raw display
$41$ \( T^{16} - 40 T^{15} + 796 T^{14} + \cdots + 6728836 \) Copy content Toggle raw display
$43$ \( T^{16} - 24 T^{15} + \cdots + 4773151744 \) Copy content Toggle raw display
$47$ \( T^{16} + 320 T^{14} + \cdots + 33926692864 \) Copy content Toggle raw display
$53$ \( T^{16} + 24 T^{15} + \cdots + 143598555136 \) Copy content Toggle raw display
$59$ \( T^{16} - 16 T^{15} + \cdots + 29302041354496 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 307387995210256 \) Copy content Toggle raw display
$67$ \( (T^{8} - 316 T^{6} - 304 T^{5} + \cdots + 2236384)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} - 16 T^{15} + \cdots + 4226040064 \) Copy content Toggle raw display
$73$ \( T^{16} - 24 T^{15} + \cdots + 130269021184 \) Copy content Toggle raw display
$79$ \( T^{16} - 96 T^{15} + \cdots + 25305004646464 \) Copy content Toggle raw display
$83$ \( T^{16} - 160 T^{13} + \cdots + 797549019136 \) Copy content Toggle raw display
$89$ \( T^{16} + 1040 T^{14} + \cdots + 47876546426944 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 343239946813696 \) Copy content Toggle raw display
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