Properties

Label 1170.2.w.i
Level $1170$
Weight $2$
Character orbit 1170.w
Analytic conductor $9.342$
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] = [1170,2,Mod(307,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.w (of order \(4\), 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: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - q^{4} - \beta_{9} q^{5} + ( - \beta_{15} - \beta_{13} - \beta_{11} - \beta_{9} - \beta_{8} + \beta_1) q^{7} + \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - q^{4} - \beta_{9} q^{5} + ( - \beta_{15} - \beta_{13} - \beta_{11} - \beta_{9} - \beta_{8} + \beta_1) q^{7} + \beta_{6} q^{8} + \beta_{11} q^{10} + (\beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3}) q^{11} + ( - \beta_{15} - \beta_{13} - \beta_{8}) q^{13} + (\beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{3} + \beta_{2}) q^{14} + q^{16} + ( - \beta_{12} + \beta_{9} - \beta_{5}) q^{17} + (\beta_{14} + \beta_{11} - \beta_{8}) q^{19} + \beta_{9} q^{20} + ( - \beta_{12} - \beta_{8} - \beta_{7} - \beta_{4} - \beta_{2} + \beta_1) q^{22} + ( - \beta_{15} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{23} + ( - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{4} - \beta_{3} + \cdots + 1) q^{25}+ \cdots + (\beta_{15} + 2 \beta_{14} + \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} + 3 \beta_{9} - 4 \beta_{8} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 4 q^{11} - 8 q^{13} + 16 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{22} + 16 q^{23} + 16 q^{25} - 4 q^{26} + 12 q^{31} - 4 q^{34} + 12 q^{35} - 32 q^{37} + 4 q^{38} - 4 q^{41} + 16 q^{43} + 4 q^{44} + 16 q^{46} - 16 q^{47} + 80 q^{49} - 8 q^{50} + 8 q^{52} + 44 q^{53} - 20 q^{55} + 12 q^{59} - 32 q^{61} + 12 q^{62} - 16 q^{64} + 28 q^{65} - 4 q^{68} - 36 q^{70} - 16 q^{71} + 4 q^{76} - 32 q^{77} - 4 q^{82} - 16 q^{83} - 40 q^{85} + 16 q^{86} - 4 q^{88} + 4 q^{89} + 76 q^{91} - 16 q^{92} + 40 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 201 \nu^{15} + 1232 \nu^{14} + 7384 \nu^{13} + 37440 \nu^{12} + 108716 \nu^{11} + 429632 \nu^{10} + 820204 \nu^{9} + 2334848 \nu^{8} + 3364836 \nu^{7} + 6288064 \nu^{6} + \cdots + 906752 ) / 113152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 319 \nu^{15} + 2136 \nu^{14} - 10920 \nu^{13} + 63232 \nu^{12} - 147956 \nu^{11} + 694432 \nu^{10} - 1015604 \nu^{9} + 3488096 \nu^{8} - 3759164 \nu^{7} + \cdots - 211712 ) / 113152 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 319 \nu^{15} - 2136 \nu^{14} - 10920 \nu^{13} - 63232 \nu^{12} - 147956 \nu^{11} - 694432 \nu^{10} - 1015604 \nu^{9} - 3488096 \nu^{8} - 3759164 \nu^{7} + \cdots + 211712 ) / 113152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 235 \nu^{15} + 44 \nu^{14} + 7176 \nu^{13} + 832 \nu^{12} + 83076 \nu^{11} + 1200 \nu^{10} + 461380 \nu^{9} - 59568 \nu^{8} + 1322476 \nu^{7} - 396752 \nu^{6} + 2066184 \nu^{5} + \cdots + 88960 ) / 56576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 235 \nu^{15} - 44 \nu^{14} + 7176 \nu^{13} - 832 \nu^{12} + 83076 \nu^{11} - 1200 \nu^{10} + 461380 \nu^{9} + 59568 \nu^{8} + 1322476 \nu^{7} + 396752 \nu^{6} + 2066184 \nu^{5} + \cdots - 88960 ) / 56576 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 725 \nu^{15} + 22776 \nu^{13} + 273788 \nu^{11} + 1589116 \nu^{9} + 4674452 \nu^{7} + 6751736 \nu^{5} + 4099732 \nu^{3} + 697504 \nu ) / 113152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 707 \nu^{15} + 1040 \nu^{14} - 21320 \nu^{13} + 30784 \nu^{12} - 240036 \nu^{11} + 337792 \nu^{10} - 1242980 \nu^{9} + 1694784 \nu^{8} - 2928076 \nu^{7} + \cdots + 126464 ) / 113152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 527 \nu^{15} - 144 \nu^{14} + 15912 \nu^{13} - 4160 \nu^{12} + 180404 \nu^{11} - 43296 \nu^{10} + 957780 \nu^{9} - 189056 \nu^{8} + 2462076 \nu^{7} - 260480 \nu^{6} + \cdots + 190208 ) / 56576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 527 \nu^{15} - 144 \nu^{14} - 15912 \nu^{13} - 4160 \nu^{12} - 180404 \nu^{11} - 43296 \nu^{10} - 957780 \nu^{9} - 189056 \nu^{8} - 2462076 \nu^{7} - 260480 \nu^{6} + \cdots + 190208 ) / 56576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 527 \nu^{15} - 1288 \nu^{14} + 15912 \nu^{13} - 39936 \nu^{12} + 180404 \nu^{11} - 470528 \nu^{10} + 957780 \nu^{9} - 2643040 \nu^{8} + 2462076 \nu^{7} - 7342048 \nu^{6} + \cdots - 312320 ) / 56576 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 87 \nu^{15} + 80 \nu^{14} - 2664 \nu^{13} + 2368 \nu^{12} - 30740 \nu^{11} + 25984 \nu^{10} - 166356 \nu^{9} + 130368 \nu^{8} - 429980 \nu^{7} + 302912 \nu^{6} - 475304 \nu^{5} + \cdots + 9728 ) / 8704 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 87 \nu^{15} - 80 \nu^{14} - 2664 \nu^{13} - 2368 \nu^{12} - 30740 \nu^{11} - 25984 \nu^{10} - 166356 \nu^{9} - 130368 \nu^{8} - 429980 \nu^{7} - 302912 \nu^{6} - 475304 \nu^{5} + \cdots - 9728 ) / 8704 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1323 \nu^{15} + 1856 \nu^{14} - 38792 \nu^{13} + 57408 \nu^{12} - 418244 \nu^{11} + 673408 \nu^{10} - 2018116 \nu^{9} + 3751744 \nu^{8} - 4204268 \nu^{7} + \cdots + 81408 ) / 113152 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2655 \nu^{15} - 1840 \nu^{14} + 80808 \nu^{13} - 54080 \nu^{12} + 926580 \nu^{11} - 586368 \nu^{10} + 5000948 \nu^{9} - 2873408 \nu^{8} + 13158844 \nu^{7} + \cdots - 109568 ) / 113152 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2655 \nu^{15} + 1840 \nu^{14} + 80808 \nu^{13} + 54080 \nu^{12} + 926580 \nu^{11} + 586368 \nu^{10} + 5000948 \nu^{9} + 2873408 \nu^{8} + 13158844 \nu^{7} + \cdots + 109568 ) / 113152 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - \beta_{13} - \beta_{11} + \beta_{5} - \beta_{4} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 4 \beta_{15} - 4 \beta_{14} - 3 \beta_{12} - 3 \beta_{11} - 4 \beta_{9} + 4 \beta_{8} + 4 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + 10 \beta_{14} + 9 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} - 11 \beta_{5} + 11 \beta_{4} + \beta_{3} - \beta_{2} - 9 \beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 37 \beta_{15} + 37 \beta_{14} + 22 \beta_{12} + 28 \beta_{11} + 38 \beta_{9} - 38 \beta_{8} - 6 \beta_{7} - 54 \beta_{6} - 36 \beta_{5} - 36 \beta_{4} - 14 \beta_{3} - 14 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20 \beta_{15} - 99 \beta_{14} - 79 \beta_{13} + 33 \beta_{12} - 112 \beta_{11} + 13 \beta_{10} + 30 \beta_{9} + 17 \beta_{8} + 107 \beta_{5} - 107 \beta_{4} - 11 \beta_{3} + 11 \beta_{2} + 79 \beta _1 - 238 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 354 \beta_{15} - 346 \beta_{14} + 8 \beta_{13} - 185 \beta_{12} - 275 \beta_{11} + 8 \beta_{10} - 371 \beta_{9} + 363 \beta_{8} + 98 \beta_{7} + 582 \beta_{6} + 336 \beta_{5} + 336 \beta_{4} + 162 \beta_{3} + 162 \beta_{2} + 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 258 \beta_{15} + 974 \beta_{14} + 716 \beta_{13} - 408 \beta_{12} + 1124 \beta_{11} - 134 \beta_{10} - 336 \beta_{9} - 202 \beta_{8} - 1020 \beta_{5} + 1020 \beta_{4} + 106 \beta_{3} - 106 \beta_{2} - 716 \beta _1 + 2168 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3422 \beta_{15} + 3262 \beta_{14} - 160 \beta_{13} + 1662 \beta_{12} + 2706 \beta_{11} - 160 \beta_{10} + 3648 \beta_{9} - 3488 \beta_{8} - 1204 \beta_{7} - 5900 \beta_{6} - 3168 \beta_{5} - 3168 \beta_{4} + \cdots - 160 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2842 \beta_{15} - 9518 \beta_{14} - 6676 \beta_{13} + 4546 \beta_{12} - 11222 \beta_{11} + 1280 \beta_{10} + 3420 \beta_{9} + 2140 \beta_{8} + 9688 \beta_{5} - 9688 \beta_{4} - 1036 \beta_{3} + 1036 \beta_{2} + \cdots - 20324 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 33218 \beta_{15} - 30982 \beta_{14} + 2236 \beta_{13} - 15394 \beta_{12} - 26542 \beta_{11} + 2236 \beta_{10} - 35986 \beta_{9} + 33750 \beta_{8} + 13384 \beta_{7} + 58488 \beta_{6} + 29980 \beta_{5} + \cdots + 2236 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 29068 \beta_{15} + 92566 \beta_{14} + 63498 \beta_{13} - 48226 \beta_{12} + 111724 \beta_{11} - 11818 \beta_{10} - 33384 \beta_{9} - 21566 \beta_{8} - 92002 \beta_{5} + 92002 \beta_{4} + \cdots + 193260 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 323184 \beta_{15} + 295960 \beta_{14} - 27224 \beta_{13} + 144802 \beta_{12} + 259686 \beta_{11} - 27224 \beta_{10} + 355650 \beta_{9} - 328426 \beta_{8} - 142108 \beta_{7} - 574140 \beta_{6} + \cdots - 27224 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 285864 \beta_{15} - 897744 \beta_{14} - 611880 \beta_{13} + 498468 \beta_{12} - 1110348 \beta_{11} + 107036 \beta_{10} + 318952 \beta_{9} + 211916 \beta_{8} + 874328 \beta_{5} + \cdots - 1852744 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3149120 \beta_{15} - 2839152 \beta_{14} + 309968 \beta_{13} - 1374128 \beta_{12} - 2537240 \beta_{11} + 309968 \beta_{10} - 3518812 \beta_{9} + 3208844 \beta_{8} + 1473080 \beta_{7} + \cdots + 309968 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
0.304860i
1.54171i
1.63314i
3.14581i
3.03462i
2.42515i
1.09662i
0.821039i
0.304860i
1.54171i
1.63314i
3.14581i
3.03462i
2.42515i
1.09662i
0.821039i
1.00000i 0 −1.00000 −2.22565 + 0.215569i 0 2.86620 1.00000i 0 0.215569 + 2.22565i
307.2 1.00000i 0 −1.00000 −1.95232 1.09015i 0 0.720033 1.00000i 0 −1.09015 + 1.95232i
307.3 1.00000i 0 −1.00000 −1.91479 + 1.15480i 0 −4.24302 1.00000i 0 1.15480 + 1.91479i
307.4 1.00000i 0 −1.00000 −0.227886 + 2.22443i 0 −4.05336 1.00000i 0 2.22443 + 0.227886i
307.5 1.00000i 0 −1.00000 0.628922 2.14580i 0 5.15192 1.00000i 0 −2.14580 0.628922i
307.6 1.00000i 0 −1.00000 1.43504 1.71484i 0 −3.30195 1.00000i 0 −1.71484 1.43504i
307.7 1.00000i 0 −1.00000 2.09731 + 0.775429i 0 −0.946681 1.00000i 0 0.775429 2.09731i
307.8 1.00000i 0 −1.00000 2.15939 + 0.580562i 0 3.80685 1.00000i 0 0.580562 2.15939i
343.1 1.00000i 0 −1.00000 −2.22565 0.215569i 0 2.86620 1.00000i 0 0.215569 2.22565i
343.2 1.00000i 0 −1.00000 −1.95232 + 1.09015i 0 0.720033 1.00000i 0 −1.09015 1.95232i
343.3 1.00000i 0 −1.00000 −1.91479 1.15480i 0 −4.24302 1.00000i 0 1.15480 1.91479i
343.4 1.00000i 0 −1.00000 −0.227886 2.22443i 0 −4.05336 1.00000i 0 2.22443 0.227886i
343.5 1.00000i 0 −1.00000 0.628922 + 2.14580i 0 5.15192 1.00000i 0 −2.14580 + 0.628922i
343.6 1.00000i 0 −1.00000 1.43504 + 1.71484i 0 −3.30195 1.00000i 0 −1.71484 + 1.43504i
343.7 1.00000i 0 −1.00000 2.09731 0.775429i 0 −0.946681 1.00000i 0 0.775429 + 2.09731i
343.8 1.00000i 0 −1.00000 2.15939 0.580562i 0 3.80685 1.00000i 0 0.580562 + 2.15939i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.w.i 16
3.b odd 2 1 390.2.t.b yes 16
5.c odd 4 1 1170.2.m.i 16
13.d odd 4 1 1170.2.m.i 16
15.d odd 2 1 1950.2.t.e 16
15.e even 4 1 390.2.j.b 16
15.e even 4 1 1950.2.j.e 16
39.f even 4 1 390.2.j.b 16
65.f even 4 1 inner 1170.2.w.i 16
195.j odd 4 1 1950.2.t.e 16
195.n even 4 1 1950.2.j.e 16
195.u odd 4 1 390.2.t.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.j.b 16 15.e even 4 1
390.2.j.b 16 39.f even 4 1
390.2.t.b yes 16 3.b odd 2 1
390.2.t.b yes 16 195.u odd 4 1
1170.2.m.i 16 5.c odd 4 1
1170.2.m.i 16 13.d odd 4 1
1170.2.w.i 16 1.a even 1 1 trivial
1170.2.w.i 16 65.f even 4 1 inner
1950.2.j.e 16 15.e even 4 1
1950.2.j.e 16 195.n even 4 1
1950.2.t.e 16 15.d odd 2 1
1950.2.t.e 16 195.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{8} - 48T_{7}^{6} - 12T_{7}^{5} + 728T_{7}^{4} + 224T_{7}^{3} - 3652T_{7}^{2} - 768T_{7} + 2176 \) Copy content Toggle raw display
\( T_{11}^{16} + 4 T_{11}^{15} + 8 T_{11}^{14} - 8 T_{11}^{13} + 988 T_{11}^{12} + 3552 T_{11}^{11} + 6336 T_{11}^{10} + 1216 T_{11}^{9} + 272304 T_{11}^{8} + 900800 T_{11}^{7} + 1463424 T_{11}^{6} + 2190720 T_{11}^{5} + \cdots + 58003456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 8 T^{14} + 8 T^{13} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} - 48 T^{6} - 12 T^{5} + 728 T^{4} + \cdots + 2176)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 4 T^{15} + 8 T^{14} + \cdots + 58003456 \) Copy content Toggle raw display
$13$ \( T^{16} + 8 T^{15} + 42 T^{14} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 4 T^{15} + 8 T^{14} + \cdots + 2534464 \) Copy content Toggle raw display
$19$ \( T^{16} + 4 T^{15} + 8 T^{14} + 80 T^{13} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{16} - 16 T^{15} + \cdots + 77275104256 \) Copy content Toggle raw display
$29$ \( T^{16} + 196 T^{14} + \cdots + 21484523776 \) Copy content Toggle raw display
$31$ \( T^{16} - 12 T^{15} + \cdots + 837825347584 \) Copy content Toggle raw display
$37$ \( (T^{8} + 16 T^{7} - 86 T^{6} + \cdots - 1045376)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 4 T^{15} + \cdots + 79370665984 \) Copy content Toggle raw display
$43$ \( T^{16} - 16 T^{15} + \cdots + 349241344 \) Copy content Toggle raw display
$47$ \( (T^{8} + 8 T^{7} - 56 T^{6} - 416 T^{5} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} - 44 T^{15} + \cdots + 50182602625024 \) Copy content Toggle raw display
$59$ \( T^{16} - 12 T^{15} + \cdots + 48729781571584 \) Copy content Toggle raw display
$61$ \( (T^{8} + 16 T^{7} - 192 T^{6} + \cdots - 3982336)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 364 T^{14} + \cdots + 48809181184 \) Copy content Toggle raw display
$71$ \( T^{16} + 16 T^{15} + \cdots + 846046756864 \) Copy content Toggle raw display
$73$ \( T^{16} + 768 T^{14} + \cdots + 8806459834624 \) Copy content Toggle raw display
$79$ \( T^{16} + 624 T^{14} + \cdots + 604860841984 \) Copy content Toggle raw display
$83$ \( (T^{8} + 8 T^{7} - 352 T^{6} + \cdots - 713728)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 4 T^{15} + \cdots + 17644340224 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 157829170008064 \) Copy content Toggle raw display
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