Properties

Label 1620.2.x.d
Level $1620$
Weight $2$
Character orbit 1620.x
Analytic conductor $12.936$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(53,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.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: \(12.9357651274\)
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: \( x^{16} - 12x^{14} + 100x^{12} - 408x^{10} + 1191x^{8} - 2040x^{6} + 2500x^{4} - 1500x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 540)
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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{5} + (\beta_{10} - \beta_{7} - \beta_{4} - \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{5} + (\beta_{10} - \beta_{7} - \beta_{4} - \beta_1 + 1) q^{7} + (\beta_{15} - \beta_{3} + \beta_{2}) q^{11} - \beta_{12} q^{13} + ( - \beta_{14} + \beta_{13} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{2}) q^{17} + ( - 2 \beta_{11} + 2 \beta_{10} - \beta_{7} + 2 \beta_{5}) q^{19} + ( - \beta_{15} - 2 \beta_{8} - \beta_{6}) q^{23} + (\beta_{12} - 2 \beta_{10} + 3 \beta_{7} + \beta_{5} - \beta_{4} + 3 \beta_1 + 1) q^{25} + (\beta_{14} - \beta_{13}) q^{29} + (\beta_{12} - \beta_{11} + 2 \beta_{4}) q^{31} + ( - \beta_{13} + \beta_{9} - \beta_{8} - \beta_{2}) q^{35} + (3 \beta_{11} - 3 \beta_{10}) q^{37} + (2 \beta_{15} - 3 \beta_{8} - \beta_{6}) q^{41} + (2 \beta_{12} - 3 \beta_{7} + 2 \beta_{5} + 3 \beta_{4} - 3 \beta_1 - 3) q^{43} + (2 \beta_{12} + 2 \beta_{11} + 2 \beta_1) q^{49} + ( - \beta_{14} - 2 \beta_{13} - 2 \beta_{8} - \beta_{6} + \beta_{2}) q^{53} + (5 \beta_{7} - 5 \beta_{5}) q^{55} + ( - 3 \beta_{15} + \beta_{9} - \beta_{8} - \beta_{6} - \beta_{3}) q^{59} + ( - 4 \beta_{12} + 4 \beta_{10} - 4 \beta_{5} + 3 \beta_{4} - 3) q^{61} + \beta_{14} q^{65} + ( - 3 \beta_{11} - 5 \beta_{4} - 5 \beta_1) q^{67} + ( - \beta_{14} - 3 \beta_{13} - 3 \beta_{8} - \beta_{6} - 2 \beta_{2}) q^{71} + ( - 5 \beta_{7} - \beta_{5} - 5) q^{73} + (4 \beta_{15} + \beta_{9} + \beta_{8} - 2 \beta_{6} - \beta_{3}) q^{77} + ( - 5 \beta_{12} - 5 \beta_{10} + 3 \beta_{7} - 5 \beta_{5} + 3 \beta_1) q^{79} + (3 \beta_{13} - \beta_{3}) q^{83} + ( - 5 \beta_{12} + 5 \beta_{11} - 5 \beta_{4} + 5 \beta_1) q^{85} + ( - \beta_{14} - \beta_{13} + \beta_{9} - \beta_{8} - \beta_{6} + 3 \beta_{2}) q^{89} + (\beta_{11} - \beta_{10} + \beta_{5} - 3) q^{91} + (\beta_{15} + 2 \beta_{9} - 2 \beta_{6} - 2 \beta_{3}) q^{95} + (\beta_{10} - 6 \beta_{7} - 6 \beta_{4} - 6 \beta_1 + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} + 8 q^{25} + 16 q^{31} - 24 q^{43} - 24 q^{61} - 40 q^{67} - 80 q^{73} - 40 q^{85} - 48 q^{91} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 100x^{12} - 408x^{10} + 1191x^{8} - 2040x^{6} + 2500x^{4} - 1500x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 162 \nu^{14} + 1386 \nu^{12} + 190 \nu^{10} + 31754 \nu^{8} + 610802 \nu^{6} - 1207925 \nu^{4} + 3359500 \nu^{2} + 1408250 ) / 3170625 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 519 \nu^{15} - 11728 \nu^{13} + 111150 \nu^{11} - 674877 \nu^{9} + 2149629 \nu^{7} - 4634385 \nu^{5} + 4860000 \nu^{3} - 3584500 \nu ) / 634125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3896 \nu^{15} + 58387 \nu^{13} - 477945 \nu^{11} + 2122518 \nu^{9} - 4421091 \nu^{7} + 2915550 \nu^{5} + 7915750 \nu^{3} - 16439750 \nu ) / 3170625 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4084 \nu^{14} + 46148 \nu^{12} - 378480 \nu^{10} + 1425447 \nu^{8} - 4067664 \nu^{6} + 6150300 \nu^{4} - 8357500 \nu^{2} + 4996625 ) / 3170625 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 831 \nu^{14} - 5647 \nu^{12} + 31350 \nu^{10} + 79902 \nu^{8} - 618129 \nu^{6} + 2318760 \nu^{4} - 2891475 \nu^{2} + 2262875 ) / 634125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5812 \nu^{15} + 75864 \nu^{13} - 662340 \nu^{11} + 3059571 \nu^{9} - 10067427 \nu^{7} + 21289500 \nu^{5} - 32848375 \nu^{3} + 22664250 \nu ) / 3170625 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -12\nu^{14} + 119\nu^{12} - 950\nu^{10} + 2996\nu^{8} - 8842\nu^{6} + 12730\nu^{4} - 19550\nu^{2} + 7750 ) / 7125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8514 \nu^{15} - 71783 \nu^{13} + 526680 \nu^{11} - 891137 \nu^{9} + 1391094 \nu^{7} + 5307650 \nu^{5} - 6466875 \nu^{3} + 12366500 \nu ) / 3170625 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1903 \nu^{15} - 23126 \nu^{13} + 204630 \nu^{11} - 910374 \nu^{9} + 3154293 \nu^{7} - 6153435 \nu^{5} + 8646325 \nu^{3} - 4035500 \nu ) / 634125 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10471 \nu^{14} + 144287 \nu^{12} - 1218945 \nu^{10} + 5567493 \nu^{8} - 15642291 \nu^{6} + 28273425 \nu^{4} - 27410125 \nu^{2} + 14057750 ) / 3170625 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12874 \nu^{14} - 123728 \nu^{12} + 939930 \nu^{10} - 2489517 \nu^{8} + 5326479 \nu^{6} - 1132875 \nu^{4} - 2093375 \nu^{2} + 7535875 ) / 3170625 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14343 \nu^{14} - 183596 \nu^{12} + 1544985 \nu^{10} - 6645669 \nu^{8} + 18837978 \nu^{6} - 30781425 \nu^{4} + 28939125 \nu^{2} - 11347625 ) / 3170625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19846 \nu^{15} - 243562 \nu^{13} + 2011245 \nu^{11} - 8254868 \nu^{9} + 22837116 \nu^{7} - 37808575 \nu^{5} + 37381375 \nu^{3} - 19270875 \nu ) / 3170625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20057 \nu^{15} - 242004 \nu^{13} + 2020365 \nu^{11} - 8294406 \nu^{9} + 24255447 \nu^{7} - 41296500 \nu^{5} + 47826875 \nu^{3} - 14753625 \nu ) / 3170625 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 26303 \nu^{15} + 315391 \nu^{13} - 2586660 \nu^{11} + 10264224 \nu^{9} - 27799788 \nu^{7} + 42033225 \nu^{5} - 40406750 \nu^{3} + 15650125 \nu ) / 3170625 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3\beta_{15} + \beta_{14} + 3\beta_{13} + 2\beta_{9} - 3\beta_{8} - \beta_{6} - 4\beta_{3} - 6\beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{12} + \beta_{11} + \beta_{10} + 3\beta_{7} - 2\beta_{5} - 9\beta_{4} - 3\beta _1 + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -24\beta_{15} - 13\beta_{14} - 9\beta_{13} + 19\beta_{9} - 36\beta_{8} - 2\beta_{6} - 8\beta_{3} - 42\beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{12} + 6\beta_{11} - 4\beta_{10} + 14\beta_{7} - 4\beta_{5} - 14\beta_{4} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 243 \beta_{15} + 14 \beta_{14} - 213 \beta_{13} + 43 \beta_{9} - 192 \beta_{8} + 106 \beta_{6} + 79 \beta_{3} - 114 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -95\beta_{12} + 38\beta_{11} - 133\beta_{10} + 135\beta_{7} + 38\beta_{5} + 270\beta _1 - 243 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 756 \beta_{15} + 778 \beta_{14} - 1296 \beta_{13} - 559 \beta_{9} + 36 \beta_{8} + 617 \beta_{6} + 818 \beta_{3} + 777 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 84\beta_{12} - 224\beta_{11} - 84\beta_{10} - 292\beta_{7} + 308\beta_{5} + 511\beta_{4} + 292\beta _1 - 511 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4938 \beta_{15} + 4111 \beta_{14} - 27 \beta_{13} - 5488 \beta_{9} + 8622 \beta_{8} - 1246 \beta_{6} + 1646 \beta_{3} + 9999 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6266\beta_{12} - 6266\beta_{11} + 4579\beta_{10} - 11514\beta_{7} + 4579\beta_{5} + 9999\beta_{4} - 5757\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 66066 \beta_{15} - 8758 \beta_{14} + 56736 \beta_{13} - 10736 \beta_{9} + 57609 \beta_{8} - 36137 \beta_{6} - 25973 \beta_{3} + 33858 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10260\beta_{12} - 3762\beta_{11} + 14022\beta_{10} - 12705\beta_{7} - 3762\beta_{5} - 25410\beta _1 + 22022 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 226122 \beta_{15} - 241966 \beta_{14} + 383712 \beta_{13} + 174193 \beta_{9} + 7653 \beta_{8} - 182384 \beta_{6} - 245066 \beta_{3} - 212619 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 75374 \beta_{12} + 205829 \beta_{11} + 75374 \beta_{10} + 253257 \beta_{7} - 281203 \beta_{5} - 438741 \beta_{4} - 253257 \beta _1 + 438741 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1411131 \beta_{15} - 1215167 \beta_{14} + 55944 \beta_{13} + 1634426 \beta_{9} - 2500344 \beta_{8} + 400367 \beta_{6} - 470377 \beta_{3} - 2919603 \beta_{2} ) / 15 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-\beta_{7}\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.44919 0.836690i
2.23560 + 1.29073i
−2.23560 1.29073i
1.44919 + 0.836690i
0.750156 + 0.433103i
−1.15723 0.668129i
1.15723 + 0.668129i
−0.750156 0.433103i
0.750156 0.433103i
−1.15723 + 0.668129i
1.15723 0.668129i
−0.750156 + 0.433103i
−1.44919 + 0.836690i
2.23560 1.29073i
−2.23560 + 1.29073i
1.44919 0.836690i
0 0 0 −1.70289 1.44919i 0 −0.307007 0.0822623i 0 0 0
53.2 0 0 0 −0.0455319 + 2.23560i 0 3.03906 + 0.814313i 0 0 0
53.3 0 0 0 0.0455319 2.23560i 0 3.03906 + 0.814313i 0 0 0
53.4 0 0 0 1.70289 + 1.44919i 0 −0.307007 0.0822623i 0 0 0
377.1 0 0 0 −2.10648 0.750156i 0 0.0822623 0.307007i 0 0 0
377.2 0 0 0 −1.91332 + 1.15723i 0 −0.814313 + 3.03906i 0 0 0
377.3 0 0 0 1.91332 1.15723i 0 −0.814313 + 3.03906i 0 0 0
377.4 0 0 0 2.10648 + 0.750156i 0 0.0822623 0.307007i 0 0 0
593.1 0 0 0 −2.10648 + 0.750156i 0 0.0822623 + 0.307007i 0 0 0
593.2 0 0 0 −1.91332 1.15723i 0 −0.814313 3.03906i 0 0 0
593.3 0 0 0 1.91332 + 1.15723i 0 −0.814313 3.03906i 0 0 0
593.4 0 0 0 2.10648 0.750156i 0 0.0822623 + 0.307007i 0 0 0
917.1 0 0 0 −1.70289 + 1.44919i 0 −0.307007 + 0.0822623i 0 0 0
917.2 0 0 0 −0.0455319 2.23560i 0 3.03906 0.814313i 0 0 0
917.3 0 0 0 0.0455319 + 2.23560i 0 3.03906 0.814313i 0 0 0
917.4 0 0 0 1.70289 1.44919i 0 −0.307007 + 0.0822623i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.x.d 16
3.b odd 2 1 inner 1620.2.x.d 16
5.c odd 4 1 inner 1620.2.x.d 16
9.c even 3 1 540.2.j.a 8
9.c even 3 1 inner 1620.2.x.d 16
9.d odd 6 1 540.2.j.a 8
9.d odd 6 1 inner 1620.2.x.d 16
15.e even 4 1 inner 1620.2.x.d 16
36.f odd 6 1 2160.2.w.e 8
36.h even 6 1 2160.2.w.e 8
45.h odd 6 1 2700.2.j.j 8
45.j even 6 1 2700.2.j.j 8
45.k odd 12 1 540.2.j.a 8
45.k odd 12 1 inner 1620.2.x.d 16
45.k odd 12 1 2700.2.j.j 8
45.l even 12 1 540.2.j.a 8
45.l even 12 1 inner 1620.2.x.d 16
45.l even 12 1 2700.2.j.j 8
180.v odd 12 1 2160.2.w.e 8
180.x even 12 1 2160.2.w.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.j.a 8 9.c even 3 1
540.2.j.a 8 9.d odd 6 1
540.2.j.a 8 45.k odd 12 1
540.2.j.a 8 45.l even 12 1
1620.2.x.d 16 1.a even 1 1 trivial
1620.2.x.d 16 3.b odd 2 1 inner
1620.2.x.d 16 5.c odd 4 1 inner
1620.2.x.d 16 9.c even 3 1 inner
1620.2.x.d 16 9.d odd 6 1 inner
1620.2.x.d 16 15.e even 4 1 inner
1620.2.x.d 16 45.k odd 12 1 inner
1620.2.x.d 16 45.l even 12 1 inner
2160.2.w.e 8 36.f odd 6 1
2160.2.w.e 8 36.h even 6 1
2160.2.w.e 8 180.v odd 12 1
2160.2.w.e 8 180.x even 12 1
2700.2.j.j 8 45.h odd 6 1
2700.2.j.j 8 45.j even 6 1
2700.2.j.j 8 45.k odd 12 1
2700.2.j.j 8 45.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} - 40T_{7}^{5} + 79T_{7}^{4} + 40T_{7}^{3} + 8T_{7}^{2} + 4T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{14} + 16 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + 8 T^{6} - 40 T^{5} + 79 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 40 T^{6} + 1350 T^{4} + \cdots + 62500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 9 T^{4} + 81)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 4400 T^{4} + 1000000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 50 T^{2} + 529)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} - 5900 T^{12} + \cdots + 3906250000 \) Copy content Toggle raw display
$29$ \( (T^{8} + 40 T^{6} + 1350 T^{4} + \cdots + 62500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + 18 T^{2} + 8 T + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 729)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 160 T^{6} + 19350 T^{4} + \cdots + 39062500)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 12 T^{7} + 72 T^{6} + 720 T^{5} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{8} + 5900 T^{4} + 62500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 160 T^{6} + 24600 T^{4} + \cdots + 1000000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} + 123 T^{2} - 522 T + 7569)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 20 T^{7} + 200 T^{6} + \cdots + 279841)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 160 T^{2} + 6250)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 20 T^{3} + 200 T^{2} + 940 T + 2209)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 318 T^{6} + 81243 T^{4} + \cdots + 395254161)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 9900 T^{12} + \cdots + 25628906250000 \) Copy content Toggle raw display
$89$ \( (T^{4} - 160 T^{2} + 1000)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} - 24 T^{7} + 288 T^{6} + \cdots + 22667121)^{2} \) Copy content Toggle raw display
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