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Properties

Label 243.2.e.a
Level $243$
Weight $2$
Character orbit 243.e
Analytic conductor $1.940$
Analytic rank $0$
Dimension $12$
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] = [243,2,Mod(28,243)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("243.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 243.e (of order \(9\), degree \(6\), 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: \(1.94036476912\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.1952986685049.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{11} - \beta_{8} - \beta_{2}) q^{2} + ( - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 - 1) q^{4} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{5} + (2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{11} + \beta_{10} - \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} - \beta_{8} - \beta_{2}) q^{2} + ( - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 - 1) q^{4} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{5} + (2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + (10 \beta_{11} + 3 \beta_{10} - \beta_{9} + 3 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 6 q^{8} - 3 q^{10} + 6 q^{11} + 3 q^{13} + 21 q^{14} + 9 q^{16} - 9 q^{17} - 3 q^{19} - 24 q^{20} + 12 q^{22} + 12 q^{23} + 12 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} + 12 q^{31} - 27 q^{32} - 12 q^{35} - 3 q^{37} + 30 q^{38} - 15 q^{40} - 6 q^{41} - 15 q^{43} - 3 q^{44} - 3 q^{46} - 12 q^{47} - 33 q^{49} - 21 q^{50} - 45 q^{52} + 18 q^{53} - 12 q^{55} - 30 q^{56} - 51 q^{58} + 3 q^{59} - 33 q^{61} + 12 q^{62} + 12 q^{64} - 21 q^{65} - 6 q^{67} - 9 q^{68} - 15 q^{70} - 27 q^{71} + 6 q^{73} + 21 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{79} - 42 q^{80} - 12 q^{82} + 6 q^{83} + 36 q^{85} + 21 q^{86} + 42 q^{88} - 9 q^{89} + 6 q^{91} + 3 q^{92} + 48 q^{94} - 3 q^{95} + 39 q^{97} + 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} - 1030 \nu^{4} + 803 \nu^{3} - 445 \nu^{2} + 155 \nu - 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( - 6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + 1762 \nu^{4} - 1285 \nu^{3} + 649 \nu^{2} - 195 \nu + 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( - 9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + 2918 \nu^{4} - 2170 \nu^{3} + 1118 \nu^{2} - 348 \nu + 49 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} - 3178 \nu^{4} + 2411 \nu^{3} - 1286 \nu^{2} + 421 \nu - 62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} - 3643 \nu^{4} + 2724 \nu^{3} - 1417 \nu^{2} + 442 \nu - 61 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - 16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + 5122 \nu^{4} - 3802 \nu^{3} + 1958 \nu^{2} - 610 \nu + 85 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( - 16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} - 12058 \nu^{4} + 9024 \nu^{3} - 4708 \nu^{2} + 1486 \nu - 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( - 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( - 42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + 14172 \nu^{4} - 10648 \nu^{3} + 5589 \nu^{2} - 1785 \nu + 257 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 8 \beta _1 - 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{11} + 17 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} - 14 \beta_{5} + 4 \beta_{4} - 14 \beta_{3} - 8 \beta_{2} + \beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19 \beta_{11} - \beta_{10} + 31 \beta_{9} - 32 \beta_{8} + 43 \beta_{7} - 20 \beta_{6} - 41 \beta_{5} - 44 \beta_{4} + 10 \beta_{3} + 25 \beta_{2} + 40 \beta _1 + 87 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 85 \beta_{11} - 97 \beta_{10} + 55 \beta_{9} + 64 \beta_{8} + 10 \beta_{7} - 101 \beta_{6} + 19 \beta_{5} - 62 \beta_{4} + 91 \beta_{3} + 70 \beta_{2} + 31 \beta _1 + 60 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 20 \beta_{11} - 118 \beta_{10} - 134 \beta_{9} + 244 \beta_{8} - 218 \beta_{7} + \beta_{6} + 232 \beta_{5} + 157 \beta_{4} + 52 \beta_{3} - 74 \beta_{2} - 179 \beta _1 - 357 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 503 \beta_{11} + 386 \beta_{10} - 440 \beta_{9} - 47 \beta_{8} - 233 \beta_{7} + 514 \beta_{6} + 163 \beta_{5} + 466 \beta_{4} - 431 \beta_{3} - 461 \beta_{2} - 329 \beta _1 - 639 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 425 \beta_{11} + 1076 \beta_{10} + 319 \beta_{9} - 1313 \beta_{8} + 955 \beta_{7} + 502 \beta_{6} - 1013 \beta_{5} - 332 \beta_{4} - 743 \beta_{3} - 59 \beta_{2} + 631 \beta _1 + 1164 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2299 \beta_{11} - 862 \beta_{10} + 2725 \beta_{9} - 1193 \beta_{8} + 2104 \beta_{7} - 2135 \beta_{6} - 1907 \beta_{5} - 2705 \beta_{4} + 1495 \beta_{3} + 2425 \beta_{2} + 2344 \beta _1 + 4356 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4708 \beta_{11} - 6628 \beta_{10} + 985 \beta_{9} + 5506 \beta_{8} - 3107 \beta_{7} - 4679 \beta_{6} + 3238 \beta_{5} - 992 \beta_{4} + 5476 \beta_{3} + 2770 \beta_{2} - 1043 \beta _1 - 1698 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{8} - \beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
0.500000 + 0.258654i
0.500000 − 2.22827i
0.500000 + 1.00210i
0.500000 − 1.68614i
0.500000 + 0.0126039i
0.500000 + 1.27297i
0.500000 − 0.0126039i
0.500000 − 1.27297i
0.500000 − 1.00210i
0.500000 + 1.68614i
0.500000 − 0.258654i
0.500000 + 2.22827i
0.0721450 + 0.409154i 0 1.71718 − 0.625003i −1.69693 − 1.42389i 0 1.24005 + 0.451340i 0.795075 + 1.37711i 0 0.460168 − 0.797034i
28.2 0.367548 + 2.08447i 0 −2.33052 + 0.848241i 2.05537 + 1.72466i 0 −0.913694 − 0.332557i −0.508086 − 0.880031i 0 −2.83955 + 4.91825i
55.1 −2.25679 + 0.821403i 0 2.88629 − 2.42189i −0.0161638 − 0.0916693i 0 −0.444200 − 0.372728i −2.12277 + 3.67675i 0 0.111776 + 0.193601i
55.2 0.990741 − 0.360600i 0 −0.680553 + 0.571052i 0.303153 + 1.71926i 0 1.88389 + 1.58077i −1.52266 + 2.63732i 0 0.920313 + 1.59403i
109.1 −1.28765 + 1.08047i 0 0.143341 − 0.812925i −1.06142 + 0.386327i 0 −0.678777 − 3.84954i −0.987144 − 1.70978i 0 0.949332 − 1.64429i
109.2 0.614005 − 0.515212i 0 −0.235737 + 1.33693i −2.58401 + 0.940501i 0 0.412733 + 2.34072i 1.34559 + 2.33062i 0 −1.10204 + 1.90878i
136.1 −1.28765 − 1.08047i 0 0.143341 + 0.812925i −1.06142 − 0.386327i 0 −0.678777 + 3.84954i −0.987144 + 1.70978i 0 0.949332 + 1.64429i
136.2 0.614005 + 0.515212i 0 −0.235737 − 1.33693i −2.58401 − 0.940501i 0 0.412733 − 2.34072i 1.34559 − 2.33062i 0 −1.10204 − 1.90878i
190.1 −2.25679 − 0.821403i 0 2.88629 + 2.42189i −0.0161638 + 0.0916693i 0 −0.444200 + 0.372728i −2.12277 − 3.67675i 0 0.111776 − 0.193601i
190.2 0.990741 + 0.360600i 0 −0.680553 − 0.571052i 0.303153 − 1.71926i 0 1.88389 − 1.58077i −1.52266 − 2.63732i 0 0.920313 − 1.59403i
217.1 0.0721450 − 0.409154i 0 1.71718 + 0.625003i −1.69693 + 1.42389i 0 1.24005 − 0.451340i 0.795075 − 1.37711i 0 0.460168 + 0.797034i
217.2 0.367548 − 2.08447i 0 −2.33052 − 0.848241i 2.05537 − 1.72466i 0 −0.913694 + 0.332557i −0.508086 + 0.880031i 0 −2.83955 − 4.91825i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.2.e.a 12
3.b odd 2 1 243.2.e.d 12
9.c even 3 1 81.2.e.a 12
9.c even 3 1 243.2.e.b 12
9.d odd 6 1 27.2.e.a ✓ 12
9.d odd 6 1 243.2.e.c 12
27.e even 9 1 81.2.e.a 12
27.e even 9 1 inner 243.2.e.a 12
27.e even 9 1 243.2.e.b 12
27.e even 9 1 729.2.a.d 6
27.e even 9 2 729.2.c.b 12
27.f odd 18 1 27.2.e.a ✓ 12
27.f odd 18 1 243.2.e.c 12
27.f odd 18 1 243.2.e.d 12
27.f odd 18 1 729.2.a.a 6
27.f odd 18 2 729.2.c.e 12
36.h even 6 1 432.2.u.c 12
45.h odd 6 1 675.2.l.c 12
45.l even 12 2 675.2.u.b 24
108.l even 18 1 432.2.u.c 12
135.n odd 18 1 675.2.l.c 12
135.q even 36 2 675.2.u.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a ✓ 12 9.d odd 6 1
27.2.e.a ✓ 12 27.f odd 18 1
81.2.e.a 12 9.c even 3 1
81.2.e.a 12 27.e even 9 1
243.2.e.a 12 1.a even 1 1 trivial
243.2.e.a 12 27.e even 9 1 inner
243.2.e.b 12 9.c even 3 1
243.2.e.b 12 27.e even 9 1
243.2.e.c 12 9.d odd 6 1
243.2.e.c 12 27.f odd 18 1
243.2.e.d 12 3.b odd 2 1
243.2.e.d 12 27.f odd 18 1
432.2.u.c 12 36.h even 6 1
432.2.u.c 12 108.l even 18 1
675.2.l.c 12 45.h odd 6 1
675.2.l.c 12 135.n odd 18 1
675.2.u.b 24 45.l even 12 2
675.2.u.b 24 135.q even 36 2
729.2.a.a 6 27.f odd 18 1
729.2.a.d 6 27.e even 9 1
729.2.c.b 12 27.e even 9 2
729.2.c.e 12 27.f odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 3 T_{2}^{11} + 3 T_{2}^{10} + 6 T_{2}^{9} + 9 T_{2}^{8} - 27 T_{2}^{7} - 21 T_{2}^{6} + 36 T_{2}^{5} + 63 T_{2}^{4} - 117 T_{2}^{3} + 81 T_{2}^{2} - 27 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(243, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + 3 T^{10} + 6 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + 12 T^{10} + 12 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + 21 T^{10} - 65 T^{9} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + 21 T^{10} + 6 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{12} - 3 T^{11} - 24 T^{10} + 79 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} + 9 T^{11} + 72 T^{10} + 189 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{12} + 3 T^{11} + 39 T^{10} - 14 T^{9} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + 138 T^{10} + \cdots + 106929 \) Copy content Toggle raw display
$29$ \( T^{12} - 24 T^{11} + 327 T^{10} + \cdots + 45369 \) Copy content Toggle raw display
$31$ \( T^{12} - 12 T^{11} + 12 T^{10} + \cdots + 26569 \) Copy content Toggle raw display
$37$ \( T^{12} + 3 T^{11} + 66 T^{10} + \cdots + 24334489 \) Copy content Toggle raw display
$41$ \( T^{12} + 6 T^{11} - 6 T^{10} + \cdots + 11229201 \) Copy content Toggle raw display
$43$ \( T^{12} + 15 T^{11} + 138 T^{10} + \cdots + 3308761 \) Copy content Toggle raw display
$47$ \( T^{12} + 12 T^{11} + 147 T^{10} + \cdots + 42732369 \) Copy content Toggle raw display
$53$ \( (T^{6} - 9 T^{5} - 108 T^{4} + 513 T^{3} + \cdots - 12393)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 3 T^{11} - 15 T^{10} + \cdots + 176384961 \) Copy content Toggle raw display
$61$ \( T^{12} + 33 T^{11} + \cdots + 273670849 \) Copy content Toggle raw display
$67$ \( T^{12} + 6 T^{11} - 6 T^{10} + \cdots + 8288641 \) Copy content Toggle raw display
$71$ \( T^{12} + 27 T^{11} + 504 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$73$ \( T^{12} - 6 T^{11} + 210 T^{10} + \cdots + 185761 \) Copy content Toggle raw display
$79$ \( T^{12} - 21 T^{11} + 255 T^{10} + \cdots + 3508129 \) Copy content Toggle raw display
$83$ \( T^{12} - 6 T^{11} + \cdots + 6951057129 \) Copy content Toggle raw display
$89$ \( T^{12} + 9 T^{11} + \cdots + 1062042921 \) Copy content Toggle raw display
$97$ \( T^{12} - 39 T^{11} + 624 T^{10} + \cdots + 66765241 \) Copy content Toggle raw display
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