Properties

Label 8022.2.a.r
Level $8022$
Weight $2$
Character orbit 8022.a
Self dual yes
Analytic conductor $64.056$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.a (trivial)

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: yes
Analytic conductor: \(64.0559925015\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 19x^{8} + 28x^{7} + 114x^{6} - 110x^{5} - 282x^{4} + 149x^{3} + 285x^{2} - 49x - 79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + ( - \beta_1 + 1) q^{5} - q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + ( - \beta_1 + 1) q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + ( - \beta_1 + 1) q^{10} + ( - \beta_{3} - \beta_{2} + 1) q^{11} - q^{12} + ( - \beta_{3} + 1) q^{13} + q^{14} + (\beta_1 - 1) q^{15} + q^{16} + (\beta_{8} + \beta_1) q^{17} + q^{18} + (\beta_{9} + \beta_{4} + 1) q^{19} + ( - \beta_1 + 1) q^{20} - q^{21} + ( - \beta_{3} - \beta_{2} + 1) q^{22} + (\beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{23} - q^{24} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_1) q^{25} + ( - \beta_{3} + 1) q^{26} - q^{27} + q^{28} + (\beta_{9} - \beta_{4} + 1) q^{29} + (\beta_1 - 1) q^{30} + ( - \beta_{4} + \beta_1 + 3) q^{31} + q^{32} + (\beta_{3} + \beta_{2} - 1) q^{33} + (\beta_{8} + \beta_1) q^{34} + ( - \beta_1 + 1) q^{35} + q^{36} + (\beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 - 1) q^{37} + (\beta_{9} + \beta_{4} + 1) q^{38} + (\beta_{3} - 1) q^{39} + ( - \beta_1 + 1) q^{40} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 + 3) q^{41} - q^{42} + ( - 2 \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{43} + ( - \beta_{3} - \beta_{2} + 1) q^{44} + ( - \beta_1 + 1) q^{45} + (\beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{46} + (\beta_{9} - \beta_{8} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{47} - q^{48} + q^{49} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_1) q^{50} + ( - \beta_{8} - \beta_1) q^{51} + ( - \beta_{3} + 1) q^{52} + ( - 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_1) q^{53} - q^{54} + ( - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{55} + q^{56} + ( - \beta_{9} - \beta_{4} - 1) q^{57} + (\beta_{9} - \beta_{4} + 1) q^{58} + ( - \beta_{9} + \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{3} - \beta_1 + 2) q^{59} + (\beta_1 - 1) q^{60} + ( - 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{61} + ( - \beta_{4} + \beta_1 + 3) q^{62} + q^{63} + q^{64} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_1 + 1) q^{65} + (\beta_{3} + \beta_{2} - 1) q^{66} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{67} + (\beta_{8} + \beta_1) q^{68} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{69} + ( - \beta_1 + 1) q^{70} + (\beta_{9} - 3 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{71} + q^{72} + ( - 2 \beta_{8} + 2 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{73} + (\beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 - 1) q^{74} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_1) q^{75} + (\beta_{9} + \beta_{4} + 1) q^{76} + ( - \beta_{3} - \beta_{2} + 1) q^{77} + (\beta_{3} - 1) q^{78} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{79} + ( - \beta_1 + 1) q^{80} + q^{81} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 + 3) q^{82} + ( - \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{83} - q^{84} + ( - 2 \beta_{9} + 3 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{2} + 3 \beta_1 - 3) q^{85} + ( - 2 \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{86} + ( - \beta_{9} + \beta_{4} - 1) q^{87} + ( - \beta_{3} - \beta_{2} + 1) q^{88} + (2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2}) q^{89} + ( - \beta_1 + 1) q^{90} + ( - \beta_{3} + 1) q^{91} + (\beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{92} + (\beta_{4} - \beta_1 - 3) q^{93} + (\beta_{9} - \beta_{8} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{94} + ( - \beta_{8} + \beta_{5} + \beta_{4} - 2 \beta_1 + 2) q^{95} - q^{96} + (2 \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{3} + 2) q^{97} + q^{98} + ( - \beta_{3} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 10 q^{3} + 10 q^{4} + 8 q^{5} - 10 q^{6} + 10 q^{7} + 10 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} - 10 q^{3} + 10 q^{4} + 8 q^{5} - 10 q^{6} + 10 q^{7} + 10 q^{8} + 10 q^{9} + 8 q^{10} + 4 q^{11} - 10 q^{12} + 6 q^{13} + 10 q^{14} - 8 q^{15} + 10 q^{16} + 5 q^{17} + 10 q^{18} + 15 q^{19} + 8 q^{20} - 10 q^{21} + 4 q^{22} + 12 q^{23} - 10 q^{24} - 2 q^{25} + 6 q^{26} - 10 q^{27} + 10 q^{28} + 7 q^{29} - 8 q^{30} + 28 q^{31} + 10 q^{32} - 4 q^{33} + 5 q^{34} + 8 q^{35} + 10 q^{36} - 3 q^{37} + 15 q^{38} - 6 q^{39} + 8 q^{40} + 24 q^{41} - 10 q^{42} + 4 q^{43} + 4 q^{44} + 8 q^{45} + 12 q^{46} + 16 q^{47} - 10 q^{48} + 10 q^{49} - 2 q^{50} - 5 q^{51} + 6 q^{52} + 5 q^{53} - 10 q^{54} - q^{55} + 10 q^{56} - 15 q^{57} + 7 q^{58} + 17 q^{59} - 8 q^{60} + 15 q^{61} + 28 q^{62} + 10 q^{63} + 10 q^{64} + 14 q^{65} - 4 q^{66} + 6 q^{67} + 5 q^{68} - 12 q^{69} + 8 q^{70} + 5 q^{71} + 10 q^{72} + 16 q^{73} - 3 q^{74} + 2 q^{75} + 15 q^{76} + 4 q^{77} - 6 q^{78} - 5 q^{79} + 8 q^{80} + 10 q^{81} + 24 q^{82} + 24 q^{83} - 10 q^{84} - 19 q^{85} + 4 q^{86} - 7 q^{87} + 4 q^{88} + 17 q^{89} + 8 q^{90} + 6 q^{91} + 12 q^{92} - 28 q^{93} + 16 q^{94} + 15 q^{95} - 10 q^{96} + 20 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 19x^{8} + 28x^{7} + 114x^{6} - 110x^{5} - 282x^{4} + 149x^{3} + 285x^{2} - 49x - 79 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 229 \nu^{9} - 2973 \nu^{8} + 1250 \nu^{7} + 49952 \nu^{6} - 46396 \nu^{5} - 247296 \nu^{4} + 163808 \nu^{3} + 461973 \nu^{2} - 136188 \nu - 238207 ) / 18602 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 489 \nu^{9} - 2587 \nu^{8} + 13902 \nu^{7} + 49542 \nu^{6} - 94908 \nu^{5} - 251022 \nu^{4} + 128906 \nu^{3} + 341569 \nu^{2} + 57434 \nu - 28117 ) / 18602 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1035 \nu^{9} + 4339 \nu^{8} + 15958 \nu^{7} - 70776 \nu^{6} - 71692 \nu^{5} + 337626 \nu^{4} + 135320 \nu^{3} - 558531 \nu^{2} - 94928 \nu + 234811 ) / 18602 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1237 \nu^{9} - 5743 \nu^{8} - 15424 \nu^{7} + 86773 \nu^{6} + 35234 \nu^{5} - 353133 \nu^{4} - 1142 \nu^{3} + 450947 \nu^{2} - 9444 \nu - 119763 ) / 9301 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1517 \nu^{9} - 5479 \nu^{8} - 23156 \nu^{7} + 84083 \nu^{6} + 95113 \nu^{5} - 362380 \nu^{4} - 138938 \nu^{3} + 507824 \nu^{2} + 75368 \nu - 159203 ) / 9301 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1818 \nu^{9} - 3335 \nu^{8} - 32398 \nu^{7} + 41662 \nu^{6} + 164831 \nu^{5} - 130262 \nu^{4} - 280558 \nu^{3} + 129262 \nu^{2} + 139568 \nu - 34183 ) / 9301 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5003 \nu^{9} - 9633 \nu^{8} - 87796 \nu^{7} + 120350 \nu^{6} + 435548 \nu^{5} - 364498 \nu^{4} - 699314 \nu^{3} + 298399 \nu^{2} + 225360 \nu - 17763 ) / 18602 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5605 \nu^{9} - 5345 \nu^{8} - 106280 \nu^{7} + 35508 \nu^{6} + 574984 \nu^{5} + 99738 \nu^{4} - 982554 \nu^{3} - 440123 \nu^{2} + 353760 \nu + 157869 ) / 18602 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - 3\beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{9} - 10 \beta_{8} - 14 \beta_{7} + 14 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21 \beta_{9} + 9 \beta_{8} - 47 \beta_{7} + 22 \beta_{6} - 19 \beta_{5} + 5 \beta_{4} + 22 \beta_{3} + 19 \beta_{2} + 96 \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 148 \beta_{9} - 100 \beta_{8} - 190 \beta_{7} + 184 \beta_{6} - 49 \beta_{5} + 57 \beta_{4} + 50 \beta_{3} + 35 \beta_{2} + 100 \beta _1 + 273 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 346 \beta_{9} + 57 \beta_{8} - 673 \beta_{7} + 389 \beta_{6} - 291 \beta_{5} + 133 \beta_{4} + 341 \beta_{3} + 291 \beta_{2} + 1137 \beta _1 + 119 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1929 \beta_{9} - 1061 \beta_{8} - 2638 \beta_{7} + 2442 \beta_{6} - 863 \beta_{5} + 873 \beta_{4} + 906 \beta_{3} + 701 \beta_{2} + 1880 \beta _1 + 2934 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5352 \beta_{9} + 54 \beta_{8} - 9607 \beta_{7} + 6287 \beta_{6} - 4221 \beta_{5} + 2427 \beta_{4} + 4896 \beta_{3} + 4173 \beta_{2} + 14314 \beta _1 + 2787 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.78925
2.49471
1.57301
1.48373
0.667588
−0.632147
−1.09237
−1.38859
−1.81749
−3.07769
1.00000 −1.00000 1.00000 −2.78925 −1.00000 1.00000 1.00000 1.00000 −2.78925
1.2 1.00000 −1.00000 1.00000 −1.49471 −1.00000 1.00000 1.00000 1.00000 −1.49471
1.3 1.00000 −1.00000 1.00000 −0.573010 −1.00000 1.00000 1.00000 1.00000 −0.573010
1.4 1.00000 −1.00000 1.00000 −0.483732 −1.00000 1.00000 1.00000 1.00000 −0.483732
1.5 1.00000 −1.00000 1.00000 0.332412 −1.00000 1.00000 1.00000 1.00000 0.332412
1.6 1.00000 −1.00000 1.00000 1.63215 −1.00000 1.00000 1.00000 1.00000 1.63215
1.7 1.00000 −1.00000 1.00000 2.09237 −1.00000 1.00000 1.00000 1.00000 2.09237
1.8 1.00000 −1.00000 1.00000 2.38859 −1.00000 1.00000 1.00000 1.00000 2.38859
1.9 1.00000 −1.00000 1.00000 2.81749 −1.00000 1.00000 1.00000 1.00000 2.81749
1.10 1.00000 −1.00000 1.00000 4.07769 −1.00000 1.00000 1.00000 1.00000 4.07769
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(191\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8022.2.a.r 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8022.2.a.r 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\( T_{5}^{10} - 8T_{5}^{9} + 8T_{5}^{8} + 76T_{5}^{7} - 180T_{5}^{6} - 98T_{5}^{5} + 486T_{5}^{4} - 69T_{5}^{3} - 321T_{5}^{2} - 10T_{5} + 36 \) Copy content Toggle raw display
\( T_{11}^{10} - 4 T_{11}^{9} - 46 T_{11}^{8} + 193 T_{11}^{7} + 437 T_{11}^{6} - 2246 T_{11}^{5} - 805 T_{11}^{4} + 8662 T_{11}^{3} - 1835 T_{11}^{2} - 8768 T_{11} + 252 \) Copy content Toggle raw display
\( T_{13}^{10} - 6 T_{13}^{9} - 46 T_{13}^{8} + 367 T_{13}^{7} - 95 T_{13}^{6} - 3938 T_{13}^{5} + 8761 T_{13}^{4} - 3052 T_{13}^{3} - 7653 T_{13}^{2} + 7258 T_{13} - 1636 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 8 T^{9} + 8 T^{8} + 76 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 4 T^{9} - 46 T^{8} + 193 T^{7} + \cdots + 252 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} - 46 T^{8} + \cdots - 1636 \) Copy content Toggle raw display
$17$ \( T^{10} - 5 T^{9} - 48 T^{8} + \cdots + 1183 \) Copy content Toggle raw display
$19$ \( T^{10} - 15 T^{9} + 47 T^{8} + \cdots - 28812 \) Copy content Toggle raw display
$23$ \( T^{10} - 12 T^{9} - 37 T^{8} + \cdots - 2224 \) Copy content Toggle raw display
$29$ \( T^{10} - 7 T^{9} - 65 T^{8} + \cdots + 11632 \) Copy content Toggle raw display
$31$ \( T^{10} - 28 T^{9} + 294 T^{8} + \cdots + 375219 \) Copy content Toggle raw display
$37$ \( T^{10} + 3 T^{9} - 166 T^{8} + \cdots - 4517068 \) Copy content Toggle raw display
$41$ \( T^{10} - 24 T^{9} + 20 T^{8} + \cdots + 1680336 \) Copy content Toggle raw display
$43$ \( T^{10} - 4 T^{9} - 269 T^{8} + \cdots + 1828932 \) Copy content Toggle raw display
$47$ \( T^{10} - 16 T^{9} + 15 T^{8} + \cdots - 219968 \) Copy content Toggle raw display
$53$ \( T^{10} - 5 T^{9} - 297 T^{8} + \cdots + 70936944 \) Copy content Toggle raw display
$59$ \( T^{10} - 17 T^{9} + \cdots + 299304512 \) Copy content Toggle raw display
$61$ \( T^{10} - 15 T^{9} - 104 T^{8} + \cdots + 106124 \) Copy content Toggle raw display
$67$ \( T^{10} - 6 T^{9} - 245 T^{8} + \cdots - 1392 \) Copy content Toggle raw display
$71$ \( T^{10} - 5 T^{9} - 507 T^{8} + \cdots - 381484599 \) Copy content Toggle raw display
$73$ \( T^{10} - 16 T^{9} + \cdots + 2382040541 \) Copy content Toggle raw display
$79$ \( T^{10} + 5 T^{9} - 325 T^{8} + \cdots + 5550421 \) Copy content Toggle raw display
$83$ \( T^{10} - 24 T^{9} + \cdots - 287072528 \) Copy content Toggle raw display
$89$ \( T^{10} - 17 T^{9} - 330 T^{8} + \cdots - 6915376 \) Copy content Toggle raw display
$97$ \( T^{10} - 20 T^{9} - 247 T^{8} + \cdots - 93971984 \) Copy content Toggle raw display
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