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Properties

Label 162.2.e.b
Level $162$
Weight $2$
Character orbit 162.e
Analytic conductor $1.294$
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] = [162,2,Mod(19,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.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: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 54)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 3\nu^{5} + 14\nu^{4} - 23\nu^{3} + 41\nu^{2} - 30\nu + 11 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060 ) / 218 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{11} - 11 \nu^{10} + 115 \nu^{9} - 435 \nu^{8} + 1781 \nu^{7} - 4226 \nu^{6} + 9493 \nu^{5} - 13637 \nu^{4} + 16775 \nu^{3} - 12275 \nu^{2} + 5163 \nu - 882 ) / 218 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263 ) / 218 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 26 \nu^{11} + 34 \nu^{10} - 187 \nu^{9} - 449 \nu^{8} + 1590 \nu^{7} - 6865 \nu^{6} + 12623 \nu^{5} - 20118 \nu^{4} + 19981 \nu^{3} - 12972 \nu^{2} + 4712 \nu - 524 ) / 218 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83 ) / 218 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 39 \nu^{11} - 58 \nu^{10} + 210 \nu^{9} - 3126 \nu^{8} + 8816 \nu^{7} - 26048 \nu^{6} + 44604 \nu^{5} - 65711 \nu^{4} + 63925 \nu^{3} - 42784 \nu^{2} + 16878 \nu - 3184 ) / 218 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26 \nu^{11} + 252 \nu^{10} - 1277 \nu^{9} + 4892 \nu^{8} - 13234 \nu^{7} + 28887 \nu^{6} - 47327 \nu^{5} + 60760 \nu^{4} - 56973 \nu^{3} + 36296 \nu^{2} - 13927 \nu + 2201 ) / 218 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19 \nu^{11} + 268 \nu^{10} - 1365 \nu^{9} + 5713 \nu^{8} - 15666 \nu^{7} + 35787 \nu^{6} - 59173 \nu^{5} + 78267 \nu^{4} - 73416 \nu^{3} + 47888 \nu^{2} - 18038 \nu + 3256 ) / 218 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 106 \nu^{11} - 583 \nu^{10} + 3043 \nu^{9} - 9321 \nu^{8} + 24960 \nu^{7} - 47943 \nu^{6} + 77593 \nu^{5} - 93068 \nu^{4} + 88252 \nu^{3} - 58487 \nu^{2} + 25228 \nu - 5544 ) / 218 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 106 \nu^{11} + 583 \nu^{10} - 3043 \nu^{9} + 9321 \nu^{8} - 24960 \nu^{7} + 47943 \nu^{6} - 77593 \nu^{5} + 92850 \nu^{4} - 87816 \nu^{3} + 56961 \nu^{2} - 23920 \nu + 4454 ) / 218 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{3} + \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{5} + 2\beta_{4} - 2\beta_{3} + 2\beta_{2} - 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - \beta_{7} - 6 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} - 8 \beta_{2} - 5 \beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{11} - 3 \beta_{9} + 8 \beta_{8} + 5 \beta_{7} - 6 \beta_{6} - 5 \beta_{5} - 14 \beta_{4} + 16 \beta_{3} - 24 \beta_{2} - 4 \beta _1 + 40 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10 \beta_{10} - 13 \beta_{9} + 21 \beta_{8} + 12 \beta_{7} + 28 \beta_{6} - 16 \beta_{5} + 3 \beta_{4} - 23 \beta_{3} + 29 \beta_{2} + 24 \beta _1 + 49 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 10 \beta_{11} + 25 \beta_{10} + 8 \beta_{9} + 3 \beta_{8} - 16 \beta_{7} + 60 \beta_{6} + 43 \beta_{5} + 84 \beta_{4} - 110 \beta_{3} + 187 \beta_{2} + 49 \beta _1 - 169 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 23 \beta_{11} - 9 \beta_{10} + 87 \beta_{9} - 142 \beta_{8} - 88 \beta_{7} - 105 \beta_{6} + 121 \beta_{5} + 70 \beta_{4} + 25 \beta_{3} + 12 \beta_{2} - 79 \beta _1 - 395 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 108 \beta_{11} - 188 \beta_{10} + 32 \beta_{9} - 297 \beta_{8} + 6 \beta_{7} - 431 \beta_{6} - 265 \beta_{5} - 429 \beta_{4} + 652 \beta_{3} - 1153 \beta_{2} - 363 \beta _1 + 607 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 133 \beta_{11} - 263 \beta_{10} - 511 \beta_{9} + 483 \beta_{8} + 533 \beta_{7} + 261 \beta_{6} - 1040 \beta_{5} - 891 \beta_{4} + 478 \beta_{3} - 1265 \beta_{2} + 49 \beta _1 + 2735 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 976 \beta_{11} + 849 \beta_{10} - 717 \beta_{9} + 2720 \beta_{8} + 476 \beta_{7} + 2778 \beta_{6} + 1021 \beta_{5} + 1633 \beta_{4} - 3353 \beta_{3} + 5760 \beta_{2} + 2135 \beta _1 - 1121 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 45 \beta_{11} + 2914 \beta_{10} + 2441 \beta_{9} + 558 \beta_{8} - 2784 \beta_{7} + 742 \beta_{6} + 7907 \beta_{5} + 7221 \beta_{4} - 6041 \beta_{3} + 13586 \beta_{2} + 2025 \beta _1 - 16958 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.500000 − 0.677980i
0.500000 + 1.96356i
0.500000 + 0.168222i
0.500000 + 1.80139i
0.500000 − 1.74095i
0.500000 + 2.42499i
0.500000 + 1.74095i
0.500000 − 2.42499i
0.500000 − 0.168222i
0.500000 − 1.80139i
0.500000 + 0.677980i
0.500000 − 1.96356i
0.939693 − 0.342020i 0 0.766044 − 0.642788i −0.617090 − 3.49969i 0 −0.244752 − 0.205371i 0.500000 − 0.866025i 0 −1.77684 − 3.07758i
19.2 0.939693 − 0.342020i 0 0.766044 − 0.642788i 0.177398 + 1.00607i 0 2.04289 + 1.71418i 0.500000 − 0.866025i 0 0.510796 + 0.884725i
37.1 −0.766044 + 0.642788i 0 0.173648 − 0.984808i −0.696050 + 0.253341i 0 0.717657 + 4.07003i 0.500000 + 0.866025i 0 0.370360 − 0.641483i
37.2 −0.766044 + 0.642788i 0 0.173648 − 0.984808i 1.96209 − 0.714144i 0 −0.696712 − 3.95125i 0.500000 + 0.866025i 0 −1.04401 + 1.80828i
73.1 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i −2.42692 + 2.03643i 0 −3.46344 + 1.26059i 0.500000 − 0.866025i 0 −1.58406 − 2.74367i
73.2 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i 3.10057 − 2.60168i 0 0.144365 − 0.0525446i 0.500000 − 0.866025i 0 2.02375 + 3.50524i
91.1 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i −2.42692 − 2.03643i 0 −3.46344 − 1.26059i 0.500000 + 0.866025i 0 −1.58406 + 2.74367i
91.2 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i 3.10057 + 2.60168i 0 0.144365 + 0.0525446i 0.500000 + 0.866025i 0 2.02375 − 3.50524i
127.1 −0.766044 − 0.642788i 0 0.173648 + 0.984808i −0.696050 − 0.253341i 0 0.717657 − 4.07003i 0.500000 − 0.866025i 0 0.370360 + 0.641483i
127.2 −0.766044 − 0.642788i 0 0.173648 + 0.984808i 1.96209 + 0.714144i 0 −0.696712 + 3.95125i 0.500000 − 0.866025i 0 −1.04401 − 1.80828i
145.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.617090 + 3.49969i 0 −0.244752 + 0.205371i 0.500000 + 0.866025i 0 −1.77684 + 3.07758i
145.2 0.939693 + 0.342020i 0 0.766044 + 0.642788i 0.177398 − 1.00607i 0 2.04289 − 1.71418i 0.500000 + 0.866025i 0 0.510796 − 0.884725i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.2
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 162.2.e.b 12
3.b odd 2 1 54.2.e.b ✓ 12
9.c even 3 1 486.2.e.e 12
9.c even 3 1 486.2.e.g 12
9.d odd 6 1 486.2.e.f 12
9.d odd 6 1 486.2.e.h 12
12.b even 2 1 432.2.u.b 12
27.e even 9 1 inner 162.2.e.b 12
27.e even 9 1 486.2.e.e 12
27.e even 9 1 486.2.e.g 12
27.e even 9 1 1458.2.a.f 6
27.e even 9 2 1458.2.c.g 12
27.f odd 18 1 54.2.e.b ✓ 12
27.f odd 18 1 486.2.e.f 12
27.f odd 18 1 486.2.e.h 12
27.f odd 18 1 1458.2.a.g 6
27.f odd 18 2 1458.2.c.f 12
108.l even 18 1 432.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.b ✓ 12 3.b odd 2 1
54.2.e.b ✓ 12 27.f odd 18 1
162.2.e.b 12 1.a even 1 1 trivial
162.2.e.b 12 27.e even 9 1 inner
432.2.u.b 12 12.b even 2 1
432.2.u.b 12 108.l even 18 1
486.2.e.e 12 9.c even 3 1
486.2.e.e 12 27.e even 9 1
486.2.e.f 12 9.d odd 6 1
486.2.e.f 12 27.f odd 18 1
486.2.e.g 12 9.c even 3 1
486.2.e.g 12 27.e even 9 1
486.2.e.h 12 9.d odd 6 1
486.2.e.h 12 27.f odd 18 1
1458.2.a.f 6 27.e even 9 1
1458.2.a.g 6 27.f odd 18 1
1458.2.c.f 12 27.f odd 18 2
1458.2.c.g 12 27.e even 9 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 3 T_{5}^{11} + 9 T_{5}^{10} - 24 T_{5}^{9} + 162 T_{5}^{8} + 27 T_{5}^{7} + 1053 T_{5}^{6} - 5184 T_{5}^{5} + 3564 T_{5}^{4} + 3672 T_{5}^{3} + 2592 T_{5}^{2} + 7776 T_{5} + 5184 \) acting on \(S_{2}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + 9 T^{10} - 24 T^{9} + \cdots + 5184 \) Copy content Toggle raw display
$7$ \( T^{12} + 3 T^{11} + 24 T^{10} + 88 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{12} - 12 T^{11} + 90 T^{10} - 537 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{12} - 12 T^{11} + 48 T^{10} + \cdots + 23104 \) Copy content Toggle raw display
$17$ \( T^{12} - 6 T^{11} + 54 T^{10} + \cdots + 110889 \) Copy content Toggle raw display
$19$ \( T^{12} + 9 T^{11} + 78 T^{10} + \cdots + 94249 \) Copy content Toggle raw display
$23$ \( T^{12} + 30 T^{11} + 414 T^{10} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{12} + 15 T^{11} + 81 T^{10} + \cdots + 5184 \) Copy content Toggle raw display
$31$ \( T^{12} + 81 T^{10} + 421 T^{9} + \cdots + 4032064 \) Copy content Toggle raw display
$37$ \( T^{12} + 15 T^{11} + \cdots + 142659136 \) Copy content Toggle raw display
$41$ \( T^{12} - 12 T^{11} + 117 T^{10} + \cdots + 2653641 \) Copy content Toggle raw display
$43$ \( T^{12} - 9 T^{11} + 36 T^{10} + \cdots + 49674304 \) Copy content Toggle raw display
$47$ \( T^{12} - 9 T^{11} + 99 T^{10} + \cdots + 419904 \) Copy content Toggle raw display
$53$ \( (T^{6} - 6 T^{5} - 63 T^{4} - 3 T^{3} + \cdots - 72)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 12 T^{11} + 9 T^{10} + \cdots + 82464561 \) Copy content Toggle raw display
$61$ \( T^{12} + 36 T^{11} + 531 T^{10} + \cdots + 1000000 \) Copy content Toggle raw display
$67$ \( T^{12} - 36 T^{11} + \cdots + 249393368449 \) Copy content Toggle raw display
$71$ \( T^{12} + 12 T^{11} + \cdots + 488586816 \) Copy content Toggle raw display
$73$ \( T^{12} + 21 T^{11} + 390 T^{10} + \cdots + 72361 \) Copy content Toggle raw display
$79$ \( T^{12} - 39 T^{11} + \cdots + 591851584 \) Copy content Toggle raw display
$83$ \( T^{12} + 18 T^{11} + \cdots + 13756474944 \) Copy content Toggle raw display
$89$ \( T^{12} + 12 T^{11} + \cdots + 126899100441 \) Copy content Toggle raw display
$97$ \( T^{12} - 39 T^{11} + \cdots + 373532435929 \) Copy content Toggle raw display
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