↑

Properties

Label 54.2.e.b
Level $54$
Weight $2$
Character orbit 54.e
Analytic conductor $0.431$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,2,Mod(7,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, 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("54.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 54.e (of order \(9\), degree \(6\), 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: \(0.431192170915\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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.

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

\(\beta_{1}\)\(=\) \( ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060 ) / 218 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} - 2069 \nu^{4} + 5579 \nu^{3} - 6002 \nu^{2} + 5080 \nu - 1706 ) / 218 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263 ) / 218 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + 20118 \nu^{4} - 19981 \nu^{3} + 12972 \nu^{2} - 4712 \nu + 524 ) / 218 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + 31404 \nu^{4} - 25822 \nu^{3} + 15545 \nu^{2} - 4618 \nu + 555 ) / 218 \) 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
\(\beta_{7}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + 279 \nu^{2} - 116 \nu + 26 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + 52308 \nu^{4} - 55710 \nu^{3} + 41571 \nu^{2} - 19689 \nu + 4350 ) / 218 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} - 60760 \nu^{4} + 56973 \nu^{3} - 36296 \nu^{2} + 13927 \nu - 2201 ) / 218 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} - 84255 \nu^{4} + 84604 \nu^{3} - 58431 \nu^{2} + 25899 \nu - 5194 ) / 218 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + 99754 \nu^{5} - 123498 \nu^{4} + 112914 \nu^{3} - 71337 \nu^{2} + 28743 \nu - 5469 ) / 218 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} - 2 \beta_{4} - 31 \beta_{3} - \beta_{2} - 7 \beta _1 + 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 29 \beta _1 + 77 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} - 17 \beta_{4} + 161 \beta_{3} + 17 \beta_{2} + 104 \beta _1 - 121 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} - 46 \beta_{5} - 56 \beta_{4} + 212 \beta_{3} + 149 \beta_{2} - 82 \beta _1 - 535 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + 515 \beta_{5} + 151 \beta_{4} - 724 \beta_{3} - 19 \beta_{2} - 829 \beta _1 + 257 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + 812 \beta_{5} + 640 \beta_{4} - 1951 \beta_{3} - 1108 \beta_{2} - 406 \beta _1 + 3359 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} - 2308 \beta_{5} - 521 \beta_{4} + 2375 \beta_{3} - 913 \beta_{2} + 4949 \beta _1 + 1451 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} - 7273 \beta_{5} - 5168 \beta_{4} + 14021 \beta_{3} + 6587 \beta_{2} + 7973 \beta _1 - 18736 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 − 1.96356i
0.500000 + 0.677980i
0.500000 + 2.42499i
0.500000 − 1.74095i
0.500000 − 2.42499i
0.500000 + 1.74095i
0.500000 + 1.96356i
0.500000 − 0.677980i
0.500000 + 1.80139i
0.500000 + 0.168222i
0.500000 − 1.80139i
0.500000 − 0.168222i
−0.939693 + 0.342020i −1.14517 + 1.29945i 0.766044 − 0.642788i 0.617090 + 3.49969i 0.631669 − 1.61276i −0.244752 − 0.205371i −0.500000 + 0.866025i −0.377165 − 2.97620i −1.77684 − 3.07758i
7.2 −0.939693 + 0.342020i 0.552775 − 1.64147i 0.766044 − 0.642788i −0.177398 − 1.00607i 0.0419788 + 1.73154i 2.04289 + 1.71418i −0.500000 + 0.866025i −2.38888 − 1.81473i 0.510796 + 0.884725i
13.1 0.173648 + 0.984808i 0.140451 − 1.72635i −0.939693 + 0.342020i 2.42692 + 2.03643i 1.72451 − 0.161460i −3.46344 − 1.26059i −0.500000 − 0.866025i −2.96055 − 0.484935i −1.58406 + 2.74367i
13.2 0.173648 + 0.984808i 1.56529 + 0.741539i −0.939693 + 0.342020i −3.10057 − 2.60168i −0.458464 + 1.67027i 0.144365 + 0.0525446i −0.500000 − 0.866025i 1.90024 + 2.32144i 2.02375 − 3.50524i
25.1 0.173648 − 0.984808i 0.140451 + 1.72635i −0.939693 − 0.342020i 2.42692 − 2.03643i 1.72451 + 0.161460i −3.46344 + 1.26059i −0.500000 + 0.866025i −2.96055 + 0.484935i −1.58406 − 2.74367i
25.2 0.173648 − 0.984808i 1.56529 − 0.741539i −0.939693 − 0.342020i −3.10057 + 2.60168i −0.458464 − 1.67027i 0.144365 − 0.0525446i −0.500000 + 0.866025i 1.90024 − 2.32144i 2.02375 + 3.50524i
31.1 −0.939693 − 0.342020i −1.14517 − 1.29945i 0.766044 + 0.642788i 0.617090 − 3.49969i 0.631669 + 1.61276i −0.244752 + 0.205371i −0.500000 − 0.866025i −0.377165 + 2.97620i −1.77684 + 3.07758i
31.2 −0.939693 − 0.342020i 0.552775 + 1.64147i 0.766044 + 0.642788i −0.177398 + 1.00607i 0.0419788 − 1.73154i 2.04289 − 1.71418i −0.500000 − 0.866025i −2.38888 + 1.81473i 0.510796 − 0.884725i
43.1 0.766044 + 0.642788i −1.36085 + 1.07149i 0.173648 + 0.984808i 0.696050 + 0.253341i −1.73121 − 0.0539310i 0.717657 − 4.07003i −0.500000 + 0.866025i 0.703829 − 2.91627i 0.370360 + 0.641483i
43.2 0.766044 + 0.642788i 0.247510 − 1.71428i 0.173648 + 0.984808i −1.96209 − 0.714144i 1.29152 − 1.15411i −0.696712 + 3.95125i −0.500000 + 0.866025i −2.87748 − 0.848600i −1.04401 − 1.80828i
49.1 0.766044 − 0.642788i −1.36085 − 1.07149i 0.173648 − 0.984808i 0.696050 − 0.253341i −1.73121 + 0.0539310i 0.717657 + 4.07003i −0.500000 − 0.866025i 0.703829 + 2.91627i 0.370360 − 0.641483i
49.2 0.766044 − 0.642788i 0.247510 + 1.71428i 0.173648 − 0.984808i −1.96209 + 0.714144i 1.29152 + 1.15411i −0.696712 − 3.95125i −0.500000 − 0.866025i −2.87748 + 0.848600i −1.04401 + 1.80828i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.2
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 54.2.e.b ✓ 12
3.b odd 2 1 162.2.e.b 12
4.b odd 2 1 432.2.u.b 12
9.c even 3 1 486.2.e.f 12
9.c even 3 1 486.2.e.h 12
9.d odd 6 1 486.2.e.e 12
9.d odd 6 1 486.2.e.g 12
27.e even 9 1 inner 54.2.e.b ✓ 12
27.e even 9 1 486.2.e.f 12
27.e even 9 1 486.2.e.h 12
27.e even 9 1 1458.2.a.g 6
27.e even 9 2 1458.2.c.f 12
27.f odd 18 1 162.2.e.b 12
27.f odd 18 1 486.2.e.e 12
27.f odd 18 1 486.2.e.g 12
27.f odd 18 1 1458.2.a.f 6
27.f odd 18 2 1458.2.c.g 12
108.j odd 18 1 432.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.b ✓ 12 1.a even 1 1 trivial
54.2.e.b ✓ 12 27.e even 9 1 inner
162.2.e.b 12 3.b odd 2 1
162.2.e.b 12 27.f odd 18 1
432.2.u.b 12 4.b odd 2 1
432.2.u.b 12 108.j odd 18 1
486.2.e.e 12 9.d odd 6 1
486.2.e.e 12 27.f odd 18 1
486.2.e.f 12 9.c even 3 1
486.2.e.f 12 27.e even 9 1
486.2.e.g 12 9.d odd 6 1
486.2.e.g 12 27.f odd 18 1
486.2.e.h 12 9.c even 3 1
486.2.e.h 12 27.e even 9 1
1458.2.a.f 6 27.f odd 18 1
1458.2.a.g 6 27.e even 9 1
1458.2.c.f 12 27.e even 9 2
1458.2.c.g 12 27.f odd 18 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 \(S_{2}^{\mathrm{new}}(54, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{10} + 18 T^{8} - 18 T^{7} + \cdots + 729 \) 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
show more
show less