↑

Properties

Label 273.2.l.c
Level $273$
Weight $2$
Character orbit 273.l
Analytic conductor $2.180$
Analytic rank $0$
Dimension $20$
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] = [273,2,Mod(16,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.l (of order \(3\), degree \(2\), 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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 18 x^{18} - 4 x^{17} + 211 x^{16} - 59 x^{15} + 1458 x^{14} - 526 x^{13} + 7324 x^{12} + \cdots + 1369 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 18 x^{18} - 4 x^{17} + 211 x^{16} - 59 x^{15} + 1458 x^{14} - 526 x^{13} + 7324 x^{12} + \cdots + 1369 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 40\!\cdots\!97 \nu^{19} + \cdots - 54\!\cdots\!82 ) / 13\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!58 \nu^{19} + \cdots + 31\!\cdots\!39 ) / 66\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 95\!\cdots\!54 \nu^{19} + \cdots + 32\!\cdots\!33 ) / 13\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 69\!\cdots\!90 \nu^{19} + \cdots + 76\!\cdots\!03 ) / 59\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22\!\cdots\!28 \nu^{19} + \cdots + 69\!\cdots\!02 ) / 17\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 87\!\cdots\!09 \nu^{19} + \cdots + 33\!\cdots\!41 ) / 51\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 76\!\cdots\!29 \nu^{19} + \cdots + 39\!\cdots\!22 ) / 31\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 54\!\cdots\!61 \nu^{19} + \cdots + 41\!\cdots\!48 ) / 22\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\!\cdots\!27 \nu^{19} + \cdots + 49\!\cdots\!78 ) / 85\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!62 \nu^{19} + \cdots - 95\!\cdots\!36 ) / 31\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 81\!\cdots\!78 \nu^{19} + \cdots + 33\!\cdots\!27 ) / 22\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 79\!\cdots\!49 \nu^{19} + \cdots - 49\!\cdots\!74 ) / 17\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 96\!\cdots\!47 \nu^{19} + \cdots + 53\!\cdots\!65 ) / 17\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!66 \nu^{19} + \cdots + 17\!\cdots\!44 ) / 17\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 17\!\cdots\!18 \nu^{19} + \cdots + 35\!\cdots\!66 ) / 25\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 16\!\cdots\!22 \nu^{19} + \cdots + 53\!\cdots\!99 ) / 22\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 17\!\cdots\!04 \nu^{19} + \cdots - 43\!\cdots\!26 ) / 22\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 14\!\cdots\!24 \nu^{19} + \cdots - 55\!\cdots\!06 ) / 17\!\cdots\!98 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{16} + 4\beta_{7} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} - \beta_{13} + 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} - 7\beta_{16} + \beta_{14} + \beta_{13} - \beta_{11} - 23\beta_{7} - \beta_{6} - 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{18} - \beta_{17} + 2 \beta_{16} + 9 \beta_{13} - \beta_{12} - \beta_{11} - 2 \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{19} - 3 \beta_{17} + \beta_{16} - 10 \beta_{15} - 10 \beta_{13} - 13 \beta_{12} - 2 \beta_{11} + \cdots + 149 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{19} - 15 \beta_{18} - 26 \beta_{16} + 67 \beta_{15} + 68 \beta_{14} + \beta_{13} + \cdots + 181 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 30 \beta_{19} - 106 \beta_{18} + 42 \beta_{17} + 289 \beta_{16} + 79 \beta_{15} - 79 \beta_{14} + \cdots - 1012 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 50 \beta_{19} - 60 \beta_{18} + 79 \beta_{17} + 50 \beta_{16} - 490 \beta_{15} - 474 \beta_{14} + \cdots + 294 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 163 \beta_{19} + 853 \beta_{18} - 2039 \beta_{16} + 3 \beta_{15} + 587 \beta_{14} + 584 \beta_{13} + \cdots + 308 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1136 \beta_{19} + 1690 \beta_{18} - 567 \beta_{17} + 1575 \beta_{16} + 179 \beta_{15} - 179 \beta_{14} + \cdots - 3004 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1560 \beta_{19} - 110 \beta_{18} - 3680 \beta_{17} + 1560 \beta_{16} - 4287 \beta_{15} - 64 \beta_{14} + \cdots + 49354 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5501 \beta_{19} - 10744 \beta_{18} - 17337 \beta_{16} + 22872 \beta_{15} + 24606 \beta_{14} + \cdots + 52573 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 27908 \beta_{19} - 46312 \beta_{18} + 30151 \beta_{17} + 81854 \beta_{16} + 30325 \beta_{15} + \cdots - 349412 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 48951 \beta_{19} - 10016 \beta_{18} + 24025 \beta_{17} + 48951 \beta_{16} - 174053 \beta_{15} + \cdots + 236300 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 119629 \beta_{19} + 337294 \beta_{18} - 668559 \beta_{16} + 9811 \beta_{15} + 227269 \beta_{14} + \cdots + 175656 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 827974 \beta_{19} + 660926 \beta_{18} - 141583 \beta_{17} + 626550 \beta_{16} + 133291 \beta_{15} + \cdots - 1955655 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 995799 \beta_{19} - 12046 \beta_{18} - 1828222 \beta_{17} + 995799 \beta_{16} - 1659313 \beta_{15} + \cdots + 17778807 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 3387188 \beta_{19} - 4486483 \beta_{18} - 7877160 \beta_{16} + 7647734 \beta_{15} + \cdots + 16936871 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7}\) \(-1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
−1.35774 − 2.35168i
−1.27537 − 2.20901i
−0.828334 − 1.43472i
−0.707433 − 1.22531i
−0.130586 − 0.226181i
0.328258 + 0.568560i
0.613260 + 1.06220i
0.904928 + 1.56738i
1.14017 + 1.97483i
1.31285 + 2.27393i
−1.35774 + 2.35168i
−1.27537 + 2.20901i
−0.828334 + 1.43472i
−0.707433 + 1.22531i
−0.130586 + 0.226181i
0.328258 − 0.568560i
0.613260 − 1.06220i
0.904928 − 1.56738i
1.14017 − 1.97483i
1.31285 − 2.27393i
−2.71549 −0.500000 + 0.866025i 5.37387 1.94413 − 3.36734i 1.35774 − 2.35168i 1.94064 − 1.79830i −9.16172 −0.500000 − 0.866025i −5.27927 + 9.14396i
16.2 −2.55074 −0.500000 + 0.866025i 4.50630 −1.40932 + 2.44101i 1.27537 − 2.20901i −2.27422 + 1.35201i −6.39292 −0.500000 − 0.866025i 3.59481 − 6.22639i
16.3 −1.65667 −0.500000 + 0.866025i 0.744548 1.05011 − 1.81885i 0.828334 − 1.43472i −1.49221 + 2.18479i 2.07987 −0.500000 − 0.866025i −1.73968 + 3.01322i
16.4 −1.41487 −0.500000 + 0.866025i 0.00184802 −1.42962 + 2.47618i 0.707433 − 1.22531i 2.56093 + 0.664549i 2.82712 −0.500000 − 0.866025i 2.02273 − 3.50347i
16.5 −0.261171 −0.500000 + 0.866025i −1.93179 0.708533 − 1.22721i 0.130586 − 0.226181i 2.55312 − 0.693954i 1.02687 −0.500000 − 0.866025i −0.185048 + 0.320513i
16.6 0.656517 −0.500000 + 0.866025i −1.56899 −0.0109774 + 0.0190133i −0.328258 + 0.568560i −1.55912 + 2.13756i −2.34310 −0.500000 − 0.866025i −0.00720682 + 0.0124826i
16.7 1.22652 −0.500000 + 0.866025i −0.495647 −2.10660 + 3.64874i −0.613260 + 1.06220i 0.113533 − 2.64331i −3.06096 −0.500000 − 0.866025i −2.58379 + 4.47526i
16.8 1.80986 −0.500000 + 0.866025i 1.27558 1.98776 − 3.44291i −0.904928 + 1.56738i 1.70815 + 2.02045i −1.31110 −0.500000 − 0.866025i 3.59756 − 6.23116i
16.9 2.28034 −0.500000 + 0.866025i 3.19995 −1.46862 + 2.54373i −1.14017 + 1.97483i 0.102378 + 2.64377i 2.73629 −0.500000 − 0.866025i −3.34896 + 5.80057i
16.10 2.62571 −0.500000 + 0.866025i 4.89433 0.734607 − 1.27238i −1.31285 + 2.27393i −2.15321 − 1.53742i 7.59966 −0.500000 − 0.866025i 1.92886 − 3.34089i
256.1 −2.71549 −0.500000 − 0.866025i 5.37387 1.94413 + 3.36734i 1.35774 + 2.35168i 1.94064 + 1.79830i −9.16172 −0.500000 + 0.866025i −5.27927 − 9.14396i
256.2 −2.55074 −0.500000 − 0.866025i 4.50630 −1.40932 − 2.44101i 1.27537 + 2.20901i −2.27422 − 1.35201i −6.39292 −0.500000 + 0.866025i 3.59481 + 6.22639i
256.3 −1.65667 −0.500000 − 0.866025i 0.744548 1.05011 + 1.81885i 0.828334 + 1.43472i −1.49221 − 2.18479i 2.07987 −0.500000 + 0.866025i −1.73968 − 3.01322i
256.4 −1.41487 −0.500000 − 0.866025i 0.00184802 −1.42962 − 2.47618i 0.707433 + 1.22531i 2.56093 − 0.664549i 2.82712 −0.500000 + 0.866025i 2.02273 + 3.50347i
256.5 −0.261171 −0.500000 − 0.866025i −1.93179 0.708533 + 1.22721i 0.130586 + 0.226181i 2.55312 + 0.693954i 1.02687 −0.500000 + 0.866025i −0.185048 − 0.320513i
256.6 0.656517 −0.500000 − 0.866025i −1.56899 −0.0109774 − 0.0190133i −0.328258 − 0.568560i −1.55912 − 2.13756i −2.34310 −0.500000 + 0.866025i −0.00720682 − 0.0124826i
256.7 1.22652 −0.500000 − 0.866025i −0.495647 −2.10660 − 3.64874i −0.613260 − 1.06220i 0.113533 + 2.64331i −3.06096 −0.500000 + 0.866025i −2.58379 − 4.47526i
256.8 1.80986 −0.500000 − 0.866025i 1.27558 1.98776 + 3.44291i −0.904928 − 1.56738i 1.70815 − 2.02045i −1.31110 −0.500000 + 0.866025i 3.59756 + 6.23116i
256.9 2.28034 −0.500000 − 0.866025i 3.19995 −1.46862 − 2.54373i −1.14017 − 1.97483i 0.102378 − 2.64377i 2.73629 −0.500000 + 0.866025i −3.34896 − 5.80057i
256.10 2.62571 −0.500000 − 0.866025i 4.89433 0.734607 + 1.27238i −1.31285 − 2.27393i −2.15321 + 1.53742i 7.59966 −0.500000 + 0.866025i 1.92886 + 3.34089i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.l.c yes 20
3.b odd 2 1 819.2.s.f 20
7.c even 3 1 273.2.j.c ✓ 20
13.c even 3 1 273.2.j.c ✓ 20
21.h odd 6 1 819.2.n.f 20
39.i odd 6 1 819.2.n.f 20
91.h even 3 1 inner 273.2.l.c yes 20
273.s odd 6 1 819.2.s.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.j.c ✓ 20 7.c even 3 1
273.2.j.c ✓ 20 13.c even 3 1
273.2.l.c yes 20 1.a even 1 1 trivial
273.2.l.c yes 20 91.h even 3 1 inner
819.2.n.f 20 21.h odd 6 1
819.2.n.f 20 39.i odd 6 1
819.2.s.f 20 3.b odd 2 1
819.2.s.f 20 273.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 18T_{2}^{8} + 2T_{2}^{7} + 113T_{2}^{6} - 23T_{2}^{5} - 290T_{2}^{4} + 76T_{2}^{3} + 271T_{2}^{2} - 81T_{2} - 37 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 18 T^{8} + \cdots - 37)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + 41 T^{18} + \cdots + 21904 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} + 8 T^{19} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( (T^{10} - 102 T^{8} + \cdots + 18804)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 4253126656 \) Copy content Toggle raw display
$23$ \( (T^{10} - 14 T^{9} + \cdots + 156324)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + 9 T^{19} + \cdots + 1882384 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 64085935104 \) Copy content Toggle raw display
$37$ \( (T^{10} + 18 T^{9} + \cdots + 17337)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 5541909136 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 29\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 6617497104 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 406192078224 \) Copy content Toggle raw display
$59$ \( (T^{10} - 15 T^{9} + \cdots - 3044508)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 83534872576 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 1342437380496 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 21167631081 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{10} - 20 T^{9} + \cdots + 548326464)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 2 T^{9} + \cdots - 451948)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 479579795289 \) Copy content Toggle raw display
show more
show less