↑

Properties

Label 1000.2.f.d
Level $1000$
Weight $2$
Character orbit 1000.f
Analytic conductor $7.985$
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] = [1000,2,Mod(749,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.f (of order \(2\), degree \(1\), 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} + 6 x^{18} - 5 x^{17} - 3 x^{16} + 20 x^{15} - 28 x^{14} + 24 x^{13} + 16 x^{12} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} + 6 x^{18} - 5 x^{17} - 3 x^{16} + 20 x^{15} - 28 x^{14} + 24 x^{13} + 16 x^{12} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{19} - 11 \nu^{18} + 32 \nu^{17} - 43 \nu^{16} - 23 \nu^{15} + 90 \nu^{14} - 200 \nu^{13} + \cdots - 10752 ) / 3072 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3 \nu^{19} + 23 \nu^{18} - 32 \nu^{17} + 23 \nu^{16} + 35 \nu^{15} - 50 \nu^{14} + 144 \nu^{13} + \cdots + 8704 ) / 3072 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11 \nu^{19} + 3 \nu^{18} + 28 \nu^{17} - 81 \nu^{16} + 55 \nu^{15} + 42 \nu^{14} - 200 \nu^{13} + \cdots - 8704 ) / 3072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7 \nu^{19} - \nu^{18} - 2 \nu^{17} + 21 \nu^{16} - 17 \nu^{15} + 8 \nu^{14} + 76 \nu^{13} + \cdots + 512 ) / 1536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{19} + 9 \nu^{18} - 24 \nu^{17} + 17 \nu^{16} + 13 \nu^{15} - 70 \nu^{14} + 92 \nu^{13} + \cdots + 3584 ) / 768 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15 \nu^{19} + 7 \nu^{18} + 28 \nu^{17} - 93 \nu^{16} + 75 \nu^{15} + 34 \nu^{14} - 296 \nu^{13} + \cdots - 14848 ) / 3072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{19} + 27 \nu^{18} - 68 \nu^{17} + 7 \nu^{16} + 95 \nu^{15} - 278 \nu^{14} + 224 \nu^{13} + \cdots + 7680 ) / 3072 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7 \nu^{19} - 19 \nu^{18} + 24 \nu^{17} + 13 \nu^{16} - 55 \nu^{15} + 114 \nu^{14} - 56 \nu^{13} + \cdots - 512 ) / 1536 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5 \nu^{19} + 4 \nu^{18} - 3 \nu^{17} - 11 \nu^{16} + 14 \nu^{15} - 37 \nu^{14} - 2 \nu^{13} + \cdots + 256 ) / 768 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 39 \nu^{19} - 55 \nu^{18} + 36 \nu^{17} + 45 \nu^{16} - 219 \nu^{15} + 254 \nu^{14} - 120 \nu^{12} + \cdots - 7680 ) / 3072 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17 \nu^{19} - 39 \nu^{18} + 70 \nu^{17} - 17 \nu^{16} - 111 \nu^{15} + 268 \nu^{14} - 256 \nu^{13} + \cdots - 10752 ) / 1536 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21 \nu^{19} + 23 \nu^{18} - 46 \nu^{17} - 31 \nu^{16} + 95 \nu^{15} - 188 \nu^{14} + 52 \nu^{13} + \cdots + 3072 ) / 1536 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 39 \nu^{19} + 107 \nu^{18} - 176 \nu^{17} + 59 \nu^{16} + 263 \nu^{15} - 698 \nu^{14} + \cdots + 27136 ) / 3072 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 45 \nu^{19} - 129 \nu^{18} + 208 \nu^{17} + 31 \nu^{16} - 341 \nu^{15} + 862 \nu^{14} - 520 \nu^{13} + \cdots - 4608 ) / 3072 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 13 \nu^{19} - 37 \nu^{18} + 46 \nu^{17} - 15 \nu^{16} - 101 \nu^{15} + 200 \nu^{14} - 182 \nu^{13} + \cdots - 7168 ) / 768 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 51 \nu^{19} + 163 \nu^{18} - 260 \nu^{17} + 143 \nu^{16} + 407 \nu^{15} - 1014 \nu^{14} + \cdots + 48640 ) / 3072 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 25 \nu^{19} + 93 \nu^{18} - 132 \nu^{17} + 97 \nu^{16} + 225 \nu^{15} - 530 \nu^{14} + \cdots + 27136 ) / 1536 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} + \beta_{18} + 2 \beta_{17} - \beta_{16} - \beta_{15} + \beta_{14} + 2 \beta_{13} - \beta_{12} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{19} + \beta_{18} - \beta_{17} - \beta_{15} - \beta_{8} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{18} + 2 \beta_{17} - \beta_{15} - \beta_{14} - \beta_{12} - \beta_{10} - \beta_{9} + 2 \beta_{7} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{19} - 3 \beta_{18} + \beta_{16} + \beta_{15} - \beta_{14} - 2 \beta_{13} - \beta_{12} - 2 \beta_{10} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{17} - \beta_{15} + 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{18} + 2 \beta_{17} - 2 \beta_{16} + 5 \beta_{15} + \beta_{14} + 6 \beta_{13} - \beta_{12} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5 \beta_{19} + 7 \beta_{18} + 2 \beta_{17} - \beta_{16} + 5 \beta_{15} - 3 \beta_{14} + 4 \beta_{13} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7 \beta_{19} - 17 \beta_{18} - 15 \beta_{17} + 4 \beta_{16} + 3 \beta_{15} - 10 \beta_{14} - 24 \beta_{13} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 18 \beta_{18} - 18 \beta_{17} + 4 \beta_{16} + \beta_{15} - 7 \beta_{14} - 28 \beta_{13} + 17 \beta_{12} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 21 \beta_{19} - 11 \beta_{18} + 16 \beta_{17} - 7 \beta_{16} + 5 \beta_{15} + 27 \beta_{14} + 10 \beta_{13} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 15 \beta_{19} + 23 \beta_{18} - 17 \beta_{17} + 8 \beta_{16} + 7 \beta_{15} + 32 \beta_{14} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 24 \beta_{19} + 10 \beta_{18} - 22 \beta_{17} + 22 \beta_{16} - 11 \beta_{15} + \beta_{14} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 47 \beta_{19} - 5 \beta_{18} + 34 \beta_{17} - 5 \beta_{16} - 47 \beta_{15} + 9 \beta_{14} - 12 \beta_{13} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 13 \beta_{19} - 37 \beta_{18} - 83 \beta_{17} + 36 \beta_{16} - 73 \beta_{15} - 18 \beta_{14} + \cdots + 50 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 64 \beta_{19} + 54 \beta_{18} + 86 \beta_{17} - 4 \beta_{16} + 21 \beta_{15} - 3 \beta_{14} + 124 \beta_{13} + \cdots + 171 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 55 \beta_{19} + 9 \beta_{18} - 24 \beta_{17} + 45 \beta_{16} + 81 \beta_{15} - 89 \beta_{14} + \cdots + 229 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
−1.39810 − 0.212863i
−1.39810 + 0.212863i
−1.29304 − 0.572751i
−1.29304 + 0.572751i
−0.689292 − 1.23486i
−0.689292 + 1.23486i
−0.0329621 − 1.41383i
−0.0329621 + 1.41383i
0.220862 − 1.39686i
0.220862 + 1.39686i
0.368522 − 1.36535i
0.368522 + 1.36535i
0.627237 − 1.26751i
0.627237 + 1.26751i
1.16607 − 0.800179i
1.16607 + 0.800179i
1.20549 − 0.739452i
1.20549 + 0.739452i
1.32522 − 0.493756i
1.32522 + 0.493756i
−1.39810 − 0.212863i −1.21236 1.90938 + 0.595207i 0 1.69500 + 0.258065i 4.63239i −2.54281 − 1.23860i −1.53019 0
749.2 −1.39810 + 0.212863i −1.21236 1.90938 − 0.595207i 0 1.69500 − 0.258065i − 4.63239i −2.54281 + 1.23860i −1.53019 0
749.3 −1.29304 − 0.572751i −0.207209 1.34391 + 1.48118i 0 0.267929 + 0.118679i − 2.73181i −0.889389 − 2.68496i −2.95706 0
749.4 −1.29304 + 0.572751i −0.207209 1.34391 − 1.48118i 0 0.267929 − 0.118679i 2.73181i −0.889389 + 2.68496i −2.95706 0
749.5 −0.689292 − 1.23486i 2.52102 −1.04975 + 1.70236i 0 −1.73772 − 3.11311i 0.987050i 2.82576 + 0.122873i 3.35556 0
749.6 −0.689292 + 1.23486i 2.52102 −1.04975 − 1.70236i 0 −1.73772 + 3.11311i − 0.987050i 2.82576 − 0.122873i 3.35556 0
749.7 −0.0329621 − 1.41383i −3.08659 −1.99783 + 0.0932055i 0 0.101740 + 4.36391i − 3.24242i 0.197629 + 2.82151i 6.52702 0
749.8 −0.0329621 + 1.41383i −3.08659 −1.99783 − 0.0932055i 0 0.101740 − 4.36391i 3.24242i 0.197629 − 2.82151i 6.52702 0
749.9 0.220862 − 1.39686i −0.939252 −1.90244 − 0.617026i 0 −0.207445 + 1.31200i 1.90117i −1.28208 + 2.52117i −2.11781 0
749.10 0.220862 + 1.39686i −0.939252 −1.90244 + 0.617026i 0 −0.207445 − 1.31200i − 1.90117i −1.28208 − 2.52117i −2.11781 0
749.11 0.368522 − 1.36535i 0.513027 −1.72838 − 1.00633i 0 0.189062 − 0.700464i 1.12889i −2.01094 + 1.98900i −2.73680 0
749.12 0.368522 + 1.36535i 0.513027 −1.72838 + 1.00633i 0 0.189062 + 0.700464i − 1.12889i −2.01094 − 1.98900i −2.73680 0
749.13 0.627237 − 1.26751i 1.69676 −1.21315 − 1.59005i 0 1.06427 − 2.15066i − 4.40040i −2.77634 + 0.540334i −0.120998 0
749.14 0.627237 + 1.26751i 1.69676 −1.21315 + 1.59005i 0 1.06427 + 2.15066i 4.40040i −2.77634 − 0.540334i −0.120998 0
749.15 1.16607 − 0.800179i 2.62662 0.719426 − 1.86613i 0 3.06282 − 2.10177i − 0.269237i −0.654336 − 2.75170i 3.89913 0
749.16 1.16607 + 0.800179i 2.62662 0.719426 + 1.86613i 0 3.06282 + 2.10177i 0.269237i −0.654336 + 2.75170i 3.89913 0
749.17 1.20549 − 0.739452i −2.07677 0.906422 − 1.78281i 0 −2.50353 + 1.53567i 2.55636i −0.225614 − 2.81941i 1.31298 0
749.18 1.20549 + 0.739452i −2.07677 0.906422 + 1.78281i 0 −2.50353 − 1.53567i − 2.55636i −0.225614 + 2.81941i 1.31298 0
749.19 1.32522 − 0.493756i −1.83526 1.51241 − 1.30867i 0 −2.43212 + 0.906170i 1.31564i 1.35811 − 2.48104i 0.368171 0
749.20 1.32522 + 0.493756i −1.83526 1.51241 + 1.30867i 0 −2.43212 − 0.906170i − 1.31564i 1.35811 + 2.48104i 0.368171 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.f.d 20
4.b odd 2 1 4000.2.f.d 20
5.b even 2 1 1000.2.f.c 20
5.c odd 4 2 1000.2.d.c ✓ 40
8.b even 2 1 1000.2.f.c 20
8.d odd 2 1 4000.2.f.c 20
20.d odd 2 1 4000.2.f.c 20
20.e even 4 2 4000.2.d.c 40
40.e odd 2 1 4000.2.f.d 20
40.f even 2 1 inner 1000.2.f.d 20
40.i odd 4 2 1000.2.d.c ✓ 40
40.k even 4 2 4000.2.d.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.d.c ✓ 40 5.c odd 4 2
1000.2.d.c ✓ 40 40.i odd 4 2
1000.2.f.c 20 5.b even 2 1
1000.2.f.c 20 8.b even 2 1
1000.2.f.d 20 1.a even 1 1 trivial
1000.2.f.d 20 40.f even 2 1 inner
4000.2.d.c 40 20.e even 4 2
4000.2.d.c 40 40.k even 4 2
4000.2.f.c 20 8.d odd 2 1
4000.2.f.c 20 20.d odd 2 1
4000.2.f.d 20 4.b odd 2 1
4000.2.f.d 20 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 2T_{3}^{9} - 16T_{3}^{8} - 32T_{3}^{7} + 77T_{3}^{6} + 168T_{3}^{5} - 94T_{3}^{4} - 292T_{3}^{3} - 61T_{3}^{2} + 76T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 3 T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( (T^{10} + 2 T^{9} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 73 T^{18} + \cdots + 119961 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 242921025 \) Copy content Toggle raw display
$13$ \( (T^{10} + 3 T^{9} + \cdots - 74475)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 93273522176 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 14131585001 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 198733991936 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 188118041760000 \) Copy content Toggle raw display
$31$ \( (T^{10} - 12 T^{9} + \cdots - 13424)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 9 T^{9} + \cdots - 97344)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 11 T^{9} + \cdots + 4900275)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 30 T^{9} + \cdots - 22343616)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 155998693721081 \) Copy content Toggle raw display
$53$ \( (T^{10} + 5 T^{9} + \cdots - 21379)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 58948955361 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 29661335302400 \) Copy content Toggle raw display
$67$ \( (T^{10} - 20 T^{9} + \cdots + 292107024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 24 T^{9} + \cdots - 2766384)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{10} + 24 T^{9} + \cdots - 2000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 20 T^{9} + \cdots - 44548096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 11 T^{9} + \cdots + 24825600)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 20198622433536 \) Copy content Toggle raw display
show more
show less