↑

Properties

Label 1155.2.q.l
Level $1155$
Weight $2$
Character orbit 1155.q
Analytic conductor $9.223$
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] = [1155,2,Mod(331,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.331");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.q (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: \(9.22272143346\)
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: \( x^{20} + 16 x^{18} + 172 x^{16} - 2 x^{15} + 1006 x^{14} - 56 x^{13} + 4224 x^{12} - 288 x^{11} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{6} - \beta_{4}) q^{4} + (\beta_{4} - 1) q^{5} - \beta_{8} q^{6} - \beta_{7} q^{7} + (\beta_{17} - \beta_{15} + \cdots + \beta_{2}) q^{8}+ \cdots + (\beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{6} - \beta_{4}) q^{4} + (\beta_{4} - 1) q^{5} - \beta_{8} q^{6} - \beta_{7} q^{7} + (\beta_{17} - \beta_{15} + \cdots + \beta_{2}) q^{8}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{3} - 12 q^{4} - 10 q^{5} - 4 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 10 q^{3} - 12 q^{4} - 10 q^{5} - 4 q^{7} - 10 q^{9} - 10 q^{11} - 12 q^{12} + 8 q^{13} + 20 q^{15} - 24 q^{16} - 12 q^{17} - 9 q^{19} + 24 q^{20} + 5 q^{21} + q^{23} - 10 q^{25} - 6 q^{26} + 20 q^{27} + 24 q^{28} + 6 q^{29} - 9 q^{31} + 10 q^{32} - 10 q^{33} + 12 q^{34} - q^{35} + 24 q^{36} - 3 q^{37} - 42 q^{38} - 4 q^{39} + 6 q^{41} - 12 q^{44} - 10 q^{45} + 22 q^{46} - 21 q^{47} + 48 q^{48} + 24 q^{49} - 12 q^{51} - 24 q^{52} + 5 q^{53} + 20 q^{55} + 4 q^{56} + 18 q^{57} + 26 q^{58} - 16 q^{59} - 12 q^{60} - 20 q^{61} - 36 q^{62} - q^{63} + 140 q^{64} - 4 q^{65} + q^{67} - 30 q^{68} - 2 q^{69} + 44 q^{71} + q^{73} + 34 q^{74} - 10 q^{75} - 56 q^{76} + 5 q^{77} + 12 q^{78} - 19 q^{79} - 24 q^{80} - 10 q^{81} - 32 q^{82} + 40 q^{83} + 12 q^{84} + 24 q^{85} - 3 q^{87} - 18 q^{89} - 18 q^{91} - 16 q^{92} - 9 q^{93} + 6 q^{94} - 9 q^{95} + 10 q^{96} + 12 q^{97} + 48 q^{98} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 16 x^{18} + 172 x^{16} - 2 x^{15} + 1006 x^{14} - 56 x^{13} + 4224 x^{12} - 288 x^{11} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 56\!\cdots\!79 \nu^{19} + \cdots + 13\!\cdots\!64 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\!\cdots\!39 \nu^{19} + \cdots + 15\!\cdots\!48 ) / 63\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27\!\cdots\!67 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 62\!\cdots\!61 \nu^{19} + \cdots + 11\!\cdots\!72 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\!\cdots\!67 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 84\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 88\!\cdots\!39 \nu^{19} + \cdots + 17\!\cdots\!76 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31\!\cdots\!62 \nu^{19} + \cdots - 70\!\cdots\!52 ) / 63\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 48\!\cdots\!21 \nu^{19} + \cdots - 44\!\cdots\!52 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 59\!\cdots\!03 \nu^{19} + \cdots + 26\!\cdots\!76 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 55\!\cdots\!19 \nu^{19} + \cdots + 91\!\cdots\!52 ) / 12\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 61\!\cdots\!27 \nu^{19} + \cdots - 42\!\cdots\!96 ) / 12\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!59 \nu^{19} + \cdots + 12\!\cdots\!52 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 95\!\cdots\!52 \nu^{19} + \cdots - 17\!\cdots\!60 ) / 63\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19\!\cdots\!47 \nu^{19} + \cdots - 10\!\cdots\!52 ) / 12\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 11\!\cdots\!58 \nu^{19} + \cdots - 43\!\cdots\!68 ) / 63\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 12\!\cdots\!27 \nu^{19} + \cdots + 77\!\cdots\!68 ) / 63\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 69\!\cdots\!69 \nu^{19} + \cdots - 71\!\cdots\!32 ) / 31\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 27\!\cdots\!87 \nu^{19} + \cdots + 31\!\cdots\!04 ) / 12\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - 6\beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} - \beta_{11} - \beta_{9} - 7 \beta_{6} - \beta_{5} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{19} - 10 \beta_{17} - 2 \beta_{11} - \beta_{10} - \beta_{9} + 31 \beta_{8} - 7 \beta_{7} + \cdots - 31 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{19} + 12 \beta_{18} - \beta_{17} - \beta_{16} + 13 \beta_{14} + \beta_{13} + \beta_{12} + \cdots + 101 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 86 \beta_{19} + 16 \beta_{18} + 16 \beta_{17} - 15 \beta_{16} + 70 \beta_{15} + 15 \beta_{14} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 112 \beta_{19} + 112 \beta_{17} - 110 \beta_{16} + 15 \beta_{15} - 110 \beta_{14} - 15 \beta_{13} + \cdots + 34 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 174 \beta_{18} + 529 \beta_{17} + 156 \beta_{16} - 529 \beta_{15} - 158 \beta_{14} - 529 \beta_{13} + \cdots - 184 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1121 \beta_{19} - 923 \beta_{18} - 767 \beta_{17} + 1079 \beta_{16} - 198 \beta_{15} - 158 \beta_{14} + \cdots - 4734 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5602 \beta_{19} - 5602 \beta_{17} + 48 \beta_{16} - 2 \beta_{15} + 48 \beta_{14} + 2 \beta_{13} + \cdots - 10411 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1405 \beta_{19} + 7452 \beta_{18} - 1981 \beta_{17} - 1459 \beta_{16} + 576 \beta_{15} + 8855 \beta_{14} + \cdots + 33849 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 44052 \beta_{19} + 14088 \beta_{18} + 14034 \beta_{17} - 12565 \beta_{16} + 29964 \beta_{15} + \cdots + 17414 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 65570 \beta_{19} + 65570 \beta_{17} - 58062 \beta_{16} + 11783 \beta_{15} - 58062 \beta_{14} + \cdots + 38720 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 892 \beta_{19} - 117312 \beta_{18} + 226181 \beta_{17} + 95880 \beta_{16} - 225289 \beta_{15} + \cdots - 156324 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 624813 \beta_{19} - 462405 \beta_{18} - 367417 \beta_{17} + 556445 \beta_{16} - 162408 \beta_{15} + \cdots - 1819464 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2654074 \beta_{19} - 2654074 \beta_{17} + 97992 \beta_{16} - 11696 \beta_{15} + 97992 \beta_{14} + \cdots - 4178551 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 748085 \beta_{19} + 3602276 \beta_{18} - 1397917 \beta_{17} - 882113 \beta_{16} + 649832 \beta_{15} + \cdots + 13561817 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 20632606 \beta_{19} + 7657128 \beta_{18} + 7523100 \beta_{17} - 6934723 \beta_{16} + 12975478 \beta_{15} + \cdots + 11603836 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
331.1
−1.34921 + 2.33690i
−1.15349 + 1.99791i
−0.753690 + 1.30543i
−0.393060 + 0.680800i
−0.371576 + 0.643589i
0.380508 − 0.659059i
0.448165 − 0.776244i
0.696463 − 1.20631i
1.10130 − 1.90751i
1.39460 − 2.41551i
−1.34921 − 2.33690i
−1.15349 − 1.99791i
−0.753690 − 1.30543i
−0.393060 − 0.680800i
−0.371576 − 0.643589i
0.380508 + 0.659059i
0.448165 + 0.776244i
0.696463 + 1.20631i
1.10130 + 1.90751i
1.39460 + 2.41551i
−1.34921 + 2.33690i −0.500000 − 0.866025i −2.64075 − 4.57391i −0.500000 + 0.866025i 2.69843 −1.99135 + 1.74199i 8.85488 −0.500000 + 0.866025i −1.34921 − 2.33690i
331.2 −1.15349 + 1.99791i −0.500000 − 0.866025i −1.66110 − 2.87710i −0.500000 + 0.866025i 2.30699 1.29052 − 2.30967i 3.05028 −0.500000 + 0.866025i −1.15349 − 1.99791i
331.3 −0.753690 + 1.30543i −0.500000 − 0.866025i −0.136098 − 0.235728i −0.500000 + 0.866025i 1.50738 −2.61351 − 0.411796i −2.60446 −0.500000 + 0.866025i −0.753690 − 1.30543i
331.4 −0.393060 + 0.680800i −0.500000 − 0.866025i 0.691007 + 1.19686i −0.500000 + 0.866025i 0.786121 2.38322 + 1.14903i −2.65867 −0.500000 + 0.866025i −0.393060 − 0.680800i
331.5 −0.371576 + 0.643589i −0.500000 − 0.866025i 0.723862 + 1.25377i −0.500000 + 0.866025i 0.743153 0.959625 + 2.46559i −2.56219 −0.500000 + 0.866025i −0.371576 − 0.643589i
331.6 0.380508 − 0.659059i −0.500000 − 0.866025i 0.710427 + 1.23050i −0.500000 + 0.866025i −0.761016 −2.11758 − 1.58615i 2.60333 −0.500000 + 0.866025i 0.380508 + 0.659059i
331.7 0.448165 − 0.776244i −0.500000 − 0.866025i 0.598297 + 1.03628i −0.500000 + 0.866025i −0.896329 1.40656 − 2.24089i 2.86520 −0.500000 + 0.866025i 0.448165 + 0.776244i
331.8 0.696463 − 1.20631i −0.500000 − 0.866025i 0.0298792 + 0.0517522i −0.500000 + 0.866025i −1.39293 −1.29684 + 2.30612i 2.86909 −0.500000 + 0.866025i 0.696463 + 1.20631i
331.9 1.10130 − 1.90751i −0.500000 − 0.866025i −1.42573 − 2.46943i −0.500000 + 0.866025i −2.20260 2.61584 + 0.396686i −1.87541 −0.500000 + 0.866025i 1.10130 + 1.90751i
331.10 1.39460 − 2.41551i −0.500000 − 0.866025i −2.88980 − 5.00528i −0.500000 + 0.866025i −2.78919 −2.63649 + 0.221131i −10.5420 −0.500000 + 0.866025i 1.39460 + 2.41551i
991.1 −1.34921 − 2.33690i −0.500000 + 0.866025i −2.64075 + 4.57391i −0.500000 − 0.866025i 2.69843 −1.99135 − 1.74199i 8.85488 −0.500000 − 0.866025i −1.34921 + 2.33690i
991.2 −1.15349 − 1.99791i −0.500000 + 0.866025i −1.66110 + 2.87710i −0.500000 − 0.866025i 2.30699 1.29052 + 2.30967i 3.05028 −0.500000 − 0.866025i −1.15349 + 1.99791i
991.3 −0.753690 − 1.30543i −0.500000 + 0.866025i −0.136098 + 0.235728i −0.500000 − 0.866025i 1.50738 −2.61351 + 0.411796i −2.60446 −0.500000 − 0.866025i −0.753690 + 1.30543i
991.4 −0.393060 − 0.680800i −0.500000 + 0.866025i 0.691007 − 1.19686i −0.500000 − 0.866025i 0.786121 2.38322 − 1.14903i −2.65867 −0.500000 − 0.866025i −0.393060 + 0.680800i
991.5 −0.371576 − 0.643589i −0.500000 + 0.866025i 0.723862 − 1.25377i −0.500000 − 0.866025i 0.743153 0.959625 − 2.46559i −2.56219 −0.500000 − 0.866025i −0.371576 + 0.643589i
991.6 0.380508 + 0.659059i −0.500000 + 0.866025i 0.710427 − 1.23050i −0.500000 − 0.866025i −0.761016 −2.11758 + 1.58615i 2.60333 −0.500000 − 0.866025i 0.380508 − 0.659059i
991.7 0.448165 + 0.776244i −0.500000 + 0.866025i 0.598297 − 1.03628i −0.500000 − 0.866025i −0.896329 1.40656 + 2.24089i 2.86520 −0.500000 − 0.866025i 0.448165 − 0.776244i
991.8 0.696463 + 1.20631i −0.500000 + 0.866025i 0.0298792 − 0.0517522i −0.500000 − 0.866025i −1.39293 −1.29684 − 2.30612i 2.86909 −0.500000 − 0.866025i 0.696463 − 1.20631i
991.9 1.10130 + 1.90751i −0.500000 + 0.866025i −1.42573 + 2.46943i −0.500000 − 0.866025i −2.20260 2.61584 − 0.396686i −1.87541 −0.500000 − 0.866025i 1.10130 − 1.90751i
991.10 1.39460 + 2.41551i −0.500000 + 0.866025i −2.88980 + 5.00528i −0.500000 − 0.866025i −2.78919 −2.63649 − 0.221131i −10.5420 −0.500000 − 0.866025i 1.39460 − 2.41551i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.q.l ✓ 20
7.c even 3 1 inner 1155.2.q.l ✓ 20
7.c even 3 1 8085.2.a.cn 10
7.d odd 6 1 8085.2.a.ck 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.q.l ✓ 20 1.a even 1 1 trivial
1155.2.q.l ✓ 20 7.c even 3 1 inner
8085.2.a.ck 10 7.d odd 6 1
8085.2.a.cn 10 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{20} + 16 T_{2}^{18} + 172 T_{2}^{16} - 2 T_{2}^{15} + 1006 T_{2}^{14} - 56 T_{2}^{13} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{13}^{10} - 4 T_{13}^{9} - 86 T_{13}^{8} + 396 T_{13}^{7} + 2016 T_{13}^{6} - 11328 T_{13}^{5} + \cdots - 1788 \) Copy content Toggle raw display
\( T_{17}^{20} + 12 T_{17}^{19} + 161 T_{17}^{18} + 1016 T_{17}^{17} + 8517 T_{17}^{16} + 41354 T_{17}^{15} + \cdots + 6056574976 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 16 T^{18} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$13$ \( (T^{10} - 4 T^{9} + \cdots - 1788)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 6056574976 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 7762776702976 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 5262631936 \) Copy content Toggle raw display
$29$ \( (T^{10} - 3 T^{9} + \cdots - 34016)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 48721740169216 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 12010826960896 \) Copy content Toggle raw display
$41$ \( (T^{10} - 3 T^{9} + \cdots - 297984)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 305 T^{8} + \cdots - 2408797)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 92115154256896 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 41\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 29400339595264 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 1195471077376 \) Copy content Toggle raw display
$71$ \( (T^{10} - 22 T^{9} + \cdots + 57359712)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 25379993076736 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{10} - 20 T^{9} + \cdots + 189485312)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 52\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{10} - 6 T^{9} + \cdots - 51240576)^{2} \) Copy content Toggle raw display
show more
show less