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Properties

Label 1089.2.e.l
Level $1089$
Weight $2$
Character orbit 1089.e
Analytic conductor $8.696$
Analytic rank $0$
Dimension $20$
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] = [1089,2,Mod(364,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1089.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.e (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: \(8.69570878012\)
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: \( x^{20} + 15 x^{18} - 2 x^{17} + 150 x^{16} - 30 x^{15} + 830 x^{14} - 321 x^{13} + 3324 x^{12} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{16} q^{3} + (\beta_{17} - \beta_{12}) q^{4} + ( - \beta_{13} - \beta_{10}) q^{5} + ( - \beta_{13} + \beta_{12} - \beta_{4} - 1) q^{6} + (\beta_{11} + \beta_{6}) q^{7} + (\beta_{15} + \beta_{14} - \beta_{13} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{19} + \beta_{13} + \cdots - \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{16} q^{3} + (\beta_{17} - \beta_{12}) q^{4} + ( - \beta_{13} - \beta_{10}) q^{5} + ( - \beta_{13} + \beta_{12} - \beta_{4} - 1) q^{6} + (\beta_{11} + \beta_{6}) q^{7} + (\beta_{15} + \beta_{14} - \beta_{13} + \cdots + 1) q^{8}+ \cdots + (3 \beta_{19} - 3 \beta_{17} + \cdots - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{3} - 10 q^{4} - 3 q^{5} - 10 q^{6} + q^{7} + 6 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - q^{3} - 10 q^{4} - 3 q^{5} - 10 q^{6} + q^{7} + 6 q^{8} + q^{9} - 5 q^{12} + q^{13} - 15 q^{14} - 7 q^{15} - 10 q^{16} + 6 q^{17} - 17 q^{18} + 4 q^{19} - 15 q^{20} - 4 q^{21} - 18 q^{23} + 27 q^{24} - 7 q^{25} + 12 q^{26} + 23 q^{27} - 8 q^{28} + 12 q^{29} + 26 q^{30} + 7 q^{31} + 24 q^{32} + 6 q^{34} + 36 q^{35} + 20 q^{36} - 2 q^{37} + 12 q^{38} - 4 q^{39} - 12 q^{40} - 6 q^{41} + 2 q^{42} - 8 q^{43} - 35 q^{45} - 36 q^{46} - 27 q^{47} + 19 q^{48} - 15 q^{49} + 15 q^{50} + 2 q^{51} - 5 q^{52} + 12 q^{53} + 44 q^{54} - 60 q^{56} - 27 q^{57} - 9 q^{58} - 21 q^{59} - 5 q^{60} - 20 q^{61} - 6 q^{62} - 74 q^{63} + 62 q^{64} + 12 q^{65} + 10 q^{67} - 69 q^{68} + 16 q^{69} - 6 q^{70} + 36 q^{71} + 102 q^{72} - 14 q^{73} - 6 q^{74} - 26 q^{75} - 2 q^{76} + 92 q^{78} + q^{79} + 102 q^{80} - 35 q^{81} - 60 q^{82} + 30 q^{83} + 70 q^{84} - 24 q^{85} - 36 q^{86} - 60 q^{87} + 108 q^{89} - 11 q^{90} + 20 q^{91} - 6 q^{92} - 19 q^{93} - 15 q^{94} + 12 q^{95} + 106 q^{96} + 16 q^{97} - 120 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 15 x^{18} - 2 x^{17} + 150 x^{16} - 30 x^{15} + 830 x^{14} - 321 x^{13} + 3324 x^{12} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!84 \nu^{19} + \cdots + 16\!\cdots\!25 ) / 55\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\!\cdots\!74 \nu^{19} + \cdots - 32\!\cdots\!40 ) / 55\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 66\!\cdots\!62 \nu^{19} + \cdots + 26\!\cdots\!87 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 71\!\cdots\!14 \nu^{19} + \cdots - 21\!\cdots\!70 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 92\!\cdots\!60 \nu^{19} + \cdots - 89\!\cdots\!76 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\!\cdots\!56 \nu^{19} + \cdots - 22\!\cdots\!17 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 27\!\cdots\!10 \nu^{19} + \cdots + 26\!\cdots\!90 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49\!\cdots\!81 \nu^{19} + \cdots + 11\!\cdots\!98 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 49\!\cdots\!66 \nu^{19} + \cdots + 17\!\cdots\!27 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!01 \nu^{19} + \cdots - 43\!\cdots\!84 ) / 36\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 36\!\cdots\!60 \nu^{19} + \cdots - 59\!\cdots\!95 ) / 55\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 74\!\cdots\!31 \nu^{19} + \cdots - 16\!\cdots\!48 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 42\!\cdots\!79 \nu^{19} + \cdots - 17\!\cdots\!82 ) / 36\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 48\!\cdots\!01 \nu^{19} + \cdots + 17\!\cdots\!52 ) / 36\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 17\!\cdots\!97 \nu^{19} + \cdots - 45\!\cdots\!55 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 10\!\cdots\!64 \nu^{19} + \cdots - 18\!\cdots\!51 ) / 55\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 22\!\cdots\!56 \nu^{19} + \cdots + 83\!\cdots\!09 ) / 11\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 24\!\cdots\!99 \nu^{19} + \cdots + 41\!\cdots\!07 ) / 11\!\cdots\!94 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} + 3\beta_{12} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{4} + 5\beta_{3} + \beta_{2} - 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} + 7 \beta_{17} + \beta_{16} - 2 \beta_{13} - 15 \beta_{12} - 2 \beta_{10} - \beta_{8} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} + 9 \beta_{18} - 12 \beta_{16} - \beta_{15} - 10 \beta_{13} + 7 \beta_{12} - \beta_{11} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{19} - 11 \beta_{17} - 11 \beta_{16} + 9 \beta_{15} - \beta_{14} + 2 \beta_{13} + 10 \beta_{11} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{19} - 69 \beta_{18} - 16 \beta_{17} + 92 \beta_{16} - 57 \beta_{15} - 69 \beta_{14} + \cdots + 174 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 147 \beta_{19} - 18 \beta_{18} - 226 \beta_{17} + 42 \beta_{16} - 80 \beta_{15} + 165 \beta_{13} + \cdots - 558 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 94 \beta_{19} + 94 \beta_{17} + 94 \beta_{16} + 504 \beta_{15} + 505 \beta_{14} - 522 \beta_{13} + \cdots + 229 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 356 \beta_{19} + 207 \beta_{18} + 2199 \beta_{17} + 294 \beta_{16} + 174 \beta_{15} + 207 \beta_{14} + \cdots - 884 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1078 \beta_{19} + 3633 \beta_{18} + 855 \beta_{17} - 5883 \beta_{16} - 777 \beta_{15} - 4711 \beta_{13} + \cdots - 1108 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5404 \beta_{19} - 5404 \beta_{17} - 5404 \beta_{16} + 2651 \beta_{15} - 2002 \beta_{14} + 2358 \beta_{13} + \cdots + 24997 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2953 \beta_{19} - 25974 \beta_{18} - 13820 \beta_{17} + 36868 \beta_{16} - 20007 \beta_{15} + \cdots + 55256 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 55941 \beta_{19} - 17784 \beta_{18} - 68509 \beta_{17} + 33033 \beta_{16} - 30367 \beta_{15} + \cdots - 171852 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 45752 \beta_{19} + 45752 \beta_{17} + 45752 \beta_{16} + 179393 \beta_{15} + 185410 \beta_{14} + \cdots + 5173 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 112683 \beta_{19} + 150561 \beta_{18} + 768605 \beta_{17} + 16829 \beta_{16} + 119922 \beta_{15} + \cdots - 474978 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 551677 \beta_{19} + 1324155 \beta_{18} + 559224 \beta_{17} - 2214990 \beta_{16} - 246554 \beta_{15} + \cdots + 132539 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2100150 \beta_{19} - 2100150 \beta_{17} - 2100150 \beta_{16} + 508624 \beta_{15} - 1236778 \beta_{14} + \cdots + 8338087 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 1532148 \beta_{19} - 9469920 \beta_{18} - 7204254 \beta_{17} + 13165230 \beta_{16} + \cdots + 19981447 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-\beta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
−1.29266 + 2.23895i
−1.09739 + 1.90074i
−0.799486 + 1.38475i
−0.637546 + 1.10426i
0.0306451 − 0.0530788i
0.342978 − 0.594056i
0.401074 − 0.694681i
0.747945 − 1.29548i
0.946004 − 1.63853i
1.35844 − 2.35288i
−1.29266 − 2.23895i
−1.09739 − 1.90074i
−0.799486 − 1.38475i
−0.637546 − 1.10426i
0.0306451 + 0.0530788i
0.342978 + 0.594056i
0.401074 + 0.694681i
0.747945 + 1.29548i
0.946004 + 1.63853i
1.35844 + 2.35288i
−1.29266 − 2.23895i 1.67052 − 0.457565i −2.34193 + 4.05633i −1.53230 + 2.65402i −3.18387 − 3.14873i −1.51305 − 2.62068i 6.93860 2.58127 − 1.52874i 7.92296
364.2 −1.09739 − 1.90074i −1.56183 + 0.748790i −1.40854 + 2.43966i 0.684725 − 1.18598i 3.13719 + 2.14691i −0.676841 − 1.17232i 1.79331 1.87863 − 2.33897i −3.00564
364.3 −0.799486 − 1.38475i 1.06054 + 1.36940i −0.278356 + 0.482127i 0.800947 − 1.38728i 1.04839 − 2.56340i 1.53735 + 2.66278i −2.30778 −0.750504 + 2.90461i −2.56138
364.4 −0.637546 − 1.10426i −0.670495 − 1.59701i 0.187069 − 0.324014i −0.743461 + 1.28771i −1.33604 + 1.75857i 0.0131769 + 0.0228230i −3.02725 −2.10087 + 2.14157i 1.89596
364.5 0.0306451 + 0.0530788i 1.27092 − 1.17676i 0.998122 − 1.72880i 0.789859 − 1.36808i 0.101408 + 0.0313974i −0.615498 − 1.06607i 0.244930 0.230495 − 2.99113i 0.0968211
364.6 0.342978 + 0.594056i −0.510856 + 1.65500i 0.764732 − 1.32455i 1.07171 − 1.85626i −1.15838 + 0.264152i −0.684647 − 1.18584i 2.42106 −2.47805 − 1.69093i 1.47030
364.7 0.401074 + 0.694681i 1.56035 + 0.751869i 0.678279 − 1.17481i −1.96023 + 3.39523i 0.103508 + 1.38550i 1.47750 + 2.55911i 2.69246 1.86939 + 2.34636i −3.14480
364.8 0.747945 + 1.29548i −1.65449 − 0.512505i −0.118843 + 0.205843i −1.28427 + 2.22442i −0.573528 − 2.52668i −2.55164 − 4.41957i 2.63623 2.47468 + 1.69587i −3.84225
364.9 0.946004 + 1.63853i −0.909580 − 1.47400i −0.789847 + 1.36806i 1.50872 − 2.61318i 1.55472 − 2.88478i 2.21759 + 3.84098i 0.795222 −1.34533 + 2.68143i 5.70903
364.10 1.35844 + 2.35288i −0.755083 + 1.55880i −2.69069 + 4.66041i −0.835704 + 1.44748i −4.69339 + 0.340905i 1.29605 + 2.24483i −9.18678 −1.85970 − 2.35404i −4.54100
727.1 −1.29266 + 2.23895i 1.67052 + 0.457565i −2.34193 − 4.05633i −1.53230 − 2.65402i −3.18387 + 3.14873i −1.51305 + 2.62068i 6.93860 2.58127 + 1.52874i 7.92296
727.2 −1.09739 + 1.90074i −1.56183 − 0.748790i −1.40854 − 2.43966i 0.684725 + 1.18598i 3.13719 − 2.14691i −0.676841 + 1.17232i 1.79331 1.87863 + 2.33897i −3.00564
727.3 −0.799486 + 1.38475i 1.06054 − 1.36940i −0.278356 − 0.482127i 0.800947 + 1.38728i 1.04839 + 2.56340i 1.53735 − 2.66278i −2.30778 −0.750504 − 2.90461i −2.56138
727.4 −0.637546 + 1.10426i −0.670495 + 1.59701i 0.187069 + 0.324014i −0.743461 − 1.28771i −1.33604 − 1.75857i 0.0131769 − 0.0228230i −3.02725 −2.10087 − 2.14157i 1.89596
727.5 0.0306451 − 0.0530788i 1.27092 + 1.17676i 0.998122 + 1.72880i 0.789859 + 1.36808i 0.101408 − 0.0313974i −0.615498 + 1.06607i 0.244930 0.230495 + 2.99113i 0.0968211
727.6 0.342978 − 0.594056i −0.510856 − 1.65500i 0.764732 + 1.32455i 1.07171 + 1.85626i −1.15838 − 0.264152i −0.684647 + 1.18584i 2.42106 −2.47805 + 1.69093i 1.47030
727.7 0.401074 − 0.694681i 1.56035 − 0.751869i 0.678279 + 1.17481i −1.96023 − 3.39523i 0.103508 − 1.38550i 1.47750 − 2.55911i 2.69246 1.86939 − 2.34636i −3.14480
727.8 0.747945 − 1.29548i −1.65449 + 0.512505i −0.118843 − 0.205843i −1.28427 − 2.22442i −0.573528 + 2.52668i −2.55164 + 4.41957i 2.63623 2.47468 − 1.69587i −3.84225
727.9 0.946004 − 1.63853i −0.909580 + 1.47400i −0.789847 − 1.36806i 1.50872 + 2.61318i 1.55472 + 2.88478i 2.21759 − 3.84098i 0.795222 −1.34533 − 2.68143i 5.70903
727.10 1.35844 − 2.35288i −0.755083 − 1.55880i −2.69069 − 4.66041i −0.835704 − 1.44748i −4.69339 − 0.340905i 1.29605 − 2.24483i −9.18678 −1.85970 + 2.35404i −4.54100
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.2.e.l ✓ 20
9.c even 3 1 inner 1089.2.e.l ✓ 20
9.c even 3 1 9801.2.a.cd 10
9.d odd 6 1 9801.2.a.cb 10
11.b odd 2 1 1089.2.e.m yes 20
99.g even 6 1 9801.2.a.cc 10
99.h odd 6 1 1089.2.e.m yes 20
99.h odd 6 1 9801.2.a.ce 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.2.e.l ✓ 20 1.a even 1 1 trivial
1089.2.e.l ✓ 20 9.c even 3 1 inner
1089.2.e.m yes 20 11.b odd 2 1
1089.2.e.m yes 20 99.h odd 6 1
9801.2.a.cb 10 9.d odd 6 1
9801.2.a.cc 10 99.g even 6 1
9801.2.a.cd 10 9.c even 3 1
9801.2.a.ce 10 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{20} + 15 T_{2}^{18} - 2 T_{2}^{17} + 150 T_{2}^{16} - 30 T_{2}^{15} + 830 T_{2}^{14} - 321 T_{2}^{13} + \cdots + 9 \) Copy content Toggle raw display
\( T_{5}^{20} + 3 T_{5}^{19} + 33 T_{5}^{18} + 46 T_{5}^{17} + 531 T_{5}^{16} + 474 T_{5}^{15} + \cdots + 2954961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 15 T^{18} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{20} + T^{19} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} + 3 T^{19} + \cdots + 2954961 \) Copy content Toggle raw display
$7$ \( T^{20} - T^{19} + \cdots + 9409 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} - T^{19} + \cdots + 6017209 \) Copy content Toggle raw display
$17$ \( (T^{10} - 3 T^{9} + \cdots + 104577)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} - 2 T^{9} + \cdots - 384912)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 9616467687849 \) Copy content Toggle raw display
$29$ \( T^{20} - 12 T^{19} + \cdots + 531441 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 833303689 \) Copy content Toggle raw display
$37$ \( (T^{10} + T^{9} + \cdots - 32157)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 63\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 110451869649 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 320683231521 \) Copy content Toggle raw display
$53$ \( (T^{10} - 6 T^{9} + \cdots + 58563)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 6253171929 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 7464976948521 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 181492008778609 \) Copy content Toggle raw display
$71$ \( (T^{10} - 18 T^{9} + \cdots + 26778357)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 7 T^{9} + \cdots + 69819137)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 29582057445721 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 202515924794481 \) Copy content Toggle raw display
$89$ \( (T^{10} - 54 T^{9} + \cdots + 26994627)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 550257074238481 \) Copy content Toggle raw display
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