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Properties

Label 95.4.e.c
Level $95$
Weight $4$
Character orbit 95.e
Analytic conductor $5.605$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [95,4,Mod(11,95)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(95, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("95.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.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: \(5.60518145055\)
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: \( x^{20} - 3 x^{19} + 66 x^{18} - 125 x^{17} + 2555 x^{16} - 3995 x^{15} + 60229 x^{14} + \cdots + 2336368896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} - \beta_{2} q^{3} + ( - \beta_{8} + \beta_{5} + 4 \beta_{4}) q^{4} + (5 \beta_{4} + 5) q^{5} + ( - \beta_{14} - \beta_{4} - \beta_1) q^{6} + (\beta_{15} + \beta_{9} + \beta_{3} + 1) q^{7} + ( - \beta_{13} - \beta_{11} + \beta_{7} + \cdots + 4) q^{8}+ \cdots + ( - \beta_{19} - 2 \beta_{18} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} - \beta_{2} q^{3} + ( - \beta_{8} + \beta_{5} + 4 \beta_{4}) q^{4} + (5 \beta_{4} + 5) q^{5} + ( - \beta_{14} - \beta_{4} - \beta_1) q^{6} + (\beta_{15} + \beta_{9} + \beta_{3} + 1) q^{7} + ( - \beta_{13} - \beta_{11} + \beta_{7} + \cdots + 4) q^{8}+ \cdots + ( - 6 \beta_{19} - 12 \beta_{18} + \cdots + 108) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 3 q^{2} - 5 q^{3} - 43 q^{4} + 50 q^{5} + 9 q^{6} + 6 q^{7} + 96 q^{8} - 97 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 3 q^{2} - 5 q^{3} - 43 q^{4} + 50 q^{5} + 9 q^{6} + 6 q^{7} + 96 q^{8} - 97 q^{9} + 15 q^{10} - 36 q^{11} + 186 q^{12} + 14 q^{13} + 68 q^{14} + 25 q^{15} + 9 q^{16} - 144 q^{17} - 22 q^{18} + 96 q^{19} - 430 q^{20} + 46 q^{21} - 136 q^{22} + 321 q^{23} - 416 q^{24} - 250 q^{25} - 46 q^{26} + 1006 q^{27} + 130 q^{28} - 178 q^{29} + 90 q^{30} + 604 q^{31} - 202 q^{32} + 1099 q^{33} - 751 q^{34} + 15 q^{35} - 526 q^{36} - 774 q^{37} - 12 q^{38} - 1216 q^{39} + 240 q^{40} - 388 q^{41} + 143 q^{42} - 514 q^{43} - 1246 q^{44} - 970 q^{45} + 3650 q^{46} + 522 q^{47} + 4 q^{48} + 1582 q^{49} + 150 q^{50} - 1080 q^{51} + 569 q^{52} - 681 q^{53} - 321 q^{54} - 90 q^{55} - 3184 q^{56} + 514 q^{57} + 1198 q^{58} - 891 q^{59} + 465 q^{60} + 1110 q^{61} - 1921 q^{62} - 727 q^{63} + 952 q^{64} + 140 q^{65} + 3312 q^{66} - 691 q^{67} - 228 q^{68} + 172 q^{69} - 340 q^{70} + 382 q^{71} - 2678 q^{72} - 797 q^{73} + 404 q^{74} + 250 q^{75} + 462 q^{76} + 2390 q^{77} + 1000 q^{78} - 660 q^{79} - 45 q^{80} - 2454 q^{81} - 1155 q^{82} - 2026 q^{83} + 10756 q^{84} + 720 q^{85} + 858 q^{86} + 312 q^{87} - 98 q^{88} - 2957 q^{89} - 55 q^{90} - 3110 q^{91} + 98 q^{92} + 1500 q^{93} - 6374 q^{94} + 945 q^{95} - 584 q^{96} - 2881 q^{97} + 4062 q^{98} + 2723 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} + 66 x^{18} - 125 x^{17} + 2555 x^{16} - 3995 x^{15} + 60229 x^{14} + \cdots + 2336368896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24\!\cdots\!39 \nu^{19} + \cdots + 14\!\cdots\!44 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!87 \nu^{19} + \cdots + 60\!\cdots\!56 ) / 45\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!33 \nu^{19} + \cdots - 21\!\cdots\!24 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 66\!\cdots\!34 \nu^{19} + \cdots + 64\!\cdots\!92 ) / 57\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 53\!\cdots\!97 \nu^{19} + \cdots - 11\!\cdots\!28 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77\!\cdots\!89 \nu^{19} + \cdots + 23\!\cdots\!56 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!33 \nu^{19} + \cdots + 29\!\cdots\!48 ) / 20\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74\!\cdots\!78 \nu^{19} + \cdots - 36\!\cdots\!08 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!94 \nu^{19} + \cdots + 41\!\cdots\!84 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!77 \nu^{19} + \cdots + 51\!\cdots\!96 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24\!\cdots\!51 \nu^{19} + \cdots + 37\!\cdots\!76 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 72\!\cdots\!93 \nu^{19} + \cdots - 70\!\cdots\!36 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29\!\cdots\!29 \nu^{19} + \cdots + 46\!\cdots\!28 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 93\!\cdots\!29 \nu^{19} + \cdots + 20\!\cdots\!08 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 89\!\cdots\!39 \nu^{19} + \cdots + 22\!\cdots\!36 ) / 34\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 68\!\cdots\!77 \nu^{19} + \cdots - 22\!\cdots\!12 ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 54\!\cdots\!07 \nu^{19} + \cdots + 58\!\cdots\!68 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 13\!\cdots\!15 \nu^{19} + \cdots - 30\!\cdots\!96 ) / 37\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 12\beta_{4} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{11} - \beta_{7} + 19\beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} + 2 \beta_{18} + \beta_{17} + \beta_{16} + 2 \beta_{15} + \beta_{12} + 2 \beta_{9} + \cdots - 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{19} + 2 \beta_{18} + 2 \beta_{16} + 2 \beta_{15} - 2 \beta_{13} + 4 \beta_{12} + 4 \beta_{10} + \cdots + 114 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 33 \beta_{19} - 41 \beta_{18} - 45 \beta_{17} - 4 \beta_{16} - 43 \beta_{15} - 6 \beta_{14} + \cdots + 4724 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 99 \beta_{19} - 198 \beta_{18} - 99 \beta_{17} - 79 \beta_{16} + 7 \beta_{15} - 210 \beta_{14} + \cdots + 754 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 457 \beta_{19} - 1312 \beta_{18} + 140 \beta_{17} - 1312 \beta_{16} - 1312 \beta_{15} + 1312 \beta_{13} + \cdots - 109559 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1110 \beta_{19} + 3654 \beta_{18} + 2510 \beta_{17} - 1144 \beta_{16} - 2588 \beta_{15} + \cdots - 145416 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 38900 \beta_{19} + 77800 \beta_{18} + 38900 \beta_{17} + 41856 \beta_{16} + 81758 \beta_{15} + \cdots + 64913 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 153702 \beta_{19} + 121061 \beta_{18} + 44752 \beta_{17} + 121061 \beta_{16} + 121061 \beta_{15} + \cdots + 3924286 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 472473 \beta_{19} - 1115054 \beta_{18} - 1150302 \beta_{17} - 35248 \beta_{16} - 1221303 \beta_{15} + \cdots + 60744609 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3788507 \beta_{19} - 7577014 \beta_{18} - 3788507 \beta_{17} - 2290847 \beta_{16} - 3089854 \beta_{15} + \cdots + 5889135 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 20725155 \beta_{19} - 31361391 \beta_{18} - 458212 \beta_{17} - 31361391 \beta_{16} + \cdots - 1538725247 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 25687413 \beta_{19} + 114315050 \beta_{18} + 68091262 \beta_{17} - 46223788 \beta_{16} + \cdots - 3349074377 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 871274048 \beta_{19} + 1742548096 \beta_{18} + 871274048 \beta_{17} + 820760560 \beta_{16} + \cdots + 1853436370 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 4060053382 \beta_{19} + 3362041236 \beta_{18} + 1361298156 \beta_{17} + 3362041236 \beta_{16} + \cdots + 91341114761 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 5072870529 \beta_{19} - 23998096885 \beta_{18} - 21672349557 \beta_{17} + 2325747328 \beta_{16} + \cdots + 896947824564 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 97018958338 \beta_{19} - 194037916676 \beta_{18} - 97018958338 \beta_{17} - 58076801194 \beta_{16} + \cdots - 31095269469 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
2.58383 + 4.47532i
2.39071 + 4.14084i
1.59029 + 2.75446i
1.35656 + 2.34963i
0.937599 + 1.62397i
−0.575519 − 0.996828i
−1.09984 − 1.90498i
−1.18523 − 2.05287i
−2.09673 − 3.63164i
−2.40167 − 4.15982i
2.58383 − 4.47532i
2.39071 − 4.14084i
1.59029 − 2.75446i
1.35656 − 2.34963i
0.937599 − 1.62397i
−0.575519 + 0.996828i
−1.09984 + 1.90498i
−1.18523 + 2.05287i
−2.09673 + 3.63164i
−2.40167 + 4.15982i
−2.58383 + 4.47532i 1.43183 − 2.47999i −9.35234 − 16.1987i 2.50000 − 4.33013i 7.39918 + 12.8158i −14.1365 55.3182 9.39975 + 16.2809i 12.9191 + 22.3766i
11.2 −2.39071 + 4.14084i −4.61551 + 7.99430i −7.43103 − 12.8709i 2.50000 − 4.33013i −22.0687 − 38.2242i 5.17574 32.8105 −29.1059 − 50.4129i 11.9536 + 20.7042i
11.3 −1.59029 + 2.75446i −0.0359133 + 0.0622037i −1.05802 − 1.83254i 2.50000 − 4.33013i −0.114225 − 0.197843i −23.3899 −18.7144 13.4974 + 23.3782i 7.95143 + 13.7723i
11.4 −1.35656 + 2.34963i 4.30093 − 7.44942i 0.319478 + 0.553352i 2.50000 − 4.33013i 11.6689 + 20.2112i 10.4999 −23.4386 −23.4959 − 40.6961i 6.78281 + 11.7482i
11.5 −0.937599 + 1.62397i −0.636211 + 1.10195i 2.24182 + 3.88294i 2.50000 − 4.33013i −1.19302 − 2.06637i 29.8625 −23.4093 12.6905 + 21.9805i 4.68799 + 8.11984i
11.6 0.575519 − 0.996828i −0.184280 + 0.319183i 3.33756 + 5.78082i 2.50000 − 4.33013i 0.212114 + 0.367391i −2.58808 16.8916 13.4321 + 23.2650i −2.87759 − 4.98414i
11.7 1.09984 − 1.90498i −4.37756 + 7.58215i 1.58071 + 2.73786i 2.50000 − 4.33013i 9.62922 + 16.6783i −33.5680 24.5515 −24.8260 − 43.0000i −5.49920 − 9.52489i
11.8 1.18523 − 2.05287i 3.70163 − 6.41141i 1.19047 + 2.06195i 2.50000 − 4.33013i −8.77455 − 15.1980i 9.37372 24.6076 −13.9041 − 24.0826i −5.92614 − 10.2644i
11.9 2.09673 − 3.63164i −3.72764 + 6.45646i −4.79255 − 8.30094i 2.50000 − 4.33013i 15.6317 + 27.0749i 33.0991 −6.64705 −14.2906 − 24.7520i −10.4836 − 18.1582i
11.10 2.40167 − 4.15982i 1.64274 − 2.84530i −7.53608 − 13.0529i 2.50000 − 4.33013i −7.89063 − 13.6670i −11.3286 −33.9701 8.10284 + 14.0345i −12.0084 − 20.7991i
26.1 −2.58383 − 4.47532i 1.43183 + 2.47999i −9.35234 + 16.1987i 2.50000 + 4.33013i 7.39918 − 12.8158i −14.1365 55.3182 9.39975 − 16.2809i 12.9191 − 22.3766i
26.2 −2.39071 − 4.14084i −4.61551 − 7.99430i −7.43103 + 12.8709i 2.50000 + 4.33013i −22.0687 + 38.2242i 5.17574 32.8105 −29.1059 + 50.4129i 11.9536 − 20.7042i
26.3 −1.59029 − 2.75446i −0.0359133 − 0.0622037i −1.05802 + 1.83254i 2.50000 + 4.33013i −0.114225 + 0.197843i −23.3899 −18.7144 13.4974 − 23.3782i 7.95143 − 13.7723i
26.4 −1.35656 − 2.34963i 4.30093 + 7.44942i 0.319478 − 0.553352i 2.50000 + 4.33013i 11.6689 − 20.2112i 10.4999 −23.4386 −23.4959 + 40.6961i 6.78281 − 11.7482i
26.5 −0.937599 − 1.62397i −0.636211 − 1.10195i 2.24182 − 3.88294i 2.50000 + 4.33013i −1.19302 + 2.06637i 29.8625 −23.4093 12.6905 − 21.9805i 4.68799 − 8.11984i
26.6 0.575519 + 0.996828i −0.184280 − 0.319183i 3.33756 − 5.78082i 2.50000 + 4.33013i 0.212114 − 0.367391i −2.58808 16.8916 13.4321 − 23.2650i −2.87759 + 4.98414i
26.7 1.09984 + 1.90498i −4.37756 − 7.58215i 1.58071 − 2.73786i 2.50000 + 4.33013i 9.62922 − 16.6783i −33.5680 24.5515 −24.8260 + 43.0000i −5.49920 + 9.52489i
26.8 1.18523 + 2.05287i 3.70163 + 6.41141i 1.19047 − 2.06195i 2.50000 + 4.33013i −8.77455 + 15.1980i 9.37372 24.6076 −13.9041 + 24.0826i −5.92614 + 10.2644i
26.9 2.09673 + 3.63164i −3.72764 − 6.45646i −4.79255 + 8.30094i 2.50000 + 4.33013i 15.6317 − 27.0749i 33.0991 −6.64705 −14.2906 + 24.7520i −10.4836 + 18.1582i
26.10 2.40167 + 4.15982i 1.64274 + 2.84530i −7.53608 + 13.0529i 2.50000 + 4.33013i −7.89063 + 13.6670i −11.3286 −33.9701 8.10284 − 14.0345i −12.0084 + 20.7991i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.4.e.c ✓ 20
19.c even 3 1 inner 95.4.e.c ✓ 20
19.c even 3 1 1805.4.a.r 10
19.d odd 6 1 1805.4.a.p 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.4.e.c ✓ 20 1.a even 1 1 trivial
95.4.e.c ✓ 20 19.c even 3 1 inner
1805.4.a.p 10 19.d odd 6 1
1805.4.a.r 10 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 3 T_{2}^{19} + 66 T_{2}^{18} + 125 T_{2}^{17} + 2555 T_{2}^{16} + 3995 T_{2}^{15} + \cdots + 2336368896 \) acting on \(S_{4}^{\mathrm{new}}(95, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 2336368896 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 147865600 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{10} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots - 163855792416)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 20\!\cdots\!39)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 61\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 38\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 24\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 83\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 41\!\cdots\!24 \) Copy content Toggle raw display
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