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Properties

Label 33.6.e.a
Level $33$
Weight $6$
Character orbit 33.e
Analytic conductor $5.293$
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] = [33,6,Mod(4,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.e (of order \(5\), degree \(4\), 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: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} + 142 x^{18} - 389 x^{17} + 14927 x^{16} - 6599 x^{15} + 1399353 x^{14} + \cdots + 25735126080400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 11^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_1) q^{2} - 9 \beta_{9} q^{3} + ( - \beta_{15} - 2 \beta_{9} + \cdots - \beta_{6}) q^{4}+ \cdots + 81 \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_1) q^{2} - 9 \beta_{9} q^{3} + ( - \beta_{15} - 2 \beta_{9} + \cdots - \beta_{6}) q^{4}+ \cdots + (405 \beta_{19} + 162 \beta_{18} + \cdots - 4617) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9} + 1952 q^{10} - 835 q^{11} + 3474 q^{12} - 959 q^{13} - 3020 q^{14} + 198 q^{15} - 1428 q^{16} - 3144 q^{17} - 567 q^{18} - 930 q^{19} + 3177 q^{20} + 8460 q^{21} + 2713 q^{22} + 1400 q^{23} + 1332 q^{24} - 9388 q^{25} + 3022 q^{26} - 3645 q^{27} - 21753 q^{28} + 4494 q^{29} - 4302 q^{30} + 9208 q^{31} + 69114 q^{32} + 6615 q^{33} - 9084 q^{34} - 31371 q^{35} - 10044 q^{36} - 3153 q^{37} + 1852 q^{38} - 8631 q^{39} - 53123 q^{40} - 7497 q^{41} + 22320 q^{42} + 63624 q^{43} + 58299 q^{44} + 1782 q^{45} - 94681 q^{46} - 43031 q^{47} - 12357 q^{48} - 5640 q^{49} - 76590 q^{50} + 459 q^{51} - 80149 q^{52} - 20452 q^{53} + 13122 q^{54} + 62974 q^{55} + 169638 q^{56} - 39375 q^{57} + 93145 q^{58} + 101730 q^{59} + 28593 q^{60} + 50745 q^{61} - 171957 q^{62} - 27135 q^{63} + 199638 q^{64} - 28210 q^{65} + 46107 q^{66} + 10730 q^{67} - 420141 q^{68} + 94365 q^{69} + 292664 q^{70} + 164895 q^{71} + 11988 q^{72} - 77236 q^{73} - 328228 q^{74} + 9063 q^{75} + 79212 q^{76} + 56087 q^{77} - 280062 q^{78} - 291931 q^{79} + 18026 q^{80} - 32805 q^{81} + 424659 q^{82} - 255451 q^{83} - 26127 q^{84} - 452271 q^{85} + 252843 q^{86} + 187056 q^{87} + 171694 q^{88} + 181632 q^{89} - 40338 q^{90} + 284615 q^{91} + 1199691 q^{92} + 82872 q^{93} - 550105 q^{94} - 511091 q^{95} - 587574 q^{96} - 145083 q^{97} - 150570 q^{98} - 83430 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} + 142 x^{18} - 389 x^{17} + 14927 x^{16} - 6599 x^{15} + 1399353 x^{14} + \cdots + 25735126080400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 43\!\cdots\!67 \nu^{19} + \cdots - 33\!\cdots\!00 ) / 78\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!83 \nu^{19} + \cdots + 41\!\cdots\!00 ) / 86\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 45\!\cdots\!18 \nu^{19} + \cdots - 64\!\cdots\!50 ) / 21\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52\!\cdots\!85 \nu^{19} + \cdots + 76\!\cdots\!80 ) / 17\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!56 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 43\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 65\!\cdots\!47 \nu^{19} + \cdots + 39\!\cdots\!00 ) / 86\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 46\!\cdots\!17 \nu^{19} + \cdots - 67\!\cdots\!40 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 59\!\cdots\!67 \nu^{19} + \cdots - 62\!\cdots\!80 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 35\!\cdots\!52 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!63 \nu^{19} + \cdots - 73\!\cdots\!40 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 62\!\cdots\!25 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!65 \nu^{19} + \cdots + 49\!\cdots\!80 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 21\!\cdots\!75 \nu^{19} + \cdots - 32\!\cdots\!00 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 61\!\cdots\!65 \nu^{19} + \cdots + 29\!\cdots\!00 ) / 99\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 31\!\cdots\!87 \nu^{19} + \cdots + 33\!\cdots\!00 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 34\!\cdots\!31 \nu^{19} + \cdots - 22\!\cdots\!60 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 18\!\cdots\!43 \nu^{19} + \cdots + 22\!\cdots\!40 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 18\!\cdots\!89 \nu^{19} + \cdots + 14\!\cdots\!20 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 40\!\cdots\!63 \nu^{19} + \cdots + 33\!\cdots\!80 ) / 39\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} - 49\beta_{9} - 2\beta_{7} + \beta_{5} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} - \beta_{16} + 2 \beta_{15} - 3 \beta_{12} - 16 \beta_{9} + 44 \beta_{8} + 16 \beta_{7} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 15 \beta_{19} - \beta_{18} + 4 \beta_{17} + \beta_{16} + 7 \beta_{15} - 9 \beta_{14} - 118 \beta_{13} + \cdots + 111 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 110 \beta_{18} - 395 \beta_{17} - 494 \beta_{16} - 505 \beta_{15} - 150 \beta_{14} - 161 \beta_{13} + \cdots - 10803 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2122 \beta_{19} + 1654 \beta_{18} - 2122 \beta_{17} + 827 \beta_{16} - 1053 \beta_{15} - 9744 \beta_{14} + \cdots - 378939 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3254 \beta_{19} - 9950 \beta_{18} + 40006 \beta_{17} + 40970 \beta_{16} - 10342 \beta_{15} + \cdots - 922644 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 177348 \beta_{19} - 31076 \beta_{18} + 50584 \beta_{16} - 1303965 \beta_{15} + 124144 \beta_{14} + \cdots - 78920 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4524927 \beta_{19} + 1170881 \beta_{18} - 3886103 \beta_{17} + 2336386 \beta_{16} - 6312007 \beta_{15} + \cdots + 126112226 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3689918 \beta_{18} + 25938091 \beta_{17} + 15102918 \beta_{16} + 22248173 \beta_{15} + \cdots + 3935325780 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 467464551 \beta_{19} - 77745526 \beta_{18} + 467464551 \beta_{17} - 38872763 \beta_{16} + \cdots + 22616158914 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2521311739 \beta_{19} - 285290813 \beta_{18} - 230635103 \beta_{17} - 2111953519 \beta_{16} + \cdots + 51454975884 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 14716314606 \beta_{19} + 10210150996 \beta_{18} - 16215654966 \beta_{16} + 94634925546 \beta_{15} + \cdots + 4355405318 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 292352283784 \beta_{19} - 34854111888 \beta_{18} - 5557400036 \beta_{17} - 252545818884 \beta_{16} + \cdots - 7544573958818 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 479735928081 \beta_{18} - 4998525075371 \beta_{17} - 5009638076492 \beta_{16} + \cdots - 316478871362913 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 31257017635655 \beta_{19} + 10444761992334 \beta_{18} - 31257017635655 \beta_{17} + \cdots - 47\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 253472038415379 \beta_{19} - 38850817478382 \beta_{18} + 267287994097648 \beta_{17} + \cdots - 25\!\cdots\!41 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 40\!\cdots\!51 \beta_{19} - 311297496607090 \beta_{18} + \cdots - 479901689565643 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 54\!\cdots\!64 \beta_{19} + \cdots + 29\!\cdots\!94 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-\beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−7.12695 + 5.17803i
−4.29115 + 3.11770i
0.218287 − 0.158595i
3.88934 − 2.82577i
8.61948 − 6.26242i
−3.00337 + 9.24343i
−2.05193 + 6.31518i
0.992840 − 3.05565i
1.17535 − 3.61737i
3.07809 − 9.47338i
−7.12695 − 5.17803i
−4.29115 − 3.11770i
0.218287 + 0.158595i
3.88934 + 2.82577i
8.61948 + 6.26242i
−3.00337 − 9.24343i
−2.05193 − 6.31518i
0.992840 + 3.05565i
1.17535 + 3.61737i
3.07809 + 9.47338i
−7.93597 − 5.76582i 2.78115 − 8.55951i 19.8464 + 61.0809i −37.2750 + 27.0819i −71.4237 + 51.8924i 37.2644 + 114.688i 97.6803 − 300.629i −65.5304 − 47.6106i 451.963
4.2 −5.10017 − 3.70549i 2.78115 − 8.55951i 2.39251 + 7.36340i 54.7018 − 39.7432i −45.9015 + 33.3494i −37.8504 − 116.492i −47.2561 + 145.439i −65.5304 − 47.6106i −426.256
4.3 −0.590730 − 0.429190i 2.78115 − 8.55951i −9.72379 − 29.9267i −77.5329 + 56.3310i −5.31657 + 3.86271i 29.5747 + 91.0216i −14.3206 + 44.0742i −65.5304 − 47.6106i 69.9777
4.4 3.08033 + 2.23799i 2.78115 − 8.55951i −5.40872 − 16.6463i 20.5684 − 14.9438i 27.7229 − 20.1419i −26.9084 − 82.8155i 58.2442 − 179.257i −65.5304 − 47.6106i 96.8014
4.5 7.81047 + 5.67464i 2.78115 − 8.55951i 18.9134 + 58.2093i 25.1385 − 18.2642i 70.2942 − 51.0717i 56.7190 + 174.563i −87.1280 + 268.152i −65.5304 − 47.6106i 299.987
16.1 −2.69436 − 8.29237i −7.28115 − 5.29007i −35.6154 + 25.8761i −23.0548 + 70.9555i −24.2492 + 74.6314i 46.3463 − 33.6726i 84.8093 + 61.6176i 25.0304 + 77.0356i 650.507
16.2 −1.74291 − 5.36413i −7.28115 − 5.29007i 0.152432 − 0.110748i 21.0541 − 64.7978i −15.6862 + 48.2771i −106.538 + 77.4041i −146.876 − 106.711i 25.0304 + 77.0356i −384.279
16.3 1.30186 + 4.00670i −7.28115 − 5.29007i 11.5297 − 8.37681i −26.9724 + 83.0125i 11.7167 − 36.0603i −114.007 + 82.8310i 157.639 + 114.532i 25.0304 + 77.0356i −367.721
16.4 1.48437 + 4.56842i −7.28115 − 5.29007i 7.22140 − 5.24666i 18.4760 − 56.8633i 13.3593 − 41.1158i 58.2735 − 42.3382i 159.044 + 115.553i 25.0304 + 77.0356i 287.201
16.5 3.38711 + 10.4244i −7.28115 − 5.29007i −71.3079 + 51.8082i 8.39633 − 25.8412i 30.4839 − 93.8199i −110.374 + 80.1917i −497.837 − 361.700i 25.0304 + 77.0356i 297.820
25.1 −7.93597 + 5.76582i 2.78115 + 8.55951i 19.8464 − 61.0809i −37.2750 − 27.0819i −71.4237 − 51.8924i 37.2644 − 114.688i 97.6803 + 300.629i −65.5304 + 47.6106i 451.963
25.2 −5.10017 + 3.70549i 2.78115 + 8.55951i 2.39251 − 7.36340i 54.7018 + 39.7432i −45.9015 − 33.3494i −37.8504 + 116.492i −47.2561 − 145.439i −65.5304 + 47.6106i −426.256
25.3 −0.590730 + 0.429190i 2.78115 + 8.55951i −9.72379 + 29.9267i −77.5329 − 56.3310i −5.31657 − 3.86271i 29.5747 − 91.0216i −14.3206 − 44.0742i −65.5304 + 47.6106i 69.9777
25.4 3.08033 − 2.23799i 2.78115 + 8.55951i −5.40872 + 16.6463i 20.5684 + 14.9438i 27.7229 + 20.1419i −26.9084 + 82.8155i 58.2442 + 179.257i −65.5304 + 47.6106i 96.8014
25.5 7.81047 − 5.67464i 2.78115 + 8.55951i 18.9134 − 58.2093i 25.1385 + 18.2642i 70.2942 + 51.0717i 56.7190 − 174.563i −87.1280 − 268.152i −65.5304 + 47.6106i 299.987
31.1 −2.69436 + 8.29237i −7.28115 + 5.29007i −35.6154 − 25.8761i −23.0548 − 70.9555i −24.2492 − 74.6314i 46.3463 + 33.6726i 84.8093 − 61.6176i 25.0304 − 77.0356i 650.507
31.2 −1.74291 + 5.36413i −7.28115 + 5.29007i 0.152432 + 0.110748i 21.0541 + 64.7978i −15.6862 − 48.2771i −106.538 − 77.4041i −146.876 + 106.711i 25.0304 − 77.0356i −384.279
31.3 1.30186 − 4.00670i −7.28115 + 5.29007i 11.5297 + 8.37681i −26.9724 − 83.0125i 11.7167 + 36.0603i −114.007 − 82.8310i 157.639 − 114.532i 25.0304 − 77.0356i −367.721
31.4 1.48437 − 4.56842i −7.28115 + 5.29007i 7.22140 + 5.24666i 18.4760 + 56.8633i 13.3593 + 41.1158i 58.2735 + 42.3382i 159.044 − 115.553i 25.0304 − 77.0356i 287.201
31.5 3.38711 − 10.4244i −7.28115 + 5.29007i −71.3079 − 51.8082i 8.39633 + 25.8412i 30.4839 + 93.8199i −110.374 − 80.1917i −497.837 + 361.700i 25.0304 − 77.0356i 297.820
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.6.e.a ✓ 20
3.b odd 2 1 99.6.f.c 20
11.c even 5 1 inner 33.6.e.a ✓ 20
11.c even 5 1 363.6.a.u 10
11.d odd 10 1 363.6.a.q 10
33.f even 10 1 1089.6.a.bl 10
33.h odd 10 1 99.6.f.c 20
33.h odd 10 1 1089.6.a.bh 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.a ✓ 20 1.a even 1 1 trivial
33.6.e.a ✓ 20 11.c even 5 1 inner
99.6.f.c 20 3.b odd 2 1
99.6.f.c 20 33.h odd 10 1
363.6.a.q 10 11.d odd 10 1
363.6.a.u 10 11.c even 5 1
1089.6.a.bh 10 33.h odd 10 1
1089.6.a.bl 10 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 2 T_{2}^{19} + 144 T_{2}^{18} + 592 T_{2}^{17} + 15553 T_{2}^{16} + 24816 T_{2}^{15} + \cdots + 327810433847296 \) acting on \(S_{6}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 327810433847296 \) Copy content Toggle raw display
$3$ \( (T^{4} + 9 T^{3} + \cdots + 6561)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 55\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 37\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 41\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 82\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 76\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 38\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 89\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 46\!\cdots\!61 \) Copy content Toggle raw display
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