↑

Properties

Label 33.6.e.b
Level $33$
Weight $6$
Character orbit 33.e
Analytic conductor $5.293$
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] = [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: \( x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} + \cdots + 25599187870096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{5}) q^{2} + 9 \beta_{7} q^{3} + (\beta_{16} + \beta_{12} + 18 \beta_{7} + \cdots - 12) q^{4}+ \cdots - 81 \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{5}) q^{2} + 9 \beta_{7} q^{3} + (\beta_{16} + \beta_{12} + 18 \beta_{7} + \cdots - 12) q^{4}+ \cdots + ( - 81 \beta_{19} + 324 \beta_{17} + \cdots - 4212) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{2} + 45 q^{3} + 8 q^{4} - 11 q^{5} - 54 q^{6} - 139 q^{7} - 76 q^{8} - 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{2} + 45 q^{3} + 8 q^{4} - 11 q^{5} - 54 q^{6} - 139 q^{7} - 76 q^{8} - 405 q^{9} - 424 q^{10} + 2289 q^{11} - 2682 q^{12} - 847 q^{13} + 2022 q^{14} - 396 q^{15} - 7148 q^{16} + 2482 q^{17} + 81 q^{18} + 2958 q^{19} + 8037 q^{20} - 1404 q^{21} + 7441 q^{22} + 8140 q^{23} - 4626 q^{24} - 13120 q^{25} - 13508 q^{26} + 3645 q^{27} + 27819 q^{28} - 20210 q^{29} + 11106 q^{30} + 5540 q^{31} - 4626 q^{32} + 639 q^{33} + 16540 q^{34} - 34101 q^{35} + 648 q^{36} - 25173 q^{37} - 55878 q^{38} + 7623 q^{39} + 22689 q^{40} + 54349 q^{41} + 32292 q^{42} - 21688 q^{43} + 24587 q^{44} - 5346 q^{45} + 44155 q^{46} - 48387 q^{47} - 43083 q^{48} - 78880 q^{49} + 19956 q^{50} + 7317 q^{51} + 109051 q^{52} - 74382 q^{53} + 10206 q^{54} - 47630 q^{55} + 168894 q^{56} + 52983 q^{57} + 1501 q^{58} - 108412 q^{59} - 72333 q^{60} + 16737 q^{61} + 132685 q^{62} - 11259 q^{63} - 157030 q^{64} - 134194 q^{65} - 127539 q^{66} + 20178 q^{67} + 387013 q^{68} + 19305 q^{69} - 411988 q^{70} + 143157 q^{71} + 41634 q^{72} + 164980 q^{73} + 286114 q^{74} - 96255 q^{75} + 46868 q^{76} - 367461 q^{77} - 152658 q^{78} + 369613 q^{79} - 107448 q^{80} - 32805 q^{81} - 184129 q^{82} + 267741 q^{83} + 279819 q^{84} + 379937 q^{85} - 290531 q^{86} + 214920 q^{87} - 740846 q^{88} + 205492 q^{89} + 117126 q^{90} + 394559 q^{91} - 483181 q^{92} - 49860 q^{93} + 561327 q^{94} + 267881 q^{95} + 228564 q^{96} - 300567 q^{97} - 2296174 q^{98} - 115506 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} + \cdots + 25599187870096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24\!\cdots\!63 \nu^{19} + \cdots - 65\!\cdots\!64 ) / 34\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 98\!\cdots\!84 \nu^{19} + \cdots + 78\!\cdots\!30 ) / 13\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!82 \nu^{19} + \cdots - 50\!\cdots\!28 ) / 27\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 68\!\cdots\!67 \nu^{19} + \cdots + 56\!\cdots\!00 ) / 54\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!65 \nu^{19} + \cdots + 20\!\cdots\!20 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!06 \nu^{19} + \cdots + 32\!\cdots\!16 ) / 69\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 56\!\cdots\!13 \nu^{19} + \cdots - 29\!\cdots\!88 ) / 54\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!73 \nu^{19} + \cdots - 39\!\cdots\!64 ) / 34\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 81\!\cdots\!62 \nu^{19} + \cdots + 18\!\cdots\!84 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!41 \nu^{19} + \cdots + 16\!\cdots\!84 ) / 31\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 56\!\cdots\!84 \nu^{19} + \cdots + 10\!\cdots\!64 ) / 69\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 40\!\cdots\!71 \nu^{19} + \cdots - 24\!\cdots\!12 ) / 40\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!25 \nu^{19} + \cdots + 40\!\cdots\!80 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23\!\cdots\!85 \nu^{19} + \cdots + 58\!\cdots\!24 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 16\!\cdots\!35 \nu^{19} + \cdots - 10\!\cdots\!84 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 61\!\cdots\!69 \nu^{19} + \cdots - 13\!\cdots\!12 ) / 40\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 62\!\cdots\!69 \nu^{19} + \cdots - 11\!\cdots\!44 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 80\!\cdots\!34 \nu^{19} + \cdots - 26\!\cdots\!28 ) / 20\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{9} - \beta_{8} - 49\beta_{7} - 6\beta_{6} + \beta_{4} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} + \beta_{16} + 2 \beta_{14} + 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{9} + \cdots - 35 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{19} - 3 \beta_{18} + 5 \beta_{17} - 109 \beta_{16} + 3 \beta_{15} - 5 \beta_{14} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 17 \beta_{19} + 17 \beta_{18} - 249 \beta_{17} - 219 \beta_{16} + 249 \beta_{13} - 408 \beta_{11} + \cdots + 7332 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 391 \beta_{18} - 943 \beta_{17} - 470 \beta_{15} + 472 \beta_{14} + 2129 \beta_{12} - 943 \beta_{11} + \cdots - 129800 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3378 \beta_{19} + 10216 \beta_{18} - 18712 \beta_{17} + 18726 \beta_{16} + 3378 \beta_{15} + \cdots + 101768 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 103772 \beta_{19} + 1140633 \beta_{16} - 45416 \beta_{15} - 74628 \beta_{14} - 33804 \beta_{13} + \cdots - 24013831 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1450621 \beta_{19} - 960777 \beta_{18} + 2035773 \beta_{17} - 2777337 \beta_{16} + 960777 \beta_{15} + \cdots + 26404325 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5314715 \beta_{19} + 5314715 \beta_{18} + 4888469 \beta_{17} + 37443407 \beta_{16} + \cdots + 2321917415 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 63162745 \beta_{18} + 291186469 \beta_{17} - 93206829 \beta_{15} + 215238939 \beta_{14} + \cdots - 5229729401 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 628941271 \beta_{19} + 732584482 \beta_{18} - 444888904 \beta_{17} - 4464778965 \beta_{16} + \cdots + 206326475808 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 17032821338 \beta_{19} + 43657317140 \beta_{16} - 7707637614 \beta_{15} + 30587060194 \beta_{14} + \cdots - 1324846024046 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 153953121956 \beta_{19} - 79410379408 \beta_{18} + 53176977104 \beta_{17} - 1277679603561 \beta_{16} + \cdots + 2160057360596 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 913604260816 \beta_{19} + 913604260816 \beta_{18} - 3235932238730 \beta_{17} + 863704227014 \beta_{16} + \cdots + 144241411965627 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 8796134072883 \beta_{18} + 3823389968623 \beta_{17} - 8576511792099 \beta_{15} + \cdots - 26\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 106507268471761 \beta_{19} + 99981415905261 \beta_{18} - 246103968451527 \beta_{17} + \cdots + 14\!\cdots\!77 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 19\!\cdots\!09 \beta_{19} + \cdots - 25\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 22\!\cdots\!26 \beta_{19} + \cdots + 55\!\cdots\!22 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1 - \beta_{2} + \beta_{6} + \beta_{7}\) \(1\)

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
−8.59311 + 6.24326i
−3.89647 + 2.83095i
−0.490010 + 0.356013i
4.70404 − 3.41769i
7.96653 − 5.78803i
−2.34585 + 7.21978i
−1.64776 + 5.07128i
0.527694 − 1.62407i
1.34436 − 4.13753i
2.93057 − 9.01936i
−8.59311 − 6.24326i
−3.89647 − 2.83095i
−0.490010 − 0.356013i
4.70404 + 3.41769i
7.96653 + 5.78803i
−2.34585 − 7.21978i
−1.64776 − 5.07128i
0.527694 + 1.62407i
1.34436 + 4.13753i
2.93057 + 9.01936i
−7.78409 − 5.65548i −2.78115 + 8.55951i 18.7192 + 57.6117i 48.1408 − 34.9763i 70.0568 − 50.8993i 9.43666 + 29.0431i 84.9654 − 261.497i −65.5304 − 47.6106i −572.540
4.2 −3.08746 − 2.24317i −2.78115 + 8.55951i −5.38796 − 16.5824i 3.77724 − 2.74432i 27.7871 − 20.1885i 49.4196 + 152.098i −58.2998 + 179.428i −65.5304 − 47.6106i −17.8180
4.3 0.319007 + 0.231772i −2.78115 + 8.55951i −9.84050 − 30.2859i −17.3054 + 12.5731i −2.87106 + 2.08595i −52.3175 − 161.017i 7.77944 − 23.9427i −65.5304 − 47.6106i −8.43465
4.4 5.51306 + 4.00547i −2.78115 + 8.55951i 4.46148 + 13.7310i −62.5986 + 45.4806i −49.6176 + 36.0493i 59.3828 + 182.761i 36.9828 − 113.821i −65.5304 − 47.6106i −527.281
4.5 8.77555 + 6.37581i −2.78115 + 8.55951i 26.4708 + 81.4687i 31.3852 − 22.8027i −78.9800 + 57.3823i −67.6896 − 208.327i −179.871 + 553.584i −65.5304 − 47.6106i 420.808
16.1 −2.65487 − 8.17084i 7.28115 + 5.29007i −33.8257 + 24.5758i −21.1802 + 65.1858i 23.8938 − 73.5376i −196.384 + 142.681i 68.1911 + 49.5438i 25.0304 + 77.0356i 588.853
16.2 −1.95678 − 6.02234i 7.28115 + 5.29007i −6.55106 + 4.75963i 8.65323 − 26.6319i 17.6110 − 54.2011i 121.539 − 88.3034i −122.450 − 88.9651i 25.0304 + 77.0356i −177.319
16.3 0.218677 + 0.673018i 7.28115 + 5.29007i 25.4834 − 18.5148i −21.0138 + 64.6739i −1.96809 + 6.05716i 98.3565 − 71.4602i 36.3535 + 26.4124i 25.0304 + 77.0356i −48.1219
16.4 1.03535 + 3.18647i 7.28115 + 5.29007i 16.8069 − 12.2109i 32.9083 − 101.281i −9.31813 + 28.6783i −32.7853 + 23.8199i 143.049 + 103.931i 25.0304 + 77.0356i 356.801
16.5 2.62155 + 8.06830i 7.28115 + 5.29007i −32.3365 + 23.4938i −8.26670 + 25.4423i −23.5940 + 72.6147i −58.4588 + 42.4728i −54.7010 − 39.7426i 25.0304 + 77.0356i −226.948
25.1 −7.78409 + 5.65548i −2.78115 − 8.55951i 18.7192 − 57.6117i 48.1408 + 34.9763i 70.0568 + 50.8993i 9.43666 − 29.0431i 84.9654 + 261.497i −65.5304 + 47.6106i −572.540
25.2 −3.08746 + 2.24317i −2.78115 − 8.55951i −5.38796 + 16.5824i 3.77724 + 2.74432i 27.7871 + 20.1885i 49.4196 − 152.098i −58.2998 − 179.428i −65.5304 + 47.6106i −17.8180
25.3 0.319007 − 0.231772i −2.78115 − 8.55951i −9.84050 + 30.2859i −17.3054 − 12.5731i −2.87106 − 2.08595i −52.3175 + 161.017i 7.77944 + 23.9427i −65.5304 + 47.6106i −8.43465
25.4 5.51306 − 4.00547i −2.78115 − 8.55951i 4.46148 − 13.7310i −62.5986 − 45.4806i −49.6176 − 36.0493i 59.3828 − 182.761i 36.9828 + 113.821i −65.5304 + 47.6106i −527.281
25.5 8.77555 − 6.37581i −2.78115 − 8.55951i 26.4708 − 81.4687i 31.3852 + 22.8027i −78.9800 − 57.3823i −67.6896 + 208.327i −179.871 − 553.584i −65.5304 + 47.6106i 420.808
31.1 −2.65487 + 8.17084i 7.28115 − 5.29007i −33.8257 − 24.5758i −21.1802 − 65.1858i 23.8938 + 73.5376i −196.384 − 142.681i 68.1911 − 49.5438i 25.0304 − 77.0356i 588.853
31.2 −1.95678 + 6.02234i 7.28115 − 5.29007i −6.55106 − 4.75963i 8.65323 + 26.6319i 17.6110 + 54.2011i 121.539 + 88.3034i −122.450 + 88.9651i 25.0304 − 77.0356i −177.319
31.3 0.218677 − 0.673018i 7.28115 − 5.29007i 25.4834 + 18.5148i −21.0138 − 64.6739i −1.96809 − 6.05716i 98.3565 + 71.4602i 36.3535 − 26.4124i 25.0304 − 77.0356i −48.1219
31.4 1.03535 − 3.18647i 7.28115 − 5.29007i 16.8069 + 12.2109i 32.9083 + 101.281i −9.31813 − 28.6783i −32.7853 − 23.8199i 143.049 − 103.931i 25.0304 − 77.0356i 356.801
31.5 2.62155 − 8.06830i 7.28115 − 5.29007i −32.3365 − 23.4938i −8.26670 − 25.4423i −23.5940 − 72.6147i −58.4588 − 42.4728i −54.7010 + 39.7426i 25.0304 − 77.0356i −226.948
\(n\): 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.b ✓ 20
3.b odd 2 1 99.6.f.b 20
11.c even 5 1 inner 33.6.e.b ✓ 20
11.c even 5 1 363.6.a.r 10
11.d odd 10 1 363.6.a.t 10
33.f even 10 1 1089.6.a.bi 10
33.h odd 10 1 99.6.f.b 20
33.h odd 10 1 1089.6.a.bk 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.b ✓ 20 1.a even 1 1 trivial
33.6.e.b ✓ 20 11.c even 5 1 inner
99.6.f.b 20 3.b odd 2 1
99.6.f.b 20 33.h odd 10 1
363.6.a.r 10 11.c even 5 1
363.6.a.t 10 11.d odd 10 1
1089.6.a.bi 10 33.f even 10 1
1089.6.a.bk 10 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 6 T_{2}^{19} + 94 T_{2}^{18} - 488 T_{2}^{17} + 12401 T_{2}^{16} - 30756 T_{2}^{15} + \cdots + 1371559530496 \) acting on \(S_{6}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 1371559530496 \) Copy content Toggle raw display
$3$ \( (T^{4} - 9 T^{3} + \cdots + 6561)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 43\!\cdots\!61 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 12\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 91\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 33\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 77\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 75\!\cdots\!61 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 11\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
show more
show less