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Properties

Label 41.3.e.b
Level $41$
Weight $3$
Character orbit 41.e
Analytic conductor $1.117$
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] = [41,3,Mod(3,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.3");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 41.e (of order \(8\), 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: \(1.11716908388\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 66 x^{18} + 1853 x^{16} + 28868 x^{14} + 272678 x^{12} + 1600296 x^{10} + 5739482 x^{8} + \cdots + 776161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{12} q^{2} - \beta_{8} q^{3} + (4 \beta_{17} - \beta_{10} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{2} - \beta_{8} q^{3} + (4 \beta_{17} - \beta_{10} + \cdots - \beta_1) q^{4}+ \cdots + ( - 5 \beta_{19} + 42 \beta_{17} + \cdots - 11 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 8 q^{2} + 4 q^{3} - 12 q^{5} - 16 q^{6} - 4 q^{7} + 36 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 8 q^{2} + 4 q^{3} - 12 q^{5} - 16 q^{6} - 4 q^{7} + 36 q^{8} + 4 q^{9} + 16 q^{10} - 12 q^{11} - 100 q^{12} - 48 q^{13} + 88 q^{14} + 40 q^{15} - 36 q^{16} - 28 q^{17} - 12 q^{18} + 76 q^{19} - 16 q^{20} - 88 q^{21} - 116 q^{22} + 268 q^{24} + 40 q^{26} - 80 q^{27} + 72 q^{28} - 24 q^{29} - 216 q^{30} + 176 q^{32} + 176 q^{33} + 80 q^{34} + 60 q^{35} + 48 q^{36} + 208 q^{37} - 380 q^{38} - 68 q^{39} - 116 q^{41} + 280 q^{42} - 40 q^{43} + 116 q^{44} - 176 q^{46} - 64 q^{47} - 480 q^{48} + 168 q^{49} - 148 q^{50} - 72 q^{51} - 184 q^{52} - 120 q^{53} - 284 q^{54} + 20 q^{55} + 188 q^{56} + 560 q^{57} + 36 q^{58} - 512 q^{59} + 500 q^{60} - 460 q^{61} + 68 q^{62} - 520 q^{63} + 432 q^{65} + 300 q^{67} + 120 q^{68} + 300 q^{69} + 308 q^{70} - 108 q^{71} + 60 q^{73} + 140 q^{74} + 52 q^{75} + 872 q^{76} + 112 q^{77} + 1072 q^{78} - 208 q^{79} - 68 q^{80} - 376 q^{82} - 120 q^{83} - 1604 q^{84} + 172 q^{85} + 200 q^{87} + 316 q^{88} + 268 q^{89} - 512 q^{90} - 800 q^{91} - 448 q^{92} + 312 q^{93} - 212 q^{94} - 184 q^{95} - 612 q^{96} - 120 q^{97} - 20 q^{98} + 164 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 66 x^{18} + 1853 x^{16} + 28868 x^{14} + 272678 x^{12} + 1600296 x^{10} + 5739482 x^{8} + \cdots + 776161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11073161236 \nu^{18} - 626579506934 \nu^{16} - 14551029156987 \nu^{14} + \cdots + 559189348481147 ) / 106150240578984 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1307437 \nu^{18} + 74855464 \nu^{16} + 1768989176 \nu^{14} + 22326313968 \nu^{12} + \cdots + 119232721616 ) / 9921046832 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1721480208401 \nu^{19} - 346760702513847 \nu^{18} + 68265928277717 \nu^{17} + \cdots - 10\!\cdots\!69 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1721480208401 \nu^{19} - 346760702513847 \nu^{18} - 68265928277717 \nu^{17} + \cdots - 10\!\cdots\!69 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 483323209483 \nu^{19} + 2025964935437 \nu^{18} - 16101907510229 \nu^{17} + \cdots + 13\!\cdots\!67 ) / 32\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 483323209483 \nu^{19} - 2025964935437 \nu^{18} - 16101907510229 \nu^{17} + \cdots - 13\!\cdots\!67 ) / 32\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6155016118801 \nu^{19} + 30172313894137 \nu^{18} - 213791143566857 \nu^{17} + \cdots - 29\!\cdots\!39 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6155016118801 \nu^{19} + 30172313894137 \nu^{18} + 213791143566857 \nu^{17} + \cdots - 29\!\cdots\!39 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24823170781 \nu^{19} - 1522357508879 \nu^{17} - 39684414007902 \nu^{15} + \cdots - 10\!\cdots\!05 \nu ) / 45\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2299619677 \nu^{19} - 17931242129 \nu^{18} + 127167617843 \nu^{17} + \cdots - 425807747554523 ) / 363943681985088 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2299619677 \nu^{19} - 17931242129 \nu^{18} - 127167617843 \nu^{17} + \cdots - 425807747554523 ) / 363943681985088 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17537145217875 \nu^{19} - 140393773054721 \nu^{18} + \cdots - 43\!\cdots\!03 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17537145217875 \nu^{19} + 140393773054721 \nu^{18} + \cdots + 43\!\cdots\!03 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1762507495379 \nu^{19} - 13642991677411 \nu^{18} + 119174469966544 \nu^{17} + \cdots + 85\!\cdots\!82 ) / 18\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1762507495379 \nu^{19} - 13642991677411 \nu^{18} - 119174469966544 \nu^{17} + \cdots + 85\!\cdots\!82 ) / 18\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 135337936 \nu^{19} + 7780451779 \nu^{17} + 184833531624 \nu^{15} + 2348456072392 \nu^{13} + \cdots - 9339283083873 \nu ) / 8740442258992 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 117455709110375 \nu^{19} - 45007179263885 \nu^{18} + \cdots - 19\!\cdots\!93 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 117455709110375 \nu^{19} + 45007179263885 \nu^{18} + \cdots + 19\!\cdots\!93 ) / 22\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} + \beta_{18} - \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} - \beta_{16} + \beta_{15} + \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} + \cdots - 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14 \beta_{19} - 14 \beta_{18} + \beta_{16} + \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{9} + \cdots + 83 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{19} - 3 \beta_{18} + 36 \beta_{17} + 18 \beta_{16} - 18 \beta_{15} + 2 \beta_{14} + \cdots + 71 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 187 \beta_{19} + 187 \beta_{18} - 25 \beta_{16} - 25 \beta_{15} - 41 \beta_{14} + 41 \beta_{13} + \cdots - 938 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 89 \beta_{19} + 89 \beta_{18} - 829 \beta_{17} - 285 \beta_{16} + 285 \beta_{15} - 53 \beta_{14} + \cdots - 668 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2530 \beta_{19} - 2530 \beta_{18} + 496 \beta_{16} + 496 \beta_{15} + 674 \beta_{14} - 674 \beta_{13} + \cdots + 11263 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1834 \beta_{19} - 1834 \beta_{18} + 15626 \beta_{17} + 4300 \beta_{16} - 4300 \beta_{15} + \cdots + 6499 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 34927 \beta_{19} + 34927 \beta_{18} - 8826 \beta_{16} - 8826 \beta_{15} - 10388 \beta_{14} + \cdots - 141994 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 32736 \beta_{19} + 32736 \beta_{18} - 266747 \beta_{17} - 63253 \beta_{16} + 63253 \beta_{15} + \cdots - 64506 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 491032 \beta_{19} - 491032 \beta_{18} + 147389 \beta_{16} + 147389 \beta_{15} + 156146 \beta_{14} + \cdots + 1861637 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 543425 \beta_{19} - 543425 \beta_{18} + 4311018 \beta_{17} + 918326 \beta_{16} - 918326 \beta_{15} + \cdots + 647849 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7000853 \beta_{19} + 7000853 \beta_{18} - 2365879 \beta_{16} - 2365879 \beta_{15} - 2320695 \beta_{14} + \cdots - 25164708 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 8653505 \beta_{19} + 8653505 \beta_{18} - 67436103 \beta_{17} - 13253881 \beta_{16} + \cdots - 6544370 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 100815580 \beta_{19} - 100815580 \beta_{18} + 37010452 \beta_{16} + 37010452 \beta_{15} + \cdots + 348101909 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 134295104 \beta_{19} - 134295104 \beta_{18} + 1033486228 \beta_{17} + 190952224 \beta_{16} + \cdots + 66129757 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1461663333 \beta_{19} + 1461663333 \beta_{18} - 568954648 \beta_{16} - 568954648 \beta_{15} + \cdots - 4897504276 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 2049475588 \beta_{19} + 2049475588 \beta_{18} - 15628316581 \beta_{17} - 2752518457 \beta_{16} + \cdots - 664260428 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).

\(n\) \(6\)
\(\chi(n)\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
− 3.83033i
− 2.46048i
− 1.38143i
0.701494i
3.14232i
3.83033i
2.46048i
1.38143i
− 0.701494i
− 3.14232i
− 3.37131i
− 2.90722i
− 0.527584i
1.98128i
2.99641i
3.37131i
2.90722i
0.527584i
− 1.98128i
− 2.99641i
−2.70845 + 2.70845i 1.91810 − 4.63070i − 10.6714i −0.877057 − 0.877057i 7.34695 + 17.7371i 1.14833 − 2.77231i 18.0692 + 18.0692i −11.4003 − 11.4003i 4.75094
3.2 −1.73982 + 1.73982i −0.978968 + 2.36344i − 2.05398i −3.70619 − 3.70619i −2.40873 − 5.81520i −3.57359 + 8.62742i −3.38573 − 3.38573i 1.73650 + 1.73650i 12.8962
3.3 −0.976817 + 0.976817i 0.355101 − 0.857291i 2.09166i 4.54303 + 4.54303i 0.490547 + 1.18428i 0.392560 − 0.947724i −5.95043 − 5.95043i 5.75511 + 5.75511i −8.87541
3.4 0.496031 − 0.496031i −1.75813 + 4.24449i 3.50791i −0.197696 − 0.197696i 1.23332 + 2.97749i 4.76097 − 11.4940i 3.72416 + 3.72416i −8.56074 − 8.56074i −0.196127
3.5 2.22195 − 2.22195i −0.657427 + 1.58717i − 5.87417i −1.34788 − 1.34788i 2.06584 + 4.98739i −2.31405 + 5.58661i −4.16431 − 4.16431i 4.27707 + 4.27707i −5.98984
14.1 −2.70845 − 2.70845i 1.91810 + 4.63070i 10.6714i −0.877057 + 0.877057i 7.34695 − 17.7371i 1.14833 + 2.77231i 18.0692 − 18.0692i −11.4003 + 11.4003i 4.75094
14.2 −1.73982 − 1.73982i −0.978968 − 2.36344i 2.05398i −3.70619 + 3.70619i −2.40873 + 5.81520i −3.57359 − 8.62742i −3.38573 + 3.38573i 1.73650 − 1.73650i 12.8962
14.3 −0.976817 − 0.976817i 0.355101 + 0.857291i − 2.09166i 4.54303 − 4.54303i 0.490547 − 1.18428i 0.392560 + 0.947724i −5.95043 + 5.95043i 5.75511 − 5.75511i −8.87541
14.4 0.496031 + 0.496031i −1.75813 − 4.24449i − 3.50791i −0.197696 + 0.197696i 1.23332 − 2.97749i 4.76097 + 11.4940i 3.72416 − 3.72416i −8.56074 + 8.56074i −0.196127
14.5 2.22195 + 2.22195i −0.657427 − 1.58717i 5.87417i −1.34788 + 1.34788i 2.06584 − 4.98739i −2.31405 − 5.58661i −4.16431 + 4.16431i 4.27707 − 4.27707i −5.98984
27.1 −2.38388 − 2.38388i 4.75273 − 1.96865i 7.36576i 2.78016 − 2.78016i −16.0230 − 6.63692i −9.79337 + 4.05655i 8.02356 − 8.02356i 12.3489 − 12.3489i −13.2551
27.2 −2.05572 − 2.05572i −2.68274 + 1.11123i 4.45194i −3.30766 + 3.30766i 7.79931 + 3.23058i −1.83339 + 0.759414i 0.929052 − 0.929052i −0.401704 + 0.401704i 13.5992
27.3 −0.373059 − 0.373059i 2.90610 − 1.20375i − 3.72165i −4.13074 + 4.13074i −1.53322 − 0.635078i 8.85589 − 3.66823i −2.88063 + 2.88063i 0.632467 − 0.632467i 3.08201
27.4 1.40098 + 1.40098i 1.39232 − 0.576719i − 0.0745260i −0.664543 + 0.664543i 2.75858 + 1.14264i −9.04050 + 3.74470i 5.70832 − 5.70832i −4.75800 + 4.75800i −1.86202
27.5 2.11878 + 2.11878i −3.24710 + 1.34499i 4.97849i 0.908569 − 0.908569i −9.72965 − 4.03015i 9.39715 − 3.89242i −2.07320 + 2.07320i 2.37070 − 2.37070i 3.85012
38.1 −2.38388 + 2.38388i 4.75273 + 1.96865i − 7.36576i 2.78016 + 2.78016i −16.0230 + 6.63692i −9.79337 − 4.05655i 8.02356 + 8.02356i 12.3489 + 12.3489i −13.2551
38.2 −2.05572 + 2.05572i −2.68274 − 1.11123i − 4.45194i −3.30766 − 3.30766i 7.79931 − 3.23058i −1.83339 − 0.759414i 0.929052 + 0.929052i −0.401704 − 0.401704i 13.5992
38.3 −0.373059 + 0.373059i 2.90610 + 1.20375i 3.72165i −4.13074 − 4.13074i −1.53322 + 0.635078i 8.85589 + 3.66823i −2.88063 − 2.88063i 0.632467 + 0.632467i 3.08201
38.4 1.40098 − 1.40098i 1.39232 + 0.576719i 0.0745260i −0.664543 − 0.664543i 2.75858 − 1.14264i −9.04050 − 3.74470i 5.70832 + 5.70832i −4.75800 − 4.75800i −1.86202
38.5 2.11878 − 2.11878i −3.24710 − 1.34499i − 4.97849i 0.908569 + 0.908569i −9.72965 + 4.03015i 9.39715 + 3.89242i −2.07320 − 2.07320i 2.37070 + 2.37070i 3.85012
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.e odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.3.e.b ✓ 20
3.b odd 2 1 369.3.l.b 20
41.e odd 8 1 inner 41.3.e.b ✓ 20
123.i even 8 1 369.3.l.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.3.e.b ✓ 20 1.a even 1 1 trivial
41.3.e.b ✓ 20 41.e odd 8 1 inner
369.3.l.b 20 3.b odd 2 1
369.3.l.b 20 123.i even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 8 T_{2}^{19} + 32 T_{2}^{18} + 52 T_{2}^{17} + 229 T_{2}^{16} + 1528 T_{2}^{15} + \cdots + 776161 \) acting on \(S_{3}^{\mathrm{new}}(41, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 8 T^{19} + \cdots + 776161 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 546217352 \) Copy content Toggle raw display
$5$ \( T^{20} + 12 T^{19} + \cdots + 8340544 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 31\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 18\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 184438225952 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 49\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 62\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 85570416372152)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 77\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 45\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 766359853741024)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 41\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 29\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 30\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 42\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 61\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 79\!\cdots\!88 \) Copy content Toggle raw display
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