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Properties

Label 171.2.e.b
Level $171$
Weight $2$
Character orbit 171.e
Analytic conductor $1.365$
Analytic rank $0$
Dimension $18$
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] = [171,2,Mod(58,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, 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("171.58");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.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: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + 2 x^{16} - 7 x^{15} + 8 x^{14} - 46 x^{13} + 58 x^{12} - 102 x^{11} + 270 x^{10} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{8}) q^{2} + (\beta_{16} - \beta_{3}) q^{3} + (\beta_{15} + \beta_{12} + \cdots + \beta_{2}) q^{4}+ \cdots + ( - \beta_{17} - \beta_{12} + 2 \beta_{10} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - \beta_{8}) q^{2} + (\beta_{16} - \beta_{3}) q^{3} + (\beta_{15} + \beta_{12} + \cdots + \beta_{2}) q^{4}+ \cdots + (2 \beta_{17} + 2 \beta_{16} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{2} - q^{3} - 9 q^{4} + 7 q^{5} + 5 q^{6} - 24 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{2} - q^{3} - 9 q^{4} + 7 q^{5} + 5 q^{6} - 24 q^{8} - 3 q^{9} + 8 q^{11} + 10 q^{12} + 17 q^{14} + q^{15} - 21 q^{16} - 8 q^{17} - 42 q^{18} - 18 q^{19} + 14 q^{20} + 4 q^{21} + 5 q^{23} + 12 q^{24} + 6 q^{25} - 20 q^{26} - 16 q^{27} + 28 q^{29} + 40 q^{30} + 3 q^{31} + 27 q^{32} - 14 q^{33} - 30 q^{35} - 45 q^{36} + 12 q^{37} - 3 q^{38} + 41 q^{39} + 16 q^{41} + 31 q^{42} - 6 q^{43} - 32 q^{44} - 14 q^{45} - 24 q^{46} + 9 q^{47} + 40 q^{48} - 3 q^{49} + q^{50} - 52 q^{51} + 33 q^{52} - 58 q^{53} - 34 q^{54} + 6 q^{55} + 45 q^{56} + q^{57} + 3 q^{58} + 23 q^{59} - q^{60} + 6 q^{61} - 16 q^{62} + 38 q^{63} + 48 q^{64} + 24 q^{65} + 85 q^{66} + 3 q^{67} - 22 q^{68} - 33 q^{69} - 39 q^{70} - 84 q^{71} - 84 q^{72} - 24 q^{73} - 37 q^{74} + 57 q^{75} + 9 q^{76} + 29 q^{77} + 14 q^{78} + 6 q^{79} + 26 q^{80} - 39 q^{81} + 6 q^{82} - 2 q^{83} + 80 q^{84} - 6 q^{85} + 3 q^{86} - 36 q^{87} - 12 q^{88} - 80 q^{89} - 41 q^{90} + 42 q^{91} + 13 q^{92} + 5 q^{93} - 6 q^{94} - 7 q^{95} - 38 q^{96} - 24 q^{97} + 122 q^{98} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 2 x^{16} - 7 x^{15} + 8 x^{14} - 46 x^{13} + 58 x^{12} - 102 x^{11} + 270 x^{10} + \cdots + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13 \nu^{17} + 2342 \nu^{16} - 10393 \nu^{15} + 26318 \nu^{14} - 54568 \nu^{13} + 98072 \nu^{12} + \cdots + 22386132 ) / 1436859 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{17} - \nu^{16} + 2 \nu^{15} - 7 \nu^{14} + 8 \nu^{13} - 46 \nu^{12} + 58 \nu^{11} - 102 \nu^{10} + \cdots - 6561 ) / 6561 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 154 \nu^{17} + 614 \nu^{16} - 2106 \nu^{15} + 5154 \nu^{14} - 9675 \nu^{13} + 20094 \nu^{12} + \cdots + 2410074 ) / 478953 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 169 \nu^{17} + 881 \nu^{16} - 3052 \nu^{15} + 9035 \nu^{14} - 19534 \nu^{13} + 31235 \nu^{12} + \cdots + 7958493 ) / 478953 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 205 \nu^{17} - 641 \nu^{16} - 291 \nu^{15} + 1026 \nu^{14} - 4503 \nu^{13} + 6804 \nu^{12} + \cdots + 8448381 ) / 478953 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1441 \nu^{17} - 2024 \nu^{16} + 18817 \nu^{15} - 45182 \nu^{14} + 120205 \nu^{13} + \cdots - 74493594 ) / 2873718 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1604 \nu^{17} - 7913 \nu^{16} + 10606 \nu^{15} - 26123 \nu^{14} + 32092 \nu^{13} - 79985 \nu^{12} + \cdots + 17445699 ) / 2873718 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 938 \nu^{17} - 269 \nu^{16} - 4958 \nu^{15} + 14113 \nu^{14} - 39209 \nu^{13} + 52285 \nu^{12} + \cdots + 26742636 ) / 1436859 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1213 \nu^{17} - 1720 \nu^{16} - 217 \nu^{15} + 665 \nu^{14} - 17401 \nu^{13} + 2804 \nu^{12} + \cdots + 21047688 ) / 1436859 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 37 \nu^{17} - 190 \nu^{16} + 407 \nu^{15} - 934 \nu^{14} + 1673 \nu^{13} - 3349 \nu^{12} + \cdots + 13122 ) / 39366 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 505 \nu^{17} - 1693 \nu^{16} + 3494 \nu^{15} - 7486 \nu^{14} + 11231 \nu^{13} - 29350 \nu^{12} + \cdots + 3004938 ) / 478953 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3041 \nu^{17} - 8648 \nu^{16} + 27601 \nu^{15} - 57350 \nu^{14} + 114625 \nu^{13} + \cdots - 37450188 ) / 2873718 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1141 \nu^{17} + 2545 \nu^{16} - 11489 \nu^{15} + 20839 \nu^{14} - 42473 \nu^{13} + \cdots + 14257053 ) / 957906 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 570 \nu^{17} - 641 \nu^{16} + 1169 \nu^{15} - 1237 \nu^{14} - 634 \nu^{13} - 11446 \nu^{12} + \cdots + 3978153 ) / 478953 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 3208 \nu^{17} - 6847 \nu^{16} + 11576 \nu^{15} - 21805 \nu^{14} + 23669 \nu^{13} - 95365 \nu^{12} + \cdots + 25791291 ) / 1436859 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 2931 \nu^{17} - 8491 \nu^{16} + 18025 \nu^{15} - 40247 \nu^{14} + 58627 \nu^{13} - 165362 \nu^{12} + \cdots + 7582329 ) / 957906 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{13} + \beta_{11} - \beta_{9} - \beta_{6} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} + \beta_{15} + \beta_{12} + 2\beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} - 2\beta_{6} - \beta_{5} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} + \beta_{16} + 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} - 4 \beta_{10} + \cdots - \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{17} + 2 \beta_{16} - 5 \beta_{15} - 3 \beta_{13} - \beta_{12} - \beta_{11} - 9 \beta_{10} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} - \beta_{16} - 3 \beta_{15} + \beta_{14} + 3 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{17} - 2 \beta_{16} + 25 \beta_{15} + 9 \beta_{14} - 15 \beta_{13} + 9 \beta_{12} + 17 \beta_{11} + \cdots - 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{17} + 9 \beta_{16} - 11 \beta_{15} - 9 \beta_{14} - 7 \beta_{13} + 6 \beta_{12} + 16 \beta_{11} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5 \beta_{17} + 3 \beta_{16} - 11 \beta_{15} + 9 \beta_{14} + 31 \beta_{12} - 13 \beta_{11} - 26 \beta_{10} + \cdots - 21 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22 \beta_{17} + 13 \beta_{16} + 11 \beta_{15} + 38 \beta_{14} - 56 \beta_{13} + 44 \beta_{12} + \cdots + 114 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 55 \beta_{17} + 17 \beta_{16} - 59 \beta_{15} + 54 \beta_{14} - 39 \beta_{13} + 74 \beta_{12} + \cdots - 92 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 20 \beta_{17} + 29 \beta_{16} - 66 \beta_{15} + 37 \beta_{14} - 69 \beta_{13} - 89 \beta_{12} + \cdots - 428 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 231 \beta_{17} + 274 \beta_{16} - 263 \beta_{15} - 144 \beta_{14} + 201 \beta_{13} + 12 \beta_{12} + \cdots + 350 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 36 \beta_{17} - 168 \beta_{16} + 349 \beta_{15} + 306 \beta_{14} + 65 \beta_{13} + 813 \beta_{12} + \cdots - 286 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 425 \beta_{17} - 441 \beta_{16} - 713 \beta_{15} - 351 \beta_{14} - 693 \beta_{13} + 34 \beta_{12} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 389 \beta_{17} + 79 \beta_{16} - 1663 \beta_{15} - 394 \beta_{14} + 727 \beta_{13} + 221 \beta_{12} + \cdots - 2085 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 326 \beta_{17} + 1940 \beta_{16} + 1552 \beta_{15} + 2511 \beta_{14} - 318 \beta_{13} - 1606 \beta_{12} + \cdots - 5696 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(-\beta_{10}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1
0.528045 − 1.64960i
−1.31818 − 1.12357i
0.269536 + 1.71095i
−0.521101 + 1.65180i
0.881767 − 1.49080i
1.69921 − 0.335679i
−1.33277 + 1.10622i
1.71282 + 0.257365i
−1.41933 − 0.992718i
0.528045 + 1.64960i
−1.31818 + 1.12357i
0.269536 − 1.71095i
−0.521101 − 1.65180i
0.881767 + 1.49080i
1.69921 + 0.335679i
−1.33277 − 1.10622i
1.71282 − 0.257365i
−1.41933 + 0.992718i
−1.20664 − 2.08996i −1.16457 + 1.28210i −1.91197 + 3.31163i 0.841188 − 1.45698i 4.08476 + 0.886877i 2.32169 + 4.02128i 4.40167 −0.287553 − 2.98619i −4.06005
58.2 −0.716420 − 1.24088i −1.63213 − 0.579790i −0.0265142 + 0.0459240i 1.17087 − 2.02801i 0.449842 + 2.44064i −0.279061 − 0.483348i −2.78970 2.32769 + 1.89258i −3.35535
58.3 −0.594252 − 1.02928i 1.61649 − 0.622050i 0.293728 − 0.508753i −0.238630 + 0.413319i −1.60087 − 1.29416i −1.39425 − 2.41491i −3.07520 2.22611 − 2.01108i 0.567225
58.4 −0.0396384 − 0.0686557i 1.16995 − 1.27719i 0.996858 − 1.72661i −0.584016 + 1.01155i −0.134061 − 0.0296983i 2.28296 + 3.95420i −0.316609 −0.262419 − 2.98850i 0.0925978
58.5 0.179588 + 0.311056i −0.850188 + 1.50903i 0.935496 − 1.62033i 1.82671 − 3.16396i −0.622078 + 0.00654869i −1.30445 − 2.25938i 1.39037 −1.55436 − 2.56592i 1.31222
58.6 0.338372 + 0.586078i 0.558899 + 1.63940i 0.771009 − 1.33543i −1.22897 + 2.12864i −0.771700 + 0.882285i 0.179375 + 0.310686i 2.39704 −2.37526 + 1.83252i −1.66339
58.7 0.937324 + 1.62349i 0.291630 − 1.70732i −0.757151 + 1.31142i 0.833426 − 1.44354i 3.04518 − 1.12685i −0.0896831 − 0.155336i 0.910513 −2.82990 − 0.995814i 3.12476
58.8 1.21030 + 2.09631i 1.07930 + 1.35467i −1.92967 + 3.34229i 1.25974 − 2.18194i −1.53352 + 3.90209i −1.43315 − 2.48228i −4.50074 −0.670240 + 2.92417i 6.09869
58.9 1.39136 + 2.40991i −1.56939 − 0.732821i −2.87179 + 4.97408i −0.380329 + 0.658748i −0.417553 − 4.80171i −0.283426 − 0.490908i −10.4173 1.92595 + 2.30016i −2.11670
115.1 −1.20664 + 2.08996i −1.16457 − 1.28210i −1.91197 − 3.31163i 0.841188 + 1.45698i 4.08476 − 0.886877i 2.32169 − 4.02128i 4.40167 −0.287553 + 2.98619i −4.06005
115.2 −0.716420 + 1.24088i −1.63213 + 0.579790i −0.0265142 − 0.0459240i 1.17087 + 2.02801i 0.449842 − 2.44064i −0.279061 + 0.483348i −2.78970 2.32769 − 1.89258i −3.35535
115.3 −0.594252 + 1.02928i 1.61649 + 0.622050i 0.293728 + 0.508753i −0.238630 − 0.413319i −1.60087 + 1.29416i −1.39425 + 2.41491i −3.07520 2.22611 + 2.01108i 0.567225
115.4 −0.0396384 + 0.0686557i 1.16995 + 1.27719i 0.996858 + 1.72661i −0.584016 − 1.01155i −0.134061 + 0.0296983i 2.28296 − 3.95420i −0.316609 −0.262419 + 2.98850i 0.0925978
115.5 0.179588 − 0.311056i −0.850188 − 1.50903i 0.935496 + 1.62033i 1.82671 + 3.16396i −0.622078 − 0.00654869i −1.30445 + 2.25938i 1.39037 −1.55436 + 2.56592i 1.31222
115.6 0.338372 − 0.586078i 0.558899 − 1.63940i 0.771009 + 1.33543i −1.22897 − 2.12864i −0.771700 − 0.882285i 0.179375 − 0.310686i 2.39704 −2.37526 − 1.83252i −1.66339
115.7 0.937324 − 1.62349i 0.291630 + 1.70732i −0.757151 − 1.31142i 0.833426 + 1.44354i 3.04518 + 1.12685i −0.0896831 + 0.155336i 0.910513 −2.82990 + 0.995814i 3.12476
115.8 1.21030 − 2.09631i 1.07930 − 1.35467i −1.92967 − 3.34229i 1.25974 + 2.18194i −1.53352 − 3.90209i −1.43315 + 2.48228i −4.50074 −0.670240 − 2.92417i 6.09869
115.9 1.39136 − 2.40991i −1.56939 + 0.732821i −2.87179 − 4.97408i −0.380329 − 0.658748i −0.417553 + 4.80171i −0.283426 + 0.490908i −10.4173 1.92595 − 2.30016i −2.11670
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.9
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 171.2.e.b ✓ 18
3.b odd 2 1 513.2.e.a 18
9.c even 3 1 inner 171.2.e.b ✓ 18
9.c even 3 1 1539.2.a.o 9
9.d odd 6 1 513.2.e.a 18
9.d odd 6 1 1539.2.a.p 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.e.b ✓ 18 1.a even 1 1 trivial
171.2.e.b ✓ 18 9.c even 3 1 inner
513.2.e.a 18 3.b odd 2 1
513.2.e.a 18 9.d odd 6 1
1539.2.a.o 9 9.c even 3 1
1539.2.a.p 9 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 3 T_{2}^{17} + 18 T_{2}^{16} - 29 T_{2}^{15} + 144 T_{2}^{14} - 195 T_{2}^{13} + 740 T_{2}^{12} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 3 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{18} + T^{17} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{18} - 7 T^{17} + \cdots + 3969 \) Copy content Toggle raw display
$7$ \( T^{18} + 33 T^{16} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 534580641 \) Copy content Toggle raw display
$13$ \( T^{18} + 54 T^{16} + \cdots + 68112009 \) Copy content Toggle raw display
$17$ \( (T^{9} + 4 T^{8} + \cdots - 22356)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{18} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 1390022089 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 458774574241 \) Copy content Toggle raw display
$31$ \( T^{18} - 3 T^{17} + \cdots + 2047761 \) Copy content Toggle raw display
$37$ \( (T^{9} - 6 T^{8} + \cdots + 2157516)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 7544366383401 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 2304594211921 \) Copy content Toggle raw display
$47$ \( T^{18} - 9 T^{17} + \cdots + 43046721 \) Copy content Toggle raw display
$53$ \( (T^{9} + 29 T^{8} + \cdots + 4747572)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 1185509393721 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 1407075121 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 4509256801 \) Copy content Toggle raw display
$71$ \( (T^{9} + 42 T^{8} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} + 12 T^{8} + \cdots + 212)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 280881980289 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 11695107275721 \) Copy content Toggle raw display
$89$ \( (T^{9} + 40 T^{8} + \cdots + 893700)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 250003000009 \) Copy content Toggle raw display
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