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Properties

Label 270.2.k.d
Level $270$
Weight $2$
Character orbit 270.k
Analytic conductor $2.156$
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] = [270,2,Mod(31,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(18))
 
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("270.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.k (of order \(9\), degree \(6\), 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: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{7} q^{3} + \beta_{3} q^{4} + ( - \beta_{8} + \beta_{3}) q^{5} + ( - \beta_{16} - \beta_{4}) q^{6} + ( - \beta_{16} + \beta_{15} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{16} + \beta_{15} - \beta_{14} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{7} q^{3} + \beta_{3} q^{4} + ( - \beta_{8} + \beta_{3}) q^{5} + ( - \beta_{16} - \beta_{4}) q^{6} + ( - \beta_{16} + \beta_{15} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{17} - 2 \beta_{16} - 2 \beta_{15} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{3} - 3 q^{7} - 9 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{3} - 3 q^{7} - 9 q^{8} - 3 q^{9} + 9 q^{10} + 9 q^{11} + 3 q^{12} - 15 q^{13} - 3 q^{14} + 6 q^{15} - 3 q^{17} + 6 q^{18} + 9 q^{19} + 36 q^{21} + 9 q^{22} - 24 q^{23} - 3 q^{24} + 12 q^{26} - 18 q^{27} + 6 q^{28} - 9 q^{29} - 6 q^{30} - 3 q^{31} - 45 q^{33} + 24 q^{34} + 3 q^{35} + 6 q^{36} - 12 q^{37} - 18 q^{38} + 6 q^{39} + 6 q^{41} + 33 q^{42} - 33 q^{43} - 9 q^{44} + 12 q^{45} - 15 q^{46} - 33 q^{47} + 51 q^{49} - 9 q^{51} - 15 q^{52} + 24 q^{53} + 18 q^{54} - 18 q^{55} - 3 q^{56} - 15 q^{57} + 9 q^{58} - 12 q^{59} + 21 q^{61} + 6 q^{62} - 15 q^{63} - 9 q^{64} - 12 q^{65} + 9 q^{66} + 27 q^{67} - 21 q^{68} + 54 q^{69} - 6 q^{70} - 36 q^{71} - 12 q^{72} - 12 q^{73} - 12 q^{74} - 18 q^{76} + 72 q^{77} - 42 q^{78} - 75 q^{79} - 18 q^{80} + 9 q^{81} + 42 q^{82} + 33 q^{83} + 3 q^{84} - 24 q^{85} - 33 q^{86} - 36 q^{87} - 9 q^{88} - 33 q^{89} - 15 q^{90} + 27 q^{91} + 12 q^{92} - 30 q^{93} + 12 q^{94} - 18 q^{95} - 6 q^{96} - 18 q^{97} - 12 q^{98} + 45 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + \cdots + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{17} - 6 \nu^{16} + 24 \nu^{15} - 66 \nu^{14} + 153 \nu^{13} - 315 \nu^{12} + 651 \nu^{11} + \cdots - 39366 ) / 6561 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 256 \nu^{17} - 252 \nu^{16} + 819 \nu^{15} - 2946 \nu^{14} + 8433 \nu^{13} - 18684 \nu^{12} + \cdots - 3726648 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 463 \nu^{17} - 1365 \nu^{16} + 5913 \nu^{15} - 26250 \nu^{14} + 52794 \nu^{13} + \cdots - 17445699 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 568 \nu^{17} - 2640 \nu^{16} + 12876 \nu^{15} - 35031 \nu^{14} + 78066 \nu^{13} - 153621 \nu^{12} + \cdots - 8562105 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 685 \nu^{17} - 8100 \nu^{16} + 31695 \nu^{15} - 100830 \nu^{14} + 219969 \nu^{13} + \cdots - 51333264 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 974 \nu^{17} + 2553 \nu^{16} - 8700 \nu^{15} + 10680 \nu^{14} - 22356 \nu^{13} + \cdots - 11291481 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1037 \nu^{17} + 5784 \nu^{16} - 23631 \nu^{15} + 90453 \nu^{14} - 185886 \nu^{13} + \cdots + 51589143 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1330 \nu^{17} - 5085 \nu^{16} + 18540 \nu^{15} - 38388 \nu^{14} + 80928 \nu^{13} - 154008 \nu^{12} + \cdots + 4494285 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1537 \nu^{17} - 10986 \nu^{16} + 39195 \nu^{15} - 107463 \nu^{14} + 225909 \nu^{13} + \cdots - 36577575 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2659 \nu^{17} - 17343 \nu^{16} + 59721 \nu^{15} - 157755 \nu^{14} + 328077 \nu^{13} + \cdots - 43735626 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1093 \nu^{17} + 8166 \nu^{16} - 29135 \nu^{15} + 79917 \nu^{14} - 167442 \nu^{13} + \cdots + 24291009 ) / 391473 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1334 \nu^{17} - 5391 \nu^{16} + 17655 \nu^{15} - 38283 \nu^{14} + 79776 \nu^{13} - 161046 \nu^{12} + \cdots - 2659392 ) / 391473 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1408 \nu^{17} - 8546 \nu^{16} + 30191 \nu^{15} - 80643 \nu^{14} + 170697 \nu^{13} - 351393 \nu^{12} + \cdots - 25194240 ) / 391473 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5171 \nu^{17} - 22098 \nu^{16} + 69939 \nu^{15} - 151422 \nu^{14} + 311850 \nu^{13} + \cdots - 10425429 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5524 \nu^{17} - 30039 \nu^{16} + 105816 \nu^{15} - 257745 \nu^{14} + 540441 \nu^{13} + \cdots - 43860285 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 5575 \nu^{17} - 28839 \nu^{16} + 100842 \nu^{15} - 250365 \nu^{14} + 530586 \nu^{13} + \cdots - 54915570 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 7070 \nu^{17} - 30168 \nu^{16} + 108249 \nu^{15} - 246966 \nu^{14} + 526941 \nu^{13} + \cdots - 28710936 ) / 1174419 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{17} - 2 \beta_{15} + \beta_{14} + 2 \beta_{13} - \beta_{12} - \beta_{11} - 2 \beta_{10} + \cdots + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{5} - \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - \beta_{13} - \beta_{12} + \beta_{11} + 2\beta_{10} + \beta_{8} + \beta_{6} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{17} + 2 \beta_{16} + 2 \beta_{15} - \beta_{14} + 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{17} + 3 \beta_{16} + 3 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} + 6 \beta_{12} + \beta_{11} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6 \beta_{17} - 9 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 5 \beta_{11} - 4 \beta_{10} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5 \beta_{17} - 6 \beta_{16} - 4 \beta_{15} + 11 \beta_{14} - 2 \beta_{13} - 11 \beta_{12} + 4 \beta_{11} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12 \beta_{17} + 24 \beta_{16} - 15 \beta_{15} + 21 \beta_{14} - 21 \beta_{13} + 18 \beta_{11} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 21 \beta_{17} + 51 \beta_{16} - 3 \beta_{15} + 9 \beta_{14} + 27 \beta_{13} - 3 \beta_{12} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 75 \beta_{17} + 12 \beta_{16} + 72 \beta_{15} - 27 \beta_{14} + 30 \beta_{13} - 48 \beta_{12} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 45 \beta_{17} - 63 \beta_{16} - 18 \beta_{15} + 27 \beta_{14} - 12 \beta_{13} - 36 \beta_{12} + \cdots - 75 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 135 \beta_{17} - 108 \beta_{16} + 189 \beta_{15} - 30 \beta_{14} - 12 \beta_{13} + 225 \beta_{12} + \cdots - 363 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3 \beta_{17} + 297 \beta_{16} + 255 \beta_{15} - 51 \beta_{14} - 471 \beta_{13} + 213 \beta_{12} + \cdots - 315 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 36 \beta_{17} + 279 \beta_{16} - 45 \beta_{15} + 108 \beta_{14} - 297 \beta_{13} - 126 \beta_{12} + \cdots + 153 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 342 \beta_{17} - 612 \beta_{16} - 36 \beta_{15} + 297 \beta_{14} - 261 \beta_{13} - 81 \beta_{11} + \cdots + 1872 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 2178 \beta_{17} - 684 \beta_{16} - 2772 \beta_{15} + 1062 \beta_{14} - 261 \beta_{13} - 351 \beta_{12} + \cdots + 648 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 621 \beta_{17} - 2646 \beta_{16} + 1026 \beta_{15} - 621 \beta_{14} + 2385 \beta_{13} - 621 \beta_{12} + \cdots - 5904 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.219955 + 1.71803i
1.68668 − 0.393823i
0.472963 − 1.66622i
−0.219955 − 1.71803i
1.68668 + 0.393823i
0.472963 + 1.66622i
1.16555 + 1.28120i
−1.29960 + 1.14501i
0.960398 − 1.44140i
0.381933 − 1.68942i
1.20201 + 1.24706i
−1.34999 + 1.08514i
0.381933 + 1.68942i
1.20201 − 1.24706i
−1.34999 − 1.08514i
1.16555 − 1.28120i
−1.29960 − 1.14501i
0.960398 + 1.44140i
−0.939693 − 0.342020i −1.37788 + 1.04950i 0.766044 + 0.642788i −0.173648 + 0.984808i 1.65373 − 0.514946i 1.95914 − 1.64391i −0.500000 − 0.866025i 0.797098 − 2.89217i 0.500000 − 0.866025i
31.2 −0.939693 − 0.342020i −0.502281 − 1.65762i 0.766044 + 0.642788i −0.173648 + 0.984808i −0.0949504 + 1.72945i −2.60035 + 2.18196i −0.500000 − 0.866025i −2.49543 + 1.66519i 0.500000 − 0.866025i
31.3 −0.939693 − 0.342020i 1.20651 − 1.24271i 0.766044 + 0.642788i −0.173648 + 0.984808i −1.55878 + 0.755115i 1.67330 − 1.40407i −0.500000 − 0.866025i −0.0886598 − 2.99869i 0.500000 − 0.866025i
61.1 −0.939693 + 0.342020i −1.37788 − 1.04950i 0.766044 − 0.642788i −0.173648 − 0.984808i 1.65373 + 0.514946i 1.95914 + 1.64391i −0.500000 + 0.866025i 0.797098 + 2.89217i 0.500000 + 0.866025i
61.2 −0.939693 + 0.342020i −0.502281 + 1.65762i 0.766044 − 0.642788i −0.173648 − 0.984808i −0.0949504 − 1.72945i −2.60035 − 2.18196i −0.500000 + 0.866025i −2.49543 − 1.66519i 0.500000 + 0.866025i
61.3 −0.939693 + 0.342020i 1.20651 + 1.24271i 0.766044 − 0.642788i −0.173648 − 0.984808i −1.55878 − 0.755115i 1.67330 + 1.40407i −0.500000 + 0.866025i −0.0886598 + 2.99869i 0.500000 + 0.866025i
121.1 0.173648 + 0.984808i −1.69233 − 0.368799i −0.939693 + 0.342020i −0.766044 − 0.642788i 0.0693253 − 1.73066i −1.17449 − 0.427479i −0.500000 − 0.866025i 2.72798 + 1.24826i 0.500000 − 0.866025i
121.2 0.173648 + 0.984808i −0.341803 + 1.69799i −0.939693 + 0.342020i −0.766044 − 0.642788i −1.73155 − 0.0417572i −4.75088 − 1.72918i −0.500000 − 0.866025i −2.76634 − 1.16076i 0.500000 − 0.866025i
121.3 0.173648 + 0.984808i 0.768090 − 1.55243i −0.939693 + 0.342020i −0.766044 − 0.642788i 1.66222 + 0.486845i 3.54598 + 1.29063i −0.500000 − 0.866025i −1.82007 − 2.38481i 0.500000 − 0.866025i
151.1 0.766044 + 0.642788i −1.65404 − 0.513944i 0.173648 + 0.984808i 0.939693 + 0.342020i −0.936714 − 1.45690i −0.500571 + 2.83888i −0.500000 + 0.866025i 2.47172 + 1.70017i 0.500000 + 0.866025i
151.2 0.766044 + 0.642788i 0.478983 + 1.66450i 0.173648 + 0.984808i 0.939693 + 0.342020i −0.703001 + 1.58297i 0.0883297 − 0.500943i −0.500000 + 0.866025i −2.54115 + 1.59454i 0.500000 + 0.866025i
151.3 0.766044 + 0.642788i 1.61475 − 0.626555i 0.173648 + 0.984808i 0.939693 + 0.342020i 1.63971 + 0.557975i 0.259537 − 1.47191i −0.500000 + 0.866025i 2.21486 − 2.02346i 0.500000 + 0.866025i
211.1 0.766044 − 0.642788i −1.65404 + 0.513944i 0.173648 − 0.984808i 0.939693 − 0.342020i −0.936714 + 1.45690i −0.500571 − 2.83888i −0.500000 − 0.866025i 2.47172 − 1.70017i 0.500000 − 0.866025i
211.2 0.766044 − 0.642788i 0.478983 − 1.66450i 0.173648 − 0.984808i 0.939693 − 0.342020i −0.703001 − 1.58297i 0.0883297 + 0.500943i −0.500000 − 0.866025i −2.54115 − 1.59454i 0.500000 − 0.866025i
211.3 0.766044 − 0.642788i 1.61475 + 0.626555i 0.173648 − 0.984808i 0.939693 − 0.342020i 1.63971 − 0.557975i 0.259537 + 1.47191i −0.500000 − 0.866025i 2.21486 + 2.02346i 0.500000 − 0.866025i
241.1 0.173648 − 0.984808i −1.69233 + 0.368799i −0.939693 − 0.342020i −0.766044 + 0.642788i 0.0693253 + 1.73066i −1.17449 + 0.427479i −0.500000 + 0.866025i 2.72798 − 1.24826i 0.500000 + 0.866025i
241.2 0.173648 − 0.984808i −0.341803 − 1.69799i −0.939693 − 0.342020i −0.766044 + 0.642788i −1.73155 + 0.0417572i −4.75088 + 1.72918i −0.500000 + 0.866025i −2.76634 + 1.16076i 0.500000 + 0.866025i
241.3 0.173648 − 0.984808i 0.768090 + 1.55243i −0.939693 − 0.342020i −0.766044 + 0.642788i 1.66222 − 0.486845i 3.54598 − 1.29063i −0.500000 + 0.866025i −1.82007 + 2.38481i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.2.k.d ✓ 18
3.b odd 2 1 810.2.k.d 18
27.e even 9 1 inner 270.2.k.d ✓ 18
27.e even 9 1 7290.2.a.r 9
27.f odd 18 1 810.2.k.d 18
27.f odd 18 1 7290.2.a.q 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.k.d ✓ 18 1.a even 1 1 trivial
270.2.k.d ✓ 18 27.e even 9 1 inner
810.2.k.d 18 3.b odd 2 1
810.2.k.d 18 27.f odd 18 1
7290.2.a.q 9 27.f odd 18 1
7290.2.a.r 9 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{18} + 3 T_{7}^{17} - 21 T_{7}^{16} - 57 T_{7}^{15} + 249 T_{7}^{14} + 291 T_{7}^{13} + \cdots + 982081 \) acting on \(S_{2}^{\mathrm{new}}(270, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} + 3 T^{17} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} + 3 T^{17} + \cdots + 982081 \) Copy content Toggle raw display
$11$ \( T^{18} - 9 T^{17} + \cdots + 2985984 \) Copy content Toggle raw display
$13$ \( T^{18} + 15 T^{17} + \cdots + 87616 \) Copy content Toggle raw display
$17$ \( T^{18} + 3 T^{17} + \cdots + 3779136 \) Copy content Toggle raw display
$19$ \( T^{18} - 9 T^{17} + \cdots + 19855936 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 297666009 \) Copy content Toggle raw display
$29$ \( T^{18} + 9 T^{17} + \cdots + 3674889 \) Copy content Toggle raw display
$31$ \( T^{18} + 3 T^{17} + \cdots + 4700224 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 578992548805696 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 269887523049 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 669790017649 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
$53$ \( (T^{9} - 12 T^{8} + \cdots - 2346408)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 1719926784 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 19813208147209 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 5542008097609 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 68\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 3169339909696 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 51843191481 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 90\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 679384765504 \) Copy content Toggle raw display
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