↑

Properties

Label 230.2.g.c
Level $230$
Weight $2$
Character orbit 230.g
Analytic conductor $1.837$
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] = [230,2,Mod(31,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), 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: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 9 x^{19} + 35 x^{18} - 66 x^{17} + 51 x^{16} - 52 x^{15} + 289 x^{14} - 451 x^{13} + 115 x^{12} + \cdots + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{18} + \beta_{5}) q^{3} + \beta_{12} q^{4} - \beta_{7} q^{5} + ( - \beta_{14} - \beta_{13}) q^{6} + (\beta_{19} - \beta_{17} + \beta_{13} + \cdots - 1) q^{7}+ \cdots + (\beta_{17} - 2 \beta_{15} + \beta_{13} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{18} + \beta_{5}) q^{3} + \beta_{12} q^{4} - \beta_{7} q^{5} + ( - \beta_{14} - \beta_{13}) q^{6} + (\beta_{19} - \beta_{17} + \beta_{13} + \cdots - 1) q^{7}+ \cdots + (3 \beta_{19} + 3 \beta_{18} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + q^{3} - 2 q^{4} + 2 q^{5} - q^{6} - 12 q^{7} + 2 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} + q^{3} - 2 q^{4} + 2 q^{5} - q^{6} - 12 q^{7} + 2 q^{8} + 17 q^{9} - 2 q^{10} - 3 q^{11} + q^{12} - 18 q^{13} + q^{14} - q^{15} - 2 q^{16} + 14 q^{17} - 17 q^{18} - 7 q^{19} + 2 q^{20} - 43 q^{21} - 8 q^{22} + 9 q^{23} + 10 q^{24} - 2 q^{25} - 4 q^{26} + 10 q^{27} - q^{28} - 28 q^{29} + q^{30} - 7 q^{31} + 2 q^{32} + 35 q^{33} - 3 q^{34} - 10 q^{35} - 27 q^{36} + 30 q^{37} + 40 q^{38} + 69 q^{39} - 2 q^{40} - 2 q^{41} + 10 q^{42} - 26 q^{43} - 3 q^{44} - 6 q^{45} + 2 q^{46} - 26 q^{47} + q^{48} - 8 q^{49} + 2 q^{50} + 20 q^{51} + 15 q^{52} - q^{53} + 23 q^{54} + 3 q^{55} + 12 q^{56} + 3 q^{57} - 27 q^{58} - 48 q^{59} - q^{60} - 7 q^{61} - 26 q^{62} + 8 q^{63} - 2 q^{64} - 15 q^{65} + 20 q^{66} + 7 q^{67} - 8 q^{68} + 12 q^{69} + 10 q^{70} + 19 q^{71} - 17 q^{72} + 2 q^{73} - 30 q^{74} + q^{75} - 7 q^{76} - 34 q^{77} - 14 q^{78} - 49 q^{79} + 2 q^{80} - 56 q^{81} - 31 q^{82} + 23 q^{83} + 12 q^{84} + 8 q^{85} + 37 q^{86} - 31 q^{87} + 14 q^{88} + 43 q^{89} - 5 q^{90} + 128 q^{91} - 13 q^{92} - 50 q^{93} - 7 q^{94} + 40 q^{95} - q^{96} - 7 q^{97} - 36 q^{98} - 85 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 9 x^{19} + 35 x^{18} - 66 x^{17} + 51 x^{16} - 52 x^{15} + 289 x^{14} - 451 x^{13} + 115 x^{12} + \cdots + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 27\!\cdots\!90 \nu^{19} + \cdots + 36\!\cdots\!08 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43\!\cdots\!40 \nu^{19} + \cdots + 35\!\cdots\!64 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 52\!\cdots\!18 \nu^{19} + \cdots + 11\!\cdots\!09 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 54\!\cdots\!41 \nu^{19} + \cdots - 42\!\cdots\!01 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 55\!\cdots\!82 \nu^{19} + \cdots + 23\!\cdots\!60 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 66\!\cdots\!89 \nu^{19} + \cdots + 34\!\cdots\!46 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 81\!\cdots\!69 \nu^{19} + \cdots - 24\!\cdots\!96 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!53 \nu^{19} + \cdots + 23\!\cdots\!52 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!46 \nu^{19} + \cdots - 23\!\cdots\!70 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!09 \nu^{19} + \cdots + 55\!\cdots\!28 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!45 \nu^{19} + \cdots - 65\!\cdots\!34 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15\!\cdots\!12 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18\!\cdots\!72 \nu^{19} + \cdots + 19\!\cdots\!35 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 71\!\cdots\!76 \nu^{19} + \cdots - 10\!\cdots\!91 ) / 35\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 24\!\cdots\!73 \nu^{19} + \cdots + 95\!\cdots\!35 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 32\!\cdots\!56 \nu^{19} + \cdots + 37\!\cdots\!41 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 32\!\cdots\!25 \nu^{19} + \cdots + 18\!\cdots\!73 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 33\!\cdots\!03 \nu^{19} + \cdots - 14\!\cdots\!82 ) / 94\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 17\!\cdots\!21 \nu^{19} + \cdots + 17\!\cdots\!95 ) / 35\!\cdots\!49 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{19} + \beta_{18} + \beta_{17} + \beta_{16} + \beta_{11} + 1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} + \beta_{18} + \beta_{16} - \beta_{12} + \beta_{11} - \beta_{7} - \beta_{5} + \beta_{3} + 2\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} + \beta_{16} - 2 \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{19} - 8 \beta_{18} - 9 \beta_{17} - 9 \beta_{16} - 11 \beta_{15} - \beta_{13} + \beta_{12} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 44 \beta_{19} - 35 \beta_{18} - 29 \beta_{17} - 56 \beta_{16} - 30 \beta_{15} + 9 \beta_{14} + 9 \beta_{13} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 124 \beta_{19} - 74 \beta_{18} - 58 \beta_{17} - 179 \beta_{16} - 65 \beta_{15} + 15 \beta_{14} + \cdots - 68 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 267 \beta_{19} - 68 \beta_{18} - 250 \beta_{17} - 404 \beta_{16} - 68 \beta_{14} + 278 \beta_{13} + \cdots - 104 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 479 \beta_{19} + 174 \beta_{18} - 1209 \beta_{17} - 479 \beta_{16} + 554 \beta_{15} - 614 \beta_{14} + \cdots - 449 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1075 \beta_{19} + 719 \beta_{18} - 5027 \beta_{17} + 270 \beta_{16} + 2365 \beta_{15} - 2345 \beta_{14} + \cdots - 3344 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4377 \beta_{19} - 482 \beta_{18} - 16474 \beta_{17} + 1360 \beta_{16} + 5813 \beta_{15} - 4822 \beta_{14} + \cdots - 16474 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18581 \beta_{19} - 12503 \beta_{18} - 38676 \beta_{17} - 6078 \beta_{16} + 6827 \beta_{15} + \cdots - 54490 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 59288 \beta_{19} - 54373 \beta_{18} - 52863 \beta_{17} - 54373 \beta_{16} - 6425 \beta_{15} + \cdots - 118077 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 117507 \beta_{19} - 119587 \beta_{18} + 30825 \beta_{17} - 180968 \beta_{16} - 25333 \beta_{15} + \cdots - 92174 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 398663 \beta_{17} - 184858 \beta_{16} + 137107 \beta_{15} + 395408 \beta_{14} + 184858 \beta_{13} + \cdots + 464204 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1087752 \beta_{19} + 1108090 \beta_{18} + 1087752 \beta_{17} + 1288711 \beta_{16} + 1258866 \beta_{15} + \cdots + 2314452 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 5070849 \beta_{19} + 4681408 \beta_{18} + 1351905 \beta_{17} + 8578799 \beta_{16} + 4908027 \beta_{15} + \cdots + 5070849 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 14434795 \beta_{19} + 9978876 \beta_{18} + 1249385 \beta_{17} + 29097149 \beta_{16} + 10382812 \beta_{15} + \cdots + 3026852 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 28785004 \beta_{19} + 3196032 \beta_{18} + 20038814 \beta_{17} + 60403537 \beta_{16} + \cdots - 11862125 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 47315859 \beta_{19} - 54892224 \beta_{18} + 173601552 \beta_{17} + 47315859 \beta_{16} + \cdots + 18550198 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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
2.84172 + 0.834404i
−1.00046 − 0.293763i
−0.577766 − 0.371308i
1.99318 + 1.28094i
−0.357206 − 0.782171i
0.702345 + 1.53792i
−0.357206 + 0.782171i
0.702345 − 1.53792i
−0.577766 + 0.371308i
1.99318 − 1.28094i
−1.12878 − 1.30269i
1.98647 + 2.29251i
0.132527 + 0.921745i
−0.0920198 − 0.640012i
2.84172 − 0.834404i
−1.00046 + 0.293763i
0.132527 − 0.921745i
−0.0920198 + 0.640012i
−1.12878 + 1.30269i
1.98647 − 2.29251i
0.142315 − 0.989821i −0.641135 − 1.40389i −0.959493 − 0.281733i 0.654861 + 0.755750i −1.48084 + 0.434815i −3.01115 − 1.93515i −0.415415 + 0.909632i 0.404729 − 0.467082i 0.841254 − 0.540641i
31.2 0.142315 − 0.989821i 0.225720 + 0.494258i −0.959493 − 0.281733i 0.654861 + 0.755750i 0.521351 − 0.153082i 2.19754 + 1.41228i −0.415415 + 0.909632i 1.77124 − 2.04412i 0.841254 − 0.540641i
41.1 0.959493 + 0.281733i −0.267311 + 0.308493i 0.841254 + 0.540641i 0.142315 − 0.989821i −0.343396 + 0.220687i 1.05127 + 2.30197i 0.654861 + 0.755750i 0.403232 + 2.80454i 0.415415 − 0.909632i
41.2 0.959493 + 0.281733i 0.922172 − 1.06424i 0.841254 + 0.540641i 0.142315 − 0.989821i 1.18465 − 0.761328i −1.77817 − 3.89366i 0.654861 + 0.755750i 0.144732 + 1.00663i 0.415415 − 0.909632i
71.1 −0.841254 − 0.540641i −0.147290 − 1.02443i 0.415415 + 0.909632i 0.959493 + 0.281733i −0.429938 + 0.941434i 0.695383 − 0.802515i 0.142315 − 0.989821i 1.85072 − 0.543421i −0.654861 − 0.755750i
71.2 −0.841254 − 0.540641i 0.289605 + 2.01425i 0.415415 + 0.909632i 0.959493 + 0.281733i 0.845353 − 1.85107i −3.30974 + 3.81964i 0.142315 − 0.989821i −1.09485 + 0.321476i −0.654861 − 0.755750i
81.1 −0.841254 + 0.540641i −0.147290 + 1.02443i 0.415415 − 0.909632i 0.959493 − 0.281733i −0.429938 − 0.941434i 0.695383 + 0.802515i 0.142315 + 0.989821i 1.85072 + 0.543421i −0.654861 + 0.755750i
81.2 −0.841254 + 0.540641i 0.289605 − 2.01425i 0.415415 − 0.909632i 0.959493 − 0.281733i 0.845353 + 1.85107i −3.30974 − 3.81964i 0.142315 + 0.989821i −1.09485 − 0.321476i −0.654861 + 0.755750i
101.1 0.959493 − 0.281733i −0.267311 − 0.308493i 0.841254 − 0.540641i 0.142315 + 0.989821i −0.343396 − 0.220687i 1.05127 − 2.30197i 0.654861 − 0.755750i 0.403232 − 2.80454i 0.415415 + 0.909632i
101.2 0.959493 − 0.281733i 0.922172 + 1.06424i 0.841254 − 0.540641i 0.142315 + 0.989821i 1.18465 + 0.761328i −1.77817 + 3.89366i 0.654861 − 0.755750i 0.144732 − 1.00663i 0.415415 + 0.909632i
121.1 −0.415415 + 0.909632i −1.26277 − 0.370783i −0.654861 − 0.755750i −0.841254 + 0.540641i 0.861850 − 0.994628i −0.120689 + 0.839414i 0.959493 − 0.281733i −1.06665 − 0.685495i −0.142315 − 0.989821i
121.2 −0.415415 + 0.909632i 2.22226 + 0.652515i −0.654861 − 0.755750i −0.841254 + 0.540641i −1.51671 + 1.75038i −0.180372 + 1.25451i 0.959493 − 0.281733i 1.98892 + 1.27820i −0.142315 − 0.989821i
131.1 0.654861 − 0.755750i −2.75233 + 1.76881i −0.142315 − 0.989821i −0.415415 − 0.909632i −0.465612 + 3.23840i 0.267094 − 0.0784258i −0.841254 − 0.540641i 3.20037 − 7.00783i −0.959493 − 0.281733i
131.2 0.654861 − 0.755750i 1.91108 − 1.22817i −0.142315 − 0.989821i −0.415415 − 0.909632i 0.323297 − 2.24858i −1.81117 + 0.531808i −0.841254 − 0.540641i 0.897555 − 1.96537i −0.959493 − 0.281733i
141.1 0.142315 + 0.989821i −0.641135 + 1.40389i −0.959493 + 0.281733i 0.654861 − 0.755750i −1.48084 − 0.434815i −3.01115 + 1.93515i −0.415415 − 0.909632i 0.404729 + 0.467082i 0.841254 + 0.540641i
141.2 0.142315 + 0.989821i 0.225720 − 0.494258i −0.959493 + 0.281733i 0.654861 − 0.755750i 0.521351 + 0.153082i 2.19754 − 1.41228i −0.415415 − 0.909632i 1.77124 + 2.04412i 0.841254 + 0.540641i
151.1 0.654861 + 0.755750i −2.75233 − 1.76881i −0.142315 + 0.989821i −0.415415 + 0.909632i −0.465612 − 3.23840i 0.267094 + 0.0784258i −0.841254 + 0.540641i 3.20037 + 7.00783i −0.959493 + 0.281733i
151.2 0.654861 + 0.755750i 1.91108 + 1.22817i −0.142315 + 0.989821i −0.415415 + 0.909632i 0.323297 + 2.24858i −1.81117 − 0.531808i −0.841254 + 0.540641i 0.897555 + 1.96537i −0.959493 + 0.281733i
211.1 −0.415415 − 0.909632i −1.26277 + 0.370783i −0.654861 + 0.755750i −0.841254 − 0.540641i 0.861850 + 0.994628i −0.120689 − 0.839414i 0.959493 + 0.281733i −1.06665 + 0.685495i −0.142315 + 0.989821i
211.2 −0.415415 − 0.909632i 2.22226 − 0.652515i −0.654861 + 0.755750i −0.841254 − 0.540641i −1.51671 − 1.75038i −0.180372 − 1.25451i 0.959493 + 0.281733i 1.98892 − 1.27820i −0.142315 + 0.989821i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.2
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.2.g.c ✓ 20
23.c even 11 1 inner 230.2.g.c ✓ 20
23.c even 11 1 5290.2.a.bg 10
23.d odd 22 1 5290.2.a.bh 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.g.c ✓ 20 1.a even 1 1 trivial
230.2.g.c ✓ 20 23.c even 11 1 inner
5290.2.a.bg 10 23.c even 11 1
5290.2.a.bh 10 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - T_{3}^{19} - 5 T_{3}^{18} + 52 T_{3}^{16} - 151 T_{3}^{15} + 378 T_{3}^{14} - 484 T_{3}^{13} + \cdots + 529 \) acting on \(S_{2}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} - T^{19} + \cdots + 529 \) Copy content Toggle raw display
$5$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + 12 T^{19} + \cdots + 94249 \) Copy content Toggle raw display
$11$ \( T^{20} + 3 T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{20} + 18 T^{19} + \cdots + 541696 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 541213696 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 98164409344 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 420115274569 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 151588750336 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 95910213477376 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 4067857373449 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 2808894001 \) Copy content Toggle raw display
$47$ \( (T^{10} + 13 T^{9} + \cdots + 590921)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 977611781702656 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 105667236214849 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 375240779761 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 65\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 1092699136 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 54315099136 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 544700098893721 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 21\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 236678358016 \) Copy content Toggle raw display
show more
show less