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Properties

Label 539.2.f.g
Level $539$
Weight $2$
Character orbit 539.f
Analytic conductor $4.304$
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] = [539,2,Mod(148,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.148");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.f (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: \(4.30393666895\)
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: \( x^{20} - 2 x^{19} + 13 x^{18} - 14 x^{17} + 75 x^{16} - 28 x^{15} + 349 x^{14} + 203 x^{13} + 1636 x^{12} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{19} - \beta_{18} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{15} - \beta_{13} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{19} - \beta_{18} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{19} + 2 \beta_{17} - 2 \beta_{16} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 3 q^{2} - 4 q^{3} + 3 q^{4} + 4 q^{5} + 8 q^{6} - 19 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 3 q^{2} - 4 q^{3} + 3 q^{4} + 4 q^{5} + 8 q^{6} - 19 q^{8} - 7 q^{9} + 14 q^{10} + 9 q^{11} - 18 q^{12} - 3 q^{13} - 7 q^{15} + 5 q^{16} - 7 q^{17} - 24 q^{18} - 4 q^{19} + 15 q^{20} + 22 q^{22} + 14 q^{23} - 12 q^{24} - 21 q^{25} + 8 q^{27} - 16 q^{30} - 17 q^{31} + 30 q^{32} - 15 q^{33} - 24 q^{34} + 7 q^{36} - 24 q^{37} + 12 q^{38} - 28 q^{39} + 10 q^{40} - 30 q^{41} - 36 q^{43} - 18 q^{44} - 16 q^{45} - 8 q^{46} + 13 q^{47} - 64 q^{48} + 3 q^{50} + 7 q^{51} + 2 q^{52} - 33 q^{53} + 34 q^{54} + 3 q^{55} + 22 q^{57} + 17 q^{58} + 21 q^{59} + 48 q^{60} + 26 q^{62} + 47 q^{64} + 40 q^{65} - 49 q^{66} + 38 q^{67} - 23 q^{68} + 62 q^{69} + 10 q^{71} + 38 q^{72} + 11 q^{73} + 41 q^{74} - 11 q^{75} + 48 q^{76} - 50 q^{78} - 21 q^{79} + 12 q^{80} + 58 q^{81} + 6 q^{82} + 23 q^{83} - 39 q^{85} - 7 q^{86} + 48 q^{87} - 32 q^{88} - 10 q^{89} + 9 q^{90} - 55 q^{92} - 12 q^{93} + 37 q^{94} - 7 q^{95} - 53 q^{96} + 27 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 13 x^{18} - 14 x^{17} + 75 x^{16} - 28 x^{15} + 349 x^{14} + 203 x^{13} + 1636 x^{12} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!50 \nu^{19} + \cdots + 86\!\cdots\!98 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\!\cdots\!56 \nu^{19} + \cdots - 39\!\cdots\!30 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36\!\cdots\!42 \nu^{19} + \cdots + 27\!\cdots\!88 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 61\!\cdots\!35 \nu^{19} + \cdots - 21\!\cdots\!64 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 74\!\cdots\!68 \nu^{19} + \cdots + 94\!\cdots\!10 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!33 \nu^{19} + \cdots + 18\!\cdots\!72 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 44\!\cdots\!09 \nu^{19} + \cdots + 77\!\cdots\!79 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 49\!\cdots\!64 \nu^{19} + \cdots + 85\!\cdots\!77 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 64\!\cdots\!01 \nu^{19} + \cdots + 13\!\cdots\!97 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 94\!\cdots\!10 \nu^{19} + \cdots - 16\!\cdots\!25 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!02 \nu^{19} + \cdots + 25\!\cdots\!72 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18\!\cdots\!72 \nu^{19} + \cdots + 13\!\cdots\!31 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19\!\cdots\!20 \nu^{19} + \cdots - 41\!\cdots\!38 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27\!\cdots\!85 \nu^{19} + \cdots - 21\!\cdots\!34 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 33\!\cdots\!54 \nu^{19} + \cdots - 38\!\cdots\!37 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 43\!\cdots\!10 \nu^{19} + \cdots + 32\!\cdots\!19 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 51\!\cdots\!65 \nu^{19} + \cdots + 57\!\cdots\!37 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 75\!\cdots\!63 \nu^{19} + \cdots + 55\!\cdots\!73 ) / 16\!\cdots\!09 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} - 3\beta_{13} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{16} - 2 \beta_{13} - 2 \beta_{12} + \beta_{10} + \beta_{8} + 4 \beta_{7} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{19} - 6 \beta_{17} + \beta_{14} - \beta_{13} - 16 \beta_{12} + \beta_{11} + 6 \beta_{10} + \cdots - 14 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{18} - 8 \beta_{17} - 18 \beta_{12} + 18 \beta_{11} + \beta_{10} + \beta_{9} - 28 \beta_{7} + \cdots - 37 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 36 \beta_{19} + 36 \beta_{18} - 2 \beta_{17} + 8 \beta_{16} - 9 \beta_{15} - 8 \beta_{14} + 12 \beta_{13} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 58 \beta_{19} - 35 \beta_{15} - 3 \beta_{14} + 140 \beta_{13} + 134 \beta_{12} - 14 \beta_{10} + \cdots - 134 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 28 \beta_{19} - 219 \beta_{18} - 53 \beta_{16} - 14 \beta_{15} - 14 \beta_{14} + 607 \beta_{13} + \cdots - 493 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5 \beta_{19} - 407 \beta_{18} + 5 \beta_{17} - 43 \beta_{16} - 169 \beta_{14} + 944 \beta_{13} + \cdots + 560 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 282 \beta_{18} + 282 \beta_{17} - 134 \beta_{16} + 134 \beta_{15} + 980 \beta_{12} - 980 \beta_{11} + \cdots + 2974 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 98 \beta_{19} - 98 \beta_{18} + 2808 \beta_{17} - 909 \beta_{16} + 1333 \beta_{15} + 909 \beta_{14} + \cdots + 6509 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2481 \beta_{19} + 8402 \beta_{17} + 3260 \beta_{15} + 1113 \beta_{14} - 26227 \beta_{13} - 7927 \beta_{12} + \cdots + 7927 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 19193 \beta_{19} + 1246 \beta_{18} + 5037 \beta_{16} + 3593 \beta_{15} + 3593 \beta_{14} - 52459 \beta_{13} + \cdots + 7987 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 52850 \beta_{19} + 20298 \beta_{18} - 52850 \beta_{17} + 8645 \beta_{16} + 13713 \beta_{14} + \cdots - 175522 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 130530 \beta_{18} - 130530 \beta_{17} + 28156 \beta_{16} - 28156 \beta_{15} - 302726 \beta_{12} + \cdots - 66652 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 335423 \beta_{19} + 335423 \beta_{18} - 158844 \beta_{17} + 88294 \beta_{16} - 153048 \beta_{15} + \cdots - 464526 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 885578 \beta_{19} - 121783 \beta_{17} - 383164 \beta_{15} - 210831 \beta_{14} + 2591178 \beta_{13} + \cdots - 2058493 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1207014 \beta_{19} - 2146327 \beta_{18} - 573286 \beta_{16} - 474628 \beta_{15} - 474628 \beta_{14} + \cdots - 4650397 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 1059189 \beta_{19} - 6003296 \beta_{18} + 1059189 \beta_{17} - 1534679 \beta_{16} + \cdots + 12617306 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(442\)
\(\chi(n)\) \(1\) \(-\beta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−1.49573 − 1.08671i
−0.545248 − 0.396146i
−0.0615029 − 0.0446845i
0.473500 + 0.344018i
2.12899 + 1.54680i
−0.691302 + 2.12761i
−0.394019 + 1.21266i
0.135108 − 0.415819i
0.693865 − 2.13550i
0.756349 − 2.32780i
−1.49573 + 1.08671i
−0.545248 + 0.396146i
−0.0615029 + 0.0446845i
0.473500 − 0.344018i
2.12899 − 1.54680i
−0.691302 − 2.12761i
−0.394019 − 1.21266i
0.135108 + 0.415819i
0.693865 + 2.13550i
0.756349 + 2.32780i
−0.571320 − 1.75834i 0.956261 + 0.694764i −1.14733 + 0.833580i 0.730114 − 2.24706i 0.675302 − 2.07837i 0 −0.870261 − 0.632281i −0.495313 − 1.52442i −4.36823
148.2 −0.208266 − 0.640978i 1.74031 + 1.26441i 1.25056 − 0.908582i −0.521958 + 1.60642i 0.448009 − 1.37883i 0 −1.93333 − 1.40464i 0.502889 + 1.54773i 1.13839
148.3 −0.0234920 − 0.0723010i −1.90529 − 1.38427i 1.61336 − 1.17217i −0.796668 + 2.45189i −0.0553254 + 0.170274i 0 −0.245656 − 0.178480i 0.786866 + 2.42172i 0.195990
148.4 0.180861 + 0.556633i −1.11859 − 0.812702i 1.34090 − 0.974224i 1.22500 − 3.77016i 0.250068 − 0.769629i 0 1.73180 + 1.25823i −0.336295 − 1.03501i 2.32015
148.5 0.813200 + 2.50277i −1.79072 − 1.30103i −3.98455 + 2.89494i 0.363514 − 1.11878i 1.79998 − 5.53977i 0 −6.22764 − 4.52465i 0.586938 + 1.80641i 3.09567
246.1 −1.80985 − 1.31494i −0.412517 + 1.26960i 0.928480 + 2.85757i −0.700878 + 0.509218i 2.41604 − 1.75535i 0 0.694498 − 2.13745i 0.985342 + 0.715893i 1.93808
246.2 −1.03155 − 0.749468i 0.0430492 − 0.132492i −0.115632 − 0.355879i 0.958742 − 0.696567i −0.143706 + 0.104408i 0 −0.935477 + 2.87910i 2.41135 + 1.75195i −1.51105
246.3 0.353717 + 0.256991i 0.798354 − 2.45708i −0.558962 − 1.72031i −0.983592 + 0.714622i 0.913838 − 0.663942i 0 0.514604 − 1.58379i −2.97283 − 2.15989i −0.531564
246.4 1.81656 + 1.31981i 0.589939 − 1.81565i 0.939965 + 2.89291i 1.99426 − 1.44891i 3.46797 − 2.51963i 0 −0.722861 + 2.22474i −0.521495 − 0.378888i 5.53498
246.5 1.98015 + 1.43866i −0.900791 + 2.77235i 1.23320 + 3.79540i −0.268531 + 0.195099i −5.77217 + 4.19373i 0 −1.50568 + 4.63401i −4.44745 − 3.23126i −0.812412
295.1 −0.571320 + 1.75834i 0.956261 − 0.694764i −1.14733 − 0.833580i 0.730114 + 2.24706i 0.675302 + 2.07837i 0 −0.870261 + 0.632281i −0.495313 + 1.52442i −4.36823
295.2 −0.208266 + 0.640978i 1.74031 − 1.26441i 1.25056 + 0.908582i −0.521958 − 1.60642i 0.448009 + 1.37883i 0 −1.93333 + 1.40464i 0.502889 − 1.54773i 1.13839
295.3 −0.0234920 + 0.0723010i −1.90529 + 1.38427i 1.61336 + 1.17217i −0.796668 − 2.45189i −0.0553254 − 0.170274i 0 −0.245656 + 0.178480i 0.786866 − 2.42172i 0.195990
295.4 0.180861 − 0.556633i −1.11859 + 0.812702i 1.34090 + 0.974224i 1.22500 + 3.77016i 0.250068 + 0.769629i 0 1.73180 − 1.25823i −0.336295 + 1.03501i 2.32015
295.5 0.813200 − 2.50277i −1.79072 + 1.30103i −3.98455 − 2.89494i 0.363514 + 1.11878i 1.79998 + 5.53977i 0 −6.22764 + 4.52465i 0.586938 − 1.80641i 3.09567
344.1 −1.80985 + 1.31494i −0.412517 − 1.26960i 0.928480 − 2.85757i −0.700878 − 0.509218i 2.41604 + 1.75535i 0 0.694498 + 2.13745i 0.985342 − 0.715893i 1.93808
344.2 −1.03155 + 0.749468i 0.0430492 + 0.132492i −0.115632 + 0.355879i 0.958742 + 0.696567i −0.143706 − 0.104408i 0 −0.935477 − 2.87910i 2.41135 − 1.75195i −1.51105
344.3 0.353717 − 0.256991i 0.798354 + 2.45708i −0.558962 + 1.72031i −0.983592 − 0.714622i 0.913838 + 0.663942i 0 0.514604 + 1.58379i −2.97283 + 2.15989i −0.531564
344.4 1.81656 − 1.31981i 0.589939 + 1.81565i 0.939965 − 2.89291i 1.99426 + 1.44891i 3.46797 + 2.51963i 0 −0.722861 − 2.22474i −0.521495 + 0.378888i 5.53498
344.5 1.98015 − 1.43866i −0.900791 − 2.77235i 1.23320 − 3.79540i −0.268531 − 0.195099i −5.77217 − 4.19373i 0 −1.50568 − 4.63401i −4.44745 + 3.23126i −0.812412
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.5
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 539.2.f.g 20
7.b odd 2 1 539.2.f.h 20
7.c even 3 2 539.2.q.h 40
7.d odd 6 2 77.2.m.b ✓ 40
11.c even 5 1 inner 539.2.f.g 20
11.c even 5 1 5929.2.a.bx 10
11.d odd 10 1 5929.2.a.bz 10
21.g even 6 2 693.2.by.b 40
77.i even 6 2 847.2.n.j 40
77.j odd 10 1 539.2.f.h 20
77.j odd 10 1 5929.2.a.bw 10
77.l even 10 1 5929.2.a.by 10
77.m even 15 2 539.2.q.h 40
77.n even 30 2 847.2.e.h 20
77.n even 30 4 847.2.n.h 40
77.n even 30 2 847.2.n.j 40
77.p odd 30 2 77.2.m.b ✓ 40
77.p odd 30 2 847.2.e.i 20
77.p odd 30 4 847.2.n.i 40
231.bc even 30 2 693.2.by.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.b ✓ 40 7.d odd 6 2
77.2.m.b ✓ 40 77.p odd 30 2
539.2.f.g 20 1.a even 1 1 trivial
539.2.f.g 20 11.c even 5 1 inner
539.2.f.h 20 7.b odd 2 1
539.2.f.h 20 77.j odd 10 1
539.2.q.h 40 7.c even 3 2
539.2.q.h 40 77.m even 15 2
693.2.by.b 40 21.g even 6 2
693.2.by.b 40 231.bc even 30 2
847.2.e.h 20 77.n even 30 2
847.2.e.i 20 77.p odd 30 2
847.2.n.h 40 77.n even 30 4
847.2.n.i 40 77.p odd 30 4
847.2.n.j 40 77.i even 6 2
847.2.n.j 40 77.n even 30 2
5929.2.a.bw 10 77.j odd 10 1
5929.2.a.bx 10 11.c even 5 1
5929.2.a.by 10 77.l even 10 1
5929.2.a.bz 10 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{20} - 3 T_{2}^{19} + 8 T_{2}^{18} - 6 T_{2}^{17} + 25 T_{2}^{16} - 52 T_{2}^{15} + 279 T_{2}^{14} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{20} + 4 T_{3}^{19} + 19 T_{3}^{18} + 50 T_{3}^{17} + 153 T_{3}^{16} + 326 T_{3}^{15} + \cdots + 2401 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 3 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{19} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{20} - 4 T^{19} + \cdots + 2401 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( T^{20} + 3 T^{19} + \cdots + 2401 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 12800885881 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 244328161 \) Copy content Toggle raw display
$23$ \( (T^{10} - 7 T^{9} + \cdots + 13711)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 1097530641 \) Copy content Toggle raw display
$31$ \( T^{20} + 17 T^{19} + \cdots + 2019241 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 5012937403681 \) Copy content Toggle raw display
$41$ \( T^{20} + 30 T^{19} + \cdots + 2401 \) Copy content Toggle raw display
$43$ \( (T^{10} + 18 T^{9} + \cdots + 181456)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 350145268684561 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 2159345019841 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 1241223361 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 12137217790201 \) Copy content Toggle raw display
$67$ \( (T^{10} - 19 T^{9} + \cdots + 2377069)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 1035165839761 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{10} + 5 T^{9} + \cdots - 112558831)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 11737688412961 \) Copy content Toggle raw display
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