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Properties

Label 806.2.k.b
Level $806$
Weight $2$
Character orbit 806.k
Analytic conductor $6.436$
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] = [806,2,Mod(157,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.k (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: \(6.43594240292\)
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: \( x^{20} - x^{19} + 9 x^{18} - 13 x^{17} + 68 x^{16} - 21 x^{15} + 395 x^{14} - 48 x^{13} + 1897 x^{12} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + \beta_{8} q^{3} + ( - \beta_{15} - \beta_{9} - \beta_{5} - 1) q^{4} - \beta_{4} q^{5} + \beta_{3} q^{6} + (\beta_{12} + \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{7} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + \beta_{8} q^{3} + ( - \beta_{15} - \beta_{9} - \beta_{5} - 1) q^{4} - \beta_{4} q^{5} + \beta_{3} q^{6} + (\beta_{12} + \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{19} - \beta_{17} - 2 \beta_{16} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{2} - q^{3} - 5 q^{4} - 4 q^{5} + 4 q^{6} - 5 q^{7} - 5 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{2} - q^{3} - 5 q^{4} - 4 q^{5} + 4 q^{6} - 5 q^{7} - 5 q^{8} - 2 q^{9} + q^{10} + 5 q^{11} - q^{12} - 5 q^{13} - 5 q^{14} - 7 q^{15} - 5 q^{16} - 13 q^{17} - 2 q^{18} + 12 q^{19} + q^{20} - 2 q^{21} + 5 q^{22} + 9 q^{23} - q^{24} - 8 q^{25} + 20 q^{26} - 7 q^{27} + 10 q^{28} - 8 q^{29} + 28 q^{30} - 38 q^{31} + 20 q^{32} - 4 q^{33} + 17 q^{34} + 28 q^{35} + 8 q^{36} - 44 q^{37} + 12 q^{38} - q^{39} + q^{40} - 2 q^{41} + 23 q^{42} + 18 q^{43} + 26 q^{45} - 6 q^{46} + 15 q^{47} - q^{48} + 6 q^{49} + 12 q^{50} + 7 q^{51} - 5 q^{52} + 2 q^{53} - 7 q^{54} + 15 q^{55} - 10 q^{56} - 46 q^{57} + 2 q^{58} - 6 q^{59} - 7 q^{60} + 22 q^{61} + 27 q^{62} - 30 q^{63} - 5 q^{64} + q^{65} + 16 q^{66} - 44 q^{67} - 8 q^{68} - 19 q^{69} + 28 q^{70} + 55 q^{71} - 2 q^{72} - q^{73} + 26 q^{74} + 19 q^{75} + 12 q^{76} - 12 q^{77} - q^{78} + 26 q^{79} + q^{80} + 35 q^{81} + 3 q^{82} - 29 q^{83} + 23 q^{84} - 10 q^{85} - 12 q^{86} - 94 q^{87} - 10 q^{88} + 26 q^{89} + 6 q^{90} + 10 q^{91} - 6 q^{92} - 7 q^{93} + 20 q^{94} + 27 q^{95} - q^{96} - 2 q^{97} - 14 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 9 x^{18} - 13 x^{17} + 68 x^{16} - 21 x^{15} + 395 x^{14} - 48 x^{13} + 1897 x^{12} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\!\cdots\!49 \nu^{19} + \cdots - 13\!\cdots\!91 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!28 \nu^{19} + \cdots - 24\!\cdots\!87 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!35 \nu^{19} + \cdots - 10\!\cdots\!53 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!91 \nu^{19} + \cdots + 24\!\cdots\!28 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24\!\cdots\!77 \nu^{19} + \cdots - 63\!\cdots\!74 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40\!\cdots\!60 \nu^{19} + \cdots - 73\!\cdots\!12 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!26 \nu^{19} + \cdots + 25\!\cdots\!71 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\!\cdots\!71 \nu^{19} + \cdots - 47\!\cdots\!80 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 44\!\cdots\!93 \nu^{19} + \cdots + 60\!\cdots\!03 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 45\!\cdots\!49 \nu^{19} + \cdots + 63\!\cdots\!56 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 50\!\cdots\!72 \nu^{19} + \cdots - 40\!\cdots\!31 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 70\!\cdots\!12 \nu^{19} + \cdots - 97\!\cdots\!19 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 75\!\cdots\!64 \nu^{19} + \cdots + 13\!\cdots\!49 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\!\cdots\!25 \nu^{19} + \cdots - 61\!\cdots\!80 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 40\!\cdots\!46 \nu^{19} + \cdots + 11\!\cdots\!23 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 41\!\cdots\!28 \nu^{19} + \cdots - 93\!\cdots\!35 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 66\!\cdots\!55 \nu^{19} + \cdots - 49\!\cdots\!17 ) / 35\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 70\!\cdots\!60 \nu^{19} + \cdots - 91\!\cdots\!96 ) / 35\!\cdots\!81 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + 3\beta_{9} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{16} - 2\beta_{15} - \beta_{11} - 5\beta_{8} - \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{19} + \beta_{18} + \beta_{17} - \beta_{16} - 15 \beta_{15} + \beta_{14} - 14 \beta_{9} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} + \beta_{17} - \beta_{15} + 9 \beta_{12} + 9 \beta_{11} - \beta_{10} - \beta_{9} + \cdots - 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{14} + 13 \beta_{12} + 13 \beta_{11} - 9 \beta_{10} - 12 \beta_{9} + 25 \beta_{7} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{19} - 13 \beta_{17} + \beta_{16} + 18 \beta_{15} - 26 \beta_{14} + 13 \beta_{13} + \cdots - 15 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 209 \beta_{19} - 66 \beta_{18} - 67 \beta_{17} + 118 \beta_{16} + 518 \beta_{15} - 209 \beta_{14} + \cdots + 109 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 242 \beta_{19} - 118 \beta_{18} - 120 \beta_{17} + 458 \beta_{16} + 982 \beta_{15} - 137 \beta_{14} + \cdots + 982 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 595 \beta_{19} - 19 \beta_{18} - 458 \beta_{17} + 44 \beta_{16} + 898 \beta_{15} - 934 \beta_{12} + \cdots + 3301 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1163 \beta_{14} - 44 \beta_{13} - 3124 \beta_{12} - 3362 \beta_{11} + 934 \beta_{10} + 1946 \beta_{9} + \cdots + 1946 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4290 \beta_{19} + 238 \beta_{18} + 3124 \beta_{17} - 599 \beta_{16} - 7062 \beta_{15} + \cdots + 3658 \beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 15259 \beta_{19} + 6930 \beta_{18} + 7529 \beta_{17} - 21217 \beta_{16} - 49239 \beta_{15} + \cdots - 16615 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 54505 \beta_{19} + 21217 \beta_{18} + 23696 \beta_{17} - 49737 \beta_{16} - 145016 \beta_{15} + \cdots - 145016 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 69579 \beta_{19} + 6530 \beta_{18} + 49737 \beta_{17} - 23238 \beta_{16} - 133670 \beta_{15} + \cdots - 346259 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 215221 \beta_{14} + 23238 \beta_{13} + 350750 \beta_{12} + 413420 \beta_{11} - 144197 \beta_{10} + \cdots - 406365 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 513926 \beta_{19} - 62670 \beta_{18} - 350750 \beta_{17} + 203408 \beta_{16} + 1035846 \beta_{15} + \cdots - 758065 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2506986 \beta_{19} - 982672 \beta_{18} - 1186080 \beta_{17} + 2451142 \beta_{16} + 6752316 \beta_{15} + \cdots + 3015623 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 5987981 \beta_{19} - 2451142 \beta_{18} - 3005736 \beta_{17} + 6720064 \beta_{16} + 17056501 \beta_{15} + \cdots + 17056501 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/806\mathbb{Z}\right)^\times\).

\(n\) \(249\) \(313\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−0.748900 + 2.30488i
−0.673865 + 2.07394i
0.0189075 − 0.0581914i
0.481408 − 1.48162i
0.613432 − 1.88795i
−1.75742 + 1.27684i
−1.04750 + 0.761053i
−0.0575513 + 0.0418135i
1.52142 − 1.10538i
2.15007 − 1.56212i
−1.75742 − 1.27684i
−1.04750 − 0.761053i
−0.0575513 − 0.0418135i
1.52142 + 1.10538i
2.15007 + 1.56212i
−0.748900 − 2.30488i
−0.673865 − 2.07394i
0.0189075 + 0.0581914i
0.481408 + 1.48162i
0.613432 + 1.88795i
−0.809017 − 0.587785i −1.96065 + 1.42449i 0.309017 + 0.951057i −0.449817 2.42349 −1.30713 − 4.02292i 0.309017 − 0.951057i 0.887901 − 2.73268i 0.363910 + 0.264396i
157.2 −0.809017 − 0.587785i −1.76420 + 1.28177i 0.309017 + 0.951057i 0.425338 2.18067 0.0929018 + 0.285922i 0.309017 − 0.951057i 0.542428 − 1.66942i −0.344106 − 0.250007i
157.3 −0.809017 − 0.587785i 0.0495006 − 0.0359643i 0.309017 + 0.951057i 2.93507 −0.0611861 0.366057 + 1.12661i 0.309017 − 0.951057i −0.925894 + 2.84961i −2.37452 − 1.72519i
157.4 −0.809017 − 0.587785i 1.26034 − 0.915693i 0.309017 + 0.951057i −0.984827 −1.55787 −1.02282 − 3.14793i 0.309017 − 0.951057i −0.177080 + 0.544996i 0.796741 + 0.578867i
157.5 −0.809017 − 0.587785i 1.60599 − 1.16682i 0.309017 + 0.951057i −2.92576 −1.98511 0.0619756 + 0.190741i 0.309017 − 0.951057i 0.290679 − 0.894617i 2.36699 + 1.71972i
287.1 0.309017 − 0.951057i −0.671276 − 2.06598i −0.809017 − 0.587785i −3.89117 −2.17230 −1.56277 − 1.13542i −0.809017 + 0.587785i −1.39060 + 1.01033i −1.20244 + 3.70072i
287.2 0.309017 − 0.951057i −0.400109 − 1.23141i −0.809017 − 0.587785i 0.0287596 −1.29478 −2.69913 − 1.96103i −0.809017 + 0.587785i 1.07077 − 0.777958i 0.00888720 − 0.0273520i
287.3 0.309017 − 0.951057i −0.0219826 − 0.0676556i −0.809017 − 0.587785i 2.92380 −0.0711373 0.264785 + 0.192378i −0.809017 + 0.587785i 2.42296 − 1.76038i 0.903505 − 2.78070i
287.4 0.309017 − 0.951057i 0.581130 + 1.78854i −0.809017 − 0.587785i 1.34401 1.88058 3.39747 + 2.46840i −0.809017 + 0.587785i −0.434095 + 0.315388i 0.415321 − 1.27823i
287.5 0.309017 − 0.951057i 0.821255 + 2.52756i −0.809017 − 0.587785i −1.40540 2.65764 −0.0913313 − 0.0663561i −0.809017 + 0.587785i −3.28707 + 2.38819i −0.434293 + 1.33662i
469.1 0.309017 + 0.951057i −0.671276 + 2.06598i −0.809017 + 0.587785i −3.89117 −2.17230 −1.56277 + 1.13542i −0.809017 − 0.587785i −1.39060 − 1.01033i −1.20244 − 3.70072i
469.2 0.309017 + 0.951057i −0.400109 + 1.23141i −0.809017 + 0.587785i 0.0287596 −1.29478 −2.69913 + 1.96103i −0.809017 − 0.587785i 1.07077 + 0.777958i 0.00888720 + 0.0273520i
469.3 0.309017 + 0.951057i −0.0219826 + 0.0676556i −0.809017 + 0.587785i 2.92380 −0.0711373 0.264785 − 0.192378i −0.809017 − 0.587785i 2.42296 + 1.76038i 0.903505 + 2.78070i
469.4 0.309017 + 0.951057i 0.581130 − 1.78854i −0.809017 + 0.587785i 1.34401 1.88058 3.39747 − 2.46840i −0.809017 − 0.587785i −0.434095 − 0.315388i 0.415321 + 1.27823i
469.5 0.309017 + 0.951057i 0.821255 − 2.52756i −0.809017 + 0.587785i −1.40540 2.65764 −0.0913313 + 0.0663561i −0.809017 − 0.587785i −3.28707 − 2.38819i −0.434293 − 1.33662i
729.1 −0.809017 + 0.587785i −1.96065 − 1.42449i 0.309017 − 0.951057i −0.449817 2.42349 −1.30713 + 4.02292i 0.309017 + 0.951057i 0.887901 + 2.73268i 0.363910 − 0.264396i
729.2 −0.809017 + 0.587785i −1.76420 − 1.28177i 0.309017 − 0.951057i 0.425338 2.18067 0.0929018 − 0.285922i 0.309017 + 0.951057i 0.542428 + 1.66942i −0.344106 + 0.250007i
729.3 −0.809017 + 0.587785i 0.0495006 + 0.0359643i 0.309017 − 0.951057i 2.93507 −0.0611861 0.366057 − 1.12661i 0.309017 + 0.951057i −0.925894 − 2.84961i −2.37452 + 1.72519i
729.4 −0.809017 + 0.587785i 1.26034 + 0.915693i 0.309017 − 0.951057i −0.984827 −1.55787 −1.02282 + 3.14793i 0.309017 + 0.951057i −0.177080 − 0.544996i 0.796741 − 0.578867i
729.5 −0.809017 + 0.587785i 1.60599 + 1.16682i 0.309017 − 0.951057i −2.92576 −1.98511 0.0619756 − 0.190741i 0.309017 + 0.951057i 0.290679 + 0.894617i 2.36699 − 1.71972i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.k.b ✓ 20
31.d even 5 1 inner 806.2.k.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.k.b ✓ 20 1.a even 1 1 trivial
806.2.k.b ✓ 20 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + T_{3}^{19} + 9 T_{3}^{18} + 13 T_{3}^{17} + 63 T_{3}^{16} + 16 T_{3}^{15} + 305 T_{3}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{10} + 2 T^{9} - 21 T^{8} + \cdots - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + 5 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{20} - 5 T^{19} + \cdots + 13140625 \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{20} + 13 T^{19} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 2806774441 \) Copy content Toggle raw display
$23$ \( T^{20} - 9 T^{19} + \cdots + 78961 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 40297167480025 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 819628286980801 \) Copy content Toggle raw display
$37$ \( (T^{10} + 22 T^{9} + \cdots + 7281)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + 2 T^{19} + \cdots + 707281 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 40\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 775124343513025 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 20261026500625 \) Copy content Toggle raw display
$61$ \( (T^{10} - 11 T^{9} + \cdots + 117055)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 22 T^{9} + \cdots + 2638899)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 128933731300201 \) Copy content Toggle raw display
$73$ \( T^{20} + T^{19} + \cdots + 85581001 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 683430567675025 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 50130294075625 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 47760469701025 \) Copy content Toggle raw display
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