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Properties

Label 418.2.h.b
Level $418$
Weight $2$
Character orbit 418.h
Analytic conductor $3.338$
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] = [418,2,Mod(65,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.h (of order \(6\), 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: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 41 x^{18} + 707 x^{16} + 6667 x^{14} + 37400 x^{12} + 126976 x^{10} + 253280 x^{8} + \cdots + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 41 x^{18} + 707 x^{16} + 6667 x^{14} + 37400 x^{12} + 126976 x^{10} + 253280 x^{8} + \cdots + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 53 \nu^{18} + 2845 \nu^{16} + 61804 \nu^{14} + 710321 \nu^{12} + 4698655 \nu^{10} + 18093662 \nu^{8} + \cdots + 1650300 ) / 47556 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2603 \nu^{18} + 96159 \nu^{16} + 1450417 \nu^{14} + 11480565 \nu^{12} + 50950576 \nu^{10} + \cdots + 1787040 ) / 126816 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2603 \nu^{18} + 96159 \nu^{16} + 1450417 \nu^{14} + 11480565 \nu^{12} + 50950576 \nu^{10} + \cdots + 1787040 ) / 126816 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 107 \nu^{19} - 97800 \nu^{18} + 73339 \nu^{17} - 3695736 \nu^{16} + 2599705 \nu^{15} + \cdots - 185170176 ) / 4565376 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 107 \nu^{19} - 97800 \nu^{18} - 73339 \nu^{17} - 3695736 \nu^{16} - 2599705 \nu^{15} + \cdots - 185170176 ) / 4565376 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1461 \nu^{18} + 50809 \nu^{16} + 698211 \nu^{14} + 4718987 \nu^{12} + 15238980 \nu^{10} + \cdots - 3576288 ) / 63408 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9151 \nu^{19} - 1696 \nu^{18} - 384143 \nu^{17} - 91040 \nu^{16} - 6770837 \nu^{15} + \cdots - 52809600 ) / 3043584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 999 \nu^{19} - 41263 \nu^{17} - 716829 \nu^{15} - 6804269 \nu^{13} - 38321328 \nu^{11} + \cdots - 507264 ) / 253632 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6205 \nu^{19} - 233581 \nu^{17} - 3617663 \nu^{15} - 29765399 \nu^{13} - 140222480 \nu^{11} + \cdots + 507264 ) / 1014528 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18615 \nu^{19} - 79232 \nu^{18} + 700743 \nu^{17} - 2877280 \nu^{16} + 10852989 \nu^{15} + \cdots + 30557568 ) / 3043584 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18615 \nu^{19} - 79232 \nu^{18} - 700743 \nu^{17} - 2877280 \nu^{16} - 10852989 \nu^{15} + \cdots + 30557568 ) / 3043584 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 75833 \nu^{19} + 124320 \nu^{18} - 2792329 \nu^{17} + 4505568 \nu^{16} - 41911795 \nu^{15} + \cdots - 121507200 ) / 9130752 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 75833 \nu^{19} - 124320 \nu^{18} - 2792329 \nu^{17} - 4505568 \nu^{16} - 41911795 \nu^{15} + \cdots + 121507200 ) / 9130752 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 35533 \nu^{19} - 225984 \nu^{18} - 1369613 \nu^{17} - 8419488 \nu^{16} - 21888863 \nu^{15} + \cdots - 272207232 ) / 4565376 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13271 \nu^{19} - 499123 \nu^{17} - 7718701 \nu^{15} - 63320081 \nu^{13} - 296279848 \nu^{11} + \cdots - 2282688 ) / 1141344 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 20051 \nu^{19} + 50132 \nu^{18} + 703651 \nu^{17} + 1841620 \nu^{16} + 9815041 \nu^{15} + \cdots + 21851520 ) / 1521792 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 145213 \nu^{19} + 346440 \nu^{18} - 5609741 \nu^{17} + 12897096 \nu^{16} - 89878367 \nu^{15} + \cdots + 275428224 ) / 9130752 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 145213 \nu^{19} - 346440 \nu^{18} - 5609741 \nu^{17} - 12897096 \nu^{16} - 89878367 \nu^{15} + \cdots - 275428224 ) / 9130752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} + \beta_{11} + 2\beta_{10} + 6\beta_{4} - 6\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 9\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} - 10 \beta_{11} - 28 \beta_{10} + 4 \beta_{9} - \beta_{6} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{16} - 2 \beta_{15} - 13 \beta_{14} + 13 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} + \beta_{9} + \cdots - 191 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{19} - 2 \beta_{18} + 4 \beta_{17} + 2 \beta_{16} - 32 \beta_{14} - 32 \beta_{13} + \cdots - 156 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{19} - 2 \beta_{18} - 15 \beta_{16} + 30 \beta_{15} + 129 \beta_{14} - 129 \beta_{13} + \cdots + 1497 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 32 \beta_{19} + 32 \beta_{18} - 76 \beta_{17} - 32 \beta_{16} + 376 \beta_{14} + 376 \beta_{13} + \cdots + 1574 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 40 \beta_{19} + 40 \beta_{18} + 159 \beta_{16} - 318 \beta_{15} - 1175 \beta_{14} + 1175 \beta_{13} + \cdots - 12219 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 358 \beta_{19} - 358 \beta_{18} + 984 \beta_{17} + 370 \beta_{16} - 3930 \beta_{14} - 3930 \beta_{13} + \cdots - 15080 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 530 \beta_{19} - 530 \beta_{18} - 1483 \beta_{16} + 2966 \beta_{15} + 10389 \beta_{14} - 10389 \beta_{13} + \cdots + 102383 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3496 \beta_{19} + 3496 \beta_{18} - 10848 \beta_{17} - 3822 \beta_{16} + 38784 \beta_{14} + \cdots + 140304 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5848 \beta_{19} + 5848 \beta_{18} + 13079 \beta_{16} - 26158 \beta_{15} - 91123 \beta_{14} + \cdots - 872755 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 32006 \beta_{19} - 32006 \beta_{18} + 109776 \beta_{17} + 37506 \beta_{16} - 370610 \beta_{14} + \cdots - 1282624 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 58066 \beta_{19} - 58066 \beta_{18} - 112531 \beta_{16} + 225062 \beta_{15} + 799765 \beta_{14} + \cdots + 7525551 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 283128 \beta_{19} + 283128 \beta_{18} - 1055440 \beta_{17} - 357554 \beta_{16} + 3472380 \beta_{14} + \cdots + 11596848 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 537344 \beta_{19} + 537344 \beta_{18} + 959079 \beta_{16} - 1918158 \beta_{15} - 7045071 \beta_{14} + \cdots - 65395747 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 2455502 \beta_{19} - 2455502 \beta_{18} + 9820672 \beta_{17} + 3343206 \beta_{16} - 32121518 \beta_{14} + \cdots - 104107396 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/418\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(343\)
\(\chi(n)\) \(\beta_{10}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
2.51965i
2.37734i
2.02516i
0.634765i
0.579050i
− 0.401261i
− 1.50467i
− 2.01980i
− 2.96323i
− 2.97905i
− 2.51965i
− 2.37734i
− 2.02516i
− 0.634765i
− 0.579050i
0.401261i
1.50467i
2.01980i
2.96323i
2.97905i
0.500000 − 0.866025i −2.18208 − 1.25983i −0.500000 − 0.866025i −2.02560 + 3.50844i −2.18208 + 1.25983i − 1.09641i −1.00000 1.67433 + 2.90002i 2.02560 + 3.50844i
65.2 0.500000 − 0.866025i −2.05883 − 1.18867i −0.500000 − 0.866025i 2.05852 − 3.56546i −2.05883 + 1.18867i − 2.89251i −1.00000 1.32587 + 2.29647i −2.05852 − 3.56546i
65.3 0.500000 − 0.866025i −1.75384 − 1.01258i −0.500000 − 0.866025i −0.202207 + 0.350232i −1.75384 + 1.01258i − 1.38521i −1.00000 0.550630 + 0.953720i 0.202207 + 0.350232i
65.4 0.500000 − 0.866025i −0.549723 − 0.317383i −0.500000 − 0.866025i −0.464143 + 0.803919i −0.549723 + 0.317383i 0.602460i −1.00000 −1.29854 − 2.24913i 0.464143 + 0.803919i
65.5 0.500000 − 0.866025i −0.501472 − 0.289525i −0.500000 − 0.866025i 1.06779 − 1.84947i −0.501472 + 0.289525i − 0.637717i −1.00000 −1.33235 − 2.30770i −1.06779 − 1.84947i
65.6 0.500000 − 0.866025i 0.347503 + 0.200631i −0.500000 − 0.866025i −1.32940 + 2.30258i 0.347503 − 0.200631i 3.90204i −1.00000 −1.41949 − 2.45864i 1.32940 + 2.30258i
65.7 0.500000 − 0.866025i 1.30309 + 0.752337i −0.500000 − 0.866025i 1.28873 − 2.23214i 1.30309 − 0.752337i − 1.03498i −1.00000 −0.367978 − 0.637356i −1.28873 − 2.23214i
65.8 0.500000 − 0.866025i 1.74919 + 1.00990i −0.500000 − 0.866025i 0.0648685 − 0.112356i 1.74919 − 1.00990i − 4.47991i −1.00000 0.539789 + 0.934941i −0.0648685 − 0.112356i
65.9 0.500000 − 0.866025i 2.56624 + 1.48162i −0.500000 − 0.866025i −1.17566 + 2.03631i 2.56624 − 1.48162i − 0.571027i −1.00000 2.89038 + 5.00628i 1.17566 + 2.03631i
65.10 0.500000 − 0.866025i 2.57993 + 1.48952i −0.500000 − 0.866025i 1.71710 − 2.97410i 2.57993 − 1.48952i 4.12917i −1.00000 2.93737 + 5.08767i −1.71710 − 2.97410i
373.1 0.500000 + 0.866025i −2.18208 + 1.25983i −0.500000 + 0.866025i −2.02560 − 3.50844i −2.18208 − 1.25983i 1.09641i −1.00000 1.67433 − 2.90002i 2.02560 − 3.50844i
373.2 0.500000 + 0.866025i −2.05883 + 1.18867i −0.500000 + 0.866025i 2.05852 + 3.56546i −2.05883 − 1.18867i 2.89251i −1.00000 1.32587 − 2.29647i −2.05852 + 3.56546i
373.3 0.500000 + 0.866025i −1.75384 + 1.01258i −0.500000 + 0.866025i −0.202207 − 0.350232i −1.75384 − 1.01258i 1.38521i −1.00000 0.550630 − 0.953720i 0.202207 − 0.350232i
373.4 0.500000 + 0.866025i −0.549723 + 0.317383i −0.500000 + 0.866025i −0.464143 − 0.803919i −0.549723 − 0.317383i − 0.602460i −1.00000 −1.29854 + 2.24913i 0.464143 − 0.803919i
373.5 0.500000 + 0.866025i −0.501472 + 0.289525i −0.500000 + 0.866025i 1.06779 + 1.84947i −0.501472 − 0.289525i 0.637717i −1.00000 −1.33235 + 2.30770i −1.06779 + 1.84947i
373.6 0.500000 + 0.866025i 0.347503 − 0.200631i −0.500000 + 0.866025i −1.32940 − 2.30258i 0.347503 + 0.200631i − 3.90204i −1.00000 −1.41949 + 2.45864i 1.32940 − 2.30258i
373.7 0.500000 + 0.866025i 1.30309 − 0.752337i −0.500000 + 0.866025i 1.28873 + 2.23214i 1.30309 + 0.752337i 1.03498i −1.00000 −0.367978 + 0.637356i −1.28873 + 2.23214i
373.8 0.500000 + 0.866025i 1.74919 − 1.00990i −0.500000 + 0.866025i 0.0648685 + 0.112356i 1.74919 + 1.00990i 4.47991i −1.00000 0.539789 − 0.934941i −0.0648685 + 0.112356i
373.9 0.500000 + 0.866025i 2.56624 − 1.48162i −0.500000 + 0.866025i −1.17566 − 2.03631i 2.56624 + 1.48162i 0.571027i −1.00000 2.89038 − 5.00628i 1.17566 − 2.03631i
373.10 0.500000 + 0.866025i 2.57993 − 1.48952i −0.500000 + 0.866025i 1.71710 + 2.97410i 2.57993 + 1.48952i − 4.12917i −1.00000 2.93737 − 5.08767i −1.71710 + 2.97410i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.h.b yes 20
11.b odd 2 1 418.2.h.a ✓ 20
19.d odd 6 1 418.2.h.a ✓ 20
209.g even 6 1 inner 418.2.h.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.h.a ✓ 20 11.b odd 2 1
418.2.h.a ✓ 20 19.d odd 6 1
418.2.h.b yes 20 1.a even 1 1 trivial
418.2.h.b yes 20 209.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{20} - 5 T_{13}^{19} + 77 T_{13}^{18} - 352 T_{13}^{17} + 3849 T_{13}^{16} - 15391 T_{13}^{15} + \cdots + 2985984 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} - 3 T^{19} + \cdots + 2304 \) Copy content Toggle raw display
$5$ \( T^{20} - 2 T^{19} + \cdots + 9216 \) Copy content Toggle raw display
$7$ \( T^{20} + 66 T^{18} + \cdots + 5184 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( T^{20} - 5 T^{19} + \cdots + 2985984 \) Copy content Toggle raw display
$17$ \( T^{20} - 6 T^{19} + \cdots + 73547776 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 22265110462464 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 29561988096 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 949829566464 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 135789302016 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 497465017344 \) Copy content Toggle raw display
$47$ \( T^{20} - T^{19} + \cdots + 112896 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 2828537856 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 3718911688704 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 36790308864 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 5707473897024 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 4811139551761 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
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