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Properties

Label 357.2.k.b
Level $357$
Weight $2$
Character orbit 357.k
Analytic conductor $2.851$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [357,2,Mod(64,357)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(357, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("357.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 357 = 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 357.k (of order \(4\), 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: \(2.85065935216\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 32 x^{18} + 426 x^{16} + 3072 x^{14} + 13121 x^{12} + 34148 x^{10} + 53608 x^{8} + 48276 x^{6} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{8} - \beta_{7} - 1) q^{4} + ( - \beta_{12} + \beta_{9}) q^{5} - \beta_{3} q^{6} - \beta_{10} q^{7} + (\beta_{18} + \beta_{17} + \cdots - 2 \beta_1) q^{8}+ \cdots - \beta_{11} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{8} - \beta_{7} - 1) q^{4} + ( - \beta_{12} + \beta_{9}) q^{5} - \beta_{3} q^{6} - \beta_{10} q^{7} + (\beta_{18} + \beta_{17} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - \beta_{18} - \beta_{16} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{4} + 8 q^{5} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 24 q^{4} + 8 q^{5} - 4 q^{6} + 16 q^{10} + 4 q^{11} - 12 q^{13} + 4 q^{14} + 40 q^{16} + 4 q^{17} + 8 q^{18} - 52 q^{20} + 20 q^{21} - 24 q^{22} - 4 q^{23} + 4 q^{24} - 8 q^{29} + 8 q^{31} - 4 q^{33} - 44 q^{34} + 12 q^{35} + 24 q^{37} - 64 q^{38} - 12 q^{39} - 52 q^{40} - 20 q^{41} + 72 q^{44} - 8 q^{45} + 28 q^{46} - 32 q^{47} + 32 q^{48} + 104 q^{50} + 8 q^{51} + 48 q^{52} - 4 q^{54} - 36 q^{55} - 4 q^{56} - 16 q^{57} - 60 q^{58} + 28 q^{61} + 36 q^{62} - 112 q^{64} - 4 q^{65} + 40 q^{67} - 52 q^{68} + 36 q^{69} + 16 q^{71} - 24 q^{72} - 72 q^{73} - 24 q^{74} - 8 q^{75} + 40 q^{78} - 8 q^{79} + 120 q^{80} - 20 q^{81} + 108 q^{82} - 24 q^{84} + 40 q^{85} - 28 q^{88} - 64 q^{89} + 16 q^{90} - 12 q^{91} - 56 q^{92} + 60 q^{95} + 16 q^{96} + 60 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 32 x^{18} + 426 x^{16} + 3072 x^{14} + 13121 x^{12} + 34148 x^{10} + 53608 x^{8} + 48276 x^{6} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17 \nu^{19} - 406 \nu^{18} - 85 \nu^{17} - 13920 \nu^{16} - 11305 \nu^{15} - 197026 \nu^{14} + \cdots - 7424 ) / 173536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17 \nu^{19} + 406 \nu^{18} - 85 \nu^{17} + 13920 \nu^{16} - 11305 \nu^{15} + 197026 \nu^{14} + \cdots + 7424 ) / 173536 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 657 \nu^{19} + 93 \nu^{18} - 22873 \nu^{17} + 811 \nu^{16} - 335075 \nu^{15} - 21013 \nu^{14} + \cdots + 183892 ) / 173536 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 657 \nu^{19} + 93 \nu^{18} + 22873 \nu^{17} + 811 \nu^{16} + 335075 \nu^{15} - 21013 \nu^{14} + \cdots + 183892 ) / 173536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 195 \nu^{19} + 5724 \nu^{17} + 68424 \nu^{15} + 434332 \nu^{13} + 1631879 \nu^{11} + \cdots + 2034172 \nu ) / 43384 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 69 \nu^{18} - 2033 \nu^{16} - 24295 \nu^{14} - 151965 \nu^{12} - 537850 \nu^{10} - 1090814 \nu^{8} + \cdots - 368 ) / 7888 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 69 \nu^{18} - 2033 \nu^{16} - 24295 \nu^{14} - 151965 \nu^{12} - 537850 \nu^{10} - 1090814 \nu^{8} + \cdots + 23296 ) / 7888 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1856 \nu^{19} - 17 \nu^{18} + 58986 \nu^{17} + 85 \nu^{16} + 776736 \nu^{15} + 11305 \nu^{14} + \cdots + 125460 ) / 173536 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1856 \nu^{19} + 17 \nu^{18} + 58986 \nu^{17} - 85 \nu^{16} + 776736 \nu^{15} - 11305 \nu^{14} + \cdots - 125460 ) / 173536 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 92 \nu^{19} - 2875 \nu^{17} - 37159 \nu^{15} - 258329 \nu^{13} - 1055167 \nu^{11} + \cdots - 247648 \nu ) / 7888 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2100 \nu^{19} + 1443 \nu^{18} - 64146 \nu^{17} + 41273 \nu^{16} - 807790 \nu^{15} + \cdots - 183588 ) / 173536 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1401 \nu^{19} - 40207 \nu^{17} - 459813 \nu^{15} - 2656383 \nu^{13} - 8010836 \nu^{11} + \cdots + 4062736 \nu ) / 86768 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2100 \nu^{19} - 1443 \nu^{18} - 64146 \nu^{17} - 41273 \nu^{16} - 807790 \nu^{15} + \cdots + 183588 ) / 173536 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 887 \nu^{18} - 25870 \nu^{16} - 303983 \nu^{14} - 1847226 \nu^{12} - 6214991 \nu^{10} + \cdots - 65534 ) / 43384 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 207 \nu^{19} - 6592 \nu^{17} - 87182 \nu^{15} - 623022 \nu^{13} - 2627651 \nu^{11} + \cdots - 734688 \nu ) / 7888 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 5179 \nu^{19} - 172 \nu^{18} + 162953 \nu^{17} - 4882 \nu^{16} + 2121877 \nu^{15} - 58518 \nu^{14} + \cdots - 369024 ) / 173536 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 5179 \nu^{19} + 172 \nu^{18} + 162953 \nu^{17} + 4882 \nu^{16} + 2121877 \nu^{15} + 58518 \nu^{14} + \cdots + 369024 ) / 173536 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 3257 \nu^{18} + 99193 \nu^{16} + 1236549 \nu^{14} + 8160989 \nu^{12} + 30879656 \nu^{10} + \cdots - 118040 ) / 86768 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} + \beta_{17} + \beta_{16} + \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{19} + \beta_{18} - \beta_{17} - 2 \beta_{15} - \beta_{10} + \beta_{9} - 8 \beta_{8} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{18} - 9 \beta_{17} - 11 \beta_{16} - 9 \beta_{13} + 11 \beta_{11} + 8 \beta_{10} + \cdots + 40 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{19} - 10 \beta_{18} + 10 \beta_{17} + 20 \beta_{15} + \beta_{14} - \beta_{12} + 10 \beta_{10} + \cdots - 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 66 \beta_{18} + 66 \beta_{17} + 93 \beta_{16} + 69 \beta_{13} - 101 \beta_{11} - 53 \beta_{10} + \cdots - 278 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 63 \beta_{19} + 80 \beta_{18} - 80 \beta_{17} - 162 \beta_{15} - 9 \beta_{14} + 9 \beta_{12} + \cdots + 666 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 456 \beta_{18} - 456 \beta_{17} - 725 \beta_{16} + 4 \beta_{14} - 509 \beta_{13} + 4 \beta_{12} + \cdots + 1970 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 403 \beta_{19} - 598 \beta_{18} + 598 \beta_{17} + 1236 \beta_{15} + 51 \beta_{14} - 51 \beta_{12} + \cdots - 4570 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3082 \beta_{18} + 3082 \beta_{17} + 5475 \beta_{16} - 80 \beta_{14} + 3715 \beta_{13} + \cdots - 14108 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2449 \beta_{19} + 4340 \beta_{18} - 4340 \beta_{17} - 9230 \beta_{15} - 171 \beta_{14} + 171 \beta_{12} + \cdots + 31876 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 20638 \beta_{18} - 20638 \beta_{17} - 40749 \beta_{16} + 1044 \beta_{14} - 27053 \beta_{13} + \cdots + 101698 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 14223 \beta_{19} - 30956 \beta_{18} + 30956 \beta_{17} + 68352 \beta_{15} - 507 \beta_{14} + \cdots - 224670 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 137544 \beta_{18} + 137544 \beta_{17} + 300979 \beta_{16} - 11344 \beta_{14} + 197103 \beta_{13} + \cdots - 736476 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 77985 \beta_{19} + 217960 \beta_{18} - 217960 \beta_{17} - 504522 \beta_{15} + 17281 \beta_{14} + \cdots + 1595596 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 913598 \beta_{18} - 913598 \beta_{17} - 2213061 \beta_{16} + 111644 \beta_{14} - 1438101 \beta_{13} + \cdots + 5352442 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 389095 \beta_{19} - 1517444 \beta_{18} + 1517444 \beta_{17} + 3719764 \beta_{15} - 226891 \beta_{14} + \cdots - 11400710 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 6048032 \beta_{18} + 6048032 \beta_{17} + 16224911 \beta_{16} - 1034152 \beta_{14} + \cdots - 39014696 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/357\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(190\) \(239\)
\(\chi(n)\) \(1\) \(-\beta_{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
− 2.75017i
− 2.65813i
− 1.71520i
− 1.51109i
− 1.23743i
0.0320370i
0.661473i
0.901740i
1.80873i
2.46803i
− 2.46803i
− 1.80873i
− 0.901740i
− 0.661473i
− 0.0320370i
1.23743i
1.51109i
1.71520i
2.65813i
2.75017i
− 2.75017i 0.707107 + 0.707107i −5.56342 2.93829 + 2.93829i 1.94466 − 1.94466i 0.707107 − 0.707107i 9.79999i 1.00000i 8.08080 − 8.08080i
64.2 − 2.65813i −0.707107 − 0.707107i −5.06565 0.521373 + 0.521373i −1.87958 + 1.87958i −0.707107 + 0.707107i 8.14889i 1.00000i 1.38588 − 1.38588i
64.3 − 1.71520i −0.707107 − 0.707107i −0.941909 2.50881 + 2.50881i −1.21283 + 1.21283i −0.707107 + 0.707107i − 1.81484i 1.00000i 4.30311 − 4.30311i
64.4 − 1.51109i −0.707107 − 0.707107i −0.283389 −1.65084 − 1.65084i −1.06850 + 1.06850i −0.707107 + 0.707107i − 2.59395i 1.00000i −2.49457 + 2.49457i
64.5 − 1.23743i 0.707107 + 0.707107i 0.468767 −1.28416 − 1.28416i 0.874995 − 0.874995i 0.707107 − 0.707107i − 3.05493i 1.00000i −1.58906 + 1.58906i
64.6 0.0320370i 0.707107 + 0.707107i 1.99897 1.76367 + 1.76367i −0.0226536 + 0.0226536i 0.707107 − 0.707107i 0.128115i 1.00000i −0.0565026 + 0.0565026i
64.7 0.661473i −0.707107 − 0.707107i 1.56245 −1.66372 − 1.66372i 0.467732 − 0.467732i −0.707107 + 0.707107i 2.35647i 1.00000i 1.10050 − 1.10050i
64.8 0.901740i 0.707107 + 0.707107i 1.18686 −0.446227 − 0.446227i −0.637627 + 0.637627i 0.707107 − 0.707107i 2.87372i 1.00000i 0.402381 − 0.402381i
64.9 1.80873i −0.707107 − 0.707107i −1.27151 0.163056 + 0.163056i 1.27897 − 1.27897i −0.707107 + 0.707107i 1.31765i 1.00000i −0.294925 + 0.294925i
64.10 2.46803i 0.707107 + 0.707107i −4.09119 1.14975 + 1.14975i −1.74516 + 1.74516i 0.707107 − 0.707107i − 5.16112i 1.00000i −2.83762 + 2.83762i
106.1 − 2.46803i 0.707107 − 0.707107i −4.09119 1.14975 − 1.14975i −1.74516 − 1.74516i 0.707107 + 0.707107i 5.16112i − 1.00000i −2.83762 − 2.83762i
106.2 − 1.80873i −0.707107 + 0.707107i −1.27151 0.163056 − 0.163056i 1.27897 + 1.27897i −0.707107 − 0.707107i − 1.31765i − 1.00000i −0.294925 − 0.294925i
106.3 − 0.901740i 0.707107 − 0.707107i 1.18686 −0.446227 + 0.446227i −0.637627 − 0.637627i 0.707107 + 0.707107i − 2.87372i − 1.00000i 0.402381 + 0.402381i
106.4 − 0.661473i −0.707107 + 0.707107i 1.56245 −1.66372 + 1.66372i 0.467732 + 0.467732i −0.707107 − 0.707107i − 2.35647i − 1.00000i 1.10050 + 1.10050i
106.5 − 0.0320370i 0.707107 − 0.707107i 1.99897 1.76367 − 1.76367i −0.0226536 − 0.0226536i 0.707107 + 0.707107i − 0.128115i − 1.00000i −0.0565026 − 0.0565026i
106.6 1.23743i 0.707107 − 0.707107i 0.468767 −1.28416 + 1.28416i 0.874995 + 0.874995i 0.707107 + 0.707107i 3.05493i − 1.00000i −1.58906 − 1.58906i
106.7 1.51109i −0.707107 + 0.707107i −0.283389 −1.65084 + 1.65084i −1.06850 − 1.06850i −0.707107 − 0.707107i 2.59395i − 1.00000i −2.49457 − 2.49457i
106.8 1.71520i −0.707107 + 0.707107i −0.941909 2.50881 − 2.50881i −1.21283 − 1.21283i −0.707107 − 0.707107i 1.81484i − 1.00000i 4.30311 + 4.30311i
106.9 2.65813i −0.707107 + 0.707107i −5.06565 0.521373 − 0.521373i −1.87958 − 1.87958i −0.707107 − 0.707107i − 8.14889i − 1.00000i 1.38588 + 1.38588i
106.10 2.75017i 0.707107 − 0.707107i −5.56342 2.93829 − 2.93829i 1.94466 + 1.94466i 0.707107 + 0.707107i − 9.79999i − 1.00000i 8.08080 + 8.08080i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 357.2.k.b ✓ 20
3.b odd 2 1 1071.2.n.b 20
17.c even 4 1 inner 357.2.k.b ✓ 20
17.d even 8 1 6069.2.a.bd 10
17.d even 8 1 6069.2.a.be 10
51.f odd 4 1 1071.2.n.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.k.b ✓ 20 1.a even 1 1 trivial
357.2.k.b ✓ 20 17.c even 4 1 inner
1071.2.n.b 20 3.b odd 2 1
1071.2.n.b 20 51.f odd 4 1
6069.2.a.bd 10 17.d even 8 1
6069.2.a.be 10 17.d even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 32 T_{2}^{18} + 426 T_{2}^{16} + 3072 T_{2}^{14} + 13121 T_{2}^{12} + 34148 T_{2}^{10} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(357, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 32 T^{18} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} - 8 T^{19} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 101687056 \) Copy content Toggle raw display
$13$ \( (T^{10} + 6 T^{9} + \cdots - 91664)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2015993900449 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 527712784 \) Copy content Toggle raw display
$23$ \( T^{20} + 4 T^{19} + \cdots + 234256 \) Copy content Toggle raw display
$29$ \( T^{20} + 8 T^{19} + \cdots + 32444416 \) Copy content Toggle raw display
$31$ \( T^{20} - 8 T^{19} + \cdots + 5456896 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 180162895936 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 69\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{20} + 182 T^{18} + \cdots + 200704 \) Copy content Toggle raw display
$47$ \( (T^{10} + 16 T^{9} + \cdots - 9831296)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 533764670464 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 1849600000000 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 67\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( (T^{10} - 20 T^{9} + \cdots + 4614400)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 114770254495744 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 39382954541056 \) Copy content Toggle raw display
$89$ \( (T^{10} + 32 T^{9} + \cdots - 4681216)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 96348160000 \) Copy content Toggle raw display
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