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Properties

Label 67.2.e.c
Level $67$
Weight $2$
Character orbit 67.e
Analytic conductor $0.535$
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] = [67,2,Mod(9,67)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(67, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("67.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 67.e (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: \(0.534997693543\)
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: \( x^{20} - 7 x^{19} + 39 x^{18} - 148 x^{17} + 492 x^{16} - 1282 x^{15} + 2921 x^{14} - 4316 x^{13} + \cdots + 4489 \) 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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 7 x^{19} + 39 x^{18} - 148 x^{17} + 492 x^{16} - 1282 x^{15} + 2921 x^{14} - 4316 x^{13} + \cdots + 4489 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 46\!\cdots\!60 \nu^{19} + \cdots + 13\!\cdots\!10 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!39 \nu^{19} + \cdots + 12\!\cdots\!12 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!64 \nu^{19} + \cdots - 10\!\cdots\!89 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35\!\cdots\!18 \nu^{19} + \cdots - 45\!\cdots\!74 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53\!\cdots\!99 \nu^{19} + \cdots + 15\!\cdots\!77 ) / 41\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 67\!\cdots\!43 \nu^{19} + \cdots - 14\!\cdots\!22 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 67\!\cdots\!22 \nu^{19} + \cdots + 41\!\cdots\!11 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!67 \nu^{19} + \cdots - 52\!\cdots\!25 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!11 \nu^{19} + \cdots + 20\!\cdots\!59 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 18\!\cdots\!36 \nu^{19} + \cdots - 56\!\cdots\!34 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 19\!\cdots\!03 \nu^{19} + \cdots - 95\!\cdots\!77 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 21\!\cdots\!24 \nu^{19} + \cdots + 17\!\cdots\!99 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 21\!\cdots\!22 \nu^{19} + \cdots + 30\!\cdots\!98 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23\!\cdots\!31 \nu^{19} + \cdots - 52\!\cdots\!45 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 24\!\cdots\!58 \nu^{19} + \cdots + 42\!\cdots\!38 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 65\!\cdots\!88 \nu^{19} + \cdots + 90\!\cdots\!00 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 68\!\cdots\!04 \nu^{19} + \cdots + 12\!\cdots\!28 ) / 28\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 69\!\cdots\!63 \nu^{19} + \cdots + 14\!\cdots\!60 ) / 28\!\cdots\!17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} + \beta_{16} - 4\beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{4} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6 \beta_{19} + \beta_{17} - \beta_{15} - 7 \beta_{14} + 8 \beta_{11} - 9 \beta_{10} - 8 \beta_{8} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{19} + \beta_{18} + 9 \beta_{17} - 7 \beta_{16} + 5 \beta_{15} + 10 \beta_{14} - 16 \beta_{13} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{18} - 2 \beta_{17} + 8 \beta_{16} + 24 \beta_{15} - 8 \beta_{14} + 24 \beta_{13} + 15 \beta_{12} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 45 \beta_{19} - 77 \beta_{18} - 19 \beta_{17} - 104 \beta_{16} - 96 \beta_{15} + 51 \beta_{14} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 325 \beta_{19} - 221 \beta_{18} + 122 \beta_{17} - 430 \beta_{16} - 355 \beta_{15} + 817 \beta_{14} + \cdots - 857 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 309 \beta_{19} + 680 \beta_{18} - 309 \beta_{17} + 1732 \beta_{16} + 585 \beta_{15} - 585 \beta_{14} + \cdots - 490 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1200 \beta_{19} + 2398 \beta_{18} - 2534 \beta_{17} + 4853 \beta_{16} + 1827 \beta_{15} - 5725 \beta_{14} + \cdots + 9052 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2760 \beta_{19} - 4199 \beta_{18} - 1672 \beta_{17} - 14575 \beta_{16} - 7226 \beta_{15} + \cdots - 4199 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7068 \beta_{19} - 12830 \beta_{18} - 24282 \beta_{16} - 24282 \beta_{15} + 19898 \beta_{14} + \cdots - 79273 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 55107 \beta_{19} + 28866 \beta_{18} + 9300 \beta_{17} + 88243 \beta_{16} + 11392 \beta_{15} + \cdots + 108726 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 62564 \beta_{19} + 87418 \beta_{18} + 166296 \beta_{17} + 73420 \beta_{16} + 272576 \beta_{15} + \cdots + 595496 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 197672 \beta_{19} + 60285 \beta_{17} + 706584 \beta_{15} + 290566 \beta_{14} - 146365 \beta_{13} + \cdots - 373004 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 433106 \beta_{19} - 900567 \beta_{18} - 1106079 \beta_{17} - 784612 \beta_{16} - 1733161 \beta_{15} + \cdots - 1999381 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2854959 \beta_{18} + 2854959 \beta_{17} - 8164703 \beta_{16} - 7847972 \beta_{15} + 8164703 \beta_{14} + \cdots - 6288794 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3628744 \beta_{19} + 11812158 \beta_{18} + 7961140 \beta_{17} + 20664705 \beta_{16} + 19773298 \beta_{15} + \cdots + 285470 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 20352800 \beta_{19} + 42193117 \beta_{18} - 41404666 \beta_{17} + 117996676 \beta_{16} + \cdots + 165405588 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 42150258 \beta_{19} - 152375454 \beta_{18} - 42150258 \beta_{17} - 336873110 \beta_{16} + \cdots - 62459923 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−1.27213 − 2.78557i
1.07319 + 2.34996i
−0.625078 + 0.721379i
1.66916 − 1.92631i
−1.27213 + 2.78557i
1.07319 − 2.34996i
−0.165981 + 0.106669i
2.28033 − 1.46548i
−0.625078 − 0.721379i
1.66916 + 1.92631i
1.58638 − 0.465803i
−1.35948 + 0.399179i
−0.127556 + 0.887172i
0.441163 − 3.06836i
1.58638 + 0.465803i
−1.35948 − 0.399179i
−0.127556 − 0.887172i
0.441163 + 3.06836i
−0.165981 − 0.106669i
2.28033 + 1.46548i
−0.239446 − 0.153882i −2.47877 − 0.727832i −0.797176 − 1.74557i 1.21919 − 1.40702i 0.481530 + 0.555715i −1.14774 − 0.737606i −0.158746 + 1.10411i 3.09080 + 1.98634i −0.508444 + 0.149293i
9.2 −0.239446 − 0.153882i 2.93826 + 0.862752i −0.797176 − 1.74557i −1.85253 + 2.13793i −0.570792 − 0.658729i −3.42222 − 2.19933i −0.158746 + 1.10411i 5.36529 + 3.44806i 0.772569 − 0.226847i
14.1 −0.797176 − 1.74557i −2.14425 − 1.37802i −1.10181 + 1.27155i 0.201991 + 1.40488i −0.696097 + 4.84146i −0.918948 − 2.01222i −0.584585 − 0.171650i 1.45260 + 3.18076i 2.29130 − 1.47253i
14.2 −0.797176 − 1.74557i 0.802994 + 0.516053i −1.10181 + 1.27155i −0.451016 − 3.13689i 0.260680 − 1.81307i 1.40141 + 3.06866i −0.584585 − 0.171650i −0.867756 − 1.90012i −5.11612 + 3.28793i
15.1 −0.239446 + 0.153882i −2.47877 + 0.727832i −0.797176 + 1.74557i 1.21919 + 1.40702i 0.481530 − 0.555715i −1.14774 + 0.737606i −0.158746 − 1.10411i 3.09080 − 1.98634i −0.508444 − 0.149293i
15.2 −0.239446 + 0.153882i 2.93826 − 0.862752i −0.797176 + 1.74557i −1.85253 − 2.13793i −0.570792 + 0.658729i −3.42222 + 2.19933i −0.158746 − 1.10411i 5.36529 − 3.44806i 0.772569 + 0.226847i
22.1 1.25667 − 0.368991i −0.385764 − 2.68305i −0.239446 + 0.153882i −1.35049 + 2.95717i −1.47480 − 3.22936i 4.00914 − 1.17719i −1.95949 + 2.26138i −4.17146 + 1.22485i −0.605953 + 4.21450i
22.2 1.25667 − 0.368991i 0.0280790 + 0.195293i −0.239446 + 0.153882i 0.681980 − 1.49333i 0.107348 + 0.235058i −3.00345 + 0.881891i −1.95949 + 2.26138i 2.84113 − 0.834230i 0.305998 − 2.12826i
24.1 −0.797176 + 1.74557i −2.14425 + 1.37802i −1.10181 − 1.27155i 0.201991 − 1.40488i −0.696097 − 4.84146i −0.918948 + 2.01222i −0.584585 + 0.171650i 1.45260 − 3.18076i 2.29130 + 1.47253i
24.2 −0.797176 + 1.74557i 0.802994 − 0.516053i −1.10181 − 1.27155i −0.451016 + 3.13689i 0.260680 + 1.81307i 1.40141 − 3.06866i −0.584585 + 0.171650i −0.867756 + 1.90012i −5.11612 − 3.28793i
25.1 −0.118239 − 0.822373i −0.927853 + 1.07080i 1.25667 − 0.368991i 1.58839 − 1.02080i 0.990306 + 0.636431i −0.0441131 − 0.306813i −1.14231 − 2.50132i 0.141244 + 0.982373i −1.02729 − 1.18555i
25.2 −0.118239 − 0.822373i 1.08271 − 1.24952i 1.25667 − 0.368991i −3.36803 + 2.16450i −1.15559 − 0.742653i 0.0592135 + 0.411839i −1.14231 − 2.50132i 0.0379173 + 0.263721i 2.17826 + 2.51385i
40.1 −1.10181 − 1.27155i −1.28775 + 2.81978i −0.118239 + 0.822373i 2.46094 + 0.722597i 5.00435 − 1.46941i 2.41674 + 2.78906i −1.65486 + 1.06351i −4.32827 − 4.99509i −1.79266 − 3.92538i
40.2 −1.10181 − 1.27155i 0.372334 − 0.815299i −0.118239 + 0.822373i 1.36958 + 0.402144i −1.44694 + 0.424859i −3.35003 − 3.86614i −1.65486 + 1.06351i 1.43850 + 1.66012i −0.997662 − 2.18457i
59.1 −0.118239 + 0.822373i −0.927853 − 1.07080i 1.25667 + 0.368991i 1.58839 + 1.02080i 0.990306 − 0.636431i −0.0441131 + 0.306813i −1.14231 + 2.50132i 0.141244 − 0.982373i −1.02729 + 1.18555i
59.2 −0.118239 + 0.822373i 1.08271 + 1.24952i 1.25667 + 0.368991i −3.36803 − 2.16450i −1.15559 + 0.742653i 0.0592135 − 0.411839i −1.14231 + 2.50132i 0.0379173 − 0.263721i 2.17826 − 2.51385i
62.1 −1.10181 + 1.27155i −1.28775 − 2.81978i −0.118239 − 0.822373i 2.46094 − 0.722597i 5.00435 + 1.46941i 2.41674 − 2.78906i −1.65486 − 1.06351i −4.32827 + 4.99509i −1.79266 + 3.92538i
62.2 −1.10181 + 1.27155i 0.372334 + 0.815299i −0.118239 − 0.822373i 1.36958 − 0.402144i −1.44694 − 0.424859i −3.35003 + 3.86614i −1.65486 − 1.06351i 1.43850 − 1.66012i −0.997662 + 2.18457i
64.1 1.25667 + 0.368991i −0.385764 + 2.68305i −0.239446 − 0.153882i −1.35049 − 2.95717i −1.47480 + 3.22936i 4.00914 + 1.17719i −1.95949 − 2.26138i −4.17146 − 1.22485i −0.605953 − 4.21450i
64.2 1.25667 + 0.368991i 0.0280790 − 0.195293i −0.239446 − 0.153882i 0.681980 + 1.49333i 0.107348 − 0.235058i −3.00345 − 0.881891i −1.95949 − 2.26138i 2.84113 + 0.834230i 0.305998 + 2.12826i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 67.2.e.c ✓ 20
3.b odd 2 1 603.2.u.c 20
67.e even 11 1 inner 67.2.e.c ✓ 20
67.e even 11 1 4489.2.a.l 10
67.f odd 22 1 4489.2.a.m 10
201.k odd 22 1 603.2.u.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
67.2.e.c ✓ 20 1.a even 1 1 trivial
67.2.e.c ✓ 20 67.e even 11 1 inner
603.2.u.c 20 3.b odd 2 1
603.2.u.c 20 201.k odd 22 1
4489.2.a.l 10 67.e even 11 1
4489.2.a.m 10 67.f odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 2T_{2}^{9} + 4T_{2}^{8} - 3T_{2}^{7} - 6T_{2}^{6} - 12T_{2}^{5} + 9T_{2}^{4} + 7T_{2}^{3} + 14T_{2}^{2} + 6T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(67, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{19} + \cdots + 4489 \) Copy content Toggle raw display
$5$ \( T^{20} - T^{19} + \cdots + 12243001 \) Copy content Toggle raw display
$7$ \( T^{20} + 8 T^{19} + \cdots + 1739761 \) Copy content Toggle raw display
$11$ \( T^{20} + 5 T^{18} + \cdots + 4190209 \) Copy content Toggle raw display
$13$ \( T^{20} - 12 T^{19} + \cdots + 4489 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 876811321 \) Copy content Toggle raw display
$19$ \( T^{20} - 4 T^{19} + \cdots + 23319241 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 1098458449 \) Copy content Toggle raw display
$29$ \( (T^{10} + 3 T^{9} + \cdots + 737)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 867288026089 \) Copy content Toggle raw display
$37$ \( (T^{10} - 12 T^{9} + \cdots - 279553)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 132185354901721 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 638971773407881 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 2571253941169 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 51457746649 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 279020843142889 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 76356191669209 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 18\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 1590305111329 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 45\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 25849243729 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 41\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( (T^{10} + 7 T^{9} + \cdots - 294045257)^{2} \) Copy content Toggle raw display
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