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Properties

Label 114.4.h.a
Level $114$
Weight $4$
Character orbit 114.h
Analytic conductor $6.726$
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] = [114,4,Mod(65,114)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(114, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("114.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 114.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: \(6.72621774065\)
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: \( x^{20} - 5 x^{19} + 10 x^{18} - 183 x^{17} + 864 x^{16} - 495 x^{15} - 1530 x^{14} + \cdots + 205891132094649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{8} \)
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + (\beta_{3} - \beta_1) q^{3} + 4 \beta_{2} q^{4} - \beta_{8} q^{5} - 2 \beta_{3} q^{6} + ( - \beta_{11} + \beta_1 - 1) q^{7} + 8 q^{8} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + (\beta_{3} - \beta_1) q^{3} + 4 \beta_{2} q^{4} - \beta_{8} q^{5} - 2 \beta_{3} q^{6} + ( - \beta_{11} + \beta_1 - 1) q^{7} + 8 q^{8} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 2) q^{9}+ \cdots + (6 \beta_{19} - 3 \beta_{18} + \cdots + 56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} - 4 q^{3} - 40 q^{4} - 2 q^{6} - 10 q^{7} + 160 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} - 4 q^{3} - 40 q^{4} - 2 q^{6} - 10 q^{7} + 160 q^{8} + 32 q^{9} + 20 q^{12} - 111 q^{13} + 10 q^{14} + 13 q^{15} - 160 q^{16} + 90 q^{17} + 10 q^{18} - 143 q^{19} - 191 q^{21} - 54 q^{22} - 32 q^{24} + 184 q^{25} + 524 q^{27} + 20 q^{28} + 96 q^{29} - 292 q^{30} - 320 q^{32} + 16 q^{33} - 180 q^{34} - 774 q^{35} - 148 q^{36} + 314 q^{38} + 1002 q^{39} - 537 q^{41} - 382 q^{42} + 571 q^{43} + 108 q^{44} + 1516 q^{45} + 126 q^{47} - 16 q^{48} + 558 q^{49} - 736 q^{50} - 1493 q^{51} + 444 q^{52} - 126 q^{53} + 340 q^{54} + 366 q^{55} - 80 q^{56} - 877 q^{57} - 384 q^{58} + 1383 q^{59} + 532 q^{60} + 149 q^{61} + 222 q^{62} + 619 q^{63} + 1280 q^{64} - 3636 q^{65} + 110 q^{66} - 1626 q^{67} - 236 q^{69} + 1548 q^{70} + 1368 q^{71} + 256 q^{72} + 946 q^{73} - 474 q^{74} + 669 q^{75} - 56 q^{76} - 1872 q^{78} - 2109 q^{79} - 340 q^{81} - 1074 q^{82} + 1528 q^{84} + 786 q^{85} + 1142 q^{86} - 816 q^{87} + 1938 q^{89} - 1798 q^{90} - 3459 q^{91} + 663 q^{93} - 2502 q^{95} + 160 q^{96} + 1791 q^{97} - 558 q^{98} - 710 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} + 10 x^{18} - 183 x^{17} + 864 x^{16} - 495 x^{15} - 1530 x^{14} + \cdots + 205891132094649 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9797277781 \nu^{19} - 36981243338 \nu^{18} + 883646227036 \nu^{17} - 7086969056793 \nu^{16} + \cdots - 27\!\cdots\!87 ) / 55\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 444635021 \nu^{19} - 29099016638 \nu^{18} + 196076563810 \nu^{17} + \cdots + 74\!\cdots\!47 ) / 20\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 86838225881 \nu^{19} + 2272745997142 \nu^{18} - 17543295716018 \nu^{17} + \cdots - 40\!\cdots\!43 ) / 55\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 59390335841 \nu^{19} - 687738856552 \nu^{18} + 2484480438212 \nu^{17} + \cdots + 22\!\cdots\!73 ) / 27\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4302544144 \nu^{19} - 122968150787 \nu^{18} + 525923475349 \nu^{17} - 3838763863209 \nu^{16} + \cdots + 45\!\cdots\!07 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{19} - 5 \nu^{18} + 10 \nu^{17} - 183 \nu^{16} + 864 \nu^{15} - 495 \nu^{14} + \cdots - 35586121596606 ) / 2541865828329 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 558507808016 \nu^{19} + 1441158732751 \nu^{18} - 21346474083893 \nu^{17} + \cdots - 11\!\cdots\!89 ) / 12\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 472949773688 \nu^{19} + 821101507766 \nu^{18} - 54822852367 \nu^{17} + \cdots - 33\!\cdots\!90 ) / 64\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1487430265531 \nu^{19} + 2204435819932 \nu^{18} + 6021055032151 \nu^{17} + \cdots - 10\!\cdots\!23 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 278574997 \nu^{19} + 795721006 \nu^{18} + 11337975349 \nu^{17} - 4324573728 \nu^{16} + \cdots - 11\!\cdots\!10 ) / 34\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 66657678464 \nu^{19} - 995624049853 \nu^{18} + 1566194883185 \nu^{17} + \cdots + 14\!\cdots\!75 ) / 61\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 236021255645 \nu^{19} + 32088093518 \nu^{18} - 8635340760124 \nu^{17} + \cdots - 23\!\cdots\!79 ) / 21\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2114519014198 \nu^{19} - 10017584347711 \nu^{18} + 61347435287453 \nu^{17} + \cdots + 25\!\cdots\!95 ) / 12\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2463462692638 \nu^{19} + 9277439631623 \nu^{18} - 21570038995093 \nu^{17} + \cdots + 82\!\cdots\!45 ) / 12\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 8051074521964 \nu^{19} + 14224226894657 \nu^{18} - 151263231836335 \nu^{17} + \cdots - 36\!\cdots\!21 ) / 38\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 8318878188367 \nu^{19} + 35425749276196 \nu^{18} - 399431735416598 \nu^{17} + \cdots - 10\!\cdots\!17 ) / 38\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 9416299579411 \nu^{19} - 41820018902800 \nu^{18} - 192198571240414 \nu^{17} + \cdots + 44\!\cdots\!15 ) / 38\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 736194028630 \nu^{19} - 1567033489739 \nu^{18} + 17666420632465 \nu^{17} + \cdots - 12\!\cdots\!97 ) / 18\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} - 2\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{19} - 4 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{11} - 5 \beta_{9} - \beta_{8} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{19} + 6 \beta_{18} - 6 \beta_{17} + \beta_{16} - 30 \beta_{15} - 4 \beta_{14} - 10 \beta_{12} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 18 \beta_{19} - 18 \beta_{18} + 6 \beta_{17} - 70 \beta_{16} - 56 \beta_{15} + 42 \beta_{14} + \cdots + 55 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 98 \beta_{19} + 138 \beta_{18} + 36 \beta_{17} - 126 \beta_{16} - 4 \beta_{15} - 68 \beta_{14} + \cdots + 9323 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 196 \beta_{19} + 492 \beta_{18} - 834 \beta_{17} - 1556 \beta_{16} - 1222 \beta_{14} - 552 \beta_{13} + \cdots - 47310 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1602 \beta_{19} - 2268 \beta_{18} + 3966 \beta_{17} - 3616 \beta_{16} + 9754 \beta_{15} + \cdots - 115631 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 20492 \beta_{19} + 12936 \beta_{18} - 9144 \beta_{17} - 33588 \beta_{16} + 17294 \beta_{15} + \cdots - 382465 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 191468 \beta_{19} - 21108 \beta_{18} - 116232 \beta_{17} + 151345 \beta_{16} + 57060 \beta_{15} + \cdots + 1716150 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 316764 \beta_{19} - 389016 \beta_{18} + 732048 \beta_{17} + 183404 \beta_{16} + 1137544 \beta_{15} + \cdots - 4523003 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2288924 \beta_{19} + 2984340 \beta_{18} - 2884536 \beta_{17} + 1663884 \beta_{16} + 4925804 \beta_{15} + \cdots - 31459771 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11479916 \beta_{19} - 20404056 \beta_{18} - 8561616 \beta_{17} + 11807704 \beta_{16} + 28706688 \beta_{15} + \cdots - 758863500 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 86466780 \beta_{19} - 2113236 \beta_{18} + 69486360 \beta_{17} - 19313980 \beta_{16} + 111053140 \beta_{15} + \cdots - 627482717 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 210272246 \beta_{19} - 75961824 \beta_{18} - 466675848 \beta_{17} - 291691188 \beta_{16} + \cdots - 8673633955 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 2675843984 \beta_{19} - 2345697894 \beta_{18} + 642999954 \beta_{17} + 9240010993 \beta_{16} + \cdots - 45521246328 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1947418398 \beta_{19} + 7503476958 \beta_{18} + 2905824534 \beta_{17} - 4174353034 \beta_{16} + \cdots - 354802256117 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 51784542134 \beta_{19} - 9502469178 \beta_{18} - 27810369972 \beta_{17} - 76669964802 \beta_{16} + \cdots + 2276458243631 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 371200748528 \beta_{19} - 383407370292 \beta_{18} + 365303008974 \beta_{17} - 277152930116 \beta_{16} + \cdots - 14249962760874 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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
2.45275 + 4.58083i
4.56617 + 2.47993i
−1.19331 + 5.05727i
−2.50060 + 4.55489i
5.00721 − 1.38847i
4.93740 − 1.61930i
−5.19612 − 0.0179453i
1.24390 − 5.04507i
−3.89429 − 3.44013i
−2.92311 − 4.29598i
2.45275 − 4.58083i
4.56617 − 2.47993i
−1.19331 − 5.05727i
−2.50060 − 4.55489i
5.00721 + 1.38847i
4.93740 + 1.61930i
−5.19612 + 0.0179453i
1.24390 + 5.04507i
−3.89429 + 3.44013i
−2.92311 + 4.29598i
−1.00000 + 1.73205i −5.19349 − 0.166266i −2.00000 − 3.46410i 1.43899 + 0.830800i 5.48147 − 8.82912i −19.8418 8.00000 26.9447 + 1.72701i −2.87798 + 1.66160i
65.2 −1.00000 + 1.73205i −4.43077 + 2.71446i −2.00000 − 3.46410i 12.0046 + 6.93085i −0.270808 − 10.3888i 28.4828 8.00000 12.2634 − 24.0543i −24.0092 + 13.8617i
65.3 −1.00000 + 1.73205i −3.78307 − 3.56207i −2.00000 − 3.46410i −16.1113 − 9.30186i 9.95276 − 2.99040i 11.9689 8.00000 1.62327 + 26.9512i 32.2226 − 18.6037i
65.4 −1.00000 + 1.73205i −2.69434 − 4.44303i −2.00000 − 3.46410i 4.92697 + 2.84459i 10.3899 − 0.223711i 0.510228 8.00000 −12.4810 + 23.9421i −9.85394 + 5.68918i
65.5 −1.00000 + 1.73205i −1.30115 + 5.03061i −2.00000 − 3.46410i −13.6046 − 7.85461i −7.41211 − 7.28427i 16.2728 8.00000 −23.6140 − 13.0912i 27.2092 − 15.7092i
65.6 −1.00000 + 1.73205i −1.06635 + 5.08556i −2.00000 − 3.46410i 5.32650 + 3.07526i −7.74210 − 6.93253i −27.9598 8.00000 −24.7258 − 10.8459i −10.6530 + 6.15052i
65.7 −1.00000 + 1.73205i 2.61360 − 4.49100i −2.00000 − 3.46410i −3.59600 − 2.07615i 5.16504 + 9.01789i −27.3337 8.00000 −13.3382 − 23.4754i 7.19199 − 4.15230i
65.8 −1.00000 + 1.73205i 3.74721 + 3.59978i −2.00000 − 3.46410i 1.54776 + 0.893602i −9.98221 + 2.89057i 19.7997 8.00000 1.08314 + 26.9783i −3.09553 + 1.78720i
65.9 −1.00000 + 1.73205i 4.92639 − 1.65249i −2.00000 − 3.46410i −9.82557 − 5.67280i −2.06419 + 10.1852i 5.23979 8.00000 21.5386 − 16.2816i 19.6511 − 11.3456i
65.10 −1.00000 + 1.73205i 5.18198 − 0.383496i −2.00000 − 3.46410i 17.8926 + 10.3303i −4.51775 + 9.35895i −12.1387 8.00000 26.7059 − 3.97454i −35.7852 + 20.6606i
107.1 −1.00000 − 1.73205i −5.19349 + 0.166266i −2.00000 + 3.46410i 1.43899 − 0.830800i 5.48147 + 8.82912i −19.8418 8.00000 26.9447 − 1.72701i −2.87798 − 1.66160i
107.2 −1.00000 − 1.73205i −4.43077 − 2.71446i −2.00000 + 3.46410i 12.0046 − 6.93085i −0.270808 + 10.3888i 28.4828 8.00000 12.2634 + 24.0543i −24.0092 − 13.8617i
107.3 −1.00000 − 1.73205i −3.78307 + 3.56207i −2.00000 + 3.46410i −16.1113 + 9.30186i 9.95276 + 2.99040i 11.9689 8.00000 1.62327 − 26.9512i 32.2226 + 18.6037i
107.4 −1.00000 − 1.73205i −2.69434 + 4.44303i −2.00000 + 3.46410i 4.92697 − 2.84459i 10.3899 + 0.223711i 0.510228 8.00000 −12.4810 − 23.9421i −9.85394 − 5.68918i
107.5 −1.00000 − 1.73205i −1.30115 − 5.03061i −2.00000 + 3.46410i −13.6046 + 7.85461i −7.41211 + 7.28427i 16.2728 8.00000 −23.6140 + 13.0912i 27.2092 + 15.7092i
107.6 −1.00000 − 1.73205i −1.06635 − 5.08556i −2.00000 + 3.46410i 5.32650 − 3.07526i −7.74210 + 6.93253i −27.9598 8.00000 −24.7258 + 10.8459i −10.6530 − 6.15052i
107.7 −1.00000 − 1.73205i 2.61360 + 4.49100i −2.00000 + 3.46410i −3.59600 + 2.07615i 5.16504 − 9.01789i −27.3337 8.00000 −13.3382 + 23.4754i 7.19199 + 4.15230i
107.8 −1.00000 − 1.73205i 3.74721 − 3.59978i −2.00000 + 3.46410i 1.54776 − 0.893602i −9.98221 − 2.89057i 19.7997 8.00000 1.08314 − 26.9783i −3.09553 − 1.78720i
107.9 −1.00000 − 1.73205i 4.92639 + 1.65249i −2.00000 + 3.46410i −9.82557 + 5.67280i −2.06419 − 10.1852i 5.23979 8.00000 21.5386 + 16.2816i 19.6511 + 11.3456i
107.10 −1.00000 − 1.73205i 5.18198 + 0.383496i −2.00000 + 3.46410i 17.8926 − 10.3303i −4.51775 − 9.35895i −12.1387 8.00000 26.7059 + 3.97454i −35.7852 − 20.6606i
\(n\): 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
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.4.h.a ✓ 20
3.b odd 2 1 114.4.h.b yes 20
19.d odd 6 1 114.4.h.b yes 20
57.f even 6 1 inner 114.4.h.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.h.a ✓ 20 1.a even 1 1 trivial
114.4.h.a ✓ 20 57.f even 6 1 inner
114.4.h.b yes 20 3.b odd 2 1
114.4.h.b yes 20 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 717 T_{5}^{18} + 361839 T_{5}^{16} + 317592 T_{5}^{15} - 88709738 T_{5}^{14} + \cdots + 16\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 205891132094649 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( (T^{10} + 5 T^{9} + \cdots + 54053096704)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 79\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 61\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 98\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 73\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 54\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 86\!\cdots\!96 \) Copy content Toggle raw display
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