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Properties

Label 114.4.h.b
Level $114$
Weight $4$
Character orbit 114.h
Analytic conductor $6.726$
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] = [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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} - \beta_{5} q^{3} + ( - 4 \beta_1 - 4) q^{4} - \beta_{6} q^{5} - 2 \beta_{4} q^{6} + (\beta_{7} - \beta_{5} + \beta_{4} - 1) q^{7} - 8 q^{8} + (\beta_{15} + \beta_{8} - \beta_{6} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} - \beta_{5} q^{3} + ( - 4 \beta_1 - 4) q^{4} - \beta_{6} q^{5} - 2 \beta_{4} q^{6} + (\beta_{7} - \beta_{5} + \beta_{4} - 1) q^{7} - 8 q^{8} + (\beta_{15} + \beta_{8} - \beta_{6} + \cdots - 1) q^{9}+ \cdots + ( - 2 \beta_{19} + 7 \beta_{18} + \cdots - 172) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} + q^{3} - 40 q^{4} - 8 q^{6} - 10 q^{7} - 160 q^{8} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} + q^{3} - 40 q^{4} - 8 q^{6} - 10 q^{7} - 160 q^{8} - 37 q^{9} - 20 q^{12} - 111 q^{13} - 10 q^{14} - 133 q^{15} - 160 q^{16} - 90 q^{17} - 10 q^{18} - 143 q^{19} + 191 q^{21} - 54 q^{22} - 8 q^{24} + 184 q^{25} - 524 q^{27} + 20 q^{28} - 96 q^{29} - 292 q^{30} + 320 q^{32} - 55 q^{33} - 180 q^{34} + 774 q^{35} + 128 q^{36} - 314 q^{38} + 1002 q^{39} + 537 q^{41} - 382 q^{42} + 571 q^{43} - 108 q^{44} + 1516 q^{45} - 126 q^{47} + 64 q^{48} + 558 q^{49} + 736 q^{50} - 757 q^{51} + 444 q^{52} + 126 q^{53} - 1388 q^{54} + 366 q^{55} + 80 q^{56} - 281 q^{57} - 384 q^{58} - 1383 q^{59} - 52 q^{60} + 149 q^{61} - 222 q^{62} + 1153 q^{63} + 1280 q^{64} + 3636 q^{65} + 32 q^{66} - 1626 q^{67} + 236 q^{69} + 1548 q^{70} - 1368 q^{71} + 296 q^{72} + 946 q^{73} + 474 q^{74} - 669 q^{75} - 56 q^{76} + 132 q^{78} - 2109 q^{79} + 311 q^{81} - 1074 q^{82} - 1528 q^{84} + 786 q^{85} - 1142 q^{86} - 816 q^{87} - 1938 q^{89} + 1234 q^{90} - 3459 q^{91} - 705 q^{93} + 2502 q^{95} + 160 q^{96} + 1791 q^{97} + 558 q^{98} - 2285 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 9797277781 \nu^{19} + 36981243338 \nu^{18} - 883646227036 \nu^{17} + \cdots - 27\!\cdots\!29 ) / 55\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13109079460 \nu^{19} + 1197000795508 \nu^{18} - 8382923947823 \nu^{17} + \cdots - 31\!\cdots\!18 ) / 27\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13109079460 \nu^{19} - 1197000795508 \nu^{18} + 8382923947823 \nu^{17} + \cdots + 28\!\cdots\!10 ) / 27\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36222620401 \nu^{19} - 83413398082 \nu^{18} + 636267366116 \nu^{17} + \cdots + 41\!\cdots\!19 ) / 55\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12109265989 \nu^{19} - 148721829974 \nu^{18} + 453923849932 \nu^{17} + \cdots + 46\!\cdots\!13 ) / 18\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 558507808016 \nu^{19} - 1441158732751 \nu^{18} + 21346474083893 \nu^{17} + \cdots + 11\!\cdots\!89 ) / 12\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2098933069 \nu^{19} - 9202698574 \nu^{18} - 97959359101 \nu^{17} - 35787104880 \nu^{16} + \cdots - 50\!\cdots\!50 ) / 31\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 472949773688 \nu^{19} - 821101507766 \nu^{18} + 54822852367 \nu^{17} + \cdots + 33\!\cdots\!90 ) / 64\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 347730504281 \nu^{19} - 2266156345111 \nu^{18} + 18266105759783 \nu^{17} + \cdots + 32\!\cdots\!22 ) / 43\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 828297783950 \nu^{19} + 1696356838748 \nu^{18} - 3870989911804 \nu^{17} + \cdots - 51\!\cdots\!64 ) / 96\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66657678464 \nu^{19} + 995624049853 \nu^{18} - 1566194883185 \nu^{17} + \cdots - 14\!\cdots\!75 ) / 61\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2208952462724 \nu^{19} + 2169117337342 \nu^{18} + 61415767342477 \nu^{17} + \cdots + 21\!\cdots\!86 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 242376286837 \nu^{19} + 1699266247148 \nu^{18} - 10792763249908 \nu^{17} + \cdots - 12\!\cdots\!57 ) / 18\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 7559910401752 \nu^{19} - 32748139252877 \nu^{18} + 97314715982557 \nu^{17} + \cdots - 26\!\cdots\!85 ) / 38\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2711948927965 \nu^{19} - 4143854988701 \nu^{18} + 42730344562543 \nu^{17} + \cdots + 18\!\cdots\!44 ) / 12\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1510928438035 \nu^{19} - 4000790559319 \nu^{18} + 56929468522154 \nu^{17} + \cdots + 16\!\cdots\!01 ) / 64\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 4757527214960 \nu^{19} + 28392684055679 \nu^{18} + 48954917059799 \nu^{17} + \cdots - 54\!\cdots\!89 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 9632005032844 \nu^{19} + 34045825068301 \nu^{18} - 24497734857677 \nu^{17} + \cdots - 78\!\cdots\!95 ) / 38\!\cdots\!12 \) 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 \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{10} + 3\beta_{8} - 3\beta_{5} - \beta_{2} + 7\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{19} + 6 \beta_{18} + 12 \beta_{14} + 3 \beta_{13} + 6 \beta_{12} + 3 \beta_{9} + 12 \beta_{8} + \cdots + 35 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{19} + 12 \beta_{18} - 18 \beta_{17} + 18 \beta_{16} - 3 \beta_{15} + 90 \beta_{14} + 18 \beta_{13} + \cdots + 67 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 54 \beta_{19} - 126 \beta_{18} + 54 \beta_{17} - 18 \beta_{16} + 210 \beta_{15} + 168 \beta_{14} + \cdots - 1662 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 294 \beta_{19} + 204 \beta_{18} - 414 \beta_{17} - 108 \beta_{16} + 378 \beta_{15} + 12 \beta_{14} + \cdots - 5077 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 588 \beta_{19} + 3666 \beta_{18} - 1476 \beta_{17} + 2502 \beta_{16} + 4668 \beta_{15} - 4086 \beta_{13} + \cdots - 120659 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4806 \beta_{19} - 7632 \beta_{18} + 6804 \beta_{17} - 11898 \beta_{16} + 10848 \beta_{15} + \cdots - 277827 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 61476 \beta_{19} + 51504 \beta_{18} - 38808 \beta_{17} + 27432 \beta_{16} + 100764 \beta_{15} + \cdots - 4055455 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 574404 \beta_{19} - 86460 \beta_{18} + 63324 \beta_{17} + 348696 \beta_{16} - 454035 \beta_{15} + \cdots - 5927675 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 950292 \beta_{19} - 3115224 \beta_{18} + 1167048 \beta_{17} - 2196144 \beta_{16} - 550212 \beta_{15} + \cdots - 66848172 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 6866772 \beta_{19} + 2447484 \beta_{18} - 8953020 \beta_{17} + 8653608 \beta_{16} - 4991652 \beta_{15} + \cdots + 49188923 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 34439748 \beta_{19} + 33056256 \beta_{18} + 61212168 \beta_{17} + 25684848 \beta_{16} + \cdots - 67284167 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 259400340 \beta_{19} - 120950892 \beta_{18} + 6339708 \beta_{17} - 208459080 \beta_{16} + \cdots + 7356352431 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 630816738 \beta_{19} - 621089058 \beta_{18} + 227885472 \beta_{17} + 1400027544 \beta_{16} + \cdots + 12416948027 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 8027531952 \beta_{19} - 8233202616 \beta_{18} + 7037093682 \beta_{17} - 1928999862 \beta_{16} + \cdots - 75533806877 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 5842255194 \beta_{19} + 18258717042 \beta_{18} - 22510430874 \beta_{17} - 8717473602 \beta_{16} + \cdots - 85353494226 ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 155353626402 \beta_{19} + 52168975008 \beta_{18} + 28507407534 \beta_{17} + 83431109916 \beta_{16} + \cdots + 16030394124515 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 1113602245584 \beta_{19} - 880922424966 \beta_{18} + 1150222110876 \beta_{17} - 1095909026922 \beta_{16} + \cdots - 3604470006011 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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
−2.50060 + 4.55489i
−1.19331 + 5.05727i
2.45275 + 4.58083i
−5.19612 − 0.0179453i
4.56617 + 2.47993i
−3.89429 − 3.44013i
−2.92311 − 4.29598i
5.00721 − 1.38847i
4.93740 − 1.61930i
1.24390 − 5.04507i
−2.50060 − 4.55489i
−1.19331 − 5.05727i
2.45275 − 4.58083i
−5.19612 + 0.0179453i
4.56617 − 2.47993i
−3.89429 + 3.44013i
−2.92311 + 4.29598i
5.00721 + 1.38847i
4.93740 + 1.61930i
1.24390 + 5.04507i
1.00000 − 1.73205i −5.19495 − 0.111856i −2.00000 − 3.46410i −4.92697 − 2.84459i −5.38869 + 8.88606i 0.510228 −8.00000 26.9750 + 1.16217i −9.85394 + 5.68918i
65.2 1.00000 − 1.73205i −4.97638 − 1.49520i −2.00000 − 3.46410i 16.1113 + 9.30186i −7.56614 + 7.12415i 11.9689 −8.00000 22.5288 + 14.8814i 32.2226 − 18.6037i
65.3 1.00000 − 1.73205i −2.74074 − 4.41456i −2.00000 − 3.46410i −1.43899 − 0.830800i −10.3870 + 0.332533i −19.8418 −8.00000 −11.9767 + 24.1983i −2.87798 + 1.66160i
65.4 1.00000 − 1.73205i −2.58252 + 4.50895i −2.00000 − 3.46410i 3.59600 + 2.07615i 5.22720 + 8.98200i −27.3337 −8.00000 −13.6612 − 23.2889i 7.19199 − 4.15230i
65.5 1.00000 − 1.73205i 0.135404 − 5.19439i −2.00000 − 3.46410i −12.0046 − 6.93085i −8.86154 − 5.42891i 28.4828 −8.00000 −26.9633 − 1.40668i −24.0092 + 13.8617i
65.6 1.00000 − 1.73205i 1.03210 + 5.09262i −2.00000 − 3.46410i 9.82557 + 5.67280i 9.85277 + 3.30498i 5.23979 −8.00000 −24.8696 + 10.5122i 19.6511 − 11.3456i
65.7 1.00000 − 1.73205i 2.25887 + 4.67948i −2.00000 − 3.46410i −17.8926 − 10.3303i 10.3640 + 0.766992i −12.1387 −8.00000 −16.7950 + 21.1407i −35.7852 + 20.6606i
65.8 1.00000 − 1.73205i 3.70606 − 3.64214i −2.00000 − 3.46410i 13.6046 + 7.85461i −2.60231 − 10.0612i 16.2728 −8.00000 0.469703 − 26.9959i 27.2092 − 15.7092i
65.9 1.00000 − 1.73205i 3.87105 − 3.46626i −2.00000 − 3.46410i −5.32650 − 3.07526i −2.13269 − 10.1711i −27.9598 −8.00000 2.97005 − 26.8361i −10.6530 + 6.15052i
65.10 1.00000 − 1.73205i 4.99111 + 1.44529i −2.00000 − 3.46410i −1.54776 − 0.893602i 7.49442 − 7.19956i 19.7997 −8.00000 22.8223 + 14.4272i −3.09553 + 1.78720i
107.1 1.00000 + 1.73205i −5.19495 + 0.111856i −2.00000 + 3.46410i −4.92697 + 2.84459i −5.38869 − 8.88606i 0.510228 −8.00000 26.9750 − 1.16217i −9.85394 − 5.68918i
107.2 1.00000 + 1.73205i −4.97638 + 1.49520i −2.00000 + 3.46410i 16.1113 − 9.30186i −7.56614 − 7.12415i 11.9689 −8.00000 22.5288 − 14.8814i 32.2226 + 18.6037i
107.3 1.00000 + 1.73205i −2.74074 + 4.41456i −2.00000 + 3.46410i −1.43899 + 0.830800i −10.3870 − 0.332533i −19.8418 −8.00000 −11.9767 − 24.1983i −2.87798 − 1.66160i
107.4 1.00000 + 1.73205i −2.58252 − 4.50895i −2.00000 + 3.46410i 3.59600 − 2.07615i 5.22720 − 8.98200i −27.3337 −8.00000 −13.6612 + 23.2889i 7.19199 + 4.15230i
107.5 1.00000 + 1.73205i 0.135404 + 5.19439i −2.00000 + 3.46410i −12.0046 + 6.93085i −8.86154 + 5.42891i 28.4828 −8.00000 −26.9633 + 1.40668i −24.0092 − 13.8617i
107.6 1.00000 + 1.73205i 1.03210 − 5.09262i −2.00000 + 3.46410i 9.82557 − 5.67280i 9.85277 − 3.30498i 5.23979 −8.00000 −24.8696 − 10.5122i 19.6511 + 11.3456i
107.7 1.00000 + 1.73205i 2.25887 − 4.67948i −2.00000 + 3.46410i −17.8926 + 10.3303i 10.3640 − 0.766992i −12.1387 −8.00000 −16.7950 − 21.1407i −35.7852 − 20.6606i
107.8 1.00000 + 1.73205i 3.70606 + 3.64214i −2.00000 + 3.46410i 13.6046 − 7.85461i −2.60231 + 10.0612i 16.2728 −8.00000 0.469703 + 26.9959i 27.2092 + 15.7092i
107.9 1.00000 + 1.73205i 3.87105 + 3.46626i −2.00000 + 3.46410i −5.32650 + 3.07526i −2.13269 + 10.1711i −27.9598 −8.00000 2.97005 + 26.8361i −10.6530 − 6.15052i
107.10 1.00000 + 1.73205i 4.99111 − 1.44529i −2.00000 + 3.46410i −1.54776 + 0.893602i 7.49442 + 7.19956i 19.7997 −8.00000 22.8223 − 14.4272i −3.09553 − 1.78720i
\(n\): 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.b yes 20
3.b odd 2 1 114.4.h.a ✓ 20
19.d odd 6 1 114.4.h.a ✓ 20
57.f even 6 1 inner 114.4.h.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.h.a ✓ 20 3.b odd 2 1
114.4.h.a ✓ 20 19.d odd 6 1
114.4.h.b yes 20 1.a even 1 1 trivial
114.4.h.b yes 20 57.f even 6 1 inner

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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