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Properties

Label 168.4.c.b
Level $168$
Weight $4$
Character orbit 168.c
Analytic conductor $9.912$
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] = [168,4,Mod(85,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.85");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.c (of order \(2\), degree \(1\), 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: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 5 x^{18} - 26 x^{17} + 122 x^{16} + 124 x^{15} - 276 x^{14} - 1376 x^{13} + \cdots + 1073741824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{4} - 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + \beta_{6} q^{6} - 7 q^{7} + (\beta_{14} + 2 \beta_{3} - \beta_{2} + 1) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{4} - 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + \beta_{6} q^{6} - 7 q^{7} + (\beta_{14} + 2 \beta_{3} - \beta_{2} + 1) q^{8} - 9 q^{9} + (\beta_{16} + \beta_{2} - \beta_1 - 4) q^{10} + (\beta_{11} - \beta_{6} + \beta_{4} + \cdots + 1) q^{11}+ \cdots + ( - 9 \beta_{11} + 9 \beta_{6} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} - 14 q^{4} + 6 q^{6} - 140 q^{7} + 10 q^{8} - 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{2} - 14 q^{4} + 6 q^{6} - 140 q^{7} + 10 q^{8} - 180 q^{9} - 84 q^{10} + 12 q^{12} - 28 q^{14} + 60 q^{15} - 134 q^{16} - 52 q^{17} - 36 q^{18} - 8 q^{20} + 8 q^{22} - 244 q^{23} - 348 q^{24} - 844 q^{25} - 332 q^{26} + 98 q^{28} + 60 q^{30} + 264 q^{31} - 46 q^{32} + 396 q^{33} - 612 q^{34} + 126 q^{36} + 884 q^{38} - 312 q^{39} - 964 q^{40} - 236 q^{41} - 42 q^{42} + 576 q^{44} + 1356 q^{46} - 432 q^{48} + 980 q^{49} - 2456 q^{50} + 1492 q^{52} - 54 q^{54} + 72 q^{55} - 70 q^{56} - 912 q^{57} - 1404 q^{58} + 684 q^{60} + 4964 q^{62} + 1260 q^{63} - 1670 q^{64} + 1744 q^{65} + 792 q^{66} + 3408 q^{68} + 588 q^{70} - 636 q^{71} - 90 q^{72} - 2784 q^{73} - 4916 q^{74} + 2664 q^{76} - 888 q^{78} + 2872 q^{79} - 1764 q^{80} + 1620 q^{81} + 2628 q^{82} - 84 q^{84} + 3328 q^{86} - 2348 q^{88} + 220 q^{89} + 756 q^{90} + 2580 q^{92} - 1704 q^{94} - 1240 q^{95} - 564 q^{96} + 4400 q^{97} + 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} - 5 x^{18} - 26 x^{17} + 122 x^{16} + 124 x^{15} - 276 x^{14} - 1376 x^{13} + \cdots + 1073741824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1837527 \nu^{19} - 8686482 \nu^{18} - 118019755 \nu^{17} + 335103006 \nu^{16} + \cdots + 67\!\cdots\!12 ) / 176305521819648 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14856601 \nu^{19} + 12558618 \nu^{18} - 216693429 \nu^{17} - 545439942 \nu^{16} + \cdots - 47\!\cdots\!96 ) / 12\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10567955 \nu^{19} + 35602606 \nu^{18} + 39792079 \nu^{17} + 378375750 \nu^{16} + \cdots + 39\!\cdots\!56 ) / 822759101825024 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{19} - 2 \nu^{18} + 13 \nu^{17} + 46 \nu^{16} - 18 \nu^{15} - 612 \nu^{14} + \cdots + 536870912 ) / 67108864 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9567315 \nu^{19} + 156208352 \nu^{18} - 2247803029 \nu^{17} + 2602794956 \nu^{16} + \cdots + 43\!\cdots\!80 ) / 617069326368768 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3 \nu^{19} + 6 \nu^{18} + 15 \nu^{17} + 78 \nu^{16} - 366 \nu^{15} - 372 \nu^{14} + \cdots + 805306368 ) / 134217728 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 82503653 \nu^{19} + 207441762 \nu^{18} - 339815159 \nu^{17} - 1191296054 \nu^{16} + \cdots - 46\!\cdots\!68 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 105970085 \nu^{19} + 210974538 \nu^{18} - 183339911 \nu^{17} - 891995582 \nu^{16} + \cdots - 33\!\cdots\!16 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10389055 \nu^{19} + 5661738 \nu^{18} - 28248893 \nu^{17} - 168567734 \nu^{16} + \cdots - 54\!\cdots\!16 ) / 176305521819648 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1077439 \nu^{19} + 2348199 \nu^{18} - 2032013 \nu^{17} - 88075663 \nu^{16} + \cdots - 12\!\cdots\!16 ) / 14692126818304 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23184885 \nu^{19} + 69189234 \nu^{18} - 479828231 \nu^{17} + 1783589306 \nu^{16} + \cdots + 23\!\cdots\!92 ) / 189867485036544 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 303123813 \nu^{19} + 2011961542 \nu^{18} - 3533300825 \nu^{17} - 5791691154 \nu^{16} + \cdots + 79\!\cdots\!84 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 53068489 \nu^{19} + 58108744 \nu^{18} + 1093370729 \nu^{17} - 1022971060 \nu^{16} + \cdots - 30\!\cdots\!64 ) / 308534663184384 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 107825611 \nu^{19} + 73278010 \nu^{18} + 295701311 \nu^{17} + 2875637962 \nu^{16} + \cdots + 24\!\cdots\!76 ) / 617069326368768 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21858805 \nu^{19} - 90318698 \nu^{18} + 244019089 \nu^{17} + 992703686 \nu^{16} + \cdots + 89\!\cdots\!36 ) / 88152760909824 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 85242403 \nu^{19} - 293667126 \nu^{18} + 597699503 \nu^{17} + 3619487658 \nu^{16} + \cdots + 39\!\cdots\!68 ) / 308534663184384 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 972557981 \nu^{19} - 2643722986 \nu^{18} - 8246150065 \nu^{17} + 5003108990 \nu^{16} + \cdots + 15\!\cdots\!60 ) / 24\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 2078083 \nu^{19} - 1786366 \nu^{18} - 2884447 \nu^{17} - 45909238 \nu^{16} + \cdots - 419323025817600 ) / 3791516598272 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 1773294843 \nu^{19} + 3998578886 \nu^{18} - 3088836153 \nu^{17} + 29426172014 \nu^{16} + \cdots + 25\!\cdots\!68 ) / 24\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + 2\beta_{7} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 4 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - 2\beta_{8} + 2\beta_{6} - 2\beta_{4} - \beta_{2} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{19} - 8 \beta_{18} - 4 \beta_{17} - 12 \beta_{16} + 12 \beta_{15} - 12 \beta_{14} + \cdots + 138 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{19} + 8 \beta_{18} - 2 \beta_{17} - 4 \beta_{16} + 12 \beta_{15} - 4 \beta_{13} + 2 \beta_{12} + \cdots - 50 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 28 \beta_{19} - 48 \beta_{18} - 4 \beta_{17} - 24 \beta_{16} + 84 \beta_{14} - 64 \beta_{13} + \cdots - 398 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6 \beta_{19} + 8 \beta_{18} - 18 \beta_{17} - 52 \beta_{16} + 60 \beta_{15} - 24 \beta_{14} - 20 \beta_{13} + \cdots + 766 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 116 \beta_{19} + 128 \beta_{18} - 4 \beta_{17} + 48 \beta_{16} + 72 \beta_{15} - 228 \beta_{14} + \cdots - 7922 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 222 \beta_{19} - 168 \beta_{18} + 90 \beta_{17} + 36 \beta_{16} - 236 \beta_{15} + 888 \beta_{14} + \cdots + 10754 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3636 \beta_{19} + 672 \beta_{18} + 2700 \beta_{17} - 576 \beta_{16} + 1704 \beta_{15} - 5868 \beta_{14} + \cdots + 17106 ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1670 \beta_{19} + 264 \beta_{18} + 1294 \beta_{17} + 1964 \beta_{16} + 1052 \beta_{15} - 2376 \beta_{14} + \cdots - 6562 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18732 \beta_{19} - 31680 \beta_{18} - 1092 \beta_{17} + 6480 \beta_{16} + 1224 \beta_{15} + \cdots - 159890 ) / 24 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 15618 \beta_{19} + 152 \beta_{18} + 4922 \beta_{17} - 13180 \beta_{16} + 25268 \beta_{15} + \cdots - 730414 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 151652 \beta_{19} + 21280 \beta_{18} + 192380 \beta_{17} - 5376 \beta_{16} + 125352 \beta_{15} + \cdots - 1630158 ) / 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 147178 \beta_{19} + 26696 \beta_{18} - 124002 \beta_{17} + 41612 \beta_{16} - 18116 \beta_{15} + \cdots + 302558 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1134692 \beta_{19} + 664320 \beta_{18} + 1017932 \beta_{17} + 669168 \beta_{16} - 2178648 \beta_{15} + \cdots - 58938578 ) / 24 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 993710 \beta_{19} - 459304 \beta_{18} + 452106 \beta_{17} - 687644 \beta_{16} - 273068 \beta_{15} + \cdots - 34739150 ) / 8 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 5020436 \beta_{19} + 14015456 \beta_{18} - 3439060 \beta_{17} + 6513312 \beta_{16} - 482424 \beta_{15} + \cdots + 59663026 ) / 24 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 4221126 \beta_{19} - 3660792 \beta_{18} + 5971918 \beta_{17} + 3773356 \beta_{16} - 14507748 \beta_{15} + \cdots - 343906818 ) / 8 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 5451828 \beta_{19} + 21509184 \beta_{18} + 66480924 \beta_{17} + 70694736 \beta_{16} + \cdots - 1939030098 ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1
−0.878338 − 2.68859i
−0.878338 + 2.68859i
1.92113 + 2.07587i
1.92113 − 2.07587i
−2.07092 − 1.92647i
−2.07092 + 1.92647i
2.78865 + 0.472666i
2.78865 − 0.472666i
−2.80433 − 0.368457i
−2.80433 + 0.368457i
2.75494 − 0.640567i
2.75494 + 0.640567i
2.32756 − 1.60701i
2.32756 + 1.60701i
−1.78337 + 2.19536i
−1.78337 − 2.19536i
−1.66031 + 2.28984i
−1.66031 − 2.28984i
0.404977 − 2.79928i
0.404977 + 2.79928i
−2.68859 − 0.878338i − 3.00000i 6.45704 + 4.72298i 8.95699i −2.63501 + 8.06577i −7.00000 −13.2120 − 18.3696i −9.00000 7.86727 − 24.0817i
85.2 −2.68859 + 0.878338i 3.00000i 6.45704 − 4.72298i − 8.95699i −2.63501 − 8.06577i −7.00000 −13.2120 + 18.3696i −9.00000 7.86727 + 24.0817i
85.3 −2.07587 − 1.92113i 3.00000i 0.618484 + 7.97606i − 6.44878i 5.76340 − 6.22761i −7.00000 14.0392 − 17.7455i −9.00000 −12.3890 + 13.3868i
85.4 −2.07587 + 1.92113i − 3.00000i 0.618484 − 7.97606i 6.44878i 5.76340 + 6.22761i −7.00000 14.0392 + 17.7455i −9.00000 −12.3890 − 13.3868i
85.5 −1.92647 − 2.07092i − 3.00000i −0.577404 + 7.97914i − 14.7392i −6.21275 + 5.77942i −7.00000 17.6365 − 14.1758i −9.00000 −30.5238 + 28.3948i
85.6 −1.92647 + 2.07092i 3.00000i −0.577404 − 7.97914i 14.7392i −6.21275 − 5.77942i −7.00000 17.6365 + 14.1758i −9.00000 −30.5238 − 28.3948i
85.7 −0.472666 − 2.78865i 3.00000i −7.55317 + 2.63620i 8.98099i 8.36596 − 1.41800i −7.00000 10.9216 + 19.8171i −9.00000 25.0449 − 4.24501i
85.8 −0.472666 + 2.78865i − 3.00000i −7.55317 − 2.63620i − 8.98099i 8.36596 + 1.41800i −7.00000 10.9216 − 19.8171i −9.00000 25.0449 + 4.24501i
85.9 −0.368457 − 2.80433i − 3.00000i −7.72848 + 2.06655i 5.44298i −8.41298 + 1.10537i −7.00000 8.64289 + 20.9117i −9.00000 15.2639 − 2.00551i
85.10 −0.368457 + 2.80433i 3.00000i −7.72848 − 2.06655i − 5.44298i −8.41298 − 1.10537i −7.00000 8.64289 − 20.9117i −9.00000 15.2639 + 2.00551i
85.11 0.640567 − 2.75494i 3.00000i −7.17935 − 3.52944i − 19.2168i 8.26481 + 1.92170i −7.00000 −14.3222 + 17.5178i −9.00000 −52.9410 − 12.3096i
85.12 0.640567 + 2.75494i − 3.00000i −7.17935 + 3.52944i 19.2168i 8.26481 − 1.92170i −7.00000 −14.3222 − 17.5178i −9.00000 −52.9410 + 12.3096i
85.13 1.60701 − 2.32756i 3.00000i −2.83505 − 7.48081i 14.0008i 6.98267 + 4.82102i −7.00000 −21.9680 − 5.42297i −9.00000 32.5876 + 22.4993i
85.14 1.60701 + 2.32756i − 3.00000i −2.83505 + 7.48081i − 14.0008i 6.98267 − 4.82102i −7.00000 −21.9680 + 5.42297i −9.00000 32.5876 − 22.4993i
85.15 2.19536 − 1.78337i − 3.00000i 1.63917 − 7.83027i 4.34928i −5.35012 − 6.58607i −7.00000 −10.3657 − 20.1135i −9.00000 7.75639 + 9.54823i
85.16 2.19536 + 1.78337i 3.00000i 1.63917 + 7.83027i − 4.34928i −5.35012 + 6.58607i −7.00000 −10.3657 + 20.1135i −9.00000 7.75639 − 9.54823i
85.17 2.28984 − 1.66031i − 3.00000i 2.48677 − 7.60368i − 16.1369i −4.98092 − 6.86953i −7.00000 −6.93012 − 21.5400i −9.00000 −26.7922 − 36.9510i
85.18 2.28984 + 1.66031i 3.00000i 2.48677 + 7.60368i 16.1369i −4.98092 + 6.86953i −7.00000 −6.93012 + 21.5400i −9.00000 −26.7922 + 36.9510i
85.19 2.79928 − 0.404977i 3.00000i 7.67199 − 2.26729i − 19.4431i 1.21493 + 8.39785i −7.00000 20.5579 − 9.45377i −9.00000 −7.87400 − 54.4267i
85.20 2.79928 + 0.404977i − 3.00000i 7.67199 + 2.26729i 19.4431i 1.21493 − 8.39785i −7.00000 20.5579 + 9.45377i −9.00000 −7.87400 + 54.4267i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.4.c.b ✓ 20
3.b odd 2 1 504.4.c.e 20
4.b odd 2 1 672.4.c.b 20
8.b even 2 1 inner 168.4.c.b ✓ 20
8.d odd 2 1 672.4.c.b 20
12.b even 2 1 2016.4.c.f 20
24.f even 2 1 2016.4.c.f 20
24.h odd 2 1 504.4.c.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.c.b ✓ 20 1.a even 1 1 trivial
168.4.c.b ✓ 20 8.b even 2 1 inner
504.4.c.e 20 3.b odd 2 1
504.4.c.e 20 24.h odd 2 1
672.4.c.b 20 4.b odd 2 1
672.4.c.b 20 8.d odd 2 1
2016.4.c.f 20 12.b even 2 1
2016.4.c.f 20 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 1672 T_{5}^{18} + 1173484 T_{5}^{16} + 450983936 T_{5}^{14} + 103974448816 T_{5}^{12} + \cdots + 23\!\cdots\!84 \) acting on \(S_{4}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 1073741824 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$7$ \( (T + 7)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 21\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 21\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 37\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 59\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 23\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 71\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 70\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 18\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 11\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 11\!\cdots\!72)^{2} \) Copy content Toggle raw display
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