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Properties

Label 360.6.k.b
Level $360$
Weight $6$
Character orbit 360.k
Analytic conductor $57.738$
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] = [360,6,Mod(181,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.181");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 360.k (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: \(57.7381751327\)
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: \( x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{8}\cdot 5^{12} \)
Twist minimal: no (minimal twist has level 40)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{4} q^{5} + (\beta_{8} - 4 \beta_1 - 10) q^{7} + ( - \beta_{5} - 2 \beta_1 - 13) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} - \beta_{4} q^{5} + (\beta_{8} - 4 \beta_1 - 10) q^{7} + ( - \beta_{5} - 2 \beta_1 - 13) q^{8} + ( - \beta_{10} - 3) q^{10} + ( - \beta_{11} + \beta_{10} - 2 \beta_{7} + \cdots - 4) q^{11}+ \cdots + (10 \beta_{19} + 65 \beta_{18} + \cdots - 10417) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} - 32 q^{4} - 196 q^{7} - 248 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} - 32 q^{4} - 196 q^{7} - 248 q^{8} - 50 q^{10} - 2708 q^{14} + 3080 q^{16} + 1900 q^{20} + 13836 q^{22} + 4676 q^{23} - 12500 q^{25} + 8084 q^{26} + 2108 q^{28} + 7160 q^{31} - 6792 q^{32} + 21132 q^{34} + 19580 q^{38} + 6200 q^{40} - 11608 q^{41} - 72296 q^{44} - 28516 q^{46} - 44180 q^{47} + 18756 q^{49} + 1250 q^{50} - 39680 q^{52} - 24200 q^{55} + 53624 q^{56} + 59496 q^{58} - 59824 q^{62} - 11264 q^{64} - 11576 q^{68} + 29800 q^{70} + 200312 q^{71} - 105136 q^{73} - 78876 q^{74} - 153872 q^{76} + 282080 q^{79} - 16000 q^{80} - 223032 q^{82} - 27452 q^{86} + 86896 q^{88} + 3160 q^{89} - 107916 q^{92} + 148820 q^{94} - 144400 q^{95} + 147376 q^{97} - 216942 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 67985 \nu^{19} + 62912 \nu^{18} + 1230021 \nu^{17} - 3396972 \nu^{16} + \cdots + 11\!\cdots\!16 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36529 \nu^{19} - 37138 \nu^{18} - 952929 \nu^{17} - 2345826 \nu^{16} + \cdots - 36\!\cdots\!00 ) / 321582991933440 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 145107079 \nu^{19} + 5441237518 \nu^{18} + 60223606263 \nu^{17} - 32567515170 \nu^{16} + \cdots + 71\!\cdots\!16 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6673115 \nu^{19} + 30164170 \nu^{18} - 153706635 \nu^{17} - 266710470 \nu^{16} + \cdots - 15\!\cdots\!20 ) / 32\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8225 \nu^{19} - 42752 \nu^{18} - 683541 \nu^{17} + 1613676 \nu^{16} + 4831769 \nu^{15} + \cdots - 58\!\cdots\!92 ) / 37833293168640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 827259823 \nu^{19} - 1584892510 \nu^{18} - 26805464991 \nu^{17} - 15478472622 \nu^{16} + \cdots - 36\!\cdots\!08 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2121323 \nu^{19} - 2680522 \nu^{18} + 3689307 \nu^{17} - 6591738 \nu^{16} + \cdots - 15\!\cdots\!28 ) / 34\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1652193653 \nu^{19} + 1541606518 \nu^{18} - 22547436165 \nu^{17} + 28897641030 \nu^{16} + \cdots - 18\!\cdots\!72 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2605175597 \nu^{19} + 3192740762 \nu^{18} + 21645982845 \nu^{17} - 5376840438 \nu^{16} + \cdots + 56\!\cdots\!56 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 339925 \nu^{19} + 314560 \nu^{18} + 6150105 \nu^{17} - 16984860 \nu^{16} + \cdots + 56\!\cdots\!24 ) / 257266393546752 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3785823301 \nu^{19} - 8542711306 \nu^{18} - 117005988501 \nu^{17} - 216596642682 \nu^{16} + \cdots - 20\!\cdots\!32 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3859078441 \nu^{19} - 18704845934 \nu^{18} + 107189740089 \nu^{17} + \cdots + 15\!\cdots\!44 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 123634871 \nu^{19} + 1141883170 \nu^{18} + 874564281 \nu^{17} - 7285412814 \nu^{16} + \cdots + 17\!\cdots\!12 ) / 62\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2108681725 \nu^{19} - 128158838 \nu^{18} + 49805269677 \nu^{17} + 48221488218 \nu^{16} + \cdots + 37\!\cdots\!68 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 5638862219 \nu^{19} - 12629066122 \nu^{18} + 152290274427 \nu^{17} + \cdots + 21\!\cdots\!04 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1481242609 \nu^{19} + 5244252734 \nu^{18} - 11063606337 \nu^{17} - 6441981618 \nu^{16} + \cdots - 22\!\cdots\!40 ) / 53\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 187911103 \nu^{19} + 84312242 \nu^{18} - 2577538575 \nu^{17} + 1469887746 \nu^{16} + \cdots - 26\!\cdots\!64 ) / 62\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 6634308877 \nu^{19} + 4996442938 \nu^{18} + 181409713821 \nu^{17} + \cdots + 22\!\cdots\!36 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 25360811 \nu^{19} - 24180656 \nu^{18} + 402177927 \nu^{17} - 38551284 \nu^{16} + \cdots + 54\!\cdots\!92 ) / 74\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - 25\beta _1 + 3 ) / 50 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{17} - \beta_{16} + \beta_{14} - \beta_{12} - \beta_{10} - 2 \beta_{8} - \beta_{7} - \beta_{6} + \cdots + 94 ) / 50 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{18} - \beta_{17} - 3 \beta_{16} - 2 \beta_{15} - \beta_{14} - 6 \beta_{13} - 9 \beta_{12} + \cdots - 647 ) / 100 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{19} - \beta_{17} - \beta_{16} + \beta_{15} - \beta_{14} - \beta_{12} - \beta_{11} + \cdots - 156 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 50 \beta_{19} + 59 \beta_{18} - 98 \beta_{17} - 64 \beta_{16} + 9 \beta_{15} + 82 \beta_{14} + \cdots - 13373 ) / 100 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 420 \beta_{19} + 321 \beta_{18} + 203 \beta_{17} - 31 \beta_{16} + 26 \beta_{15} - 417 \beta_{14} + \cdots - 154243 ) / 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 450 \beta_{19} - 188 \beta_{18} + 815 \beta_{17} + 1799 \beta_{16} + 1397 \beta_{15} + \cdots - 607096 ) / 100 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2 \beta_{19} + 11 \beta_{18} - 58 \beta_{17} + 168 \beta_{16} - 279 \beta_{15} - 102 \beta_{14} + \cdots + 54371 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9500 \beta_{19} + 8969 \beta_{18} - 6317 \beta_{17} + 62665 \beta_{16} + 5514 \beta_{15} + \cdots + 7524877 ) / 100 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 17230 \beta_{19} - 17716 \beta_{18} + 58639 \beta_{17} + 11879 \beta_{16} - 90011 \beta_{15} + \cdots - 22748136 ) / 100 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 467550 \beta_{19} - 19549 \beta_{18} + 383022 \beta_{17} + 452640 \beta_{16} - 266239 \beta_{15} + \cdots + 104573835 ) / 100 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 44124 \beta_{19} - 90535 \beta_{18} - 57389 \beta_{17} - 32807 \beta_{16} + 10026 \beta_{15} + \cdots - 19852251 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 6631650 \beta_{19} + 5852340 \beta_{18} - 5174193 \beta_{17} - 11279897 \beta_{16} + \cdots + 2126677896 ) / 100 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 25095470 \beta_{19} + 872211 \beta_{18} - 22835066 \beta_{17} - 30676072 \beta_{16} + \cdots - 2874458293 ) / 100 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 9611100 \beta_{19} + 24695161 \beta_{18} + 79597763 \beta_{17} - 149617991 \beta_{16} + \cdots - 25663135475 ) / 100 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 11428830 \beta_{19} - 257348 \beta_{18} - 3041913 \beta_{17} - 112353 \beta_{16} - 5510547 \beta_{15} + \cdots + 2466646456 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 526429250 \beta_{19} + 52573091 \beta_{18} + 134431070 \beta_{17} + 692622672 \beta_{16} + \cdots + 416879144235 ) / 100 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 807274300 \beta_{19} + 2663673185 \beta_{18} + 1589925323 \beta_{17} + 7518933857 \beta_{16} + \cdots + 505426823773 ) / 100 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 31238833150 \beta_{19} - 6137817148 \beta_{18} + 40320191407 \beta_{17} + 24662528967 \beta_{16} + \cdots + 2126725973864 ) / 100 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
2.93366 − 2.71913i
2.93366 + 2.71913i
3.72553 − 1.45618i
3.72553 + 1.45618i
0.236693 + 3.99299i
0.236693 − 3.99299i
3.46430 + 1.99965i
3.46430 − 1.99965i
3.18502 + 2.41984i
3.18502 − 2.41984i
−2.80358 + 2.85306i
−2.80358 − 2.85306i
−3.80026 + 1.24819i
−3.80026 − 1.24819i
0.593959 + 3.95566i
0.593959 − 3.95566i
−3.90102 − 0.884346i
−3.90102 + 0.884346i
−2.63430 + 3.01006i
−2.63430 − 3.01006i
−5.65278 − 0.214529i 0 31.9080 + 2.42537i − 25.0000i 0 −107.536 −179.848 − 20.5552i 0 −5.36321 + 141.320i
181.2 −5.65278 + 0.214529i 0 31.9080 − 2.42537i 25.0000i 0 −107.536 −179.848 + 20.5552i 0 −5.36321 − 141.320i
181.3 −5.18171 − 2.26935i 0 21.7001 + 23.5182i − 25.0000i 0 163.706 −59.0729 − 171.109i 0 −56.7336 + 129.543i
181.4 −5.18171 + 2.26935i 0 21.7001 − 23.5182i 25.0000i 0 163.706 −59.0729 + 171.109i 0 −56.7336 − 129.543i
181.5 −4.22968 − 3.75630i 0 3.78045 + 31.7759i 25.0000i 0 103.624 103.370 − 148.603i 0 93.9075 − 105.742i
181.6 −4.22968 + 3.75630i 0 3.78045 − 31.7759i − 25.0000i 0 103.624 103.370 + 148.603i 0 93.9075 + 105.742i
181.7 −1.46465 − 5.46395i 0 −27.7096 + 16.0056i − 25.0000i 0 −168.173 128.039 + 127.961i 0 −136.599 + 36.6164i
181.8 −1.46465 + 5.46395i 0 −27.7096 − 16.0056i 25.0000i 0 −168.173 128.039 − 127.961i 0 −136.599 − 36.6164i
181.9 −0.765181 − 5.60486i 0 −30.8290 + 8.57748i − 25.0000i 0 −9.19080 71.6654 + 166.229i 0 −140.122 + 19.1295i
181.10 −0.765181 + 5.60486i 0 −30.8290 − 8.57748i 25.0000i 0 −9.19080 71.6654 − 166.229i 0 −140.122 − 19.1295i
181.11 −0.0494789 − 5.65664i 0 −31.9951 + 0.559768i 25.0000i 0 198.733 4.74949 + 180.957i 0 141.416 − 1.23697i
181.12 −0.0494789 + 5.65664i 0 −31.9951 − 0.559768i − 25.0000i 0 198.733 4.74949 − 180.957i 0 141.416 + 1.23697i
181.13 2.55207 − 5.04846i 0 −18.9739 − 25.7680i 25.0000i 0 −231.529 −178.512 + 30.0270i 0 126.211 + 63.8017i
181.14 2.55207 + 5.04846i 0 −18.9739 + 25.7680i − 25.0000i 0 −231.529 −178.512 − 30.0270i 0 126.211 − 63.8017i
181.15 3.36170 − 4.54962i 0 −9.39799 − 30.5888i − 25.0000i 0 47.1406 −170.761 − 60.0732i 0 −113.740 − 84.0424i
181.16 3.36170 + 4.54962i 0 −9.39799 + 30.5888i 25.0000i 0 47.1406 −170.761 + 60.0732i 0 −113.740 + 84.0424i
181.17 4.78536 − 3.01667i 0 13.7994 − 28.8717i 25.0000i 0 −56.4938 −21.0614 − 179.790i 0 75.4168 + 119.634i
181.18 4.78536 + 3.01667i 0 13.7994 + 28.8717i − 25.0000i 0 −56.4938 −21.0614 + 179.790i 0 75.4168 − 119.634i
181.19 5.64436 − 0.375761i 0 31.7176 − 4.24186i − 25.0000i 0 −38.2812 177.432 − 35.8608i 0 −9.39401 − 141.109i
181.20 5.64436 + 0.375761i 0 31.7176 + 4.24186i 25.0000i 0 −38.2812 177.432 + 35.8608i 0 −9.39401 + 141.109i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.20
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 360.6.k.b 20
3.b odd 2 1 40.6.d.a ✓ 20
8.b even 2 1 inner 360.6.k.b 20
12.b even 2 1 160.6.d.a 20
15.d odd 2 1 200.6.d.b 20
15.e even 4 1 200.6.f.b 20
15.e even 4 1 200.6.f.c 20
24.f even 2 1 160.6.d.a 20
24.h odd 2 1 40.6.d.a ✓ 20
60.h even 2 1 800.6.d.c 20
60.l odd 4 1 800.6.f.b 20
60.l odd 4 1 800.6.f.c 20
120.i odd 2 1 200.6.d.b 20
120.m even 2 1 800.6.d.c 20
120.q odd 4 1 800.6.f.b 20
120.q odd 4 1 800.6.f.c 20
120.w even 4 1 200.6.f.b 20
120.w even 4 1 200.6.f.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.d.a ✓ 20 3.b odd 2 1
40.6.d.a ✓ 20 24.h odd 2 1
160.6.d.a 20 12.b even 2 1
160.6.d.a 20 24.f even 2 1
200.6.d.b 20 15.d odd 2 1
200.6.d.b 20 120.i odd 2 1
200.6.f.b 20 15.e even 4 1
200.6.f.b 20 120.w even 4 1
200.6.f.c 20 15.e even 4 1
200.6.f.c 20 120.w even 4 1
360.6.k.b 20 1.a even 1 1 trivial
360.6.k.b 20 8.b even 2 1 inner
800.6.d.c 20 60.h even 2 1
800.6.d.c 20 120.m even 2 1
800.6.f.b 20 60.l odd 4 1
800.6.f.b 20 120.q odd 4 1
800.6.f.c 20 60.l odd 4 1
800.6.f.c 20 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 98 T_{7}^{9} - 83922 T_{7}^{8} - 6806560 T_{7}^{7} + 2129001128 T_{7}^{6} + \cdots + 13\!\cdots\!08 \) acting on \(S_{6}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{2} + 625)^{10} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 13\!\cdots\!08)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 29\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 88\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 42\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 16\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 82\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 50\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 22\!\cdots\!48)^{2} \) Copy content Toggle raw display
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