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Properties

Label 200.6.f.c
Level $200$
Weight $6$
Character orbit 200.f
Analytic conductor $32.077$
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] = [200,6,Mod(149,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.149");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.f (of order \(2\), degree \(1\), not 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: \(32.0767639626\)
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: \( 2^{45}\cdot 3^{4}\cdot 5^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + 2) q^{3} + (\beta_{5} + 2) q^{4} + ( - \beta_{10} + \beta_{2} + 2 \beta_1 + 10) q^{6} + (\beta_{6} - \beta_{5} - \beta_{2} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{15} - \beta_{14} + \cdots + 82) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + 2) q^{3} + (\beta_{5} + 2) q^{4} + ( - \beta_{10} + \beta_{2} + 2 \beta_1 + 10) q^{6} + (\beta_{6} - \beta_{5} - \beta_{2} + \cdots - 1) q^{7}+ \cdots + ( - 58 \beta_{19} + 291 \beta_{18} + \cdots + 472) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 36 q^{3} + 32 q^{4} + 204 q^{6} + 248 q^{8} + 1620 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} + 36 q^{3} + 32 q^{4} + 204 q^{6} + 248 q^{8} + 1620 q^{9} + 1252 q^{12} - 2708 q^{14} + 3080 q^{16} + 2070 q^{18} + 8244 q^{22} - 1032 q^{24} - 8084 q^{26} + 11664 q^{27} + 22924 q^{28} + 7160 q^{31} + 14792 q^{32} - 21132 q^{34} + 18344 q^{36} - 3608 q^{37} - 16884 q^{38} + 44904 q^{39} + 11608 q^{41} - 49444 q^{42} - 51772 q^{43} - 72296 q^{44} - 28516 q^{46} - 85048 q^{48} - 18756 q^{49} - 111624 q^{52} + 928 q^{53} + 100584 q^{54} - 53624 q^{56} + 152344 q^{58} + 228648 q^{62} + 11264 q^{64} - 56688 q^{66} - 161604 q^{67} + 359040 q^{68} - 200312 q^{71} + 563448 q^{72} - 78876 q^{74} - 153872 q^{76} + 26008 q^{77} - 624640 q^{78} - 282080 q^{79} + 65172 q^{81} - 410576 q^{82} - 99092 q^{83} + 297128 q^{84} + 27452 q^{86} - 464496 q^{88} + 3160 q^{89} - 519244 q^{92} + 293472 q^{93} - 148820 q^{94} + 395168 q^{96} + 663674 q^{98}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( - 1334623 \nu^{19} - 6032834 \nu^{18} + 30741327 \nu^{17} + 53342094 \nu^{16} + \cdots + 31\!\cdots\!84 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7166555 \nu^{19} - 46758710 \nu^{18} - 1326206475 \nu^{17} + 2375023674 \nu^{16} + \cdots - 93\!\cdots\!16 ) / 82\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 192881341 \nu^{19} + 14688393370 \nu^{18} + 34351336269 \nu^{17} - 97539248310 \nu^{16} + \cdots + 53\!\cdots\!36 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 36529 \nu^{19} + 37138 \nu^{18} + 952929 \nu^{17} + 2345826 \nu^{16} + \cdots + 36\!\cdots\!00 ) / 321582991933440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 299251771 \nu^{19} + 3783277174 \nu^{18} + 21114888939 \nu^{17} + 1756737030 \nu^{16} + \cdots + 30\!\cdots\!00 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1159161 \nu^{19} - 876270 \nu^{18} + 12454985 \nu^{17} - 11755262 \nu^{16} + \cdots + 12\!\cdots\!68 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 156457 \nu^{19} + 977582 \nu^{18} + 2983239 \nu^{17} - 17626242 \nu^{16} + \cdots + 28\!\cdots\!24 ) / 605332690698240 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 213569915 \nu^{19} - 3356881930 \nu^{18} - 18244966869 \nu^{17} - 19203947130 \nu^{16} + \cdots - 22\!\cdots\!76 ) / 53\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 180646075 \nu^{19} + 1159891466 \nu^{18} - 400600683 \nu^{17} - 3359147142 \nu^{16} + \cdots - 37\!\cdots\!20 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1103951461 \nu^{19} - 10117315594 \nu^{18} - 26494398261 \nu^{17} + 135432087558 \nu^{16} + \cdots - 70\!\cdots\!68 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 93706979 \nu^{19} - 141114154 \nu^{18} + 1115274483 \nu^{17} + 149641926 \nu^{16} + \cdots + 78\!\cdots\!80 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 67985 \nu^{19} - 62912 \nu^{18} - 1230021 \nu^{17} + 3396972 \nu^{16} + \cdots - 11\!\cdots\!92 ) / 128633196773376 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1346839351 \nu^{19} - 1644923758 \nu^{18} - 43011703719 \nu^{17} - 15233921598 \nu^{16} + \cdots - 41\!\cdots\!68 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 505812019 \nu^{19} + 4224516506 \nu^{18} - 6123990627 \nu^{17} - 35257502838 \nu^{16} + \cdots - 12\!\cdots\!12 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1853756651 \nu^{19} + 1747045642 \nu^{18} + 9318978981 \nu^{17} - 49450238790 \nu^{16} + \cdots - 49\!\cdots\!44 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 3850944367 \nu^{19} - 1017879362 \nu^{18} + 45482103711 \nu^{17} + 174754088526 \nu^{16} + \cdots + 85\!\cdots\!76 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 130011995 \nu^{19} - 169747150 \nu^{18} - 3380145243 \nu^{17} + 415414866 \nu^{16} + \cdots - 57\!\cdots\!04 ) / 59\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 1224412387 \nu^{19} - 847266530 \nu^{18} + 17618041827 \nu^{17} - 19790632194 \nu^{16} + \cdots + 13\!\cdots\!48 ) / 53\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + 10\beta _1 + 1 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} - 2\beta_{7} - \beta_{2} + \beta _1 + 19 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{19} + \beta_{18} - \beta_{17} + \beta_{16} - 3 \beta_{15} + \beta_{12} + \beta_{10} + 3 \beta_{8} + \cdots - 123 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{19} - \beta_{18} + \beta_{16} + \beta_{15} - 2 \beta_{14} - 3 \beta_{13} + \beta_{12} + \cdots - 147 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10 \beta_{19} - 40 \beta_{18} - 31 \beta_{17} + 10 \beta_{16} + 10 \beta_{15} + 122 \beta_{14} + \cdots - 2530 ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5 \beta_{19} - 11 \beta_{18} - 45 \beta_{17} - 7 \beta_{16} + 145 \beta_{15} - 248 \beta_{14} + \cdots - 31087 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 257 \beta_{19} - 413 \beta_{18} - 188 \beta_{17} - 403 \beta_{16} - 491 \beta_{15} + 958 \beta_{14} + \cdots - 122399 ) / 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 126 \beta_{19} + 344 \beta_{18} + 189 \beta_{17} - 110 \beta_{16} + 2 \beta_{15} - 558 \beta_{14} + \cdots + 52990 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1659 \beta_{19} + 2021 \beta_{18} - 2349 \beta_{17} - 8039 \beta_{16} - 2703 \beta_{15} + \cdots + 1453393 ) / 20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 113 \beta_{19} + 26739 \beta_{18} + 14572 \beta_{17} - 14739 \beta_{16} + 7525 \beta_{15} + \cdots - 4767607 ) / 20 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 147162 \beta_{19} + 238392 \beta_{18} - 9527 \beta_{17} - 37158 \beta_{16} - 196966 \beta_{15} + \cdots + 20624462 ) / 20 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 86847 \beta_{19} + 50961 \beta_{18} + 52727 \beta_{17} + 31301 \beta_{16} - 78563 \beta_{15} + \cdots - 19463843 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 969663 \beta_{19} - 189853 \beta_{18} + 974964 \beta_{17} + 1126797 \beta_{16} + 139669 \beta_{15} + \cdots + 435249937 ) / 20 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 7200106 \beta_{19} - 2130120 \beta_{18} - 6637631 \beta_{17} - 378214 \beta_{16} + 2461290 \beta_{15} + \cdots - 528732762 ) / 20 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3690981 \beta_{19} - 30470619 \beta_{18} + 5305331 \beta_{17} + 5479801 \beta_{16} + 8408337 \beta_{15} + \cdots - 5009602511 ) / 20 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 30094205 \beta_{19} - 17270761 \beta_{18} - 540148 \beta_{17} - 2613911 \beta_{16} - 11151071 \beta_{15} + \cdots + 2735351589 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 532611398 \beta_{19} - 437359688 \beta_{18} + 226848345 \beta_{17} + 283715482 \beta_{16} + \cdots + 83066270718 ) / 20 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 300117019 \beta_{19} - 2853763499 \beta_{18} + 592592179 \beta_{17} - 524067847 \beta_{16} + \cdots + 91076734833 ) / 20 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 2877301505 \beta_{19} + 5664859555 \beta_{18} - 333178812 \beta_{17} + 7928989165 \beta_{16} + \cdots + 375256454785 ) / 20 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−2.80358 − 2.85306i
−2.80358 + 2.85306i
−3.80026 + 1.24819i
−3.80026 − 1.24819i
0.236693 − 3.99299i
0.236693 + 3.99299i
−3.90102 − 0.884346i
−3.90102 + 0.884346i
2.93366 − 2.71913i
2.93366 + 2.71913i
−2.63430 − 3.01006i
−2.63430 + 3.01006i
3.72553 − 1.45618i
3.72553 + 1.45618i
0.593959 − 3.95566i
0.593959 + 3.95566i
3.46430 + 1.99965i
3.46430 − 1.99965i
3.18502 + 2.41984i
3.18502 − 2.41984i
−5.65664 − 0.0494789i −10.7455 31.9951 + 0.559768i 0 60.7833 + 0.531674i − 198.733i −180.957 − 4.74949i −127.535 0
149.2 −5.65664 + 0.0494789i −10.7455 31.9951 − 0.559768i 0 60.7833 − 0.531674i 198.733i −180.957 + 4.74949i −127.535 0
149.3 −5.04846 − 2.55207i 11.5927 18.9739 + 25.7680i 0 −58.5253 − 29.5854i − 231.529i −30.0270 − 178.512i −108.609 0
149.4 −5.04846 + 2.55207i 11.5927 18.9739 − 25.7680i 0 −58.5253 + 29.5854i 231.529i −30.0270 + 178.512i −108.609 0
149.5 −3.75630 − 4.22968i 25.0521 −3.78045 + 31.7759i 0 −94.1031 − 105.962i − 103.624i 148.603 − 103.370i 384.607 0
149.6 −3.75630 + 4.22968i 25.0521 −3.78045 − 31.7759i 0 −94.1031 + 105.962i 103.624i 148.603 + 103.370i 384.607 0
149.7 −3.01667 − 4.78536i −25.4343 −13.7994 + 28.8717i 0 76.7270 + 121.712i − 56.4938i 179.790 − 21.0614i 403.904 0
149.8 −3.01667 + 4.78536i −25.4343 −13.7994 − 28.8717i 0 76.7270 − 121.712i 56.4938i 179.790 + 21.0614i 403.904 0
149.9 0.214529 − 5.65278i −18.7876 −31.9080 − 2.42537i 0 −4.03048 + 106.202i 107.536i −20.5552 + 179.848i 109.975 0
149.10 0.214529 + 5.65278i −18.7876 −31.9080 + 2.42537i 0 −4.03048 − 106.202i − 107.536i −20.5552 − 179.848i 109.975 0
149.11 0.375761 − 5.64436i 6.67450 −31.7176 − 4.24186i 0 2.50801 − 37.6733i − 38.2812i −35.8608 + 177.432i −198.451 0
149.12 0.375761 + 5.64436i 6.67450 −31.7176 + 4.24186i 0 2.50801 + 37.6733i 38.2812i −35.8608 − 177.432i −198.451 0
149.13 2.26935 − 5.18171i 10.8240 −21.7001 − 23.5182i 0 24.5634 − 56.0868i − 163.706i −171.109 + 59.0729i −125.841 0
149.14 2.26935 + 5.18171i 10.8240 −21.7001 + 23.5182i 0 24.5634 + 56.0868i 163.706i −171.109 − 59.0729i −125.841 0
149.15 4.54962 − 3.36170i 6.93089 9.39799 − 30.5888i 0 31.5329 − 23.2996i 47.1406i −60.0732 − 170.761i −194.963 0
149.16 4.54962 + 3.36170i 6.93089 9.39799 + 30.5888i 0 31.5329 + 23.2996i − 47.1406i −60.0732 + 170.761i −194.963 0
149.17 5.46395 − 1.46465i 29.2080 27.7096 − 16.0056i 0 159.591 − 42.7797i 168.173i 127.961 − 128.039i 610.110 0
149.18 5.46395 + 1.46465i 29.2080 27.7096 + 16.0056i 0 159.591 + 42.7797i − 168.173i 127.961 + 128.039i 610.110 0
149.19 5.60486 − 0.765181i −17.3148 30.8290 − 8.57748i 0 −97.0471 + 13.2490i 9.19080i 166.229 − 71.6654i 56.8021 0
149.20 5.60486 + 0.765181i −17.3148 30.8290 + 8.57748i 0 −97.0471 − 13.2490i − 9.19080i 166.229 + 71.6654i 56.8021 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.6.f.c 20
4.b odd 2 1 800.6.f.b 20
5.b even 2 1 200.6.f.b 20
5.c odd 4 1 40.6.d.a ✓ 20
5.c odd 4 1 200.6.d.b 20
8.b even 2 1 200.6.f.b 20
8.d odd 2 1 800.6.f.c 20
15.e even 4 1 360.6.k.b 20
20.d odd 2 1 800.6.f.c 20
20.e even 4 1 160.6.d.a 20
20.e even 4 1 800.6.d.c 20
40.e odd 2 1 800.6.f.b 20
40.f even 2 1 inner 200.6.f.c 20
40.i odd 4 1 40.6.d.a ✓ 20
40.i odd 4 1 200.6.d.b 20
40.k even 4 1 160.6.d.a 20
40.k even 4 1 800.6.d.c 20
120.w even 4 1 360.6.k.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.d.a ✓ 20 5.c odd 4 1
40.6.d.a ✓ 20 40.i odd 4 1
160.6.d.a 20 20.e even 4 1
160.6.d.a 20 40.k even 4 1
200.6.d.b 20 5.c odd 4 1
200.6.d.b 20 40.i odd 4 1
200.6.f.b 20 5.b even 2 1
200.6.f.b 20 8.b even 2 1
200.6.f.c 20 1.a even 1 1 trivial
200.6.f.c 20 40.f even 2 1 inner
360.6.k.b 20 15.e even 4 1
360.6.k.b 20 120.w even 4 1
800.6.d.c 20 20.e even 4 1
800.6.d.c 20 40.k even 4 1
800.6.f.b 20 4.b odd 2 1
800.6.f.b 20 40.e odd 2 1
800.6.f.c 20 8.d odd 2 1
800.6.f.c 20 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 18 T_{3}^{9} - 1458 T_{3}^{8} + 23328 T_{3}^{7} + 690600 T_{3}^{6} - 10222224 T_{3}^{5} + \cdots + 377626901472 \) acting on \(S_{6}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots + 377626901472)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 78\!\cdots\!56 \) 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
$37$ \( (T^{10} + \cdots + 42\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 49\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 90\!\cdots\!88)^{2} \) 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
$67$ \( (T^{10} + \cdots + 71\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 61\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
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