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Properties

Label 200.6.d.b
Level $200$
Weight $6$
Character orbit 200.d
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(101,200)]
 
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, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.101");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.d (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: \(32.0767639626\)
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: \( 2^{45}\cdot 3^{4}\cdot 5^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + ( - \beta_{2} - 2) q^{4} + (\beta_{10} - \beta_{3} + 10) q^{6} + (\beta_{11} + 4 \beta_1 + 10) q^{7} + (\beta_{13} - 2 \beta_1 - 13) q^{8} + ( - \beta_{13} - \beta_{11} + \cdots - 80) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + ( - \beta_{2} - 2) q^{4} + (\beta_{10} - \beta_{3} + 10) q^{6} + (\beta_{11} + 4 \beta_1 + 10) q^{7} + (\beta_{13} - 2 \beta_1 - 13) q^{8} + ( - \beta_{13} - \beta_{11} + \cdots - 80) q^{9}+ \cdots + (276 \beta_{19} + 70 \beta_{18} + \cdots - 1294) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} - 32 q^{4} + 204 q^{6} + 196 q^{7} - 248 q^{8} - 1620 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} - 32 q^{4} + 204 q^{6} + 196 q^{7} - 248 q^{8} - 1620 q^{9} + 1876 q^{12} + 2708 q^{14} + 3080 q^{16} + 5294 q^{18} - 13836 q^{22} + 4676 q^{23} + 1032 q^{24} - 8084 q^{26} - 2108 q^{28} + 7160 q^{31} - 6792 q^{32} - 5672 q^{33} + 21132 q^{34} + 18344 q^{36} + 19580 q^{38} - 44904 q^{39} + 11608 q^{41} + 17116 q^{42} + 72296 q^{44} - 28516 q^{46} - 44180 q^{47} + 88856 q^{48} + 18756 q^{49} + 39680 q^{52} - 100584 q^{54} - 53624 q^{56} - 5032 q^{57} - 59496 q^{58} - 59824 q^{62} - 240620 q^{63} - 11264 q^{64} - 56688 q^{66} - 11576 q^{68} - 200312 q^{71} - 235912 q^{72} + 105136 q^{73} + 78876 q^{74} - 153872 q^{76} - 95864 q^{78} + 282080 q^{79} + 65172 q^{81} + 223032 q^{82} - 297128 q^{84} + 27452 q^{86} + 332592 q^{87} - 86896 q^{88} - 3160 q^{89} - 107916 q^{92} + 148820 q^{94} + 395168 q^{96} - 147376 q^{97} - 216942 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 6907849 \nu^{19} + 7538270 \nu^{18} - 20908089 \nu^{17} + 46950990 \nu^{16} + \cdots - 67\!\cdots\!04 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 476587 \nu^{19} + 9976912 \nu^{18} + 32762439 \nu^{17} + 2473356 \nu^{16} + \cdots + 54\!\cdots\!88 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79348087 \nu^{19} - 164467438 \nu^{18} - 2750967783 \nu^{17} - 2347892670 \nu^{16} + \cdots - 38\!\cdots\!76 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10630051 \nu^{19} + 157584599 \nu^{18} - 82595421 \nu^{17} - 1520129955 \nu^{16} + \cdots + 82\!\cdots\!16 ) / 44\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6673115 \nu^{19} + 30164170 \nu^{18} - 153706635 \nu^{17} - 266710470 \nu^{16} + \cdots - 15\!\cdots\!20 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44108477 \nu^{19} - 45460682 \nu^{18} - 832724973 \nu^{17} + 2303349222 \nu^{16} + \cdots - 79\!\cdots\!08 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 442744159 \nu^{19} - 276284890 \nu^{18} - 15613759575 \nu^{17} + 2762546766 \nu^{16} + \cdots - 73\!\cdots\!40 ) / 53\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 647072803 \nu^{19} + 3413538794 \nu^{18} + 7527762765 \nu^{17} - 75303740550 \nu^{16} + \cdots + 46\!\cdots\!64 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1994492033 \nu^{19} - 8991709778 \nu^{18} - 8905173489 \nu^{17} + 137083395486 \nu^{16} + \cdots + 26\!\cdots\!20 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 260736601 \nu^{19} - 849515726 \nu^{18} + 4127295465 \nu^{17} + 1651959906 \nu^{16} + \cdots + 43\!\cdots\!16 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2463909961 \nu^{19} - 1514854286 \nu^{18} + 31821676569 \nu^{17} - 53949338142 \nu^{16} + \cdots + 26\!\cdots\!76 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35865137 \nu^{19} + 71898944 \nu^{18} + 1371849381 \nu^{17} - 770940396 \nu^{16} + \cdots + 24\!\cdots\!32 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 801925 \nu^{19} + 748552 \nu^{18} + 20349945 \nu^{17} - 16727460 \nu^{16} + \cdots + 24\!\cdots\!72 ) / 605332690698240 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3144463637 \nu^{19} + 3824873306 \nu^{18} + 81086729733 \nu^{17} - 65335326870 \nu^{16} + \cdots + 13\!\cdots\!80 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4455171577 \nu^{19} - 6109584946 \nu^{18} + 318951927 \nu^{17} + 298766553630 \nu^{16} + \cdots + 50\!\cdots\!08 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 43886245 \nu^{19} + 10414306 \nu^{18} + 1202085189 \nu^{17} + 632094114 \nu^{16} + \cdots + 12\!\cdots\!88 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 1513955615 \nu^{19} + 2916367138 \nu^{18} - 6105531087 \nu^{17} + 58135936530 \nu^{16} + \cdots - 12\!\cdots\!76 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 3519126037 \nu^{19} - 12872853826 \nu^{18} - 104803905717 \nu^{17} + 172605077502 \nu^{16} + \cdots - 15\!\cdots\!32 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1927639321 \nu^{19} - 2114930182 \nu^{18} - 51541339329 \nu^{17} + 61140611634 \nu^{16} + \cdots - 40\!\cdots\!36 ) / 53\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{19} - \beta_{18} - 6 \beta_{17} + 3 \beta_{16} - 4 \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 92 ) / 800 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{19} - 9 \beta_{18} - 4 \beta_{17} - 3 \beta_{16} + 16 \beta_{15} - 4 \beta_{14} + 5 \beta_{13} + \cdots + 1532 ) / 800 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24 \beta_{19} + 29 \beta_{18} - 11 \beta_{17} + 68 \beta_{16} - 4 \beta_{15} + 38 \beta_{14} + \cdots - 2514 ) / 400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{19} - \beta_{18} - 6 \beta_{17} + 3 \beta_{16} - 8 \beta_{15} - 9 \beta_{13} - 7 \beta_{12} + \cdots - 1216 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 439 \beta_{19} - 1089 \beta_{18} + 1516 \beta_{17} - 883 \beta_{16} - 784 \beta_{15} + \cdots - 105388 ) / 800 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1920 \beta_{19} + 245 \beta_{18} - 835 \beta_{17} - 1880 \beta_{16} + 812 \beta_{15} + \cdots - 618682 ) / 400 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6023 \beta_{19} - 9657 \beta_{18} + 24618 \beta_{17} - 3109 \beta_{16} + 6520 \beta_{15} + \cdots - 5056976 ) / 800 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 127 \beta_{19} + 2311 \beta_{18} + 1244 \beta_{17} - 739 \beta_{16} - 464 \beta_{15} - 1628 \beta_{14} + \cdots + 407252 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5864 \beta_{19} + 10509 \beta_{18} - 78411 \beta_{17} + 55868 \beta_{16} - 25268 \beta_{15} + \cdots + 30624430 ) / 400 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 518503 \beta_{19} - 747417 \beta_{18} - 652822 \beta_{17} - 1130069 \beta_{16} + 469112 \beta_{15} + \cdots - 183111488 ) / 800 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2560265 \beta_{19} + 4084735 \beta_{18} - 1224180 \beta_{17} + 1715965 \beta_{16} + 3064176 \beta_{15} + \cdots + 837343444 ) / 800 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 427344 \beta_{19} + 247373 \beta_{18} - 285355 \beta_{17} + 547312 \beta_{16} - 229556 \beta_{15} + \cdots - 77873786 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 30454297 \beta_{19} - 4653017 \beta_{18} + 32076138 \beta_{17} - 57568869 \beta_{16} + \cdots + 17485710288 ) / 800 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 137965863 \beta_{19} - 9081873 \beta_{18} - 3134308 \beta_{17} - 60326971 \beta_{16} + \cdots - 19384640396 ) / 800 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 434822344 \beta_{19} - 309375139 \beta_{18} + 160875781 \beta_{17} - 521770428 \beta_{16} + \cdots - 103724035074 ) / 400 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 19403233 \beta_{19} + 31416991 \beta_{18} + 374493498 \beta_{17} + 18199651 \beta_{16} + \cdots + 19039296352 ) / 32 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 18688921079 \beta_{19} + 14403021951 \beta_{18} - 12525090004 \beta_{17} + 22759992877 \beta_{16} + \cdots + 3380795933588 ) / 800 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 209060384 \beta_{19} - 52243254891 \beta_{18} + 7538953949 \beta_{17} - 10855035592 \beta_{16} + \cdots + 1979801570150 ) / 400 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 18673340473 \beta_{19} - 9510462073 \beta_{18} - 202474007638 \beta_{17} - 99189196581 \beta_{16} + \cdots + 15777456688240 ) / 800 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
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 − 18.7876i 31.9080 + 2.42537i 0 −4.03048 + 106.202i 107.536 −179.848 − 20.5552i −109.975 0
101.2 −5.65278 + 0.214529i 18.7876i 31.9080 − 2.42537i 0 −4.03048 − 106.202i 107.536 −179.848 + 20.5552i −109.975 0
101.3 −5.18171 − 2.26935i 10.8240i 21.7001 + 23.5182i 0 24.5634 − 56.0868i −163.706 −59.0729 − 171.109i 125.841 0
101.4 −5.18171 + 2.26935i − 10.8240i 21.7001 − 23.5182i 0 24.5634 + 56.0868i −163.706 −59.0729 + 171.109i 125.841 0
101.5 −4.22968 − 3.75630i − 25.0521i 3.78045 + 31.7759i 0 −94.1031 + 105.962i −103.624 103.370 − 148.603i −384.607 0
101.6 −4.22968 + 3.75630i 25.0521i 3.78045 − 31.7759i 0 −94.1031 − 105.962i −103.624 103.370 + 148.603i −384.607 0
101.7 −1.46465 − 5.46395i 29.2080i −27.7096 + 16.0056i 0 159.591 − 42.7797i 168.173 128.039 + 127.961i −610.110 0
101.8 −1.46465 + 5.46395i − 29.2080i −27.7096 − 16.0056i 0 159.591 + 42.7797i 168.173 128.039 − 127.961i −610.110 0
101.9 −0.765181 − 5.60486i − 17.3148i −30.8290 + 8.57748i 0 −97.0471 + 13.2490i 9.19080 71.6654 + 166.229i −56.8021 0
101.10 −0.765181 + 5.60486i 17.3148i −30.8290 − 8.57748i 0 −97.0471 − 13.2490i 9.19080 71.6654 − 166.229i −56.8021 0
101.11 −0.0494789 − 5.65664i 10.7455i −31.9951 + 0.559768i 0 60.7833 − 0.531674i −198.733 4.74949 + 180.957i 127.535 0
101.12 −0.0494789 + 5.65664i − 10.7455i −31.9951 − 0.559768i 0 60.7833 + 0.531674i −198.733 4.74949 − 180.957i 127.535 0
101.13 2.55207 − 5.04846i − 11.5927i −18.9739 − 25.7680i 0 −58.5253 − 29.5854i 231.529 −178.512 + 30.0270i 108.609 0
101.14 2.55207 + 5.04846i 11.5927i −18.9739 + 25.7680i 0 −58.5253 + 29.5854i 231.529 −178.512 − 30.0270i 108.609 0
101.15 3.36170 − 4.54962i 6.93089i −9.39799 − 30.5888i 0 31.5329 + 23.2996i −47.1406 −170.761 − 60.0732i 194.963 0
101.16 3.36170 + 4.54962i − 6.93089i −9.39799 + 30.5888i 0 31.5329 − 23.2996i −47.1406 −170.761 + 60.0732i 194.963 0
101.17 4.78536 − 3.01667i 25.4343i 13.7994 − 28.8717i 0 76.7270 + 121.712i 56.4938 −21.0614 − 179.790i −403.904 0
101.18 4.78536 + 3.01667i − 25.4343i 13.7994 + 28.8717i 0 76.7270 − 121.712i 56.4938 −21.0614 + 179.790i −403.904 0
101.19 5.64436 − 0.375761i 6.67450i 31.7176 − 4.24186i 0 2.50801 + 37.6733i 38.2812 177.432 − 35.8608i 198.451 0
101.20 5.64436 + 0.375761i − 6.67450i 31.7176 + 4.24186i 0 2.50801 − 37.6733i 38.2812 177.432 + 35.8608i 198.451 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.20
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 200.6.d.b 20
4.b odd 2 1 800.6.d.c 20
5.b even 2 1 40.6.d.a ✓ 20
5.c odd 4 1 200.6.f.b 20
5.c odd 4 1 200.6.f.c 20
8.b even 2 1 inner 200.6.d.b 20
8.d odd 2 1 800.6.d.c 20
15.d odd 2 1 360.6.k.b 20
20.d odd 2 1 160.6.d.a 20
20.e even 4 1 800.6.f.b 20
20.e even 4 1 800.6.f.c 20
40.e odd 2 1 160.6.d.a 20
40.f even 2 1 40.6.d.a ✓ 20
40.i odd 4 1 200.6.f.b 20
40.i odd 4 1 200.6.f.c 20
40.k even 4 1 800.6.f.b 20
40.k even 4 1 800.6.f.c 20
120.i odd 2 1 360.6.k.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.d.a ✓ 20 5.b even 2 1
40.6.d.a ✓ 20 40.f even 2 1
160.6.d.a 20 20.d odd 2 1
160.6.d.a 20 40.e odd 2 1
200.6.d.b 20 1.a even 1 1 trivial
200.6.d.b 20 8.b even 2 1 inner
200.6.f.b 20 5.c odd 4 1
200.6.f.b 20 40.i odd 4 1
200.6.f.c 20 5.c odd 4 1
200.6.f.c 20 40.i odd 4 1
360.6.k.b 20 15.d odd 2 1
360.6.k.b 20 120.i odd 2 1
800.6.d.c 20 4.b odd 2 1
800.6.d.c 20 8.d odd 2 1
800.6.f.b 20 20.e even 4 1
800.6.f.b 20 40.k even 4 1
800.6.f.c 20 20.e even 4 1
800.6.f.c 20 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(200, [\chi])\):

\( T_{3}^{20} + 3240 T_{3}^{18} + 4346772 T_{3}^{16} + 3151305344 T_{3}^{14} + 1355603009184 T_{3}^{12} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
\( T_{7}^{10} - 98 T_{7}^{9} - 83922 T_{7}^{8} + 6806560 T_{7}^{7} + 2129001128 T_{7}^{6} + \cdots + 13\!\cdots\!08 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{20} \) 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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