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Properties

Label 200.6.d.c
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: \( x^{20} - x^{19} - 130 x^{17} + 144 x^{16} + 1560 x^{15} - 12320 x^{14} - 56128 x^{13} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{49}\cdot 5^{4}\cdot 31 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{5} q^{3} + \beta_{2} q^{4} + ( - \beta_{7} + 2) q^{6} + (\beta_{3} - \beta_{2} + \beta_1 - 10) q^{7} + (\beta_{5} - \beta_{4} - 20) q^{8} + (\beta_{11} + \beta_{2} + 5 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{5} q^{3} + \beta_{2} q^{4} + ( - \beta_{7} + 2) q^{6} + (\beta_{3} - \beta_{2} + \beta_1 - 10) q^{7} + (\beta_{5} - \beta_{4} - 20) q^{8} + (\beta_{11} + \beta_{2} + 5 \beta_1 - 81) q^{9} + ( - \beta_{10} + \beta_{7} - \beta_{5} + \cdots - 1) q^{11}+ \cdots + ( - 4 \beta_{19} - 20 \beta_{18} + \cdots + 215) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} + q^{4} + 33 q^{6} - 196 q^{7} - 391 q^{8} - 1620 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - q^{2} + q^{4} + 33 q^{6} - 196 q^{7} - 391 q^{8} - 1620 q^{9} - 241 q^{12} - 424 q^{14} - 55 q^{16} - 3368 q^{18} + 1197 q^{22} - 7184 q^{23} + 9459 q^{24} + 9172 q^{26} - 13492 q^{28} + 7160 q^{31} + 7869 q^{32} - 2836 q^{33} - 9591 q^{34} + 14828 q^{36} + 21505 q^{38} + 22452 q^{39} - 5804 q^{41} - 14272 q^{42} - 11593 q^{44} - 37612 q^{46} + 44180 q^{47} + 66571 q^{48} + 62652 q^{49} + 6136 q^{52} + 88947 q^{54} - 36908 q^{56} + 43696 q^{57} - 84012 q^{58} + 87460 q^{62} - 1240 q^{63} + 115177 q^{64} + 131439 q^{66} - 143341 q^{68} - 7724 q^{71} - 25772 q^{72} - 105136 q^{73} + 2112 q^{74} + 55951 q^{76} - 10948 q^{78} - 7780 q^{79} + 96984 q^{81} + 117501 q^{82} - 97556 q^{84} - 65986 q^{86} - 106188 q^{87} - 122597 q^{88} - 3160 q^{89} + 88908 q^{92} - 58540 q^{94} + 57791 q^{96} - 73688 q^{97} + 164655 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 130 x^{17} + 144 x^{16} + 1560 x^{15} - 12320 x^{14} - 56128 x^{13} + \cdots + 11\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 483 \nu^{19} + 764979 \nu^{18} - 4445008 \nu^{17} + 8154694 \nu^{16} + \cdots + 22\!\cdots\!72 ) / 64\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4229911 \nu^{19} + 16221639 \nu^{18} + 26166864 \nu^{17} - 1019195218 \nu^{16} + \cdots - 39\!\cdots\!60 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4229911 \nu^{19} + 16221639 \nu^{18} + 26166864 \nu^{17} - 1019195218 \nu^{16} + \cdots + 73\!\cdots\!20 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8184925 \nu^{19} + 200668563 \nu^{18} + 1522508560 \nu^{17} - 3755295802 \nu^{16} + \cdots - 14\!\cdots\!96 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 749483 \nu^{19} - 1635429 \nu^{18} + 98067728 \nu^{17} - 275463210 \nu^{16} + \cdots - 29\!\cdots\!16 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37460587 \nu^{19} + 1494251867 \nu^{18} - 4475752944 \nu^{17} + 4745284950 \nu^{16} + \cdots + 73\!\cdots\!04 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2398297 \nu^{19} + 28980305 \nu^{18} - 422469496 \nu^{17} - 3174987598 \nu^{16} + \cdots + 48\!\cdots\!04 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21942713 \nu^{19} + 53322057 \nu^{18} - 662660240 \nu^{17} + 1215292658 \nu^{16} + \cdots + 20\!\cdots\!68 ) / 28\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 310971 \nu^{19} - 3945067 \nu^{18} + 9206448 \nu^{17} + 50179594 \nu^{16} + \cdots - 67\!\cdots\!60 ) / 38\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{19} + \nu^{18} + 130 \nu^{16} - 144 \nu^{15} - 1560 \nu^{14} + 12320 \nu^{13} + \cdots + 32985348833280 ) / 1099511627776 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26933865 \nu^{19} + 267036039 \nu^{18} - 2862289648 \nu^{17} + 10871437742 \nu^{16} + \cdots + 85\!\cdots\!72 ) / 28\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 70351621 \nu^{19} - 907077867 \nu^{18} - 5836560656 \nu^{17} + 32934036874 \nu^{16} + \cdots + 17\!\cdots\!56 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 155908709 \nu^{19} + 852814731 \nu^{18} - 2662156528 \nu^{17} - 28225934410 \nu^{16} + \cdots + 44\!\cdots\!24 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 115985345 \nu^{19} + 354763953 \nu^{18} + 606126096 \nu^{17} - 7014813950 \nu^{16} + \cdots - 87\!\cdots\!28 ) / 14\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 552184661 \nu^{19} + 2441369189 \nu^{18} + 13939980272 \nu^{17} - 16030306774 \nu^{16} + \cdots - 12\!\cdots\!60 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 622153303 \nu^{19} - 3858273543 \nu^{18} + 7062341040 \nu^{17} + 15396948178 \nu^{16} + \cdots - 88\!\cdots\!72 ) / 56\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 490769197 \nu^{19} + 55925949 \nu^{18} + 18301120880 \nu^{17} - 34290812454 \nu^{16} + \cdots - 11\!\cdots\!00 ) / 28\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{7} + 10\beta_{5} - \beta_{3} + 20\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{18} + \beta_{17} - 5 \beta_{16} + \beta_{15} - 2 \beta_{14} + \beta_{13} + 3 \beta_{12} + \cdots - 379 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{19} - 13 \beta_{18} - 5 \beta_{17} - 11 \beta_{16} + 31 \beta_{15} + 18 \beta_{14} + \cdots + 5795 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 52 \beta_{19} - 61 \beta_{18} - 13 \beta_{17} + 53 \beta_{16} - 17 \beta_{15} - 62 \beta_{14} + \cdots + 22379 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 372 \beta_{19} - 19 \beta_{18} - 123 \beta_{17} + 475 \beta_{16} + 33 \beta_{15} - 66 \beta_{14} + \cdots - 325595 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 148 \beta_{19} - 2365 \beta_{18} - 853 \beta_{17} - 1579 \beta_{16} - 2033 \beta_{15} - 382 \beta_{14} + \cdots + 2593339 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1516 \beta_{19} - 13983 \beta_{18} - 1847 \beta_{17} - 29337 \beta_{16} - 13851 \beta_{15} + \cdots - 6436631 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 29212 \beta_{19} - 38685 \beta_{18} + 34875 \beta_{17} - 74955 \beta_{16} + 5999 \beta_{15} + \cdots - 6061605 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 88148 \beta_{19} - 347407 \beta_{18} + 120345 \beta_{17} - 93801 \beta_{16} + 463669 \beta_{15} + \cdots + 68329369 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 251812 \beta_{19} + 706667 \beta_{18} + 785827 \beta_{17} + 941677 \beta_{16} + 1073303 \beta_{15} + \cdots + 882294883 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3699892 \beta_{19} - 3989959 \beta_{18} + 12215521 \beta_{17} - 13099761 \beta_{16} + \cdots + 442797409 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 52870084 \beta_{19} + 28724851 \beta_{18} - 12855509 \beta_{17} - 31791451 \beta_{16} + 177659839 \beta_{15} + \cdots - 283897941 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 100359532 \beta_{19} - 114645551 \beta_{18} - 30515911 \beta_{17} - 444063433 \beta_{16} + \cdots + 209078563897 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 334614236 \beta_{19} + 1952445339 \beta_{18} + 618769939 \beta_{17} - 1053903683 \beta_{16} + \cdots + 1003771339283 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 3422819084 \beta_{19} - 6029586487 \beta_{18} + 3786227441 \beta_{17} - 3782299393 \beta_{16} + \cdots + 20802737295857 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 28307421700 \beta_{19} - 24408905085 \beta_{18} - 13937123781 \beta_{17} - 119534120683 \beta_{16} + \cdots - 50750283611333 \) 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
5.63982 + 0.438664i
5.63982 − 0.438664i
5.22286 + 2.17296i
5.22286 − 2.17296i
4.12326 + 3.87282i
4.12326 − 3.87282i
2.74854 + 4.94424i
2.74854 − 4.94424i
0.858917 + 5.59127i
0.858917 − 5.59127i
−1.78549 + 5.36768i
−1.78549 − 5.36768i
−2.73024 + 4.95437i
−2.73024 − 4.95437i
−2.94053 + 4.83252i
−2.94053 − 4.83252i
−5.25771 + 2.08722i
−5.25771 − 2.08722i
−5.37943 + 1.74979i
−5.37943 − 1.74979i
−5.63982 − 0.438664i 17.0419i 31.6151 + 4.94797i 0 7.47566 − 96.1131i −191.207 −176.133 − 41.7741i −47.4256 0
101.2 −5.63982 + 0.438664i − 17.0419i 31.6151 − 4.94797i 0 7.47566 + 96.1131i −191.207 −176.133 + 41.7741i −47.4256 0
101.3 −5.22286 − 2.17296i 8.95730i 22.5565 + 22.6981i 0 19.4639 − 46.7827i 179.198 −68.4872 − 167.563i 162.767 0
101.4 −5.22286 + 2.17296i − 8.95730i 22.5565 − 22.6981i 0 19.4639 + 46.7827i 179.198 −68.4872 + 167.563i 162.767 0
101.5 −4.12326 − 3.87282i − 18.2848i 2.00256 + 31.9373i 0 −70.8136 + 75.3929i −2.25573 115.430 − 139.441i −91.3327 0
101.6 −4.12326 + 3.87282i 18.2848i 2.00256 − 31.9373i 0 −70.8136 − 75.3929i −2.25573 115.430 + 139.441i −91.3327 0
101.7 −2.74854 − 4.94424i 23.5470i −16.8910 + 27.1789i 0 116.422 − 64.7200i −39.5751 180.805 + 8.81071i −311.462 0
101.8 −2.74854 + 4.94424i − 23.5470i −16.8910 − 27.1789i 0 116.422 + 64.7200i −39.5751 180.805 − 8.81071i −311.462 0
101.9 −0.858917 − 5.59127i − 2.58812i −30.5245 + 9.60487i 0 −14.4709 + 2.22298i −27.4015 79.9214 + 162.421i 236.302 0
101.10 −0.858917 + 5.59127i 2.58812i −30.5245 − 9.60487i 0 −14.4709 − 2.22298i −27.4015 79.9214 − 162.421i 236.302 0
101.11 1.78549 − 5.36768i − 29.6034i −25.6240 − 19.1679i 0 −158.902 − 52.8566i 90.3364 −148.639 + 103.318i −633.360 0
101.12 1.78549 + 5.36768i 29.6034i −25.6240 + 19.1679i 0 −158.902 + 52.8566i 90.3364 −148.639 − 103.318i −633.360 0
101.13 2.73024 − 4.95437i 18.7068i −17.0916 − 27.0532i 0 92.6806 + 51.0741i 207.924 −180.696 + 10.8164i −106.946 0
101.14 2.73024 + 4.95437i − 18.7068i −17.0916 + 27.0532i 0 92.6806 − 51.0741i 207.924 −180.696 − 10.8164i −106.946 0
101.15 2.94053 − 4.83252i 0.457817i −14.7065 − 28.4204i 0 2.21241 + 1.34622i −195.837 −180.587 − 12.5013i 242.790 0
101.16 2.94053 + 4.83252i − 0.457817i −14.7065 + 28.4204i 0 2.21241 − 1.34622i −195.837 −180.587 + 12.5013i 242.790 0
101.17 5.25771 − 2.08722i − 10.4354i 23.2871 − 21.9480i 0 −21.7810 − 54.8666i 66.0164 76.6266 − 164.001i 134.101 0
101.18 5.25771 + 2.08722i 10.4354i 23.2871 + 21.9480i 0 −21.7810 + 54.8666i 66.0164 76.6266 + 164.001i 134.101 0
101.19 5.37943 − 1.74979i 25.2673i 25.8765 − 18.8258i 0 44.2124 + 135.923i −185.199 106.259 − 146.550i −395.434 0
101.20 5.37943 + 1.74979i − 25.2673i 25.8765 + 18.8258i 0 44.2124 − 135.923i −185.199 106.259 + 146.550i −395.434 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.c ✓ 20
4.b odd 2 1 800.6.d.d 20
5.b even 2 1 200.6.d.d yes 20
5.c odd 4 2 200.6.f.d 40
8.b even 2 1 inner 200.6.d.c ✓ 20
8.d odd 2 1 800.6.d.d 20
20.d odd 2 1 800.6.d.b 20
20.e even 4 2 800.6.f.d 40
40.e odd 2 1 800.6.d.b 20
40.f even 2 1 200.6.d.d yes 20
40.i odd 4 2 200.6.f.d 40
40.k even 4 2 800.6.f.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.6.d.c ✓ 20 1.a even 1 1 trivial
200.6.d.c ✓ 20 8.b even 2 1 inner
200.6.d.d yes 20 5.b even 2 1
200.6.d.d yes 20 40.f even 2 1
200.6.f.d 40 5.c odd 4 2
200.6.f.d 40 40.i odd 4 2
800.6.d.b 20 20.d odd 2 1
800.6.d.b 20 40.e odd 2 1
800.6.d.d 20 4.b odd 2 1
800.6.d.d 20 8.d odd 2 1
800.6.f.d 40 20.e even 4 2
800.6.f.d 40 40.k even 4 2

Hecke kernels

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

\( T_{3}^{20} + 3240 T_{3}^{18} + 4338819 T_{3}^{16} + 3123786848 T_{3}^{14} + 1315141752042 T_{3}^{12} + \cdots + 12\!\cdots\!83 \) Copy content Toggle raw display
\( T_{7}^{10} + 98 T_{7}^{9} - 94896 T_{7}^{8} - 7891024 T_{7}^{7} + 2920561232 T_{7}^{6} + \cdots + 37\!\cdots\!52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 12\!\cdots\!83 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 37\!\cdots\!52)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 32\!\cdots\!75 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 29\!\cdots\!99)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 38\!\cdots\!23 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 20\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 85\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 49\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 13\!\cdots\!75)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 51\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 56\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 17\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 30\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 67\!\cdots\!63 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 49\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 32\!\cdots\!17)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 50\!\cdots\!47 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 15\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 76\!\cdots\!96)^{2} \) Copy content Toggle raw display
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