↑

Properties

Label 129.4.e.a
Level $129$
Weight $4$
Character orbit 129.e
Analytic conductor $7.611$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [129,4,Mod(49,129)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(129, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 129.e (of order \(3\), degree \(2\), 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: \(7.61124639074\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 55 x^{18} - 16 x^{17} + 1984 x^{16} - 207 x^{15} + 40040 x^{14} + 12618 x^{13} + \cdots + 2286144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} - 3 \beta_{4} q^{3} + (\beta_{2} + 3) q^{4} + (\beta_{12} - \beta_{9} + \beta_{4}) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{15} - \beta_{10} - 3 \beta_{4} + \cdots + 3) q^{7}+ \cdots + (9 \beta_{4} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} - 3 \beta_{4} q^{3} + (\beta_{2} + 3) q^{4} + (\beta_{12} - \beta_{9} + \beta_{4}) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{15} - \beta_{10} - 3 \beta_{4} + \cdots + 3) q^{7}+ \cdots + ( - 9 \beta_{15} - 9 \beta_{10} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 30 q^{3} + 58 q^{4} + 6 q^{5} - 3 q^{6} + 24 q^{7} + 84 q^{8} - 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} - 30 q^{3} + 58 q^{4} + 6 q^{5} - 3 q^{6} + 24 q^{7} + 84 q^{8} - 90 q^{9} + 36 q^{10} - 68 q^{11} - 87 q^{12} + 116 q^{13} - 48 q^{14} + 18 q^{15} + 130 q^{16} + 92 q^{17} - 9 q^{18} + 76 q^{19} + 151 q^{20} - 144 q^{21} + 312 q^{22} + 254 q^{23} - 126 q^{24} - 352 q^{25} - 103 q^{26} + 540 q^{27} + 497 q^{28} - 180 q^{29} + 108 q^{30} - 30 q^{31} - 36 q^{32} + 102 q^{33} - 106 q^{34} - 820 q^{35} - 261 q^{36} + 580 q^{37} + 182 q^{38} - 696 q^{39} - 900 q^{40} - 1204 q^{41} + 288 q^{42} + 188 q^{43} - 802 q^{44} - 108 q^{45} + 61 q^{46} + 896 q^{47} - 195 q^{48} - 184 q^{49} - 381 q^{50} - 552 q^{51} + 730 q^{52} - 36 q^{53} + 54 q^{54} + 364 q^{55} + 1282 q^{56} + 228 q^{57} + 315 q^{58} + 2200 q^{59} + 453 q^{60} + 696 q^{61} - 1951 q^{62} + 216 q^{63} - 1540 q^{64} - 640 q^{65} - 468 q^{66} - 780 q^{67} - 527 q^{68} + 762 q^{69} - 540 q^{70} + 814 q^{71} - 378 q^{72} + 26 q^{73} + 1034 q^{74} + 2112 q^{75} + 1555 q^{76} - 3264 q^{77} + 618 q^{78} + 686 q^{79} + 3826 q^{80} - 810 q^{81} - 6118 q^{82} + 838 q^{83} - 2982 q^{84} - 952 q^{85} - 3661 q^{86} + 1080 q^{87} + 2106 q^{88} + 490 q^{89} - 648 q^{90} - 566 q^{91} + 35 q^{92} - 90 q^{93} - 9598 q^{94} + 2384 q^{95} + 54 q^{96} + 3768 q^{97} + 1821 q^{98} + 306 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 55 x^{18} - 16 x^{17} + 1984 x^{16} - 207 x^{15} + 40040 x^{14} + 12618 x^{13} + \cdots + 2286144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!35 \nu^{19} + \cdots - 13\!\cdots\!98 ) / 12\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 46\!\cdots\!53 \nu^{19} + \cdots - 61\!\cdots\!20 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24\!\cdots\!70 \nu^{19} + \cdots - 82\!\cdots\!48 ) / 27\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 44\!\cdots\!20 \nu^{19} + \cdots - 17\!\cdots\!68 ) / 27\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 51\!\cdots\!92 \nu^{19} + \cdots - 55\!\cdots\!48 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 97\!\cdots\!97 \nu^{19} + \cdots - 26\!\cdots\!64 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!94 \nu^{19} + \cdots + 90\!\cdots\!64 ) / 27\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!77 \nu^{19} + \cdots - 38\!\cdots\!20 ) / 26\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27\!\cdots\!30 \nu^{19} + \cdots + 91\!\cdots\!16 ) / 27\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 32\!\cdots\!23 \nu^{19} + \cdots + 66\!\cdots\!92 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 97\!\cdots\!03 \nu^{19} + \cdots - 12\!\cdots\!80 ) / 54\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 98\!\cdots\!57 \nu^{19} + \cdots - 12\!\cdots\!36 ) / 54\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!95 \nu^{19} + \cdots - 28\!\cdots\!24 ) / 54\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 48\!\cdots\!90 \nu^{19} + \cdots - 74\!\cdots\!24 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 10\!\cdots\!18 \nu^{19} + \cdots - 11\!\cdots\!80 ) / 26\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 82\!\cdots\!00 \nu^{19} + \cdots + 13\!\cdots\!00 ) / 18\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 36\!\cdots\!34 \nu^{19} + \cdots + 51\!\cdots\!88 ) / 54\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 13\!\cdots\!64 \nu^{19} + \cdots + 97\!\cdots\!32 ) / 18\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 11\beta_{4} - \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{7} - 19\beta_{3} - 19\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} - \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} - 25 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{18} + 2 \beta_{17} + \beta_{15} + 2 \beta_{14} - 34 \beta_{13} + 30 \beta_{12} - 4 \beta_{10} + \cdots + 94 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{19} + 37 \beta_{16} + 31 \beta_{11} - 78 \beta_{9} - 14 \beta_{8} - 39 \beta_{7} - 30 \beta_{6} + \cdots + 4524 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 50 \beta_{19} + 50 \beta_{18} - 79 \beta_{17} - 79 \beta_{16} - 61 \beta_{15} - 85 \beta_{14} + \cdots + 965 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 69 \beta_{18} + 1060 \beta_{17} + 753 \beta_{15} + 760 \beta_{14} + 1259 \beta_{13} + 2399 \beta_{12} + \cdots - 103542 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1755 \beta_{19} + 2316 \beta_{16} + 2706 \beta_{11} - 17403 \beta_{9} + 7086 \beta_{8} + 25977 \beta_{7} + \cdots - 96204 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3015 \beta_{19} - 3015 \beta_{18} - 27984 \beta_{17} - 27984 \beta_{16} - 17799 \beta_{15} + \cdots - 38280 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 53847 \beta_{18} + 60957 \beta_{17} + 72822 \beta_{15} + 77799 \beta_{14} - 684136 \beta_{13} + \cdots + 1794915 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 108330 \beta_{19} + 714562 \beta_{16} + 389296 \beta_{11} - 1827347 \beta_{9} - 940290 \beta_{8} + \cdots + 60799327 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1546522 \beta_{19} + 1546522 \beta_{18} - 1526957 \beta_{17} - 1526957 \beta_{16} - 2097307 \beta_{15} + \cdots + 17817406 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3501697 \beta_{18} + 17975923 \beta_{17} + 9231540 \beta_{15} + 8631295 \beta_{14} + 32121756 \beta_{13} + \cdots - 1458896976 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 42799130 \beta_{19} + 37284592 \beta_{16} + 56986150 \beta_{11} - 210997402 \beta_{9} + \cdots - 1636966786 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 106131888 \beta_{19} - 106131888 \beta_{18} - 449129080 \beta_{17} - 449129080 \beta_{16} + \cdots - 900334052 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1157743680 \beta_{18} + 897242256 \beta_{17} + 1537101264 \beta_{15} + 1497649536 \beta_{14} + \cdots + 30642865347 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 3084072552 \beta_{19} + 11187484017 \beta_{16} + 4204773369 \beta_{11} - 32327121018 \beta_{9} + \cdots + 912826933376 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 30852572481 \beta_{19} + 30852572481 \beta_{18} - 21392622666 \beta_{17} - 21392622666 \beta_{16} + \cdots + 305094318850 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/129\mathbb{Z}\right)^\times\).

\(n\) \(44\) \(46\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−2.47458 + 4.28609i
−1.78336 + 3.08887i
−1.58363 + 2.74293i
−0.633781 + 1.09774i
−0.304861 + 0.528035i
−0.113444 + 0.196491i
1.29167 − 2.23724i
1.29675 − 2.24603i
2.26918 − 3.93033i
2.53606 − 4.39258i
−2.47458 − 4.28609i
−1.78336 − 3.08887i
−1.58363 − 2.74293i
−0.633781 − 1.09774i
−0.304861 − 0.528035i
−0.113444 − 0.196491i
1.29167 + 2.23724i
1.29675 + 2.24603i
2.26918 + 3.93033i
2.53606 + 4.39258i
−4.94915 −1.50000 + 2.59808i 16.4941 5.97490 − 10.3488i 7.42373 − 12.8583i 3.91029 + 6.77282i −42.0388 −4.50000 − 7.79423i −29.5707 + 51.2179i
49.2 −3.56672 −1.50000 + 2.59808i 4.72150 0.745098 − 1.29055i 5.35008 − 9.26661i −2.93348 − 5.08094i 11.6935 −4.50000 − 7.79423i −2.65756 + 4.60302i
49.3 −3.16726 −1.50000 + 2.59808i 2.03153 −10.2937 + 17.8292i 4.75089 − 8.22878i 14.3776 + 24.9028i 18.9037 −4.50000 − 7.79423i 32.6028 − 56.4697i
49.4 −1.26756 −1.50000 + 2.59808i −6.39329 −3.76862 + 6.52744i 1.90134 − 3.29322i −16.5615 − 28.6854i 18.2444 −4.50000 − 7.79423i 4.77696 − 8.27394i
49.5 −0.609722 −1.50000 + 2.59808i −7.62824 −1.54840 + 2.68190i 0.914583 − 1.58410i 6.21780 + 10.7696i 9.52888 −4.50000 − 7.79423i 0.944091 − 1.63521i
49.6 −0.226888 −1.50000 + 2.59808i −7.94852 7.46984 − 12.9381i 0.340332 − 0.589473i 11.0673 + 19.1692i 3.61853 −4.50000 − 7.79423i −1.69482 + 2.93551i
49.7 2.58334 −1.50000 + 2.59808i −1.32635 −4.01128 + 6.94774i −3.87501 + 6.71172i −5.34657 − 9.26053i −24.0931 −4.50000 − 7.79423i −10.3625 + 17.9484i
49.8 2.59350 −1.50000 + 2.59808i −1.27377 9.07900 − 15.7253i −3.89025 + 6.73810i −11.1104 − 19.2439i −24.0515 −4.50000 − 7.79423i 23.5464 − 40.7835i
49.9 4.53835 −1.50000 + 2.59808i 12.5967 −6.92490 + 11.9943i −6.80753 + 11.7910i 8.28809 + 14.3554i 20.8613 −4.50000 − 7.79423i −31.4277 + 54.4343i
49.10 5.07211 −1.50000 + 2.59808i 17.7263 6.27805 − 10.8739i −7.60817 + 13.1777i 4.09088 + 7.08561i 49.3331 −4.50000 − 7.79423i 31.8430 − 55.1537i
79.1 −4.94915 −1.50000 − 2.59808i 16.4941 5.97490 + 10.3488i 7.42373 + 12.8583i 3.91029 − 6.77282i −42.0388 −4.50000 + 7.79423i −29.5707 − 51.2179i
79.2 −3.56672 −1.50000 − 2.59808i 4.72150 0.745098 + 1.29055i 5.35008 + 9.26661i −2.93348 + 5.08094i 11.6935 −4.50000 + 7.79423i −2.65756 − 4.60302i
79.3 −3.16726 −1.50000 − 2.59808i 2.03153 −10.2937 − 17.8292i 4.75089 + 8.22878i 14.3776 − 24.9028i 18.9037 −4.50000 + 7.79423i 32.6028 + 56.4697i
79.4 −1.26756 −1.50000 − 2.59808i −6.39329 −3.76862 − 6.52744i 1.90134 + 3.29322i −16.5615 + 28.6854i 18.2444 −4.50000 + 7.79423i 4.77696 + 8.27394i
79.5 −0.609722 −1.50000 − 2.59808i −7.62824 −1.54840 − 2.68190i 0.914583 + 1.58410i 6.21780 − 10.7696i 9.52888 −4.50000 + 7.79423i 0.944091 + 1.63521i
79.6 −0.226888 −1.50000 − 2.59808i −7.94852 7.46984 + 12.9381i 0.340332 + 0.589473i 11.0673 − 19.1692i 3.61853 −4.50000 + 7.79423i −1.69482 − 2.93551i
79.7 2.58334 −1.50000 − 2.59808i −1.32635 −4.01128 − 6.94774i −3.87501 − 6.71172i −5.34657 + 9.26053i −24.0931 −4.50000 + 7.79423i −10.3625 − 17.9484i
79.8 2.59350 −1.50000 − 2.59808i −1.27377 9.07900 + 15.7253i −3.89025 − 6.73810i −11.1104 + 19.2439i −24.0515 −4.50000 + 7.79423i 23.5464 + 40.7835i
79.9 4.53835 −1.50000 − 2.59808i 12.5967 −6.92490 − 11.9943i −6.80753 − 11.7910i 8.28809 − 14.3554i 20.8613 −4.50000 + 7.79423i −31.4277 − 54.4343i
79.10 5.07211 −1.50000 − 2.59808i 17.7263 6.27805 + 10.8739i −7.60817 − 13.1777i 4.09088 − 7.08561i 49.3331 −4.50000 + 7.79423i 31.8430 + 55.1537i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.4.e.a ✓ 20
43.c even 3 1 inner 129.4.e.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.4.e.a ✓ 20 1.a even 1 1 trivial
129.4.e.a ✓ 20 43.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - T_{2}^{9} - 54 T_{2}^{8} + 35 T_{2}^{7} + 967 T_{2}^{6} - 251 T_{2}^{5} - 6576 T_{2}^{4} + \cdots + 1512 \) acting on \(S_{4}^{\mathrm{new}}(129, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - T^{9} + \cdots + 1512)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 12061552481649)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 62\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 65\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 66\!\cdots\!29 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 24\!\cdots\!07)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 69\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 68\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 43\!\cdots\!21)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 76\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 15\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 38\!\cdots\!47)^{2} \) Copy content Toggle raw display
show more
show less