↑

Properties

Label 115.3.c.c
Level $115$
Weight $3$
Character orbit 115.c
Analytic conductor $3.134$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,3,Mod(114,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.114");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 115.c (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: \(3.13352304014\)
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: \( x^{20} + 6 x^{18} - 827 x^{16} - 12720 x^{14} + 346250 x^{12} + 9668500 x^{10} + 216406250 x^{8} + \cdots + 95367431640625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{4} q^{3} + ( - \beta_{8} + \beta_{7} - 3) q^{4} + \beta_{13} q^{5} + (\beta_{8} + \beta_{2}) q^{6} - \beta_{12} q^{7} + (\beta_{6} - 2 \beta_{3}) q^{8} + (\beta_{9} - \beta_{7} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{4} q^{3} + ( - \beta_{8} + \beta_{7} - 3) q^{4} + \beta_{13} q^{5} + (\beta_{8} + \beta_{2}) q^{6} - \beta_{12} q^{7} + (\beta_{6} - 2 \beta_{3}) q^{8} + (\beta_{9} - \beta_{7} - 4) q^{9} + ( - \beta_{14} + \beta_{12}) q^{10} + (\beta_{18} - \beta_{13}) q^{11} + (\beta_{10} - \beta_{5} + 2 \beta_{4}) q^{12} + ( - \beta_{6} + \beta_{5} - \beta_{4}) q^{13} + ( - \beta_{19} + \beta_{16} + \cdots - \beta_{11}) q^{14}+ \cdots + (\beta_{19} - 4 \beta_{18} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 56 q^{4} - 8 q^{6} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 56 q^{4} - 8 q^{6} - 72 q^{9} + 88 q^{16} + 44 q^{24} - 12 q^{25} - 56 q^{26} + 236 q^{31} + 92 q^{35} - 32 q^{36} - 168 q^{39} + 124 q^{41} - 248 q^{46} + 88 q^{49} + 200 q^{50} - 196 q^{54} + 268 q^{55} + 56 q^{59} - 28 q^{64} + 376 q^{69} - 636 q^{70} - 196 q^{71} + 428 q^{75} - 988 q^{81} - 284 q^{85} + 276 q^{94} + 184 q^{95} - 264 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 6 x^{18} - 827 x^{16} - 12720 x^{14} + 346250 x^{12} + 9668500 x^{10} + 216406250 x^{8} + \cdots + 95367431640625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 93313 \nu^{19} - 91514253 \nu^{17} - 465822024 \nu^{15} + 39948006360 \nu^{13} + \cdots + 17\!\cdots\!75 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12892 \nu^{18} - 3304523 \nu^{16} - 33609184 \nu^{14} + 2216496260 \nu^{12} + \cdots + 39\!\cdots\!25 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1472283 \nu^{18} - 63086823 \nu^{16} + 625262416 \nu^{14} - 789210240 \nu^{12} + \cdots + 10\!\cdots\!25 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 840741 \nu^{18} - 12400071 \nu^{16} + 659830932 \nu^{14} - 18961656980 \nu^{12} + \cdots - 67\!\cdots\!75 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1557503 \nu^{18} - 13771857 \nu^{16} + 1444103144 \nu^{14} - 64591673160 \nu^{12} + \cdots - 14\!\cdots\!25 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7642359 \nu^{18} - 133729779 \nu^{16} + 16940242768 \nu^{14} + 135820276480 \nu^{12} + \cdots - 20\!\cdots\!75 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 293852 \nu^{18} + 5490013 \nu^{16} + 48565604 \nu^{14} - 885615060 \nu^{12} + \cdots - 49\!\cdots\!75 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 393036 \nu^{18} - 1027591 \nu^{16} + 125680772 \nu^{14} + 2527975420 \nu^{12} + \cdots - 84\!\cdots\!75 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 40772 \nu^{18} - 887243 \nu^{16} - 23978444 \nu^{14} + 178737660 \nu^{12} + \cdots + 12\!\cdots\!25 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 740133 \nu^{18} - 20325077 \nu^{16} - 793107116 \nu^{14} + 4327433740 \nu^{12} + \cdots + 24\!\cdots\!75 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 658377 \nu^{19} + 23203763 \nu^{17} - 1503363696 \nu^{15} + 7163348640 \nu^{13} + \cdots - 15\!\cdots\!25 \nu ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 31550041 \nu^{19} - 325579621 \nu^{17} + 3558192032 \nu^{15} + 419123861520 \nu^{13} + \cdots + 73\!\cdots\!75 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{19} + 6 \nu^{17} - 827 \nu^{15} - 12720 \nu^{13} + 346250 \nu^{11} + \cdots + 915527343750 \nu ) / 3814697265625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19160719 \nu^{19} - 643192439 \nu^{17} - 15135563512 \nu^{15} + \cdots + 17\!\cdots\!25 \nu ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21754533 \nu^{19} - 740551573 \nu^{17} + 21853899416 \nu^{15} + \cdots - 11\!\cdots\!25 \nu ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12484537 \nu^{19} + 292486597 \nu^{17} - 2338220224 \nu^{15} - 116555750640 \nu^{13} + \cdots + 88\!\cdots\!25 \nu ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 64997549 \nu^{19} - 905247831 \nu^{17} - 98713043648 \nu^{15} + 1384098736720 \nu^{13} + \cdots + 37\!\cdots\!25 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 7017709 \nu^{19} - 135470621 \nu^{17} - 228090968 \nu^{15} + 98071676520 \nu^{13} + \cdots + 18\!\cdots\!75 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 211396133 \nu^{19} - 2024318827 \nu^{17} - 82806947616 \nu^{15} - 627531951760 \nu^{13} + \cdots + 11\!\cdots\!25 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{17} + \beta_{16} - \beta_{15} + \beta_{14} + 2\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -3\beta_{9} + \beta_{8} - 3\beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 20 \beta_{18} + 24 \beta_{17} + 16 \beta_{16} + 14 \beta_{15} - 24 \beta_{14} - 15 \beta_{13} + \cdots + 22 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 14 \beta_{10} - 10 \beta_{9} - 24 \beta_{8} + 12 \beta_{7} - 10 \beta_{6} + 6 \beta_{5} + 58 \beta_{4} + \cdots + 177 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 460 \beta_{19} - 160 \beta_{18} - 291 \beta_{17} - 429 \beta_{16} - 731 \beta_{15} - 179 \beta_{14} + \cdots - 208 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 384 \beta_{10} - 371 \beta_{9} + 1381 \beta_{8} - 343 \beta_{7} - 67 \beta_{6} + 671 \beta_{5} + \cdots + 2745 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1860 \beta_{19} - 6980 \beta_{18} + 6324 \beta_{17} - 8724 \beta_{16} - 6156 \beta_{15} + \cdots + 26532 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 13132 \beta_{10} + 16096 \beta_{9} + 50136 \beta_{8} - 39128 \beta_{7} - 13788 \beta_{6} + \cdots - 17071 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 18420 \beta_{19} - 36340 \beta_{18} + 137679 \beta_{17} - 249419 \beta_{16} - 121521 \beta_{15} + \cdots - 430618 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 108720 \beta_{10} + 508157 \beta_{9} + 189741 \beta_{8} + 1142497 \beta_{7} - 7381 \beta_{6} + \cdots - 556519 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5826500 \beta_{19} + 7047880 \beta_{18} - 1870396 \beta_{17} - 5052364 \beta_{16} + \cdots + 6719162 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 7114 \beta_{10} + 323830 \beta_{9} + 15563096 \beta_{8} - 10632548 \beta_{7} + 5170370 \beta_{6} + \cdots - 163410703 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 149146040 \beta_{19} + 222018940 \beta_{18} + 312520189 \beta_{17} - 168860609 \beta_{16} + \cdots - 309792268 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 137321296 \beta_{10} + 696625229 \beta_{9} + 78577401 \beta_{8} + 299611397 \beta_{7} + \cdots + 1229580385 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 3469239160 \beta_{19} + 9591773320 \beta_{18} - 8963219496 \beta_{17} + 872126696 \beta_{16} + \cdots - 939961128 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 118904152 \beta_{10} - 5340663424 \beta_{9} - 1342035744 \beta_{8} + 4767860512 \beta_{7} + \cdots - 46334847871 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 251826960120 \beta_{19} + 108956599560 \beta_{18} + 129429472159 \beta_{17} + \cdots - 80591201478 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 100993234960 \beta_{10} + 163348625277 \beta_{9} - 648230056039 \beta_{8} + 52548058437 \beta_{7} + \cdots - 1006743753039 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 1628423366600 \beta_{19} + 4691192382580 \beta_{18} - 1950656254816 \beta_{17} + \cdots - 6561788783698 \beta_1 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
114.1
3.24366 − 3.80508i
−3.24366 + 3.80508i
4.94294 + 0.753214i
−4.94294 − 0.753214i
4.79117 + 1.42993i
−4.79117 − 1.42993i
0.993286 − 4.90035i
−0.993286 + 4.90035i
1.45055 + 4.78497i
−1.45055 − 4.78497i
1.45055 − 4.78497i
−1.45055 + 4.78497i
0.993286 + 4.90035i
−0.993286 − 4.90035i
4.79117 − 1.42993i
−4.79117 + 1.42993i
4.94294 − 0.753214i
−4.94294 + 0.753214i
3.24366 + 3.80508i
−3.24366 − 3.80508i
− 3.64975i − 3.06547i −9.32065 −3.24366 − 3.80508i −11.1882 12.1877 19.4190i −0.397130 −13.8876 + 11.8385i
114.2 − 3.64975i − 3.06547i −9.32065 3.24366 + 3.80508i −11.1882 −12.1877 19.4190i −0.397130 13.8876 − 11.8385i
114.3 − 3.38977i 3.81595i −7.49055 −4.94294 + 0.753214i 12.9352 −7.85531 11.8322i −5.56149 2.55323 + 16.7554i
114.4 − 3.38977i 3.81595i −7.49055 4.94294 − 0.753214i 12.9352 7.85531 11.8322i −5.56149 −2.55323 − 16.7554i
114.5 − 2.13699i − 3.84414i −0.566730 −4.79117 + 1.42993i −8.21490 −2.33853 − 7.33687i −5.77745 3.05575 + 10.2387i
114.6 − 2.13699i − 3.84414i −0.566730 4.79117 − 1.42993i −8.21490 2.33853 − 7.33687i −5.77745 −3.05575 − 10.2387i
114.7 − 2.10506i 1.12745i −0.431264 −0.993286 − 4.90035i 2.37335 −4.72229 − 7.51239i 7.72885 −10.3155 + 2.09092i
114.8 − 2.10506i 1.12745i −0.431264 0.993286 + 4.90035i 2.37335 4.72229 − 7.51239i 7.72885 10.3155 − 2.09092i
114.9 − 0.436812i 4.79508i 3.80920 −1.45055 + 4.78497i 2.09455 −5.38385 − 3.41115i −13.9928 2.09013 + 0.633615i
114.10 − 0.436812i 4.79508i 3.80920 1.45055 − 4.78497i 2.09455 5.38385 − 3.41115i −13.9928 −2.09013 − 0.633615i
114.11 0.436812i − 4.79508i 3.80920 −1.45055 − 4.78497i 2.09455 −5.38385 3.41115i −13.9928 2.09013 − 0.633615i
114.12 0.436812i − 4.79508i 3.80920 1.45055 + 4.78497i 2.09455 5.38385 3.41115i −13.9928 −2.09013 + 0.633615i
114.13 2.10506i − 1.12745i −0.431264 −0.993286 + 4.90035i 2.37335 −4.72229 7.51239i 7.72885 −10.3155 − 2.09092i
114.14 2.10506i − 1.12745i −0.431264 0.993286 − 4.90035i 2.37335 4.72229 7.51239i 7.72885 10.3155 + 2.09092i
114.15 2.13699i 3.84414i −0.566730 −4.79117 − 1.42993i −8.21490 −2.33853 7.33687i −5.77745 3.05575 − 10.2387i
114.16 2.13699i 3.84414i −0.566730 4.79117 + 1.42993i −8.21490 2.33853 7.33687i −5.77745 −3.05575 + 10.2387i
114.17 3.38977i − 3.81595i −7.49055 −4.94294 − 0.753214i 12.9352 −7.85531 − 11.8322i −5.56149 2.55323 − 16.7554i
114.18 3.38977i − 3.81595i −7.49055 4.94294 + 0.753214i 12.9352 7.85531 − 11.8322i −5.56149 −2.55323 + 16.7554i
114.19 3.64975i 3.06547i −9.32065 −3.24366 + 3.80508i −11.1882 12.1877 − 19.4190i −0.397130 −13.8876 − 11.8385i
114.20 3.64975i 3.06547i −9.32065 3.24366 − 3.80508i −11.1882 −12.1877 − 19.4190i −0.397130 13.8876 + 11.8385i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 114.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.b odd 2 1 inner
115.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.3.c.c ✓ 20
5.b even 2 1 inner 115.3.c.c ✓ 20
5.c odd 4 2 575.3.d.i 20
23.b odd 2 1 inner 115.3.c.c ✓ 20
115.c odd 2 1 inner 115.3.c.c ✓ 20
115.e even 4 2 575.3.d.i 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.c.c ✓ 20 1.a even 1 1 trivial
115.3.c.c ✓ 20 5.b even 2 1 inner
115.3.c.c ✓ 20 23.b odd 2 1 inner
115.3.c.c ✓ 20 115.c odd 2 1 inner
575.3.d.i 20 5.c odd 4 2
575.3.d.i 20 115.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(115, [\chi])\):

\( T_{2}^{10} + 34T_{2}^{8} + 403T_{2}^{6} + 1955T_{2}^{4} + 3456T_{2}^{2} + 591 \) Copy content Toggle raw display
\( T_{7}^{10} - 267T_{7}^{8} + 22025T_{7}^{6} - 718600T_{7}^{4} + 9238500T_{7}^{2} - 32400000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 34 T^{8} + \cdots + 591)^{2} \) Copy content Toggle raw display
$3$ \( (T^{10} + 63 T^{8} + \cdots + 59100)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( (T^{10} - 267 T^{8} + \cdots - 32400000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 733 T^{8} + \cdots + 51431775000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 927 T^{8} + \cdots + 42880787484)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 1213 T^{8} + \cdots - 16900000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 5091006975000)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{5} - 1249 T^{3} + \cdots - 1258776)^{4} \) Copy content Toggle raw display
$31$ \( (T^{5} - 59 T^{4} + \cdots - 124900)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 14923065600000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 31 T^{4} + \cdots + 1141128)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 15\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 14 T^{4} + \cdots - 9754080)^{4} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 49 T^{4} + \cdots - 32741802)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 20\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less