Properties

Label 9.10.c.a
Level $9$
Weight $10$
Character orbit 9.c
Analytic conductor $4.635$
Analytic rank $0$
Dimension $16$
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] = [9,10,Mod(4,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 9.c (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: \(4.63532252547\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 1984 x^{14} - 13748 x^{13} + 1552498 x^{12} - 9136628 x^{11} + 609566956 x^{10} - 2964409064 x^{9} + 126210674407 x^{8} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{28}\cdot 17^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} + \beta_{2} + 2) q^{2} + (\beta_{5} - 18 \beta_{3} + 9) q^{3} + (\beta_{11} + \beta_{5} - 224 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{4} + (\beta_{13} - \beta_{11} - \beta_{8} - 2 \beta_{7} + 57 \beta_{3} - 1) q^{5} + ( - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + 9 \beta_{3} + \cdots + 144) q^{6}+ \cdots + ( - 2 \beta_{15} + 7 \beta_{13} - 2 \beta_{12} + 10 \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 769) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} + \beta_{2} + 2) q^{2} + (\beta_{5} - 18 \beta_{3} + 9) q^{3} + (\beta_{11} + \beta_{5} - 224 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{4} + (\beta_{13} - \beta_{11} - \beta_{8} - 2 \beta_{7} + 57 \beta_{3} - 1) q^{5} + ( - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + 9 \beta_{3} + \cdots + 144) q^{6}+ \cdots + (27477 \beta_{15} + 30459 \beta_{14} + 219963 \beta_{13} + \cdots + 129253530) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 15 q^{2} - 3 q^{3} - 1793 q^{4} + 453 q^{5} + 2439 q^{6} - 343 q^{7} - 14478 q^{8} - 15669 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 15 q^{2} - 3 q^{3} - 1793 q^{4} + 453 q^{5} + 2439 q^{6} - 343 q^{7} - 14478 q^{8} - 15669 q^{9} + 1020 q^{10} + 99150 q^{11} - 241212 q^{12} + 32435 q^{13} + 394824 q^{14} + 723843 q^{15} - 328193 q^{16} - 831078 q^{17} - 1039500 q^{18} - 170554 q^{19} + 1855164 q^{20} - 503475 q^{21} + 529359 q^{22} + 1064559 q^{23} - 2686059 q^{24} - 2293229 q^{25} + 2436312 q^{26} - 6176520 q^{27} + 1225724 q^{28} - 1309053 q^{29} + 30713544 q^{30} - 2359819 q^{31} + 5760063 q^{32} - 19931472 q^{33} + 981801 q^{34} - 31066554 q^{35} - 48894093 q^{36} + 16391516 q^{37} + 39490203 q^{38} + 74528535 q^{39} - 16760496 q^{40} + 54747318 q^{41} + 25242354 q^{42} + 15249608 q^{43} - 332509926 q^{44} - 139865967 q^{45} + 2390520 q^{46} + 156295545 q^{47} + 312045027 q^{48} + 15239583 q^{49} + 315590163 q^{50} + 3098385 q^{51} - 19773358 q^{52} - 525516228 q^{53} - 591710103 q^{54} - 7579770 q^{55} + 470339790 q^{56} + 581271465 q^{57} + 55408560 q^{58} + 307774074 q^{59} - 430082172 q^{60} + 69192125 q^{61} - 914436924 q^{62} - 693544725 q^{63} - 403588478 q^{64} + 482470359 q^{65} + 1912534074 q^{66} + 14328044 q^{67} + 915409575 q^{68} - 724756329 q^{69} - 229271934 q^{70} - 1239601392 q^{71} - 1729340037 q^{72} + 598613198 q^{73} + 1022736000 q^{74} + 1477690413 q^{75} + 119954093 q^{76} + 717995541 q^{77} + 243023958 q^{78} + 30257531 q^{79} - 2927826528 q^{80} - 1055378241 q^{81} - 202376022 q^{82} + 1176168291 q^{83} + 1383634866 q^{84} + 4818366 q^{85} + 1426944009 q^{86} + 617958567 q^{87} + 911312427 q^{88} - 3317041296 q^{89} - 981417492 q^{90} - 739230122 q^{91} - 76813998 q^{92} - 552731577 q^{93} - 1954316784 q^{94} - 391400652 q^{95} + 1854064512 q^{96} - 267311278 q^{97} + 4827300318 q^{98} + 1672014609 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 1984 x^{14} - 13748 x^{13} + 1552498 x^{12} - 9136628 x^{11} + 609566956 x^{10} - 2964409064 x^{9} + 126210674407 x^{8} + \cdots + 13\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5356486831063 \nu^{14} + 37495407817441 \nu^{13} + \cdots + 13\!\cdots\!25 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 96\!\cdots\!97 \nu^{15} + \cdots - 19\!\cdots\!75 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\!\cdots\!94 \nu^{15} + \cdots - 44\!\cdots\!25 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!89 \nu^{15} + \cdots - 70\!\cdots\!70 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!78 \nu^{15} + \cdots + 28\!\cdots\!25 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 65\!\cdots\!08 \nu^{15} + \cdots + 51\!\cdots\!25 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!68 \nu^{15} + \cdots - 34\!\cdots\!75 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!25 \nu^{15} + \cdots - 46\!\cdots\!25 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31\!\cdots\!39 \nu^{15} + \cdots - 15\!\cdots\!25 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\!\cdots\!75 \nu^{15} + \cdots - 83\!\cdots\!95 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 53\!\cdots\!97 \nu^{15} + \cdots - 12\!\cdots\!25 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 83\!\cdots\!37 \nu^{15} + \cdots + 15\!\cdots\!75 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 86\!\cdots\!95 \nu^{15} + \cdots - 24\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!02 \nu^{15} + \cdots + 33\!\cdots\!25 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 49\!\cdots\!17 \nu^{15} + \cdots - 14\!\cdots\!50 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{3} - 2\beta_{2} - 2\beta _1 - 732 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + 3 \beta_{14} - \beta_{13} - \beta_{11} + \beta_{10} + 3 \beta_{9} - \beta_{8} + 21 \beta_{7} + 5 \beta_{6} - 35 \beta_{5} + 639 \beta_{3} + 2451 \beta_{2} + 1224 \beta _1 - 3617 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 71 \beta_{15} + 15 \beta_{14} - 11 \beta_{13} - 18 \beta_{12} + 28 \beta_{11} - 79 \beta_{10} + 19 \beta_{9} - 49 \beta_{8} - 2918 \beta_{7} - 1646 \beta_{6} + 636 \beta_{5} + 22 \beta_{4} + 1470 \beta_{3} + 4974 \beta_{2} + \cdots + 897175 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2597 \beta_{15} - 5538 \beta_{14} + 2565 \beta_{13} + 1173 \beta_{12} + 13765 \beta_{11} - 2552 \beta_{10} - 7470 \beta_{9} + 1788 \beta_{8} - 75694 \beta_{7} - 19214 \beta_{6} + 103638 \beta_{5} + \cdots + 8273308 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 59362 \beta_{15} - 14301 \beta_{14} + 11318 \beta_{13} + 18669 \beta_{12} - 5322 \beta_{11} + 55799 \beta_{10} - 19485 \beta_{9} + 43469 \beta_{8} + 1829285 \beta_{7} + 859917 \beta_{6} + \cdots - 415708453 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 523291 \beta_{15} + 981438 \beta_{14} - 622367 \beta_{13} - 346479 \beta_{12} - 4411878 \beta_{11} + 597784 \beta_{10} + 1581226 \beta_{9} - 238192 \beta_{8} + 19136342 \beta_{7} + \cdots - 2077255935 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 116464225 \beta_{15} + 30786999 \beta_{14} - 26436357 \beta_{13} - 42038088 \beta_{12} - 20267384 \beta_{11} - 97943099 \beta_{10} + 43904691 \beta_{9} + \cdots + 611209242436 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 279246943 \beta_{15} - 511110492 \beta_{14} + 377511959 \beta_{13} + 219217281 \beta_{12} + 3306495657 \beta_{11} - 388643470 \beta_{10} - 915326232 \beta_{9} + \cdots + 1468767926033 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 23382044870 \beta_{15} - 6731658981 \beta_{14} + 6052768810 \beta_{13} + 9399390081 \beta_{12} + 10162381806 \beta_{11} + 18218914951 \beta_{10} + \cdots - 102684755382075 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 421814837794 \beta_{15} + 800497372317 \beta_{14} - 611284386222 \beta_{13} - 359552917437 \beta_{12} - 6718896428063 \beta_{11} + 734819209033 \beta_{10} + \cdots - 29\!\cdots\!48 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 13590951687380 \beta_{15} + 4255343129922 \beta_{14} - 3869425371628 \beta_{13} - 5982991732530 \beta_{12} - 9606125843292 \beta_{11} + \cdots + 52\!\cdots\!97 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 68035040830958 \beta_{15} - 139988293379052 \beta_{14} + 100962598252486 \beta_{13} + 59618592120006 \beta_{12} + \cdots + 62\!\cdots\!31 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 23\!\cdots\!22 \beta_{15} + \cdots - 81\!\cdots\!68 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 31\!\cdots\!09 \beta_{15} + \cdots - 39\!\cdots\!01 ) / 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 23.8209i
0.500000 + 15.6774i
0.500000 + 13.2694i
0.500000 0.103648i
0.500000 3.55897i
0.500000 9.36376i
0.500000 19.2639i
0.500000 22.2094i
0.500000 23.8209i
0.500000 15.6774i
0.500000 13.2694i
0.500000 + 0.103648i
0.500000 + 3.55897i
0.500000 + 9.36376i
0.500000 + 19.2639i
0.500000 + 22.2094i
−19.8795 34.4322i −14.1649 + 139.579i −534.387 + 925.585i 423.223 733.045i 5087.62 2287.03i 3521.99 + 6100.27i 22136.7 −19281.7 3954.24i −33653.8
4.2 −12.8270 22.2170i −90.7998 106.950i −73.0637 + 126.550i −353.226 + 611.805i −1211.43 + 3389.15i 2284.21 + 3956.36i −9386.09 −3193.79 + 19422.2i 18123.3
4.3 −10.7416 18.6050i 137.668 27.0295i 25.2357 43.7094i 22.4201 38.8327i −1981.66 2270.97i −4672.78 8093.49i −12083.7 18221.8 7442.17i −963.310
4.4 0.839762 + 1.45451i −121.664 + 69.8637i 254.590 440.962i 769.303 1332.47i −203.786 118.292i −2828.47 4899.05i 1715.10 9921.13 16999.8i 2584.13
4.5 3.83216 + 6.63749i 58.6695 + 127.440i 226.629 392.533i −1105.48 + 1914.74i −621.050 + 877.788i 3116.58 + 5398.07i 7398.05 −12798.8 + 14953.6i −16945.4
4.6 8.85925 + 15.3447i 64.7679 124.451i 99.0272 171.520i 369.939 640.753i 2483.46 108.703i 3013.67 + 5219.83i 12581.1 −11293.2 16120.9i 13109.5
4.7 17.4330 + 30.1949i −128.285 56.7966i −351.819 + 609.369i −1003.26 + 1737.70i −521.436 4863.69i −2433.94 4215.71i −6681.68 13231.3 + 14572.3i −69959.4
4.8 19.9839 + 34.6131i 92.3087 + 105.651i −542.712 + 940.005i 1103.58 1911.45i −1812.22 + 5306.41i −2172.76 3763.33i −22918.5 −2641.20 + 19505.0i 88215.0
7.1 −19.8795 + 34.4322i −14.1649 139.579i −534.387 925.585i 423.223 + 733.045i 5087.62 + 2287.03i 3521.99 6100.27i 22136.7 −19281.7 + 3954.24i −33653.8
7.2 −12.8270 + 22.2170i −90.7998 + 106.950i −73.0637 126.550i −353.226 611.805i −1211.43 3389.15i 2284.21 3956.36i −9386.09 −3193.79 19422.2i 18123.3
7.3 −10.7416 + 18.6050i 137.668 + 27.0295i 25.2357 + 43.7094i 22.4201 + 38.8327i −1981.66 + 2270.97i −4672.78 + 8093.49i −12083.7 18221.8 + 7442.17i −963.310
7.4 0.839762 1.45451i −121.664 69.8637i 254.590 + 440.962i 769.303 + 1332.47i −203.786 + 118.292i −2828.47 + 4899.05i 1715.10 9921.13 + 16999.8i 2584.13
7.5 3.83216 6.63749i 58.6695 127.440i 226.629 + 392.533i −1105.48 1914.74i −621.050 877.788i 3116.58 5398.07i 7398.05 −12798.8 14953.6i −16945.4
7.6 8.85925 15.3447i 64.7679 + 124.451i 99.0272 + 171.520i 369.939 + 640.753i 2483.46 + 108.703i 3013.67 5219.83i 12581.1 −11293.2 + 16120.9i 13109.5
7.7 17.4330 30.1949i −128.285 + 56.7966i −351.819 609.369i −1003.26 1737.70i −521.436 + 4863.69i −2433.94 + 4215.71i −6681.68 13231.3 14572.3i −69959.4
7.8 19.9839 34.6131i 92.3087 105.651i −542.712 940.005i 1103.58 + 1911.45i −1812.22 5306.41i −2172.76 + 3763.33i −22918.5 −2641.20 19505.0i 88215.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.10.c.a 16
3.b odd 2 1 27.10.c.a 16
9.c even 3 1 inner 9.10.c.a 16
9.c even 3 1 81.10.a.c 8
9.d odd 6 1 27.10.c.a 16
9.d odd 6 1 81.10.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.10.c.a 16 1.a even 1 1 trivial
9.10.c.a 16 9.c even 3 1 inner
27.10.c.a 16 3.b odd 2 1
27.10.c.a 16 9.d odd 6 1
81.10.a.c 8 9.c even 3 1
81.10.a.d 8 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 15 T^{15} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{15} + \cdots + 22\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} - 453 T^{15} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + 343 T^{15} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{16} - 99150 T^{15} + \cdots + 21\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{16} - 32435 T^{15} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( (T^{8} + 415539 T^{7} + \cdots - 82\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 85277 T^{7} + \cdots - 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 1064559 T^{15} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + 1309053 T^{15} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{16} + 2359819 T^{15} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} - 8195758 T^{7} + \cdots + 99\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 54747318 T^{15} + \cdots + 66\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} - 15249608 T^{15} + \cdots + 14\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{16} - 156295545 T^{15} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{8} + 262758114 T^{7} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 307774074 T^{15} + \cdots + 15\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{16} - 69192125 T^{15} + \cdots + 77\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{16} - 14328044 T^{15} + \cdots + 88\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( (T^{8} + 619800696 T^{7} + \cdots - 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 299306599 T^{7} + \cdots - 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} - 30257531 T^{15} + \cdots + 38\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{16} - 1176168291 T^{15} + \cdots + 79\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + 1658520648 T^{7} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 267311278 T^{15} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
show more
show less