Properties

Label 4923.2.a.l
Level $4923$
Weight $2$
Character orbit 4923.a
Self dual yes
Analytic conductor $39.310$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4923,2,Mod(1,4923)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4923, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4923.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4923 = 3^{2} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4923.a (trivial)

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: yes
Analytic conductor: \(39.3103529151\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + \cdots + 328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 547)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{3} + 2) q^{5} + ( - \beta_{17} + \beta_{16} + \beta_{15} - \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{14} - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{3} + 2) q^{5} + ( - \beta_{17} + \beta_{16} + \beta_{15} - \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots + \beta_1) q^{7}+ \cdots + (2 \beta_{17} + 3 \beta_{16} - \beta_{15} + 5 \beta_{14} - \beta_{13} - 8 \beta_{12} - 2 \beta_{10} + \cdots - 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + \cdots + 328 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 73030826 \nu^{17} - 256120405 \nu^{16} - 1356797574 \nu^{15} + 5113465236 \nu^{14} + 9822519058 \nu^{13} - 40388724797 \nu^{12} + \cdots - 59655036146 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 197952229 \nu^{17} - 3918595304 \nu^{16} + 22600044356 \nu^{15} + 63480744292 \nu^{14} - 406133708186 \nu^{13} + \cdots + 1867879847532 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 377442591 \nu^{17} - 3691417402 \nu^{16} + 2195758068 \nu^{15} + 67864507870 \nu^{14} - 136147624382 \nu^{13} - 463002782106 \nu^{12} + \cdots + 875081238676 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 280914143 \nu^{17} - 701368576 \nu^{16} - 6767277373 \nu^{15} + 16268509010 \nu^{14} + 67383774797 \nu^{13} - 153950006775 \nu^{12} + \cdots - 245184253071 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 918896525 \nu^{17} - 5543735306 \nu^{16} - 8092976468 \nu^{15} + 105501268198 \nu^{14} - 60933186040 \nu^{13} - 765818676414 \nu^{12} + \cdots + 742560783748 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1002969289 \nu^{17} - 5357577374 \nu^{16} - 11353710084 \nu^{15} + 102147035436 \nu^{14} - 15077828450 \nu^{13} - 742460530448 \nu^{12} + \cdots + 677849390104 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 685736770 \nu^{17} + 2668474422 \nu^{16} + 11910611780 \nu^{15} - 53542188699 \nu^{14} - 72646310628 \nu^{13} + 421926099202 \nu^{12} + \cdots - 42018281445 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1386154783 \nu^{17} + 6430527688 \nu^{16} + 19874972614 \nu^{15} - 125741499426 \nu^{14} - 62775758566 \nu^{13} + \cdots - 374413808574 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1665088329 \nu^{17} - 6609375872 \nu^{16} - 28840683926 \nu^{15} + 133345072732 \nu^{14} + 175522841140 \nu^{13} - 1060816401736 \nu^{12} + \cdots - 107954465010 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1832524509 \nu^{17} - 6639698786 \nu^{16} - 33972458906 \nu^{15} + 135523924258 \nu^{14} + 236411293988 \nu^{13} - 1095061846114 \nu^{12} + \cdots - 180269480248 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1216895826 \nu^{17} - 5844979198 \nu^{16} - 17002403395 \nu^{15} + 114758539026 \nu^{14} + 46183633539 \nu^{13} - 876187343173 \nu^{12} + \cdots + 306757426577 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2694174873 \nu^{17} + 8898905406 \nu^{16} + 54157234818 \nu^{15} - 186122321980 \nu^{14} - 433999335340 \nu^{13} + \cdots + 952441749148 ) / 4676065726 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1410676829 \nu^{17} - 5054336188 \nu^{16} - 26608639913 \nu^{15} + 103627914575 \nu^{14} + 192515702928 \nu^{13} - 843825093768 \nu^{12} + \cdots - 361693721127 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1422023073 \nu^{17} - 4606595223 \nu^{16} - 28914727965 \nu^{15} + 96834645667 \nu^{14} + 234949310539 \nu^{13} - 814866392181 \nu^{12} + \cdots - 485286282032 ) / 2338032863 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 3038182645 \nu^{17} - 11663900722 \nu^{16} - 54084465652 \nu^{15} + 236680213784 \nu^{14} + 347894361946 \nu^{13} - 1897444758324 \nu^{12} + \cdots - 317577900352 ) / 4676065726 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{15} + 2\beta_{11} - \beta_{10} + \beta_{9} - \beta_{5} - \beta_{3} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} - \beta_{16} - \beta_{15} - 9 \beta_{14} - \beta_{13} + \beta_{12} - 5 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 7 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + 10 \beta_{2} + 21 \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{17} + \beta_{16} - 11 \beta_{15} - 2 \beta_{13} - 3 \beta_{12} + 24 \beta_{11} - 13 \beta_{10} + 12 \beta_{9} - \beta_{8} - 2 \beta_{6} - 12 \beta_{5} + \beta_{4} - 15 \beta_{3} + 57 \beta_{2} + 2 \beta _1 + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{17} - 14 \beta_{16} - 11 \beta_{15} - 69 \beta_{14} - 16 \beta_{13} + 18 \beta_{12} - 15 \beta_{11} + 6 \beta_{10} + 31 \beta_{9} + 2 \beta_{8} + 41 \beta_{7} - 31 \beta_{6} - 21 \beta_{5} + 13 \beta_{4} - 38 \beta_{3} + 84 \beta_{2} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 19 \beta_{17} + 10 \beta_{16} - 88 \beta_{15} - 5 \beta_{14} - 35 \beta_{13} - 27 \beta_{12} + 218 \beta_{11} - 125 \beta_{10} + 117 \beta_{9} - 10 \beta_{8} - 5 \beta_{7} - 22 \beta_{6} - 104 \beta_{5} + 19 \beta_{4} - 154 \beta_{3} + 401 \beta_{2} + \cdots + 532 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 129 \beta_{17} - 142 \beta_{16} - 90 \beta_{15} - 504 \beta_{14} - 177 \beta_{13} + 215 \beta_{12} + 34 \beta_{11} - 82 \beta_{10} + 337 \beta_{9} + 38 \beta_{8} + 222 \beta_{7} - 142 \beta_{6} - 170 \beta_{5} + 128 \beta_{4} + \cdots + 177 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 110 \beta_{17} + 52 \beta_{16} - 631 \beta_{15} - 100 \beta_{14} - 405 \beta_{13} - 123 \beta_{12} + 1793 \beta_{11} - 1069 \beta_{10} + 1050 \beta_{9} - 61 \beta_{8} - 96 \beta_{7} - 157 \beta_{6} - 807 \beta_{5} + 233 \beta_{4} + \cdots + 3349 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1143 \beta_{17} - 1271 \beta_{16} - 668 \beta_{15} - 3612 \beta_{14} - 1672 \beta_{13} + 2131 \beta_{12} + 1154 \beta_{11} - 1273 \beta_{10} + 3144 \beta_{9} + 452 \beta_{8} + 1111 \beta_{7} - 513 \beta_{6} + \cdots + 1668 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 363 \beta_{17} + 9 \beta_{16} - 4324 \beta_{15} - 1303 \beta_{14} - 3942 \beta_{13} + 112 \beta_{12} + 14074 \beta_{11} - 8613 \beta_{10} + 8939 \beta_{9} - 226 \beta_{8} - 1201 \beta_{7} - 839 \beta_{6} - 5980 \beta_{5} + \cdots + 21527 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9516 \beta_{17} - 10684 \beta_{16} - 4773 \beta_{15} - 25720 \beta_{14} - 14522 \beta_{13} + 19102 \beta_{12} + 14051 \beta_{11} - 12989 \beta_{10} + 27005 \beta_{9} + 4393 \beta_{8} + 4867 \beta_{7} + \cdots + 14428 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1271 \beta_{17} - 3636 \beta_{16} - 29054 \beta_{15} - 14050 \beta_{14} - 35039 \beta_{13} + 9835 \beta_{12} + 107749 \beta_{11} - 67069 \beta_{10} + 73347 \beta_{9} + 373 \beta_{8} - 12477 \beta_{7} + \cdots + 140724 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 76316 \beta_{17} - 86672 \beta_{16} - 33603 \beta_{15} - 183102 \beta_{14} - 120050 \beta_{13} + 161457 \beta_{12} + 137444 \beta_{11} - 115631 \beta_{10} + 220799 \beta_{9} + 38344 \beta_{8} + \cdots + 118635 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 39322 \beta_{17} - 57635 \beta_{16} - 193666 \beta_{15} - 136266 \beta_{14} - 295219 \beta_{13} + 142040 \beta_{12} + 813230 \beta_{11} - 511554 \beta_{10} + 586247 \beta_{9} + 18257 \beta_{8} + \cdots + 933162 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 597910 \beta_{17} - 688126 \beta_{16} - 235431 \beta_{15} - 1307213 \beta_{14} - 962160 \beta_{13} + 1316142 \beta_{12} + 1220338 \beta_{11} - 964358 \beta_{10} + 1749909 \beta_{9} + \cdots + 944762 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.50138
−2.24960
−1.87675
−1.52216
−1.15793
−0.957552
−0.924759
−0.735255
0.523506
0.763493
0.826129
1.04467
1.35726
1.74487
1.98431
2.35947
2.59964
2.72204
−2.50138 0 4.25691 3.57921 0 1.44216 −5.64540 0 −8.95298
1.2 −2.24960 0 3.06069 3.96974 0 −4.97706 −2.38611 0 −8.93031
1.3 −1.87675 0 1.52220 1.30620 0 −1.71403 0.896704 0 −2.45142
1.4 −1.52216 0 0.316965 −1.24712 0 −0.899316 2.56184 0 1.89831
1.5 −1.15793 0 −0.659200 0.421419 0 −0.645304 3.07917 0 −0.487974
1.6 −0.957552 0 −1.08309 4.10274 0 4.97202 2.95222 0 −3.92859
1.7 −0.924759 0 −1.14482 3.95421 0 −3.08028 2.90820 0 −3.65669
1.8 −0.735255 0 −1.45940 −0.962787 0 −3.25298 2.54354 0 0.707894
1.9 0.523506 0 −1.72594 1.35183 0 3.60927 −1.95055 0 0.707691
1.10 0.763493 0 −1.41708 1.51218 0 1.20167 −2.60892 0 1.15454
1.11 0.826129 0 −1.31751 0.786316 0 −5.06179 −2.74069 0 0.649598
1.12 1.04467 0 −0.908666 −0.714085 0 −2.03236 −3.03859 0 −0.745983
1.13 1.35726 0 −0.157846 1.46929 0 0.194696 −2.92876 0 1.99421
1.14 1.74487 0 1.04455 3.61409 0 4.28084 −1.66712 0 6.30610
1.15 1.98431 0 1.93749 −2.87852 0 −2.68467 −0.124041 0 −5.71188
1.16 2.35947 0 3.56710 4.18174 0 −2.82979 3.69754 0 9.86669
1.17 2.59964 0 4.75812 3.02323 0 −0.561390 7.17011 0 7.85930
1.18 2.72204 0 5.40952 −0.469688 0 1.03831 9.28087 0 −1.27851
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(547\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4923.2.a.l 18
3.b odd 2 1 547.2.a.b 18
12.b even 2 1 8752.2.a.s 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.a.b 18 3.b odd 2 1
4923.2.a.l 18 1.a even 1 1 trivial
8752.2.a.s 18 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4923))\):

\( T_{2}^{18} - 4 T_{2}^{17} - 18 T_{2}^{16} + 84 T_{2}^{15} + 116 T_{2}^{14} - 708 T_{2}^{13} - 282 T_{2}^{12} + 3104 T_{2}^{11} - 137 T_{2}^{10} - 7703 T_{2}^{9} + 2068 T_{2}^{8} + 11068 T_{2}^{7} - 4274 T_{2}^{6} - 9021 T_{2}^{5} + \cdots + 328 \) Copy content Toggle raw display
\( T_{11}^{18} + 2 T_{11}^{17} - 113 T_{11}^{16} - 224 T_{11}^{15} + 4921 T_{11}^{14} + 9574 T_{11}^{13} - 104309 T_{11}^{12} - 191844 T_{11}^{11} + 1134032 T_{11}^{10} + 1787123 T_{11}^{9} - 6381183 T_{11}^{8} - 6865320 T_{11}^{7} + \cdots - 82432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 4 T^{17} - 18 T^{16} + 84 T^{15} + \cdots + 328 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 27 T^{17} + 304 T^{16} + \cdots - 15872 \) Copy content Toggle raw display
$7$ \( T^{18} + 11 T^{17} - 17 T^{16} + \cdots - 58576 \) Copy content Toggle raw display
$11$ \( T^{18} + 2 T^{17} - 113 T^{16} + \cdots - 82432 \) Copy content Toggle raw display
$13$ \( T^{18} + 25 T^{17} + \cdots - 160401143 \) Copy content Toggle raw display
$17$ \( T^{18} - 30 T^{17} + 278 T^{16} + \cdots - 5744344 \) Copy content Toggle raw display
$19$ \( T^{18} - 4 T^{17} - 212 T^{16} + \cdots - 584801324 \) Copy content Toggle raw display
$23$ \( T^{18} - 26 T^{17} + \cdots + 8189636608 \) Copy content Toggle raw display
$29$ \( T^{18} - 18 T^{17} + 9 T^{16} + \cdots - 13479926 \) Copy content Toggle raw display
$31$ \( T^{18} + 5 T^{17} + \cdots + 1756132711264 \) Copy content Toggle raw display
$37$ \( T^{18} + 18 T^{17} + \cdots + 5114786922536 \) Copy content Toggle raw display
$41$ \( T^{18} - 17 T^{17} + \cdots + 341990380184 \) Copy content Toggle raw display
$43$ \( T^{18} - 8 T^{17} + \cdots + 23810212304 \) Copy content Toggle raw display
$47$ \( T^{18} - 52 T^{17} + \cdots + 35899903339 \) Copy content Toggle raw display
$53$ \( T^{18} - 60 T^{17} + \cdots - 971885299094 \) Copy content Toggle raw display
$59$ \( T^{18} - 8 T^{17} + \cdots - 1458822574688 \) Copy content Toggle raw display
$61$ \( T^{18} + 26 T^{17} + \cdots + 75258329816 \) Copy content Toggle raw display
$67$ \( T^{18} - 12 T^{17} + \cdots + 12533716163723 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 368406703332424 \) Copy content Toggle raw display
$73$ \( T^{18} + 2 T^{17} + \cdots - 12033371727622 \) Copy content Toggle raw display
$79$ \( T^{18} - 18 T^{17} + \cdots + 1079280536 \) Copy content Toggle raw display
$83$ \( T^{18} - 43 T^{17} + \cdots - 94\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{18} - 28 T^{17} + \cdots + 4077961316512 \) Copy content Toggle raw display
$97$ \( T^{18} + 34 T^{17} + \cdots - 28869767458 \) Copy content Toggle raw display
show more
show less