Properties

Label 4620.2.h.d
Level $4620$
Weight $2$
Character orbit 4620.h
Analytic conductor $36.891$
Analytic rank $0$
Dimension $14$
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] = [4620,2,Mod(1849,4620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("4620.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4620.h (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: \(36.8908857338\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 8 x^{12} - 2 x^{11} + 2 x^{10} + 72 x^{9} + 81 x^{8} - 588 x^{7} + 405 x^{6} + 1800 x^{5} + 250 x^{4} - 1250 x^{3} - 25000 x^{2} + 78125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{4} q^{5} + \beta_{3} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{4} q^{5} + \beta_{3} q^{7} - q^{9} - q^{11} + (\beta_{13} - \beta_{11} - \beta_{9} - \beta_{6} - \beta_{5} + \beta_{3}) q^{13} + \beta_{2} q^{15} + (\beta_{8} + \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{17} + ( - \beta_{13} - \beta_{11} + \beta_{8} + \beta_{7} + \beta_{4} - 1) q^{19} - q^{21} + ( - \beta_{8} - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{23} + ( - \beta_{13} + \beta_{12} + \beta_{9} + \beta_{4} - \beta_{3} - 1) q^{25} - \beta_{3} q^{27} + ( - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{8} + \beta_{4} + \beta_{2} - \beta_1) q^{29} + (\beta_{10} - \beta_{8} - \beta_{4} + 3) q^{31} - \beta_{3} q^{33} + \beta_{2} q^{35} + ( - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{3}) q^{37} + (\beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} - 1) q^{39} + (\beta_{12} - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{4} + 1) q^{41} + ( - 2 \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4}) q^{43} + \beta_{4} q^{45} + ( - \beta_{8} + 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{47} - q^{49} + ( - \beta_{12} - 2 \beta_{10} - \beta_{8} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{51} + ( - \beta_{9} - \beta_{8} - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{53} + \beta_{4} q^{55} + (\beta_{13} - \beta_{11} - \beta_{9} - \beta_{3} - \beta_{2} - \beta_1) q^{57} + (\beta_{13} + \beta_{11} - \beta_{10} - \beta_{7} + 2 \beta_{2} - 2 \beta_1 - 2) q^{59} + (\beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{8} - 2 \beta_{4} + 1) q^{61} - \beta_{3} q^{63} + ( - 2 \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5}) q^{65} + ( - \beta_{13} + \beta_{11} + \beta_{8} + 4 \beta_{5} - \beta_{4}) q^{67} + (\beta_{12} + 2 \beta_{10} - \beta_{8} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{69} + (\beta_{13} + 3 \beta_{12} + \beta_{11} + 3 \beta_{10} + \beta_{7} - \beta_{2} + \beta_1 + 1) q^{71} + ( - \beta_{8} + \beta_{4}) q^{73} + ( - \beta_{11} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{75} - \beta_{3} q^{77} + ( - \beta_{13} - \beta_{11} + \beta_{10} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{4} - \beta_{2} + \cdots + 3) q^{79}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{5} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{5} - 14 q^{9} - 14 q^{11} - 18 q^{19} - 14 q^{21} - 16 q^{25} - 6 q^{29} + 36 q^{31} + 4 q^{41} + 4 q^{45} - 14 q^{49} + 2 q^{51} + 4 q^{55} - 18 q^{59} + 10 q^{61} - 22 q^{65} - 18 q^{69} + 32 q^{71} + 8 q^{75} + 16 q^{79} + 14 q^{81} - 20 q^{85} + 2 q^{89} - 24 q^{95} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 8 x^{12} - 2 x^{11} + 2 x^{10} + 72 x^{9} + 81 x^{8} - 588 x^{7} + 405 x^{6} + 1800 x^{5} + 250 x^{4} - 1250 x^{3} - 25000 x^{2} + 78125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{13} - 8 \nu^{11} - 2 \nu^{10} + 2 \nu^{9} + 72 \nu^{8} + 81 \nu^{7} - 588 \nu^{6} + 405 \nu^{5} + 1800 \nu^{4} + 250 \nu^{3} - 1250 \nu^{2} - 25000 \nu ) / 15625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37 \nu^{13} - 85 \nu^{12} + 129 \nu^{11} - 19 \nu^{10} - 31 \nu^{9} + 3519 \nu^{8} - 10398 \nu^{7} + 3834 \nu^{6} + 20015 \nu^{5} - 33975 \nu^{4} + 127125 \nu^{3} - 349375 \nu^{2} + \cdots + 1203125 ) / 500000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 77 \nu^{13} - 185 \nu^{12} - 191 \nu^{11} - 799 \nu^{10} + 249 \nu^{9} + 5699 \nu^{8} - 11358 \nu^{7} + 6714 \nu^{6} + 12015 \nu^{5} + 38525 \nu^{4} + 189125 \nu^{3} + \cdots - 859375 ) / 500000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53 \nu^{13} - 255 \nu^{12} + \nu^{11} + 59 \nu^{10} + 341 \nu^{9} + 8081 \nu^{8} - 18842 \nu^{7} - 3094 \nu^{6} + 48955 \nu^{5} - 94025 \nu^{4} + 172625 \nu^{3} - 768125 \nu^{2} + \cdots + 2609375 ) / 500000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 88 \nu^{13} + 205 \nu^{12} - 146 \nu^{11} - 839 \nu^{10} + 1214 \nu^{9} - 6101 \nu^{8} + 13432 \nu^{7} + 9024 \nu^{6} - 71280 \nu^{5} + 136925 \nu^{4} - 169750 \nu^{3} + \cdots - 2453125 ) / 500000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24 \nu^{13} + 405 \nu^{12} + 8 \nu^{11} - 1913 \nu^{10} - 2362 \nu^{9} - 5737 \nu^{8} + 28754 \nu^{7} - 4782 \nu^{6} - 119470 \nu^{5} + 160375 \nu^{4} + 138750 \nu^{3} + \cdots - 7046875 ) / 500000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17 \nu^{13} + 85 \nu^{12} + 11 \nu^{11} - 21 \nu^{10} + 171 \nu^{9} - 2679 \nu^{8} + 5118 \nu^{7} + 1006 \nu^{6} - 20115 \nu^{5} + 23575 \nu^{4} - 60625 \nu^{3} + 219375 \nu^{2} + \cdots - 578125 ) / 100000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 88 \nu^{13} + 515 \nu^{12} - 146 \nu^{11} - 569 \nu^{10} + 594 \nu^{9} - 8731 \nu^{8} + 30252 \nu^{7} - 16616 \nu^{6} - 93060 \nu^{5} + 116475 \nu^{4} - 66250 \nu^{3} + \cdots - 1921875 ) / 500000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 53 \nu^{13} - 10 \nu^{12} - 224 \nu^{11} - 26 \nu^{10} - 849 \nu^{9} + 3396 \nu^{8} - 1027 \nu^{7} - 18824 \nu^{6} + 29170 \nu^{5} + 18750 \nu^{4} + 126875 \nu^{3} - 13750 \nu^{2} + \cdots + 375000 ) / 125000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 94 \nu^{13} + 155 \nu^{12} + 677 \nu^{11} - 802 \nu^{10} + 1352 \nu^{9} - 11433 \nu^{8} + 8521 \nu^{7} + 54177 \nu^{6} - 139785 \nu^{5} + 57300 \nu^{4} - 186250 \nu^{3} + \cdots - 4656250 ) / 250000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 228 \nu^{13} + 485 \nu^{12} + 1024 \nu^{11} - 2049 \nu^{10} + 2474 \nu^{9} - 21721 \nu^{8} + 32102 \nu^{7} + 82874 \nu^{6} - 292070 \nu^{5} + 147175 \nu^{4} + \cdots - 9046875 ) / 500000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 138 \nu^{13} + 370 \nu^{12} + 279 \nu^{11} - 1059 \nu^{10} + 1834 \nu^{9} - 14296 \nu^{8} + 24937 \nu^{7} + 28089 \nu^{6} - 149525 \nu^{5} + 135675 \nu^{4} + \cdots - 4765625 ) / 250000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{12} + 3 \beta_{11} + 2 \beta_{10} + \beta_{9} - 5 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} + 3 \beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{13} - 4 \beta_{12} - 7 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} - 9 \beta_{8} - 8 \beta_{6} - 20 \beta_{5} + 5 \beta_{4} + 22 \beta_{3} - 4 \beta_{2} + 5 \beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 19 \beta_{13} - 8 \beta_{12} + 9 \beta_{11} - 18 \beta_{10} + 26 \beta_{9} - 7 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 30 \beta_{5} + 29 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} - 27 \beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9 \beta_{13} + 36 \beta_{12} - 45 \beta_{11} - 32 \beta_{10} - 2 \beta_{9} - 15 \beta_{8} - 68 \beta_{6} - 76 \beta_{5} + 43 \beta_{4} + 18 \beta_{3} + 51 \beta_{2} + 32 \beta _1 + 80 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 137 \beta_{13} - 41 \beta_{12} + 76 \beta_{11} - 110 \beta_{10} + 139 \beta_{9} - 54 \beta_{8} + 54 \beta_{7} + 90 \beta_{6} - 14 \beta_{5} - \beta_{4} - 7 \beta_{3} + 181 \beta_{2} + 99 \beta _1 - 95 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 156 \beta_{13} - 67 \beta_{12} - 31 \beta_{11} - 90 \beta_{10} + 175 \beta_{9} - 107 \beta_{8} - 90 \beta_{7} - 306 \beta_{6} - 134 \beta_{5} + 230 \beta_{4} + 165 \beta_{3} + 183 \beta_{2} - 210 \beta _1 - 564 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 619 \beta_{13} + 247 \beta_{12} + 356 \beta_{11} + 144 \beta_{10} + 427 \beta_{9} + 84 \beta_{8} - 336 \beta_{7} + 68 \beta_{6} - 204 \beta_{5} + 35 \beta_{4} - 643 \beta_{3} + 392 \beta_{2} - 280 \beta _1 - 735 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1028 \beta_{13} + 392 \beta_{12} - 748 \beta_{11} - 140 \beta_{10} - 1020 \beta_{9} - 656 \beta_{8} + 128 \beta_{7} - 892 \beta_{6} - 1136 \beta_{5} - 1072 \beta_{4} + 3264 \beta_{3} + 1783 \beta_{2} - 148 \beta _1 - 876 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1932 \beta_{13} - 392 \beta_{12} - 436 \beta_{11} - 1808 \beta_{10} + 1020 \beta_{9} + 2796 \beta_{8} + 1272 \beta_{7} + 2572 \beta_{6} + 300 \beta_{5} + 372 \beta_{4} + 5036 \beta_{3} - 904 \beta_{2} + \cdots - 9031 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 68 \beta_{13} + 1016 \beta_{12} - 3060 \beta_{11} - 3204 \beta_{10} - 2204 \beta_{9} + 8016 \beta_{8} + 1920 \beta_{7} - 6044 \beta_{6} - 304 \beta_{5} + 6000 \beta_{4} + 4704 \beta_{3} + 3324 \beta_{2} + \cdots + 5564 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4620\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(1541\) \(2311\) \(2521\) \(3697\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
−0.763299 + 2.10176i
1.32090 + 1.80422i
2.13383 + 0.668411i
−2.18371 + 0.481066i
−2.20270 0.384828i
2.18211 0.488265i
−0.487126 2.18236i
−0.763299 2.10176i
1.32090 1.80422i
2.13383 0.668411i
−2.18371 0.481066i
−2.20270 + 0.384828i
2.18211 + 0.488265i
−0.487126 + 2.18236i
0 1.00000i 0 −2.10176 + 0.763299i 0 1.00000i 0 −1.00000 0
1849.2 0 1.00000i 0 −1.80422 1.32090i 0 1.00000i 0 −1.00000 0
1849.3 0 1.00000i 0 −0.668411 2.13383i 0 1.00000i 0 −1.00000 0
1849.4 0 1.00000i 0 −0.481066 + 2.18371i 0 1.00000i 0 −1.00000 0
1849.5 0 1.00000i 0 0.384828 + 2.20270i 0 1.00000i 0 −1.00000 0
1849.6 0 1.00000i 0 0.488265 2.18211i 0 1.00000i 0 −1.00000 0
1849.7 0 1.00000i 0 2.18236 + 0.487126i 0 1.00000i 0 −1.00000 0
1849.8 0 1.00000i 0 −2.10176 0.763299i 0 1.00000i 0 −1.00000 0
1849.9 0 1.00000i 0 −1.80422 + 1.32090i 0 1.00000i 0 −1.00000 0
1849.10 0 1.00000i 0 −0.668411 + 2.13383i 0 1.00000i 0 −1.00000 0
1849.11 0 1.00000i 0 −0.481066 2.18371i 0 1.00000i 0 −1.00000 0
1849.12 0 1.00000i 0 0.384828 2.20270i 0 1.00000i 0 −1.00000 0
1849.13 0 1.00000i 0 0.488265 + 2.18211i 0 1.00000i 0 −1.00000 0
1849.14 0 1.00000i 0 2.18236 0.487126i 0 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4620.2.h.d 14
5.b even 2 1 inner 4620.2.h.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4620.2.h.d 14 1.a even 1 1 trivial
4620.2.h.d 14 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4620, [\chi])\):

\( T_{13}^{14} + 106T_{13}^{12} + 3861T_{13}^{10} + 56772T_{13}^{8} + 285500T_{13}^{6} + 124980T_{13}^{4} + 15440T_{13}^{2} + 256 \) Copy content Toggle raw display
\( T_{17}^{14} + 151 T_{17}^{12} + 8663 T_{17}^{10} + 232981 T_{17}^{8} + 2909060 T_{17}^{6} + 14251968 T_{17}^{4} + 17484032 T_{17}^{2} + 262144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + 4 T^{13} + 16 T^{12} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$11$ \( (T + 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 106 T^{12} + 3861 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{14} + 151 T^{12} + 8663 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$19$ \( (T^{7} + 9 T^{6} - 22 T^{5} - 216 T^{4} + \cdots - 704)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 215 T^{12} + \cdots + 717382656 \) Copy content Toggle raw display
$29$ \( (T^{7} + 3 T^{6} - 84 T^{5} - 278 T^{4} + \cdots - 4124)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 18 T^{6} + 84 T^{5} + 170 T^{4} + \cdots + 2432)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 178 T^{12} + \cdots + 43454464 \) Copy content Toggle raw display
$41$ \( (T^{7} - 2 T^{6} - 130 T^{5} - 50 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 407 T^{12} + \cdots + 207294447616 \) Copy content Toggle raw display
$47$ \( T^{14} + 398 T^{12} + \cdots + 426607535104 \) Copy content Toggle raw display
$53$ \( T^{14} + 391 T^{12} + \cdots + 3615376384 \) Copy content Toggle raw display
$59$ \( (T^{7} + 9 T^{6} - 128 T^{5} - 1578 T^{4} + \cdots - 368)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 5 T^{6} - 157 T^{5} + 613 T^{4} + \cdots - 9088)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 530 T^{12} + \cdots + 229540642816 \) Copy content Toggle raw display
$71$ \( (T^{7} - 16 T^{6} - 214 T^{5} + \cdots + 683008)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 86 T^{12} + 2953 T^{10} + \cdots + 1982464 \) Copy content Toggle raw display
$79$ \( (T^{7} - 8 T^{6} - 224 T^{5} + \cdots - 188416)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 739 T^{12} + \cdots + 12977516314624 \) Copy content Toggle raw display
$89$ \( (T^{7} - T^{6} - 393 T^{5} + 169 T^{4} + \cdots - 3815424)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 795 T^{12} + \cdots + 6393696587776 \) Copy content Toggle raw display
show more
show less