Properties

Label 1089.3.c.l
Level $1089$
Weight $3$
Character orbit 1089.c
Analytic conductor $29.673$
Analytic rank $0$
Dimension $16$
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] = [1089,3,Mod(604,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.604");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 11^{4} \)
Twist minimal: no (minimal twist has level 99)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{2} - 3) q^{4} + \beta_{8} q^{5} + ( - \beta_{7} - \beta_1) q^{7} + ( - \beta_{10} + 4 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{2} - 3) q^{4} + \beta_{8} q^{5} + ( - \beta_{7} - \beta_1) q^{7} + ( - \beta_{10} + 4 \beta_{3}) q^{8} + ( - \beta_{12} - \beta_{5} + \beta_1) q^{10} + (4 \beta_{7} + \beta_{5} + \beta_1) q^{13} + (\beta_{9} + 2 \beta_{6}) q^{14} + (\beta_{11} + 2 \beta_{4} - 3 \beta_{2} + 17) q^{16} + ( - 2 \beta_{13} + \beta_{10} + 3 \beta_{3}) q^{17} + (\beta_{7} - 3 \beta_{5} + \beta_1) q^{19} + ( - \beta_{14} - 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{6}) q^{20} + ( - 2 \beta_{9} + 2 \beta_{8} - \beta_{6}) q^{23} + (2 \beta_{11} + \beta_{4} + 2 \beta_{2} + 1) q^{25} + (\beta_{14} + \beta_{8} - 5 \beta_{6}) q^{26} + ( - 4 \beta_{12} + 4 \beta_{7} - \beta_{5} + 10 \beta_1) q^{28} + ( - \beta_{15} - 3 \beta_{13} + \beta_{10} + \beta_{3}) q^{29} + (4 \beta_{11} + \beta_{4} - 2 \beta_{2} - 4) q^{31} + ( - 2 \beta_{15} - \beta_{13} + 3 \beta_{10} - 16 \beta_{3}) q^{32} + (\beta_{11} - 2 \beta_{4} - 2 \beta_{2} + 20) q^{34} + (2 \beta_{13} - 2 \beta_{10} + 3 \beta_{3}) q^{35} + ( - \beta_{11} + 2 \beta_{2} - 18) q^{37} + ( - 3 \beta_{14} - 4 \beta_{9} - 3 \beta_{8} - 2 \beta_{6}) q^{38} + ( - \beta_{12} - 6 \beta_{7} + 4 \beta_{5} - 14 \beta_1) q^{40} + (\beta_{13} + 3 \beta_{10} + 5 \beta_{3}) q^{41} + ( - 6 \beta_{12} - \beta_{5} + \beta_1) q^{43} + ( - \beta_{7} + 3 \beta_{5} - 11 \beta_1) q^{46} + ( - 3 \beta_{14} - 3 \beta_{8} + 2 \beta_{6}) q^{47} + ( - \beta_{11} - \beta_{4} + 3 \beta_{2} + 26) q^{49} + ( - \beta_{15} + \beta_{13} + 12 \beta_{3}) q^{50} + (8 \beta_{12} - 9 \beta_{7} - 8 \beta_{5} - 18 \beta_1) q^{52} + ( - 5 \beta_{9} - 4 \beta_{8} + 3 \beta_{6}) q^{53} + ( - \beta_{14} - 7 \beta_{9} - 5 \beta_{8} - 10 \beta_{6}) q^{56} + (2 \beta_{11} - 9 \beta_{4} - 3 \beta_{2} + 5) q^{58} + (4 \beta_{14} - 5 \beta_{9} - 6 \beta_{8} - 6 \beta_{6}) q^{59} + ( - 2 \beta_{12} - 11 \beta_{7} + 5 \beta_{5} + 7 \beta_1) q^{61} + ( - \beta_{15} + 3 \beta_{13} + 4 \beta_{10} + \beta_{3}) q^{62} + (2 \beta_{11} - 12 \beta_{4} + 8 \beta_{2} - 49) q^{64} + ( - 9 \beta_{13} + 4 \beta_{10}) q^{65} + (2 \beta_{11} - 2 \beta_{4} + 7 \beta_{2} + 16) q^{67} + (2 \beta_{15} - 5 \beta_{13} + 2 \beta_{10} - 14 \beta_{3}) q^{68} + (4 \beta_{4} - 7 \beta_{2} + 23) q^{70} + (5 \beta_{14} - 4 \beta_{9} + \beta_{8} - \beta_{6}) q^{71} + ( - 11 \beta_{12} + 13 \beta_{7} - \beta_{5} + 5 \beta_1) q^{73} + ( - \beta_{13} - 2 \beta_{10} + 26 \beta_{3}) q^{74} + (10 \beta_{12} + 2 \beta_{7} + 19 \beta_{5} - 31 \beta_1) q^{76} + ( - 7 \beta_{12} - 13 \beta_{7} - 14 \beta_{5} - 16 \beta_1) q^{79} + (10 \beta_{9} + 11 \beta_{8} + 11 \beta_{6}) q^{80} + ( - 4 \beta_{11} - 6 \beta_{4} + 5 \beta_{2} + 32) q^{82} + ( - 3 \beta_{15} + 4 \beta_{13} + 4 \beta_{10} - 6 \beta_{3}) q^{83} + (16 \beta_{12} + 24 \beta_{7} - \beta_{5} - 5 \beta_1) q^{85} + ( - \beta_{14} - 2 \beta_{9} - 7 \beta_{8} - 7 \beta_{6}) q^{86} + (6 \beta_{9} - 2 \beta_{8} + 11 \beta_{6}) q^{89} + (3 \beta_{11} - 2 \beta_{4} - 4 \beta_{2} + 48) q^{91} + (3 \beta_{14} + 6 \beta_{9} + 11 \beta_{8} + 8 \beta_{6}) q^{92} + (2 \beta_{12} + 10 \beta_{7} + 23 \beta_{5} + \beta_1) q^{94} + ( - 8 \beta_{13} + 5 \beta_{10} - 21 \beta_{3}) q^{95} + (6 \beta_{11} - 9 \beta_{4} - 10 \beta_{2} - 47) q^{97} + (\beta_{15} - 5 \beta_{10} - 12 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 44 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 44 q^{4} + 244 q^{16} + 16 q^{25} - 80 q^{31} + 328 q^{34} - 280 q^{37} + 436 q^{49} + 140 q^{58} - 656 q^{64} + 300 q^{67} + 308 q^{70} + 580 q^{82} + 768 q^{91} - 720 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 45667752444 \nu^{14} - 597804978248 \nu^{12} + 3528390713769 \nu^{10} - 621669021475 \nu^{8} + 721124010273630 \nu^{6} + \cdots + 332158447600837 ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5084684619 \nu^{14} - 102502036623 \nu^{12} + 1068420907817 \nu^{10} - 6488335560210 \nu^{8} + 114684022273170 \nu^{6} + \cdots + 22\!\cdots\!21 ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7530891822 \nu^{15} - 161922628410 \nu^{13} + 1773071437239 \nu^{11} - 11760003820535 \nu^{9} + 185084618505135 \nu^{7} + \cdots + 243765923221832 \nu ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6262229615 \nu^{14} + 123470462925 \nu^{12} - 1253712964279 \nu^{10} + 7604818484385 \nu^{8} - 143729875320340 \nu^{6} + \cdots + 159603221202733 ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 226078922543 \nu^{14} + 5080279956993 \nu^{12} - 57615885824889 \nu^{10} + 397910034267221 \nu^{8} + \cdots + 619468120074969 ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 445551515192 \nu^{15} - 9307380902916 \nu^{13} + 100246591559245 \nu^{11} - 655811636243113 \nu^{9} + \cdots - 11\!\cdots\!28 \nu ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 373009540956 \nu^{14} - 8272667397206 \nu^{12} + 93216689770491 \nu^{10} - 641417043065372 \nu^{8} + \cdots - 908176758810299 ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 598241835319 \nu^{15} + 13289199325505 \nu^{13} - 150985563543754 \nu^{11} + \cdots + 26\!\cdots\!82 \nu ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 675341589602 \nu^{15} + 13868921723197 \nu^{13} - 147009169987501 \nu^{11} + 940994096476483 \nu^{9} + \cdots + 13\!\cdots\!50 \nu ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 87882372397 \nu^{15} - 1883160071640 \nu^{13} + 20577181071988 \nu^{11} - 136163991479220 \nu^{9} + \cdots + 28\!\cdots\!24 \nu ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 76482029149 \nu^{14} - 1530855816173 \nu^{12} + 15742685210216 \nu^{10} - 96069570646780 \nu^{8} + \cdots + 40\!\cdots\!68 ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 914548814197 \nu^{14} - 18896431939389 \nu^{12} + 202148975636349 \nu^{10} + \cdots - 757739325471612 ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 125817353665 \nu^{15} + 2586721454850 \nu^{13} - 27473712050138 \nu^{11} + 176068897871470 \nu^{9} + \cdots - 39\!\cdots\!64 \nu ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1590497768908 \nu^{15} - 33743007445253 \nu^{13} + 368452116183311 \nu^{11} + \cdots - 47\!\cdots\!68 \nu ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26847260804 \nu^{15} - 566324648655 \nu^{13} + 6124941929985 \nu^{11} - 40072005667775 \nu^{9} + 649352121603960 \nu^{7} + \cdots + 860414508504650 \nu ) / 32325050848145 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} - 2\beta_{13} + 2\beta_{10} + 4\beta_{9} + 6\beta_{6} + 2\beta_{3} ) / 22 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{2} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} + 2\beta_{14} - \beta_{13} + 3\beta_{10} + 3\beta_{9} + 2\beta_{8} - 12\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{12} + \beta_{11} + 27\beta_{7} - 6\beta_{5} + 11\beta_{4} - 3\beta_{2} + 52\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{15} - \beta_{13} + 19\beta_{10} - 160\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -215\beta_{12} - 15\beta_{11} + 354\beta_{7} - 137\beta_{5} - 127\beta_{4} + 70\beta_{2} + 735\beta _1 - 225 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 179 \beta_{15} - 272 \beta_{14} - 146 \beta_{13} + 640 \beta_{10} - 589 \beta_{9} - 354 \beta_{8} - 397 \beta_{6} - 2889 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1420 \beta_{12} - 494 \beta_{11} + 807 \beta_{7} - 3356 \beta_{5} - 2533 \beta_{4} + 4305 \beta_{2} + 5239 \beta _1 - 21042 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1537 \beta_{15} - 2486 \beta_{14} - 1325 \beta_{13} + 4352 \beta_{10} - 13120 \beta_{9} - 5583 \beta_{8} - 18507 \beta_{6} - 12658 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -4305\beta_{11} - 17010\beta_{4} + 43805\beta_{2} - 226193 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19463 \beta_{15} - 41352 \beta_{14} + 16436 \beta_{13} - 60241 \beta_{10} - 196922 \beta_{9} - 86435 \beta_{8} - 266948 \beta_{6} + 214479 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 362330 \beta_{12} - 108925 \beta_{11} - 379152 \beta_{7} + 578566 \beta_{5} - 531443 \beta_{4} + 980208 \beta_{2} - 1290483 \beta _1 - 4851218 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 531443 \beta_{15} - 578566 \beta_{14} + 422518 \beta_{13} - 2043094 \beta_{10} - 1869049 \beta_{9} - 940896 \beta_{8} - 2031965 \beta_{6} + 10065851 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 9766705 \beta_{12} - 640368 \beta_{11} - 13570866 \beta_{7} + 10225888 \beta_{5} - 3503633 \beta_{4} + 5282864 \beta_{2} - 33907236 \beta _1 - 25255599 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 4302656\beta_{15} + 3322448\beta_{13} - 18032177\beta_{10} + 97256460\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
604.1
−3.67414 1.19380i
3.67414 1.19380i
1.83190 + 2.52140i
−1.83190 + 2.52140i
1.32111 0.429256i
−1.32111 0.429256i
−0.386583 + 0.532086i
0.386583 + 0.532086i
−0.386583 0.532086i
0.386583 0.532086i
1.32111 + 0.429256i
−1.32111 + 0.429256i
1.83190 2.52140i
−1.83190 2.52140i
−3.67414 + 1.19380i
3.67414 + 1.19380i
3.86322i 0 −10.9245 −4.15040 0 7.44465i 26.7509i 0 16.0339i
604.2 3.86322i 0 −10.9245 4.15040 0 7.44465i 26.7509i 0 16.0339i
604.3 3.11662i 0 −5.71334 −1.42146 0 3.80859i 5.33982i 0 4.43016i
604.4 3.11662i 0 −5.71334 1.42146 0 3.80859i 5.33982i 0 4.43016i
604.5 1.38910i 0 2.07040 −4.37284 0 0.612830i 8.43240i 0 6.07432i
604.6 1.38910i 0 2.07040 4.37284 0 0.612830i 8.43240i 0 6.07432i
604.7 0.657695i 0 3.56744 −8.10135 0 4.08611i 4.97707i 0 5.32822i
604.8 0.657695i 0 3.56744 8.10135 0 4.08611i 4.97707i 0 5.32822i
604.9 0.657695i 0 3.56744 −8.10135 0 4.08611i 4.97707i 0 5.32822i
604.10 0.657695i 0 3.56744 8.10135 0 4.08611i 4.97707i 0 5.32822i
604.11 1.38910i 0 2.07040 −4.37284 0 0.612830i 8.43240i 0 6.07432i
604.12 1.38910i 0 2.07040 4.37284 0 0.612830i 8.43240i 0 6.07432i
604.13 3.11662i 0 −5.71334 −1.42146 0 3.80859i 5.33982i 0 4.43016i
604.14 3.11662i 0 −5.71334 1.42146 0 3.80859i 5.33982i 0 4.43016i
604.15 3.86322i 0 −10.9245 −4.15040 0 7.44465i 26.7509i 0 16.0339i
604.16 3.86322i 0 −10.9245 4.15040 0 7.44465i 26.7509i 0 16.0339i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 604.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.c.l 16
3.b odd 2 1 inner 1089.3.c.l 16
11.b odd 2 1 inner 1089.3.c.l 16
11.c even 5 1 99.3.k.b 16
11.d odd 10 1 99.3.k.b 16
33.d even 2 1 inner 1089.3.c.l 16
33.f even 10 1 99.3.k.b 16
33.h odd 10 1 99.3.k.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.k.b 16 11.c even 5 1
99.3.k.b 16 11.d odd 10 1
99.3.k.b 16 33.f even 10 1
99.3.k.b 16 33.h odd 10 1
1089.3.c.l 16 1.a even 1 1 trivial
1089.3.c.l 16 3.b odd 2 1 inner
1089.3.c.l 16 11.b odd 2 1 inner
1089.3.c.l 16 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 27T_{2}^{6} + 204T_{2}^{4} + 363T_{2}^{2} + 121 \) acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 27 T^{6} + 204 T^{4} + 363 T^{2} + \cdots + 121)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 104 T^{6} + 2921 T^{4} + \cdots + 43681)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 87 T^{6} + 2004 T^{4} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + 822 T^{6} + 199029 T^{4} + \cdots + 441168016)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1177 T^{6} + 275249 T^{4} + \cdots + 28344976)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1545 T^{6} + \cdots + 5628000400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 1430 T^{6} + \cdots + 1301766400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 3795 T^{6} + \cdots + 136766832400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 20 T^{3} - 2045 T^{2} + \cdots + 592295)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 70 T^{3} + 1535 T^{2} + \cdots - 21780)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 5410 T^{6} + \cdots + 328443610000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 5102 T^{6} + \cdots + 431696305296)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 7755 T^{6} + \cdots + 1065065280400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 15780 T^{6} + \cdots + 130872341603025)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 22220 T^{6} + \cdots + 640510301973025)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 14865 T^{6} + \cdots + 3114589632400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 75 T^{3} - 3655 T^{2} + \cdots - 3482380)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 21555 T^{6} + \cdots + 128324037120400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 30217 T^{6} + \cdots + 494466201667216)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 39735 T^{6} + \cdots + 436852010010025)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 35437 T^{6} + \cdots + 14\!\cdots\!01)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 29246 T^{6} + \cdots + 33083847444736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 180 T^{3} - 7305 T^{2} + \cdots - 45053405)^{4} \) Copy content Toggle raw display
show more
show less