Properties

Label 441.3.n.e
Level $441$
Weight $3$
Character orbit 441.n
Analytic conductor $12.016$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,3,Mod(128,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.128");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.n (of order \(6\), degree \(2\), not 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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 30x^{14} + 699x^{12} + 5328x^{10} + 29790x^{8} + 65691x^{6} + 106920x^{4} + 28431x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{9} + \beta_{6}) q^{3} + (\beta_{8} - 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 2) q^{4} + (\beta_{10} + \beta_{6}) q^{5} + (\beta_{10} - \beta_{2}) q^{6} + ( - \beta_{12} - \beta_{7} + 4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 2) q^{8} + ( - \beta_{12} + \beta_{8} - \beta_{7} - 5 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{9} + \beta_{6}) q^{3} + (\beta_{8} - 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 2) q^{4} + (\beta_{10} + \beta_{6}) q^{5} + (\beta_{10} - \beta_{2}) q^{6} + ( - \beta_{12} - \beta_{7} + 4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 2) q^{8} + ( - \beta_{12} + \beta_{8} - \beta_{7} - 5 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 + 4) q^{9} + (\beta_{15} + \beta_{13} - \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{6}) q^{10} + (\beta_{12} + \beta_{7} - 10 \beta_{5} - \beta_{4} - \beta_{3} + 5) q^{11} + ( - \beta_{14} + \beta_{13} - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9}) q^{12} + (2 \beta_{15} + \beta_{13} - \beta_{11} - 2 \beta_{10} - \beta_{2}) q^{13} + ( - \beta_{12} + \beta_{8} - 2 \beta_{7} + 7 \beta_{5} - 3 \beta_{4} + 2 \beta_1 - 7) q^{15} + (\beta_{8} - 16 \beta_{5} + 7 \beta_{4} + 3 \beta_{3}) q^{16} + ( - \beta_{15} + 2 \beta_{14} + \beta_{13} + 2 \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{6} + \beta_{2}) q^{17} + ( - \beta_{12} + \beta_{8} - \beta_{7} + 4 \beta_{5} - 7 \beta_{4} - 8 \beta_{3} + \beta_1 + 4) q^{18} + (2 \beta_{14} + 2 \beta_{11} - \beta_{10} - 3 \beta_{9} - 3 \beta_{6} + \beta_{2}) q^{19} + ( - 5 \beta_{15} - \beta_{14} - 2 \beta_{13} + 3 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + \cdots + 3 \beta_{2}) q^{20}+ \cdots + ( - 6 \beta_{12} + 6 \beta_{8} + 4 \beta_{7} - 39 \beta_{5} - 7 \beta_{4} + \cdots + 29) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 20 q^{4} + 36 q^{9} - 42 q^{15} - 124 q^{16} + 108 q^{18} - 56 q^{22} - 140 q^{25} - 54 q^{29} - 366 q^{30} + 306 q^{32} - 240 q^{36} - 42 q^{37} + 12 q^{39} + 60 q^{43} - 432 q^{44} + 164 q^{46} + 270 q^{50} - 24 q^{51} - 36 q^{53} + 24 q^{57} + 20 q^{58} + 582 q^{60} + 124 q^{64} + 486 q^{65} + 322 q^{67} + 150 q^{72} + 1488 q^{78} + 98 q^{79} - 804 q^{81} + 198 q^{85} + 816 q^{88} - 90 q^{92} + 678 q^{93} + 432 q^{95} + 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 30x^{14} + 699x^{12} + 5328x^{10} + 29790x^{8} + 65691x^{6} + 106920x^{4} + 28431x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 21282484 \nu^{14} + 804608211 \nu^{12} + 20583395502 \nu^{10} + 244848110778 \nu^{8} + 1848412844307 \nu^{6} + \cdots - 16574313025098 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25408145 \nu^{15} + 368916351 \nu^{13} + 5397186864 \nu^{11} - 155765402118 \nu^{9} - 1703184785172 \nu^{7} + \cdots - 54703903774725 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 65563387 \nu^{14} - 2078101884 \nu^{12} - 49233800652 \nu^{10} - 427260977505 \nu^{8} - 2543724164628 \nu^{6} + \cdots - 5672088198792 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 170837740 \nu^{14} - 5067470046 \nu^{12} - 117347878302 \nu^{10} - 860933175282 \nu^{8} - 4586705751150 \nu^{6} + \cdots + 2065344345864 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9104111 \nu^{14} - 270218880 \nu^{12} - 6282065280 \nu^{10} - 46627480908 \nu^{8} - 259246736640 \nu^{6} - 533883381120 \nu^{4} + \cdots - 34104136320 ) / 212504811471 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 225560263 \nu^{15} - 7143676722 \nu^{13} - 168456107625 \nu^{11} - 1449853791948 \nu^{9} - 8375188599459 \nu^{7} + \cdots - 30395289498354 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 395998759 \nu^{14} + 13011964563 \nu^{12} + 310136238771 \nu^{10} + 2878954356330 \nu^{8} + 17299824562278 \nu^{6} + \cdots + 43019718213789 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25713893 \nu^{14} - 759898914 \nu^{12} - 17678635986 \nu^{10} - 130360980789 \nu^{8} - 736475955066 \nu^{6} + \cdots - 169011884880 ) / 120610838943 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1096139321 \nu^{15} - 32371666410 \nu^{13} - 750998975241 \nu^{11} - 5488186667382 \nu^{9} - 30071190846744 \nu^{7} + \cdots + 4613913244629 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1296291439 \nu^{15} - 39146426781 \nu^{13} - 914057896002 \nu^{11} - 7093805861448 \nu^{9} - 40149564231375 \nu^{7} + \cdots - 67097476905777 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 469378687 \nu^{15} - 13890156838 \nu^{13} - 322385937462 \nu^{11} - 2367505338378 \nu^{9} - 12969123173226 \nu^{7} + \cdots + 10622180893239 \nu ) / 1487533680297 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1466984245 \nu^{14} + 43310985330 \nu^{12} + 1003079833071 \nu^{10} + 7290823118904 \nu^{8} + 39144254738508 \nu^{6} + \cdots - 15222494380671 ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1791818123 \nu^{15} - 54977990757 \nu^{13} - 1287846205536 \nu^{11} - 10363064355684 \nu^{9} - 58992973152840 \nu^{7} + \cdots - 112110494969529 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 611596542 \nu^{15} - 18097719049 \nu^{13} - 419884986432 \nu^{11} - 3081094359999 \nu^{9} - 16830347926293 \nu^{7} + \cdots - 6195946171653 \nu ) / 1487533680297 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2181847172 \nu^{15} - 65469020325 \nu^{13} - 1525986845787 \nu^{11} - 11645666506404 \nu^{9} - 65318287312629 \nu^{7} + \cdots - 55381211181522 \nu ) / 4462601040891 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} - \beta_{6} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 8\beta_{5} + 3\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{15} - 6\beta_{14} - 6\beta_{13} + 16\beta_{11} + 5\beta_{10} + 13\beta_{9} + 32\beta_{6} + 14\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{12} - 18\beta_{8} - 4\beta_{7} + 132\beta_{5} - 47\beta_{4} - 76\beta_{3} - 18\beta _1 - 114 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 87\beta_{15} + 51\beta_{13} - 114\beta_{11} - 151\beta_{10} + 91\beta_{9} - 110\beta_{6} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -51\beta_{12} + 51\beta_{7} - 669\beta_{4} + 1026\beta_{3} + 357\beta _1 + 2274 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -83\beta_{15} + 1128\beta_{14} + 344\beta_{11} + 2398\beta_{10} - 4405\beta_{9} - 3086\beta_{6} - 1697\beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2256\beta_{12} + 7443\beta_{8} + 1128\beta_{7} - 55125\beta_{5} + 36627\beta_{4} + 14592\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 35922 \beta_{15} - 24291 \beta_{14} - 24291 \beta_{13} + 43365 \beta_{11} + 16668 \beta_{10} + 52821 \beta_{9} + 119055 \beta_{6} + 37524 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 24291 \beta_{12} - 157770 \beta_{8} - 48582 \beta_{7} + 1170603 \beta_{5} - 471159 \beta_{4} - 784548 \beta_{3} - 157770 \beta _1 - 1012833 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 798099 \beta_{15} + 519741 \beta_{13} - 1087197 \beta_{11} - 1442505 \beta_{10} + 896322 \beta_{9} - 1131042 \beta_{6} - 33105 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -519741\beta_{12} + 519741\beta_{7} - 6701148\beta_{4} + 10062054\beta_{3} + 3360906\beta _1 + 21589848 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 699876 \beta_{15} + 11101536 \beta_{14} + 3520377 \beta_{11} + 23199786 \beta_{10} - 43303230 \beta_{9} - 30240540 \beta_{6} - 16318503 \beta_{2} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 22203072 \beta_{12} + 71698986 \beta_{8} + 11101536 \beta_{7} - 532367181 \beta_{5} + 357920694 \beta_{4} + 143110854 \beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 348262461 \beta_{15} - 237012912 \beta_{14} - 237012912 \beta_{13} + 419961447 \beta_{11} + 161804826 \beta_{10} + 515929302 \beta_{9} + 1161515007 \beta_{6} + \cdots + 363160215 \beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1
2.31006 4.00114i
−2.31006 + 4.00114i
0.758328 1.31346i
−0.758328 + 1.31346i
−0.260383 + 0.450996i
0.260383 0.450996i
−1.23319 + 2.13594i
1.23319 2.13594i
2.31006 + 4.00114i
−2.31006 4.00114i
0.758328 + 1.31346i
−0.758328 1.31346i
−0.260383 0.450996i
0.260383 + 0.450996i
−1.23319 2.13594i
1.23319 + 2.13594i
−3.22764 + 1.86348i −2.42048 + 1.77236i 4.94513 8.56521i 6.87761i 4.50970 10.2311i 0 21.9528i 2.71748 8.57994i 12.8163 + 22.1985i
128.2 −3.22764 + 1.86348i 2.42048 1.77236i 4.94513 8.56521i 6.87761i −4.50970 + 10.2311i 0 21.9528i 2.71748 8.57994i −12.8163 22.1985i
128.3 −0.490034 + 0.282921i −1.95832 + 2.27266i −1.83991 + 3.18682i 0.799133i 0.316661 1.66773i 0 4.34557i −1.32995 8.90119i −0.226092 0.391602i
128.4 −0.490034 + 0.282921i 1.95832 2.27266i −1.83991 + 3.18682i 0.799133i −0.316661 + 1.66773i 0 4.34557i −1.32995 8.90119i 0.226092 + 0.391602i
128.5 0.876086 0.505809i −2.16654 2.07512i −1.48832 + 2.57784i 9.07591i −2.94769 0.722129i 0 7.05768i 0.387771 + 8.99164i −4.59068 7.95128i
128.6 0.876086 0.505809i 2.16654 + 2.07512i −1.48832 + 2.57784i 9.07591i 2.94769 + 0.722129i 0 7.05768i 0.387771 + 8.99164i 4.59068 + 7.95128i
128.7 2.84159 1.64059i −2.84822 + 0.942154i 3.38310 5.85970i 2.16509i −6.54778 + 7.34999i 0 9.07642i 7.22469 5.36692i 3.55204 + 6.15231i
128.8 2.84159 1.64059i 2.84822 0.942154i 3.38310 5.85970i 2.16509i 6.54778 7.34999i 0 9.07642i 7.22469 5.36692i −3.55204 6.15231i
410.1 −3.22764 1.86348i −2.42048 1.77236i 4.94513 + 8.56521i 6.87761i 4.50970 + 10.2311i 0 21.9528i 2.71748 + 8.57994i 12.8163 22.1985i
410.2 −3.22764 1.86348i 2.42048 + 1.77236i 4.94513 + 8.56521i 6.87761i −4.50970 10.2311i 0 21.9528i 2.71748 + 8.57994i −12.8163 + 22.1985i
410.3 −0.490034 0.282921i −1.95832 2.27266i −1.83991 3.18682i 0.799133i 0.316661 + 1.66773i 0 4.34557i −1.32995 + 8.90119i −0.226092 + 0.391602i
410.4 −0.490034 0.282921i 1.95832 + 2.27266i −1.83991 3.18682i 0.799133i −0.316661 1.66773i 0 4.34557i −1.32995 + 8.90119i 0.226092 0.391602i
410.5 0.876086 + 0.505809i −2.16654 + 2.07512i −1.48832 2.57784i 9.07591i −2.94769 + 0.722129i 0 7.05768i 0.387771 8.99164i −4.59068 + 7.95128i
410.6 0.876086 + 0.505809i 2.16654 2.07512i −1.48832 2.57784i 9.07591i 2.94769 0.722129i 0 7.05768i 0.387771 8.99164i 4.59068 7.95128i
410.7 2.84159 + 1.64059i −2.84822 0.942154i 3.38310 + 5.85970i 2.16509i −6.54778 7.34999i 0 9.07642i 7.22469 + 5.36692i 3.55204 6.15231i
410.8 2.84159 + 1.64059i 2.84822 + 0.942154i 3.38310 + 5.85970i 2.16509i 6.54778 + 7.34999i 0 9.07642i 7.22469 + 5.36692i −3.55204 + 6.15231i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 128.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.n.e 16
7.b odd 2 1 inner 441.3.n.e 16
7.c even 3 1 441.3.j.e 16
7.c even 3 1 441.3.r.e 16
7.d odd 6 1 441.3.j.e 16
7.d odd 6 1 441.3.r.e 16
9.d odd 6 1 441.3.j.e 16
63.i even 6 1 441.3.r.e 16
63.j odd 6 1 441.3.r.e 16
63.n odd 6 1 inner 441.3.n.e 16
63.o even 6 1 441.3.j.e 16
63.s even 6 1 inner 441.3.n.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.j.e 16 7.c even 3 1
441.3.j.e 16 7.d odd 6 1
441.3.j.e 16 9.d odd 6 1
441.3.j.e 16 63.o even 6 1
441.3.n.e 16 1.a even 1 1 trivial
441.3.n.e 16 7.b odd 2 1 inner
441.3.n.e 16 63.n odd 6 1 inner
441.3.n.e 16 63.s even 6 1 inner
441.3.r.e 16 7.c even 3 1
441.3.r.e 16 7.d odd 6 1
441.3.r.e 16 63.i even 6 1
441.3.r.e 16 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{8} - 13T_{2}^{6} + 162T_{2}^{4} - 117T_{2}^{3} - 64T_{2}^{2} + 63T_{2} + 49 \) Copy content Toggle raw display
\( T_{13}^{16} + 945 T_{13}^{14} + 622674 T_{13}^{12} + 199146033 T_{13}^{10} + 45700667370 T_{13}^{8} + 6159153715641 T_{13}^{6} + 585149499691425 T_{13}^{4} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 13 T^{6} + 162 T^{4} - 117 T^{3} + \cdots + 49)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 18 T^{14} + 363 T^{12} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{8} + 135 T^{6} + 4590 T^{4} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 533 T^{6} + 82191 T^{4} + \cdots + 13344409)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 945 T^{14} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{16} - 1035 T^{14} + \cdots + 29\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{16} + 1818 T^{14} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( (T^{8} + 1190 T^{6} + 257865 T^{4} + \cdots + 77123524)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 27 T^{7} + \cdots + 310953447424)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + 4761 T^{14} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{8} + 21 T^{7} + \cdots + 711337054464)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 5157 T^{14} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{8} - 30 T^{7} + \cdots + 160551674721)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 4869 T^{14} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + 18 T^{7} - 205 T^{6} + \cdots + 1882384)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 8784 T^{14} + \cdots + 32\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{16} + 26532 T^{14} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{8} - 161 T^{7} + \cdots + 6699591959449)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 36515 T^{6} + \cdots + 78136795608004)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 13806 T^{14} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( (T^{8} - 49 T^{7} + \cdots + 83599422865984)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 51858 T^{14} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} - 52776 T^{14} + \cdots + 85\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{16} + 48996 T^{14} + \cdots + 23\!\cdots\!61 \) Copy content Toggle raw display
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