Properties

Label 384.5.b.d
Level $384$
Weight $5$
Character orbit 384.b
Analytic conductor $39.694$
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] = [384,5,Mod(319,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.319");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.b (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: \(39.6940658242\)
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: \( x^{16} + 46 x^{14} + 1311 x^{12} + 24382 x^{10} + 338077 x^{8} + 3338772 x^{6} + 24662556 x^{4} + 120434256 x^{2} + 362673936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{84}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{11} q^{5} - \beta_{5} q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{11} q^{5} - \beta_{5} q^{7} + 27 q^{9} + ( - \beta_{9} - 7 \beta_{3}) q^{11} + ( - \beta_{14} - 3 \beta_{11} + \beta_{7}) q^{13} + ( - \beta_{5} + \beta_{4}) q^{15} + (\beta_{6} + 30) q^{17} + ( - \beta_{13} + \beta_{9} + 18 \beta_{3}) q^{19} + ( - \beta_{14} + \beta_{12} + 5 \beta_{11} - \beta_{7}) q^{21} + (5 \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1) q^{23} + (\beta_{8} - \beta_{6} + 9) q^{25} - 27 \beta_{3} q^{27} + ( - \beta_{12} + 21 \beta_{11} + 3 \beta_{7}) q^{29} + (3 \beta_{5} - 6 \beta_{4} - 9 \beta_{2} - 2 \beta_1) q^{31} + ( - \beta_{10} - 2 \beta_{8} + 180) q^{33} + (7 \beta_{15} - \beta_{13} + 115 \beta_{3}) q^{35} + ( - \beta_{14} - 6 \beta_{12} - 28 \beta_{11} - 12 \beta_{7}) q^{37} + ( - 2 \beta_{5} - 7 \beta_{4} - 9 \beta_{2} - 6 \beta_1) q^{39} + ( - 5 \beta_{10} - 3 \beta_{8} - \beta_{6} - 222) q^{41} + ( - \beta_{15} + 2 \beta_{13} - 2 \beta_{9} - 144 \beta_{3}) q^{43} + 27 \beta_{11} q^{45} + (11 \beta_{5} + 7 \beta_{4} + 12 \beta_{2} + 19 \beta_1) q^{47} + ( - 6 \beta_{10} + 3 \beta_{8} - 3 \beta_{6} - 1255) q^{49} + ( - 6 \beta_{15} - 3 \beta_{13} + 3 \beta_{9} - 28 \beta_{3}) q^{51} + ( - 12 \beta_{14} - \beta_{12} + 29 \beta_{11} - 39 \beta_{7}) q^{53} + ( - 32 \beta_{5} + 24 \beta_{4} + 21 \beta_{2} - 12 \beta_1) q^{55} + ( - 2 \beta_{10} + 2 \beta_{8} + 3 \beta_{6} - 468) q^{57} + ( - 5 \beta_{15} - \beta_{13} - 11 \beta_{9} + 158 \beta_{3}) q^{59} + ( - 11 \beta_{14} - 12 \beta_{12} + 58 \beta_{11} + 12 \beta_{7}) q^{61} - 27 \beta_{5} q^{63} + ( - 9 \beta_{10} - 3 \beta_{8} - 5 \beta_{6} + 1440) q^{65} + ( - 17 \beta_{15} + 7 \beta_{13} + 5 \beta_{9} - 186 \beta_{3}) q^{67} + (2 \beta_{14} - 8 \beta_{12} + 11 \beta_{11} - 7 \beta_{7}) q^{69} + (26 \beta_{5} - 8 \beta_{4} + 10 \beta_{2} - 20 \beta_1) q^{71} + ( - 12 \beta_{10} - 15 \beta_{8} + 9 \beta_{6} + 2150) q^{73} + (9 \beta_{15} + 9 \beta_{9} - 6 \beta_{3}) q^{75} + ( - 42 \beta_{14} - 7 \beta_{12} + 72 \beta_{11} + 21 \beta_{7}) q^{77} + (11 \beta_{5} - 30 \beta_{4} + 36 \beta_{2} + 30 \beta_1) q^{79} + 729 q^{81} + (16 \beta_{15} + 14 \beta_{13} - 11 \beta_{9} - 223 \beta_{3}) q^{83} + ( - 48 \beta_{14} + 12 \beta_{12} + 265 \beta_{11} - 41 \beta_{7}) q^{85} + ( - 4 \beta_{5} + 22 \beta_{4} - 9 \beta_{2} - 9 \beta_1) q^{87} + ( - 13 \beta_{10} + 15 \beta_{8} + 6 \beta_{6} - 978) q^{89} + (4 \beta_{15} + 7 \beta_{13} + 53 \beta_{9} + 714 \beta_{3}) q^{91} + ( - 21 \beta_{14} - 6 \beta_{12} - 84 \beta_{11} - 3 \beta_{7}) q^{93} + (45 \beta_{5} - 21 \beta_{4} - 32 \beta_{2} + 51 \beta_1) q^{95} + ( - 24 \beta_{10} + 2 \beta_{8} + 16 \beta_{6} + 318) q^{97} + ( - 27 \beta_{9} - 189 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 432 q^{9} + 480 q^{17} + 144 q^{25} + 2880 q^{33} - 3552 q^{41} - 20080 q^{49} - 7488 q^{57} + 23040 q^{65} + 34400 q^{73} + 11664 q^{81} - 15648 q^{89} + 5088 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 46 x^{14} + 1311 x^{12} + 24382 x^{10} + 338077 x^{8} + 3338772 x^{6} + 24662556 x^{4} + 120434256 x^{2} + 362673936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2266283 \nu^{14} - 66378230 \nu^{12} - 1517851305 \nu^{10} - 18296855030 \nu^{8} - 146533806035 \nu^{6} - 185743103760 \nu^{4} + \cdots + 76779634811400 ) / 728558119080 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4108 \nu^{14} + 76360 \nu^{12} + 1460868 \nu^{10} + 9399208 \nu^{8} + 100952428 \nu^{6} + 194090928 \nu^{4} + 9371901600 \nu^{2} + \cdots + 19234287648 ) / 752642685 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 556 \nu^{15} - 18699 \nu^{13} - 428444 \nu^{11} - 5765809 \nu^{9} - 58126704 \nu^{7} - 271876859 \nu^{5} - 903142338 \nu^{3} + \cdots + 4265951220 \nu ) / 3179967120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5339189 \nu^{14} - 164490066 \nu^{12} - 4390456039 \nu^{10} - 65458015394 \nu^{8} - 801533739069 \nu^{6} + \cdots - 121687906180248 ) / 769033570140 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57146483 \nu^{14} + 2115064406 \nu^{12} + 52703126049 \nu^{10} + 806141495414 \nu^{8} + 9307681589819 \nu^{6} + \cdots + 673855067434968 ) / 4614201420840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 72757 \nu^{14} + 4260382 \nu^{12} + 121527975 \nu^{10} + 2311227190 \nu^{8} + 30370027885 \nu^{6} + 289010652420 \nu^{4} + \cdots + 7044868877688 ) / 2955295530 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1601473 \nu^{15} + 98155168 \nu^{13} + 2885978475 \nu^{11} + 54364746640 \nu^{9} + 719902590145 \nu^{7} + 6465681409230 \nu^{5} + \cdots + 118184013684432 \nu ) / 2285023191660 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31371 \nu^{14} - 1579778 \nu^{12} - 43486169 \nu^{10} - 829237674 \nu^{8} - 10759730579 \nu^{6} - 101770416284 \nu^{4} + \cdots - 2463773203656 ) / 985098510 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 510748 \nu^{15} + 36077731 \nu^{13} + 876218028 \nu^{11} + 14129962873 \nu^{9} + 125697707008 \nu^{7} + 672547885683 \nu^{5} + \cdots - 4704024865428 \nu ) / 543774377520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 44399 \nu^{14} + 1891290 \nu^{12} + 45803557 \nu^{10} + 744244322 \nu^{8} + 8371524567 \nu^{6} + 68379174892 \nu^{4} + 366832939368 \nu^{2} + \cdots + 1592996663592 ) / 985098510 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6590197 \nu^{15} + 282332912 \nu^{13} + 7259403175 \nu^{11} + 122888600320 \nu^{9} + 1494048049205 \nu^{7} + \cdots + 276231209943888 \nu ) / 4823937849060 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 116582967 \nu^{15} - 6636652714 \nu^{13} - 198004463629 \nu^{11} - 3806456973654 \nu^{9} - 49210879046239 \nu^{7} + \cdots - 30\!\cdots\!60 \nu ) / 53063316339660 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 6838460 \nu^{15} + 179593223 \nu^{13} + 3674734380 \nu^{11} + 27146644325 \nu^{9} - 2549711920 \nu^{7} - 4975055150985 \nu^{5} + \cdots - 307694922226308 \nu ) / 1631323132560 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2256580781 \nu^{15} - 111612544475 \nu^{13} - 3164107444263 \nu^{11} - 58666117370813 \nu^{9} - 758281783113113 \nu^{7} + \cdots - 15\!\cdots\!88 \nu ) / 477569847056940 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 893983 \nu^{15} - 28765249 \nu^{13} - 726540813 \nu^{11} - 10463518723 \nu^{9} - 124685999683 \nu^{7} - 859282193763 \nu^{5} + \cdots - 7232245440804 \nu ) / 101957695785 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 6 \beta_{15} - 16 \beta_{14} + 6 \beta_{13} + 16 \beta_{12} - 31 \beta_{11} - 6 \beta_{9} + 17 \beta_{7} - 132 \beta_{3} ) / 1536 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{10} - \beta_{8} - 12\beta_{6} + 11\beta_{5} - 59\beta_{4} - 36\beta_{2} + 129\beta _1 - 8832 ) / 1536 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{15} - 2\beta_{14} - 12\beta_{13} - \beta_{12} - 28\beta_{11} + 18\beta_{9} - 37\beta_{7} + 10\beta_{3} ) / 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 76 \beta_{10} + 128 \beta_{8} + 276 \beta_{6} - 1207 \beta_{5} + 103 \beta_{4} + 720 \beta_{2} - 1797 \beta _1 - 97152 ) / 1536 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 93 \beta_{15} + 5608 \beta_{14} + 2667 \beta_{13} - 2716 \beta_{12} + 29710 \beta_{11} - 2109 \beta_{9} + 3154 \beta_{7} + 175304 \beta_{3} ) / 3072 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -555\beta_{10} - 111\beta_{8} + 3523\beta_{5} + 2925\beta_{4} + 816\beta_{2} + 2025\beta _1 + 333952 ) / 256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 34281 \beta_{15} - 27752 \beta_{14} - 7809 \beta_{13} + 39068 \beta_{12} - 248798 \beta_{11} - 62985 \beta_{9} + 235486 \beta_{7} - 2985688 \beta_{3} ) / 3072 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 23246 \beta_{10} - 54302 \beta_{8} - 76164 \beta_{6} - 95783 \beta_{5} - 208873 \beta_{4} - 196200 \beta_{2} + 30507 \beta _1 - 9105792 ) / 1536 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5510 \beta_{15} - 34866 \beta_{14} - 11856 \beta_{13} - 17433 \beta_{12} - 57196 \beta_{11} + 30894 \beta_{9} - 214093 \beta_{7} + 14250 \beta_{3} ) / 128 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 142961 \beta_{10} + 1052881 \beta_{8} + 1127460 \beta_{6} - 2151179 \beta_{5} - 2358661 \beta_{4} + 185580 \beta_{2} - 2759169 \beta _1 - 58564992 ) / 1536 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7730301 \beta_{15} + 8830808 \beta_{14} + 8432997 \beta_{13} + 3010012 \beta_{12} + 23747966 \beta_{11} + 5904093 \beta_{9} + 34545146 \beta_{7} + 669116968 \beta_{3} ) / 3072 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 57750 \beta_{10} - 11550 \beta_{8} + 7709435 \beta_{5} + 13461045 \beta_{4} + 5349720 \beta_{2} + 9319185 \beta _1 - 50199424 ) / 256 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 110384487 \beta_{15} + 141741608 \beta_{14} - 130968543 \beta_{13} + 29702692 \beta_{12} + 274610738 \beta_{11} - 92312007 \beta_{9} + 489737654 \beta_{7} + \cdots - 9266543864 \beta_{3} ) / 3072 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 35312813 \beta_{10} - 236287741 \beta_{8} - 212340612 \beta_{6} - 363202879 \beta_{5} - 486159569 \beta_{4} - 4520916 \beta_{2} - 635930781 \beta _1 + 15566656128 ) / 1536 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 19515260 \beta_{15} - 120946422 \beta_{14} + 38717958 \beta_{13} - 60473211 \beta_{12} - 50067578 \beta_{11} - 96326052 \beta_{9} - 594326477 \beta_{7} - 45014670 \beta_{3} ) / 128 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
1.09182 2.89109i
−2.31712 + 3.01338i
1.45110 + 3.51338i
−1.95785 + 2.39109i
−1.95785 2.39109i
1.45110 3.51338i
−2.31712 3.01338i
1.09182 + 2.89109i
−1.09182 2.89109i
2.31712 + 3.01338i
−1.45110 + 3.51338i
1.95785 + 2.39109i
1.95785 2.39109i
−1.45110 3.51338i
2.31712 3.01338i
−1.09182 + 2.89109i
0 −5.19615 0 39.0495i 0 95.6459i 0 27.0000 0
319.2 0 −5.19615 0 23.5183i 0 40.3412i 0 27.0000 0
319.3 0 −5.19615 0 19.1142i 0 6.40010i 0 27.0000 0
319.4 0 −5.19615 0 4.54640i 0 61.7048i 0 27.0000 0
319.5 0 −5.19615 0 4.54640i 0 61.7048i 0 27.0000 0
319.6 0 −5.19615 0 19.1142i 0 6.40010i 0 27.0000 0
319.7 0 −5.19615 0 23.5183i 0 40.3412i 0 27.0000 0
319.8 0 −5.19615 0 39.0495i 0 95.6459i 0 27.0000 0
319.9 0 5.19615 0 39.0495i 0 95.6459i 0 27.0000 0
319.10 0 5.19615 0 23.5183i 0 40.3412i 0 27.0000 0
319.11 0 5.19615 0 19.1142i 0 6.40010i 0 27.0000 0
319.12 0 5.19615 0 4.54640i 0 61.7048i 0 27.0000 0
319.13 0 5.19615 0 4.54640i 0 61.7048i 0 27.0000 0
319.14 0 5.19615 0 19.1142i 0 6.40010i 0 27.0000 0
319.15 0 5.19615 0 23.5183i 0 40.3412i 0 27.0000 0
319.16 0 5.19615 0 39.0495i 0 95.6459i 0 27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.5.b.d 16
3.b odd 2 1 1152.5.b.m 16
4.b odd 2 1 inner 384.5.b.d 16
8.b even 2 1 inner 384.5.b.d 16
8.d odd 2 1 inner 384.5.b.d 16
12.b even 2 1 1152.5.b.m 16
16.e even 4 1 768.5.g.h 8
16.e even 4 1 768.5.g.j 8
16.f odd 4 1 768.5.g.h 8
16.f odd 4 1 768.5.g.j 8
24.f even 2 1 1152.5.b.m 16
24.h odd 2 1 1152.5.b.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.d 16 1.a even 1 1 trivial
384.5.b.d 16 4.b odd 2 1 inner
384.5.b.d 16 8.b even 2 1 inner
384.5.b.d 16 8.d odd 2 1 inner
768.5.g.h 8 16.e even 4 1
768.5.g.h 8 16.f odd 4 1
768.5.g.j 8 16.e even 4 1
768.5.g.j 8 16.f odd 4 1
1152.5.b.m 16 3.b odd 2 1
1152.5.b.m 16 12.b even 2 1
1152.5.b.m 16 24.f even 2 1
1152.5.b.m 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2464T_{5}^{6} + 1653120T_{5}^{4} + 341272576T_{5}^{2} + 6369316864 \) acting on \(S_{5}^{\mathrm{new}}(384, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + 2464 T^{6} + \cdots + 6369316864)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 14624 T^{6} + \cdots + 2321893298176)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 85184 T^{6} + \cdots + 47\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 148672 T^{6} + \cdots + 42\!\cdots\!64)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 120 T^{3} + \cdots + 16654958608)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} - 420032 T^{6} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 503936 T^{6} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 1506976 T^{6} + \cdots + 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 2826528 T^{6} + \cdots + 76\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 10909504 T^{6} + \cdots + 47\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 888 T^{3} + \cdots + 4399896700944)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 3851456 T^{6} + \cdots + 21\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 15831168 T^{6} + \cdots + 16\!\cdots\!16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 38624160 T^{6} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 23504064 T^{6} + \cdots + 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 58033984 T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 85260992 T^{6} + \cdots + 73\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 39923840 T^{6} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 8600 T^{3} + \cdots - 12\!\cdots\!76)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 160497440 T^{6} + \cdots + 23\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 172743360 T^{6} + \cdots + 27\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 3912 T^{3} + \cdots - 21\!\cdots\!48)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1272 T^{3} + \cdots + 319894196086800)^{4} \) Copy content Toggle raw display
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