Properties

Label 252.4.b.g
Level $252$
Weight $4$
Character orbit 252.b
Analytic conductor $14.868$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,4,Mod(55,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.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: \(14.8684813214\)
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: \( x^{16} - 4 x^{15} + 358 x^{14} - 2828 x^{13} + 52557 x^{12} - 549972 x^{11} + 4434734 x^{10} - 37785264 x^{9} + 272741368 x^{8} - 1739202044 x^{7} + \cdots + 52705588025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{5} + 3) q^{4} + \beta_{11} q^{5} + (\beta_{9} + \beta_{5} + 1) q^{7} + ( - \beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{5} + 3) q^{4} + \beta_{11} q^{5} + (\beta_{9} + \beta_{5} + 1) q^{7} + ( - \beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{2}) q^{8} - \beta_1 q^{10} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{2}) q^{11} - \beta_{14} q^{13} + ( - \beta_{10} - \beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 2 \beta_{2}) q^{14} + ( - 2 \beta_{9} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 18) q^{16} + ( - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{7}) q^{17} + ( - \beta_{13} + 2 \beta_{9} + 2 \beta_{3} + \beta_1 + 2) q^{19} + (\beta_{15} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{7}) q^{20} + ( - \beta_{9} + 2 \beta_{5} + 3 \beta_{4} + \beta_{3} - 17) q^{22} + ( - 7 \beta_{7} - 13 \beta_{6} + 13 \beta_{2}) q^{23} + (13 \beta_{5} - 13 \beta_{4} - 86) q^{25} + ( - \beta_{15} - 3 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{7}) q^{26} + (\beta_{14} + \beta_{13} + 4 \beta_{9} + 9 \beta_{5} - 15 \beta_{4} + 2 \beta_{3} - 51) q^{28} + (2 \beta_{8} + 7 \beta_{6} + 21 \beta_{2}) q^{29} + (3 \beta_{13} + 3 \beta_{9} + 3 \beta_{3} - 3 \beta_1 + 3) q^{31} + ( - 2 \beta_{8} + 18 \beta_{7} + 4 \beta_{6} - 12 \beta_{2}) q^{32} + ( - 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{9} + 2 \beta_{3} + \beta_1 + 2) q^{34} + (2 \beta_{15} + \beta_{12} + 3 \beta_{10} - 2 \beta_{8} - 14 \beta_{7} + \beta_{6} - \beta_{2}) q^{35} + (11 \beta_{5} - 11 \beta_{4} - 51) q^{37} + ( - \beta_{15} + \beta_{12} + 7 \beta_{11} - 5 \beta_{10} + 2 \beta_{8} - 3 \beta_{7}) q^{38} + ( - 4 \beta_{14} + 4 \beta_{13} + 4 \beta_{9} + 4 \beta_{3} - 2 \beta_1 + 4) q^{40} + (7 \beta_{12} - \beta_{11} - 7 \beta_{10} - 7 \beta_{7}) q^{41} + ( - 13 \beta_{9} + 2 \beta_{5} + 28 \beta_{4} + 13 \beta_{3} + 15) q^{43} + ( - \beta_{8} + 9 \beta_{7} - 14 \beta_{6} - 22 \beta_{2}) q^{44} + (7 \beta_{9} + 40 \beta_{5} - 21 \beta_{4} - 7 \beta_{3} - 151) q^{46} + ( - 4 \beta_{15} + 2 \beta_{12} + 6 \beta_{10} - 4 \beta_{8} + 2 \beta_{7}) q^{47} + ( - \beta_{14} - 4 \beta_{13} + 31 \beta_{5} - 31 \beta_{4} - 4 \beta_1 - 40) q^{49} + ( - 13 \beta_{8} + 13 \beta_{7} + 26 \beta_{6} - 73 \beta_{2}) q^{50} + ( - 2 \beta_{14} - 10 \beta_{13} + 10 \beta_{9} + 10 \beta_{3} + 2 \beta_1 + 10) q^{52} + (22 \beta_{8} + 9 \beta_{6} + 27 \beta_{2}) q^{53} + (3 \beta_{13} + 4 \beta_{9} + 4 \beta_{3} - 3 \beta_1 + 4) q^{55} + (\beta_{15} + 3 \beta_{12} - 7 \beta_{11} - 7 \beta_{10} - 8 \beta_{8} - 3 \beta_{7} + \cdots - 36 \beta_{2}) q^{56}+ \cdots + (3 \beta_{15} - 7 \beta_{12} + 35 \beta_{11} + 5 \beta_{10} - 30 \beta_{8} + 37 \beta_{7} + \cdots - 9 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 40 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 40 q^{4} - 304 q^{16} - 312 q^{22} - 1376 q^{25} - 816 q^{28} - 816 q^{37} - 2568 q^{46} - 640 q^{49} + 2336 q^{58} + 1120 q^{64} - 424 q^{70} + 5072 q^{85} - 3536 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 358 x^{14} - 2828 x^{13} + 52557 x^{12} - 549972 x^{11} + 4434734 x^{10} - 37785264 x^{9} + 272741368 x^{8} - 1739202044 x^{7} + \cdots + 52705588025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 43\!\cdots\!52 \nu^{15} + \cdots + 10\!\cdots\!25 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\!\cdots\!11 \nu^{15} + \cdots - 71\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 96\!\cdots\!34 \nu^{15} + \cdots + 12\!\cdots\!00 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 42\!\cdots\!00 \nu^{15} + \cdots - 30\!\cdots\!55 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42\!\cdots\!20 \nu^{15} + \cdots - 48\!\cdots\!75 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41\!\cdots\!21 \nu^{15} + \cdots + 32\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 69\!\cdots\!10 \nu^{15} + \cdots + 45\!\cdots\!00 ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!97 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!06 \nu^{15} + \cdots - 72\!\cdots\!25 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 94\!\cdots\!79 \nu^{15} + \cdots + 28\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 62\!\cdots\!59 \nu^{15} + \cdots - 32\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17\!\cdots\!49 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!36 \nu^{15} + \cdots + 13\!\cdots\!75 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24\!\cdots\!56 \nu^{15} + \cdots - 24\!\cdots\!00 ) / 21\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 69\!\cdots\!57 \nu^{15} + \cdots - 15\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{12} - 2\beta_{11} - \beta_{10} + \beta_{7} - 3\beta_{6} - 2\beta_{5} + 2\beta_{4} - 5\beta_{2} + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{14} - 6 \beta_{13} - 3 \beta_{12} + 10 \beta_{11} + 3 \beta_{10} - 10 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + 9 \beta_{6} - 50 \beta_{5} + 42 \beta_{4} - 18 \beta_{3} - \beta_{2} - 2 \beta _1 - 368 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{15} + 36 \beta_{14} + 66 \beta_{13} - 215 \beta_{12} + 462 \beta_{11} + 47 \beta_{10} + 114 \beta_{9} - 224 \beta_{8} - 841 \beta_{7} + 1564 \beta_{6} + 508 \beta_{5} - 388 \beta_{4} + 186 \beta_{3} + 1628 \beta_{2} + \cdots + 4462 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 36 \beta_{15} + 804 \beta_{14} + 1196 \beta_{13} + 820 \beta_{12} - 1636 \beta_{11} - 220 \beta_{10} + 12 \beta_{9} + 748 \beta_{8} + 3026 \beta_{7} - 5621 \beta_{6} + 4782 \beta_{5} + 482 \beta_{4} + 4124 \beta_{3} + \cdots + 39046 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3240 \beta_{15} - 15240 \beta_{14} - 22450 \beta_{13} + 25851 \beta_{12} - 33578 \beta_{11} + 23669 \beta_{10} - 2210 \beta_{9} + 31972 \beta_{8} + 91613 \beta_{7} - 227570 \beta_{6} + \cdots - 773946 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 18750 \beta_{15} - 141186 \beta_{14} - 173172 \beta_{13} - 161927 \beta_{12} + 233324 \beta_{11} - 118573 \beta_{10} + 329416 \beta_{9} - 193358 \beta_{8} - 579503 \beta_{7} + 1383882 \beta_{6} + \cdots - 2110944 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 266168 \beta_{15} + 2077460 \beta_{14} + 2633750 \beta_{13} - 805420 \beta_{12} - 489540 \beta_{11} - 3194548 \beta_{10} - 4009054 \beta_{9} - 2303488 \beta_{8} - 199550 \beta_{7} + \cdots + 43322176 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4847584 \beta_{15} + 17684688 \beta_{14} + 16913192 \beta_{13} + 21114556 \beta_{12} - 7210088 \beta_{11} + 51939460 \beta_{10} - 77871896 \beta_{9} + 44097328 \beta_{8} + \cdots - 366498678 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 48703440 \beta_{15} - 883011696 \beta_{14} - 946725174 \beta_{13} - 212327525 \beta_{12} + 1001801038 \beta_{11} + 946190405 \beta_{10} + 3072715818 \beta_{9} + \cdots + 4634714062 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 904335150 \beta_{15} - 834604638 \beta_{14} - 11176738 \beta_{13} - 284385156 \beta_{12} - 7899378702 \beta_{11} - 13336545744 \beta_{10} + 9554100242 \beta_{9} + \cdots + 127307491120 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1709516380 \beta_{15} + 140781230964 \beta_{14} + 118134494434 \beta_{13} + 78604043615 \beta_{12} - 182664208222 \beta_{11} - 52853769879 \beta_{10} + \cdots - 5309532618958 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 59587475706 \beta_{15} - 133295785114 \beta_{14} - 202402059216 \beta_{13} - 366139102959 \beta_{12} + 1494678265780 \beta_{11} + 1263659388555 \beta_{10} + \cdots - 7757738479936 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 849995603080 \beta_{15} - 7099600357616 \beta_{14} - 2257804253224 \beta_{13} - 4421761574763 \beta_{12} + 3090593743942 \beta_{11} + \cdots + 791181188266770 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 6936535769800 \beta_{15} + 85734367697660 \beta_{14} + 85620421150406 \beta_{13} + 242965843464895 \beta_{12} - 681091059946938 \beta_{11} + \cdots - 12\!\cdots\!76 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 345292836168636 \beta_{15} - 220267946319092 \beta_{14} + \cdots - 31\!\cdots\!18 ) / 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.929907 + 6.47387i
0.929907 12.9280i
0.929907 + 12.9280i
0.929907 6.47387i
4.06402 0.600106i
4.06402 + 1.75957i
4.06402 1.75957i
4.06402 + 0.600106i
0.467111 0.600106i
0.467111 + 1.75957i
0.467111 1.75957i
0.467111 + 0.600106i
−4.46104 + 6.47387i
−4.46104 12.9280i
−4.46104 + 12.9280i
−4.46104 6.47387i
−2.69547 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 −16.6272 8.15690i −13.6452 18.0502i 0 −8.83121 + 27.7765i
55.2 −2.69547 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 16.6272 8.15690i −13.6452 18.0502i 0 8.83121 27.7765i
55.3 −2.69547 + 0.856992i 0 6.53113 4.62000i 10.3049i 0 16.6272 + 8.15690i −13.6452 + 18.0502i 0 8.83121 + 27.7765i
55.4 −2.69547 + 0.856992i 0 6.53113 4.62000i 10.3049i 0 −16.6272 + 8.15690i −13.6452 + 18.0502i 0 −8.83121 27.7765i
55.5 −1.79845 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 5.15121 + 17.7895i 19.8950 10.7792i 0 −38.7945 + 31.9604i
55.6 −1.79845 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 −5.15121 + 17.7895i 19.8950 10.7792i 0 38.7945 31.9604i
55.7 −1.79845 + 2.18302i 0 −1.53113 7.85211i 17.7710i 0 −5.15121 17.7895i 19.8950 + 10.7792i 0 38.7945 + 31.9604i
55.8 −1.79845 + 2.18302i 0 −1.53113 7.85211i 17.7710i 0 5.15121 17.7895i 19.8950 + 10.7792i 0 −38.7945 31.9604i
55.9 1.79845 2.18302i 0 −1.53113 7.85211i 17.7710i 0 5.15121 17.7895i −19.8950 10.7792i 0 −38.7945 31.9604i
55.10 1.79845 2.18302i 0 −1.53113 7.85211i 17.7710i 0 −5.15121 17.7895i −19.8950 10.7792i 0 38.7945 + 31.9604i
55.11 1.79845 + 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 −5.15121 + 17.7895i −19.8950 + 10.7792i 0 38.7945 31.9604i
55.12 1.79845 + 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 5.15121 + 17.7895i −19.8950 + 10.7792i 0 −38.7945 + 31.9604i
55.13 2.69547 0.856992i 0 6.53113 4.62000i 10.3049i 0 −16.6272 + 8.15690i 13.6452 18.0502i 0 −8.83121 27.7765i
55.14 2.69547 0.856992i 0 6.53113 4.62000i 10.3049i 0 16.6272 + 8.15690i 13.6452 18.0502i 0 8.83121 + 27.7765i
55.15 2.69547 + 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 16.6272 8.15690i 13.6452 + 18.0502i 0 8.83121 27.7765i
55.16 2.69547 + 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 −16.6272 8.15690i 13.6452 + 18.0502i 0 −8.83121 + 27.7765i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.b.g 16
3.b odd 2 1 inner 252.4.b.g 16
4.b odd 2 1 inner 252.4.b.g 16
7.b odd 2 1 inner 252.4.b.g 16
12.b even 2 1 inner 252.4.b.g 16
21.c even 2 1 inner 252.4.b.g 16
28.d even 2 1 inner 252.4.b.g 16
84.h odd 2 1 inner 252.4.b.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.b.g 16 1.a even 1 1 trivial
252.4.b.g 16 3.b odd 2 1 inner
252.4.b.g 16 4.b odd 2 1 inner
252.4.b.g 16 7.b odd 2 1 inner
252.4.b.g 16 12.b even 2 1 inner
252.4.b.g 16 21.c even 2 1 inner
252.4.b.g 16 28.d even 2 1 inner
252.4.b.g 16 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{4} + 422T_{5}^{2} + 33536 \) Copy content Toggle raw display
\( T_{11}^{4} + 442T_{11}^{2} + 37856 \) Copy content Toggle raw display
\( T_{19}^{4} - 12028T_{19}^{2} + 23005696 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 10 T^{6} + 88 T^{4} - 640 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + 422 T^{2} + 33536)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} + 160 T^{6} + \cdots + 13841287201)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 442 T^{2} + 37856)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 7648 T^{2} + 12609536)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 6118 T^{2} + 8585216)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 12028 T^{2} + 23005696)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 26842 T^{2} + 29353856)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 10448 T^{2} + 18866176)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 64080 T^{2} + \cdots + 950745600)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 102 T - 5264)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 255782 T^{2} + \cdots + 14785888256)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 222172 T^{2} + \cdots + 115302656)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 535680 T^{2} + \cdots + 70613401600)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 293712 T^{2} + \cdots + 10418845696)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 536192 T^{2} + \cdots + 11298144256)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 436192 T^{2} + \cdots + 46920083456)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 503548 T^{2} + \cdots + 25756457216)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1021658 T^{2} + \cdots + 30079110656)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1988480 T^{2} + \cdots + 852404633600)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 508828 T^{2} + \cdots + 21224784896)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1392768 T^{2} + \cdots + 477346594816)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 5270 T^{2} + 3353600)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 919168 T^{2} + \cdots + 39543504896)^{4} \) Copy content Toggle raw display
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