Properties

Label 252.2.bf.f
Level $252$
Weight $2$
Character orbit 252.bf
Analytic conductor $2.012$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.562828176.1
Defining polynomial: \(x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( -\beta_{4} + \beta_{6} - \beta_{7} ) q^{4} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{7} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{8} +O(q^{10})\) \( q + \beta_{5} q^{2} + ( -\beta_{4} + \beta_{6} - \beta_{7} ) q^{4} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{7} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{8} + ( -2 - \beta_{4} - \beta_{6} + \beta_{7} ) q^{10} + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{13} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{14} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} ) q^{17} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{19} + ( 4 - \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{20} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{22} + ( 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{23} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{25} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{26} + ( 2 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{28} + ( 2 - 3 \beta_{1} + \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{29} + ( -3 + 4 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{31} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{32} + ( 4 - 2 \beta_{2} - 2 \beta_{4} ) q^{34} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{35} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{38} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{40} + ( -2 - 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{7} ) q^{41} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{43} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{44} + ( -4 \beta_{3} - 4 \beta_{4} ) q^{46} + ( 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{47} + ( 1 - 5 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{49} + ( -4 - \beta_{1} - \beta_{2} + 6 \beta_{3} - \beta_{4} - 2 \beta_{6} + 4 \beta_{7} ) q^{50} + ( 2 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{52} + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{53} + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{55} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{56} + ( -6 + \beta_{2} + 6 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{58} + ( -2 + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{59} + ( 4 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{61} + ( -8 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{62} + ( -4 + \beta_{2} + 6 \beta_{3} - \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{64} + ( -2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{65} + ( -3 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{67} + ( 4 + 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{68} + ( 2 - 2 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{70} + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} ) q^{71} + ( -1 - 4 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{73} + ( 6 + \beta_{1} - 4 \beta_{3} + 3 \beta_{4} + \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{74} + ( -4 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{76} + ( -2 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{77} + ( 2 - 2 \beta_{1} - \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{79} + ( 2 + 2 \beta_{1} + 4 \beta_{3} - \beta_{4} + 5 \beta_{6} - 3 \beta_{7} ) q^{80} + ( 4 \beta_{1} + 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{82} + ( \beta_{1} - \beta_{2} - \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{83} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{85} + ( 6 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{86} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{88} + ( 2 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{89} + ( 4 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{91} + ( 8 - 4 \beta_{1} - 8 \beta_{3} - 4 \beta_{5} ) q^{92} + ( -4 + 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{94} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{95} + ( 2 - \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{97} + ( 2 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{2} - q^{4} - 2q^{7} - 4q^{8} + O(q^{10}) \) \( 8q - q^{2} - q^{4} - 2q^{7} - 4q^{8} - 13q^{10} + 6q^{11} + 10q^{14} + 7q^{16} + 6q^{19} + 22q^{20} - 6q^{22} + 2q^{25} - 12q^{26} - 7q^{28} + 16q^{29} - 6q^{31} - 21q^{32} + 28q^{34} - 12q^{35} + 6q^{37} - 8q^{38} - 13q^{40} - 19q^{44} - 12q^{46} + 4q^{47} + 4q^{49} - 2q^{50} + 20q^{52} + 4q^{53} + 8q^{55} + q^{56} - 23q^{58} - 14q^{59} + 12q^{61} - 48q^{62} + 2q^{64} - 4q^{65} - 42q^{67} + 10q^{68} + 35q^{70} - 18q^{73} + 28q^{74} - 44q^{76} - 8q^{77} + 6q^{79} + 33q^{80} - 14q^{82} + 4q^{83} - 32q^{85} + 42q^{86} + 11q^{88} + 34q^{91} + 28q^{92} - 16q^{94} - 24q^{95} + 19q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + \nu^{5} + 4 \nu^{4} + 2 \nu^{3} + 8 \nu^{2} + 4 \nu - 8 \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + \nu^{5} + 4 \nu^{4} - 6 \nu^{3} + 4 \nu - 8 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - \nu^{5} - 4 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} - 4 \nu + 8 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - \nu^{5} + 2 \nu^{3} - 4 \nu + 8 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} + 4 \nu^{6} - 3 \nu^{5} - 8 \nu^{4} + 10 \nu^{3} - 12 \nu + 40 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} + 2 \nu^{6} + \nu^{5} - 6 \nu^{4} + 10 \nu^{3} + 4 \nu^{2} - 12 \nu + 32 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{4} - 2 \beta_{3} + \beta_{2}\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2}\)
\(\nu^{5}\)\(=\)\(2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2}\)
\(\nu^{6}\)\(=\)\(4 \beta_{6} - 2 \beta_{5} - \beta_{4} + 6 \beta_{3} + \beta_{2} - 4\)
\(\nu^{7}\)\(=\)\(-2 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} - \beta_{4} - 6 \beta_{3} + 3 \beta_{2} - 4 \beta_{1} + 8\)

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.856419 + 1.12541i
1.40376 0.171630i
−1.33790 + 0.458297i
0.0777157 1.41208i
0.856419 1.12541i
1.40376 + 0.171630i
−1.33790 0.458297i
0.0777157 + 1.41208i
−1.40284 + 0.178976i 0 1.93594 0.502151i 3.33878 + 1.92764i 0 −1.59285 + 2.11254i −2.62594 + 1.05092i 0 −5.02878 2.10662i
19.2 −0.553244 + 1.30151i 0 −1.38784 1.44010i −0.834598 0.481855i 0 1.20103 2.35744i 2.64212 1.00956i 0 1.08887 0.819652i
19.3 0.272050 1.38780i 0 −1.85198 0.755103i −2.12403 1.22631i 0 −2.63169 + 0.272415i −1.55176 + 2.36475i 0 −2.27971 + 2.61411i
19.4 1.18404 + 0.773342i 0 0.803884 + 1.83133i −0.380152 0.219481i 0 2.02350 + 1.70453i −0.464416 + 2.79004i 0 −0.280380 0.553861i
199.1 −1.40284 0.178976i 0 1.93594 + 0.502151i 3.33878 1.92764i 0 −1.59285 2.11254i −2.62594 1.05092i 0 −5.02878 + 2.10662i
199.2 −0.553244 1.30151i 0 −1.38784 + 1.44010i −0.834598 + 0.481855i 0 1.20103 + 2.35744i 2.64212 + 1.00956i 0 1.08887 + 0.819652i
199.3 0.272050 + 1.38780i 0 −1.85198 + 0.755103i −2.12403 + 1.22631i 0 −2.63169 0.272415i −1.55176 2.36475i 0 −2.27971 2.61411i
199.4 1.18404 0.773342i 0 0.803884 1.83133i −0.380152 + 0.219481i 0 2.02350 1.70453i −0.464416 2.79004i 0 −0.280380 + 0.553861i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.bf.f 8
3.b odd 2 1 84.2.o.b yes 8
4.b odd 2 1 252.2.bf.g 8
7.c even 3 1 1764.2.b.j 8
7.d odd 6 1 252.2.bf.g 8
7.d odd 6 1 1764.2.b.i 8
12.b even 2 1 84.2.o.a 8
21.c even 2 1 588.2.o.b 8
21.g even 6 1 84.2.o.a 8
21.g even 6 1 588.2.b.b 8
21.h odd 6 1 588.2.b.a 8
21.h odd 6 1 588.2.o.d 8
24.f even 2 1 1344.2.bl.j 8
24.h odd 2 1 1344.2.bl.i 8
28.f even 6 1 inner 252.2.bf.f 8
28.f even 6 1 1764.2.b.j 8
28.g odd 6 1 1764.2.b.i 8
84.h odd 2 1 588.2.o.d 8
84.j odd 6 1 84.2.o.b yes 8
84.j odd 6 1 588.2.b.a 8
84.n even 6 1 588.2.b.b 8
84.n even 6 1 588.2.o.b 8
168.ba even 6 1 1344.2.bl.j 8
168.be odd 6 1 1344.2.bl.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.o.a 8 12.b even 2 1
84.2.o.a 8 21.g even 6 1
84.2.o.b yes 8 3.b odd 2 1
84.2.o.b yes 8 84.j odd 6 1
252.2.bf.f 8 1.a even 1 1 trivial
252.2.bf.f 8 28.f even 6 1 inner
252.2.bf.g 8 4.b odd 2 1
252.2.bf.g 8 7.d odd 6 1
588.2.b.a 8 21.h odd 6 1
588.2.b.a 8 84.j odd 6 1
588.2.b.b 8 21.g even 6 1
588.2.b.b 8 84.n even 6 1
588.2.o.b 8 21.c even 2 1
588.2.o.b 8 84.n even 6 1
588.2.o.d 8 21.h odd 6 1
588.2.o.d 8 84.h odd 2 1
1344.2.bl.i 8 24.h odd 2 1
1344.2.bl.i 8 168.be odd 6 1
1344.2.bl.j 8 24.f even 2 1
1344.2.bl.j 8 168.ba even 6 1
1764.2.b.i 8 7.d odd 6 1
1764.2.b.i 8 28.g odd 6 1
1764.2.b.j 8 7.c even 3 1
1764.2.b.j 8 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{8} - 11 T_{5}^{6} + 125 T_{5}^{4} + 264 T_{5}^{3} + 236 T_{5}^{2} + 96 T_{5} + 16 \)
\(T_{11}^{8} - \cdots\)
\(T_{19}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 8 T + 4 T^{2} + 4 T^{3} + 2 T^{5} + T^{6} + T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 16 + 96 T + 236 T^{2} + 264 T^{3} + 125 T^{4} - 11 T^{6} + T^{8} \)
$7$ \( 2401 + 686 T + 112 T^{3} + 65 T^{4} + 16 T^{5} + 2 T^{7} + T^{8} \)
$11$ \( 400 + 240 T - 212 T^{2} - 156 T^{3} + 125 T^{4} + 78 T^{5} - T^{6} - 6 T^{7} + T^{8} \)
$13$ \( 256 + 1936 T^{2} + 473 T^{4} + 38 T^{6} + T^{8} \)
$17$ \( 1024 - 1536 T - 128 T^{2} + 1344 T^{3} + 752 T^{4} - 28 T^{6} + T^{8} \)
$19$ \( 16 + 240 T + 3628 T^{2} - 372 T^{3} + 405 T^{4} - 78 T^{5} + 43 T^{6} - 6 T^{7} + T^{8} \)
$23$ \( 16384 + 12288 T - 2048 T^{2} - 3840 T^{3} + 1472 T^{4} - 40 T^{6} + T^{8} \)
$29$ \( ( -512 + 352 T - 45 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$31$ \( 4173849 - 551610 T + 244512 T^{2} - 1836 T^{3} + 6633 T^{4} + 36 T^{5} + 120 T^{6} + 6 T^{7} + T^{8} \)
$37$ \( 355216 - 214560 T + 103972 T^{2} - 22632 T^{3} + 4605 T^{4} - 462 T^{5} + 79 T^{6} - 6 T^{7} + T^{8} \)
$41$ \( 350464 + 311552 T^{2} + 14048 T^{4} + 208 T^{6} + T^{8} \)
$43$ \( 1073296 + 140152 T^{2} + 6593 T^{4} + 134 T^{6} + T^{8} \)
$47$ \( 4096 + 1024 T + 2048 T^{2} + 64 T^{3} + 784 T^{4} + 80 T^{5} + 44 T^{6} - 4 T^{7} + T^{8} \)
$53$ \( 64 + 928 T + 12968 T^{2} + 7012 T^{3} + 3265 T^{4} + 476 T^{5} + 77 T^{6} - 4 T^{7} + T^{8} \)
$59$ \( 1420864 - 548320 T + 243784 T^{2} - 20956 T^{3} + 5977 T^{4} + 542 T^{5} + 223 T^{6} + 14 T^{7} + T^{8} \)
$61$ \( 1048576 - 1572864 T + 901120 T^{2} - 172032 T^{3} + 7424 T^{4} + 1344 T^{5} - 64 T^{6} - 12 T^{7} + T^{8} \)
$67$ \( 4129024 - 146304 T - 337616 T^{2} + 12024 T^{3} + 30929 T^{4} + 7014 T^{5} + 755 T^{6} + 42 T^{7} + T^{8} \)
$71$ \( 200704 + 173312 T^{2} + 19856 T^{4} + 280 T^{6} + T^{8} \)
$73$ \( 952576 + 1124352 T + 494096 T^{2} + 61056 T^{3} - 3127 T^{4} - 954 T^{5} + 55 T^{6} + 18 T^{7} + T^{8} \)
$79$ \( 241081 + 67758 T - 89888 T^{2} - 27048 T^{3} + 37649 T^{4} + 1176 T^{5} - 184 T^{6} - 6 T^{7} + T^{8} \)
$83$ \( ( 196 - 304 T - 103 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$89$ \( 4096 + 18432 T + 33536 T^{2} + 26496 T^{3} + 8528 T^{4} - 92 T^{6} + T^{8} \)
$97$ \( 246016 + 86176 T^{2} + 8249 T^{4} + 182 T^{6} + T^{8} \)
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