Properties

Label 460.2.i.b
Level $460$
Weight $2$
Character orbit 460.i
Analytic conductor $3.673$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 18 x^{14} + 146 x^{12} - 798 x^{10} + 3934 x^{8} - 19950 x^{6} + 91250 x^{4} - 281250 x^{2} + 390625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} -\beta_{3} q^{5} + ( -\beta_{3} + \beta_{15} ) q^{7} + ( -1 + \beta_{2} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{12} ) q^{9} +O(q^{10})\) \( q + \beta_{6} q^{3} -\beta_{3} q^{5} + ( -\beta_{3} + \beta_{15} ) q^{7} + ( -1 + \beta_{2} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{12} ) q^{9} + ( \beta_{1} - \beta_{4} - \beta_{7} + \beta_{14} + \beta_{15} ) q^{11} + ( 1 - \beta_{6} + \beta_{9} + \beta_{10} ) q^{13} + ( -\beta_{4} + 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{15} + ( \beta_{4} - \beta_{11} ) q^{17} + ( \beta_{3} + \beta_{11} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - \beta_{13} + 2 \beta_{14} ) q^{21} + ( 1 - \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{14} ) q^{23} + ( 3 + \beta_{8} - \beta_{10} - \beta_{12} ) q^{25} + ( 4 + \beta_{5} + 3 \beta_{8} - 4 \beta_{9} ) q^{27} + ( 1 - \beta_{2} + 3 \beta_{9} - \beta_{12} ) q^{29} + ( -1 + \beta_{6} - \beta_{8} ) q^{31} + ( -2 \beta_{4} - \beta_{7} - 2 \beta_{11} ) q^{33} + ( 1 - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} ) q^{35} + ( -\beta_{1} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{37} + ( \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{39} + ( -3 + \beta_{5} - \beta_{10} ) q^{41} + ( -\beta_{4} + \beta_{7} - \beta_{11} ) q^{43} + ( 4 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{7} + \beta_{11} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{45} + ( -1 + 2 \beta_{2} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{47} + ( -\beta_{5} - \beta_{6} - \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{49} + ( -\beta_{1} - 2 \beta_{4} + \beta_{7} + 2 \beta_{14} - \beta_{15} ) q^{51} + ( -\beta_{1} - 3 \beta_{4} - 2 \beta_{7} - 3 \beta_{11} - \beta_{13} ) q^{53} + ( \beta_{2} - 2 \beta_{5} - 2 \beta_{9} + 2 \beta_{12} ) q^{55} + ( \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{11} - \beta_{13} + \beta_{15} ) q^{57} + ( 1 - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} - \beta_{12} ) q^{59} + ( -2 \beta_{1} - \beta_{3} + 3 \beta_{4} + \beta_{7} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{61} + ( 2 \beta_{1} + 5 \beta_{3} + 5 \beta_{4} + 5 \beta_{11} + 2 \beta_{13} - 5 \beta_{14} ) q^{63} + ( \beta_{1} + \beta_{3} + 3 \beta_{4} + \beta_{7} - \beta_{11} - \beta_{14} - 2 \beta_{15} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{11} + 2 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} + \beta_{10} + 4 \beta_{11} - \beta_{13} + \beta_{14} ) q^{69} + ( -\beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{71} + ( -3 - \beta_{6} - 3 \beta_{9} + 2 \beta_{10} ) q^{73} + ( -3 - \beta_{2} - \beta_{5} + 3 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{75} + ( -3 - 2 \beta_{2} - \beta_{5} + 5 \beta_{9} + \beta_{10} ) q^{77} + ( -\beta_{1} - 2 \beta_{3} + 2 \beta_{7} - \beta_{11} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{79} + ( -12 + \beta_{2} + 2 \beta_{5} + 5 \beta_{6} - 5 \beta_{8} - 2 \beta_{10} - \beta_{12} ) q^{81} + ( -4 \beta_{1} - \beta_{3} + 2 \beta_{7} - 4 \beta_{13} + \beta_{14} ) q^{83} + ( 2 - 3 \beta_{6} + \beta_{10} + \beta_{12} ) q^{85} + ( 3 - 2 \beta_{2} - 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{87} + ( -\beta_{3} - 2 \beta_{7} - 3 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{89} + ( 3 \beta_{1} + 3 \beta_{3} + 5 \beta_{4} + 4 \beta_{13} - \beta_{14} ) q^{91} + ( 5 - \beta_{5} - 3 \beta_{6} + 5 \beta_{9} + \beta_{10} + 2 \beta_{12} ) q^{93} + ( -5 + \beta_{6} - 2 \beta_{8} ) q^{95} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{97} + ( -3 \beta_{1} + 2 \beta_{3} + \beta_{7} + 3 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{3} + O(q^{10}) \) \( 16q + 4q^{3} + 12q^{13} + 12q^{23} + 36q^{25} + 52q^{27} - 8q^{31} + 16q^{35} - 48q^{41} + 4q^{47} + 24q^{55} + 8q^{71} - 52q^{73} - 56q^{75} - 64q^{77} - 152q^{81} + 28q^{85} + 28q^{87} + 84q^{93} - 68q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 18 x^{14} + 146 x^{12} - 798 x^{10} + 3934 x^{8} - 19950 x^{6} + 91250 x^{4} - 281250 x^{2} + 390625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -28 \nu^{14} + 9729 \nu^{12} - 78013 \nu^{10} + 385944 \nu^{8} - 1646452 \nu^{6} + 8912125 \nu^{4} - 43639625 \nu^{2} + 123225000 \)\()/11356250\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} - 18 \nu^{13} + 146 \nu^{11} - 798 \nu^{9} + 3934 \nu^{7} - 19950 \nu^{5} + 91250 \nu^{3} - 281250 \nu \)\()/78125\)
\(\beta_{4}\)\(=\)\((\)\( 6206 \nu^{15} - 145108 \nu^{13} + 1091651 \nu^{11} - 5206913 \nu^{9} + 25292604 \nu^{7} - 120536550 \nu^{5} + 599261875 \nu^{3} - 1562984375 \nu \)\()/ 283906250 \)
\(\beta_{5}\)\(=\)\((\)\( -9539 \nu^{14} + 106427 \nu^{12} - 622119 \nu^{10} + 2938847 \nu^{8} - 15928851 \nu^{6} + 75874325 \nu^{4} - 260211875 \nu^{2} + 434828125 \)\()/56781250\)
\(\beta_{6}\)\(=\)\((\)\( -9679 \nu^{14} + 155072 \nu^{12} - 1012184 \nu^{10} + 4868567 \nu^{8} - 24161111 \nu^{6} + 120434950 \nu^{4} - 535191250 \nu^{2} + 1164515625 \)\()/56781250\)
\(\beta_{7}\)\(=\)\((\)\( 1768 \nu^{15} - 26839 \nu^{13} + 159273 \nu^{11} - 818804 \nu^{9} + 3607532 \nu^{7} - 21256985 \nu^{5} + 87834625 \nu^{3} - 119118750 \nu \)\()/56781250\)
\(\beta_{8}\)\(=\)\((\)\( 13152 \nu^{14} - 165036 \nu^{12} + 932717 \nu^{10} - 4530221 \nu^{8} + 23029618 \nu^{6} - 120333350 \nu^{4} + 471120625 \nu^{2} - 766046875 \)\()/56781250\)
\(\beta_{9}\)\(=\)\((\)\( -14766 \nu^{14} + 182013 \nu^{12} - 1107886 \nu^{10} + 5441493 \nu^{8} - 26865744 \nu^{6} + 137700225 \nu^{4} - 545257500 \nu^{2} + 983109375 \)\()/56781250\)
\(\beta_{10}\)\(=\)\((\)\( -37979 \nu^{14} + 527497 \nu^{12} - 3336559 \nu^{10} + 15924867 \nu^{8} - 79211011 \nu^{6} + 411359675 \nu^{4} - 1676681875 \nu^{2} + 3273890625 \)\()/56781250\)
\(\beta_{11}\)\(=\)\((\)\( -39617 \nu^{15} + 550556 \nu^{13} - 3408182 \nu^{11} + 16828941 \nu^{9} - 84516503 \nu^{7} + 433599950 \nu^{5} - 1809035000 \nu^{3} + 3434765625 \nu \)\()/ 283906250 \)
\(\beta_{12}\)\(=\)\((\)\( 10953 \nu^{14} - 151589 \nu^{12} + 959968 \nu^{10} - 4671004 \nu^{8} + 23307357 \nu^{6} - 120838265 \nu^{4} + 495881000 \nu^{2} - 978331250 \)\()/11356250\)
\(\beta_{13}\)\(=\)\((\)\( 62919 \nu^{15} - 763392 \nu^{13} + 4635849 \nu^{11} - 22512212 \nu^{9} + 111486021 \nu^{7} - 583590450 \nu^{5} + 2298853125 \nu^{3} - 4064531250 \nu \)\()/ 283906250 \)
\(\beta_{14}\)\(=\)\((\)\( 14766 \nu^{15} - 182013 \nu^{13} + 1107886 \nu^{11} - 5441493 \nu^{9} + 26865744 \nu^{7} - 137700225 \nu^{5} + 545257500 \nu^{3} - 983109375 \nu \)\()/56781250\)
\(\beta_{15}\)\(=\)\((\)\( 99216 \nu^{15} - 1413363 \nu^{13} + 9018211 \nu^{11} - 43634468 \nu^{9} + 219432044 \nu^{7} - 1129291475 \nu^{5} + 4663698125 \nu^{3} - 9530000000 \nu \)\()/ 283906250 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{5} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{15} - 3 \beta_{14} + 4 \beta_{13} - 4 \beta_{11} - \beta_{7} + 3 \beta_{4} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{12} + \beta_{10} - 5 \beta_{9} - \beta_{8} - 9 \beta_{6} + 2 \beta_{2} + 6\)
\(\nu^{5}\)\(=\)\(-8 \beta_{15} + 3 \beta_{14} + \beta_{13} - 13 \beta_{11} - 9 \beta_{7} + 16 \beta_{4} - 7 \beta_{3} - \beta_{1}\)
\(\nu^{6}\)\(=\)\(11 \beta_{12} + 18 \beta_{10} + 28 \beta_{9} - \beta_{8} - 24 \beta_{6} - 29 \beta_{5} + 8 \beta_{2} + 39\)
\(\nu^{7}\)\(=\)\(7 \beta_{15} + 22 \beta_{14} - 49 \beta_{13} - 45 \beta_{11} - 89 \beta_{7} - 35 \beta_{4} - 4 \beta_{3} + 65 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-6 \beta_{12} - 109 \beta_{10} + 294 \beta_{9} + 6 \beta_{8} - 38 \beta_{6} - 11 \beta_{5} + 154 \beta_{2} - 49\)
\(\nu^{9}\)\(=\)\(63 \beta_{15} - 426 \beta_{14} + 389 \beta_{13} - 129 \beta_{11} - 381 \beta_{7} + 3 \beta_{4} - 285 \beta_{3} + 60 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-360 \beta_{12} - 867 \beta_{10} + 164 \beta_{9} - 744 \beta_{8} - 3 \beta_{6} + 102 \beta_{5} + 441 \beta_{2} + 594\)
\(\nu^{11}\)\(=\)\(471 \beta_{15} - 809 \beta_{14} + 1593 \beta_{13} + 2235 \beta_{11} - 612 \beta_{7} + 1380 \beta_{4} - 507 \beta_{3} + 2058 \beta_{1}\)
\(\nu^{12}\)\(=\)\(1042 \beta_{12} - 512 \beta_{10} - 1083 \beta_{9} - 5507 \beta_{8} + 1173 \beta_{6} + 875 \beta_{5} + 2670 \beta_{2} + 4014\)
\(\nu^{13}\)\(=\)\(-437 \beta_{15} - 3337 \beta_{14} + 8644 \beta_{13} + 6996 \beta_{11} - 4706 \beta_{7} + 2753 \beta_{4} + 13080 \beta_{3} + 19038 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-1021 \beta_{12} - 4853 \beta_{10} - 18728 \beta_{9} - 17355 \beta_{8} - 2656 \beta_{6} + 14916 \beta_{5} + 23744 \beta_{2} - 35425\)
\(\nu^{15}\)\(=\)\(-30996 \beta_{15} - 34848 \beta_{14} + 81152 \beta_{13} - 20644 \beta_{11} - 37568 \beta_{7} + 33608 \beta_{4} + 36243 \beta_{3} + 4436 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(1\) \(-\beta_{9}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
−2.19448 + 0.429243i
2.19448 0.429243i
−2.23266 0.123447i
2.23266 + 0.123447i
−1.88618 1.20098i
1.88618 + 1.20098i
−1.06856 + 1.96422i
1.06856 1.96422i
−2.19448 0.429243i
2.19448 + 0.429243i
−2.23266 + 0.123447i
2.23266 0.123447i
−1.88618 + 1.20098i
1.88618 1.20098i
−1.06856 1.96422i
1.06856 + 1.96422i
0 −2.36693 + 2.36693i 0 −2.19448 0.429243i 0 −2.93008 + 2.93008i 0 8.20472i 0
137.2 0 −2.36693 + 2.36693i 0 2.19448 + 0.429243i 0 2.93008 2.93008i 0 8.20472i 0
137.3 0 0.237125 0.237125i 0 −2.23266 + 0.123447i 0 1.69421 1.69421i 0 2.88754i 0
137.4 0 0.237125 0.237125i 0 2.23266 0.123447i 0 −1.69421 + 1.69421i 0 2.88754i 0
137.5 0 1.09396 1.09396i 0 −1.88618 + 1.20098i 0 −2.67082 + 2.67082i 0 0.606488i 0
137.6 0 1.09396 1.09396i 0 1.88618 1.20098i 0 2.67082 2.67082i 0 0.606488i 0
137.7 0 2.03584 2.03584i 0 −1.06856 1.96422i 0 0.641104 0.641104i 0 5.28931i 0
137.8 0 2.03584 2.03584i 0 1.06856 + 1.96422i 0 −0.641104 + 0.641104i 0 5.28931i 0
413.1 0 −2.36693 2.36693i 0 −2.19448 + 0.429243i 0 −2.93008 2.93008i 0 8.20472i 0
413.2 0 −2.36693 2.36693i 0 2.19448 0.429243i 0 2.93008 + 2.93008i 0 8.20472i 0
413.3 0 0.237125 + 0.237125i 0 −2.23266 0.123447i 0 1.69421 + 1.69421i 0 2.88754i 0
413.4 0 0.237125 + 0.237125i 0 2.23266 + 0.123447i 0 −1.69421 1.69421i 0 2.88754i 0
413.5 0 1.09396 + 1.09396i 0 −1.88618 1.20098i 0 −2.67082 2.67082i 0 0.606488i 0
413.6 0 1.09396 + 1.09396i 0 1.88618 + 1.20098i 0 2.67082 + 2.67082i 0 0.606488i 0
413.7 0 2.03584 + 2.03584i 0 −1.06856 + 1.96422i 0 0.641104 + 0.641104i 0 5.28931i 0
413.8 0 2.03584 + 2.03584i 0 1.06856 1.96422i 0 −0.641104 0.641104i 0 5.28931i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 413.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.i.b 16
5.b even 2 1 2300.2.i.d 16
5.c odd 4 1 inner 460.2.i.b 16
5.c odd 4 1 2300.2.i.d 16
23.b odd 2 1 inner 460.2.i.b 16
115.c odd 2 1 2300.2.i.d 16
115.e even 4 1 inner 460.2.i.b 16
115.e even 4 1 2300.2.i.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.i.b 16 1.a even 1 1 trivial
460.2.i.b 16 5.c odd 4 1 inner
460.2.i.b 16 23.b odd 2 1 inner
460.2.i.b 16 115.e even 4 1 inner
2300.2.i.d 16 5.b even 2 1
2300.2.i.d 16 5.c odd 4 1
2300.2.i.d 16 115.c odd 2 1
2300.2.i.d 16 115.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 25 - 130 T + 338 T^{2} - 270 T^{3} + 110 T^{4} - 6 T^{5} + 2 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$5$ \( 390625 - 281250 T^{2} + 91250 T^{4} - 19950 T^{6} + 3934 T^{8} - 798 T^{10} + 146 T^{12} - 18 T^{14} + T^{16} \)
$7$ \( 1336336 + 2029264 T^{4} + 76792 T^{8} + 532 T^{12} + T^{16} \)
$11$ \( ( 2500 + 2596 T^{2} + 672 T^{4} + 50 T^{6} + T^{8} )^{2} \)
$13$ \( ( 13225 - 15870 T + 9522 T^{2} - 2898 T^{3} + 486 T^{4} - 42 T^{5} + 18 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$17$ \( 16 + 10288 T^{4} + 71020 T^{8} + 880 T^{12} + T^{16} \)
$19$ \( ( 100 - 244 T^{2} + 178 T^{4} - 36 T^{6} + T^{8} )^{2} \)
$23$ \( 78310985281 - 40857905364 T + 10658584008 T^{2} - 3939041916 T^{3} + 1120483364 T^{4} - 218275980 T^{5} + 60445656 T^{6} - 13578372 T^{7} + 2322310 T^{8} - 590364 T^{9} + 114264 T^{10} - 17940 T^{11} + 4004 T^{12} - 612 T^{13} + 72 T^{14} - 12 T^{15} + T^{16} \)
$29$ \( ( 28561 + 16680 T^{2} + 2110 T^{4} + 88 T^{6} + T^{8} )^{2} \)
$31$ \( ( 11 + 10 T - 20 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$37$ \( 456976 + 21869584 T^{4} + 458296 T^{8} + 1348 T^{12} + T^{16} \)
$41$ \( ( -107 - 188 T + 10 T^{2} + 12 T^{3} + T^{4} )^{4} \)
$43$ \( 127880620816 + 3092373184 T^{4} + 11231212 T^{8} + 6892 T^{12} + T^{16} \)
$47$ \( ( 24025 + 49910 T + 51842 T^{2} + 24162 T^{3} + 6086 T^{4} + 474 T^{5} + 2 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$53$ \( 533794816 + 2911069184 T^{4} + 584919936 T^{8} + 54928 T^{12} + T^{16} \)
$59$ \( ( 110224 + 167728 T^{2} + 9244 T^{4} + 168 T^{6} + T^{8} )^{2} \)
$61$ \( ( 100 + 3044 T^{2} + 5698 T^{4} + 224 T^{6} + T^{8} )^{2} \)
$67$ \( 3675002150265616 + 2504414676496 T^{4} + 498843384 T^{8} + 38308 T^{12} + T^{16} \)
$71$ \( ( 169 - 162 T - 74 T^{2} - 2 T^{3} + T^{4} )^{4} \)
$73$ \( ( 25 - 390 T + 3042 T^{2} - 3874 T^{3} + 2294 T^{4} + 1326 T^{5} + 338 T^{6} + 26 T^{7} + T^{8} )^{2} \)
$79$ \( ( 115562500 - 4549996 T^{2} + 66346 T^{4} - 424 T^{6} + T^{8} )^{2} \)
$83$ \( 17799252215056 + 1094073550768 T^{4} + 527018796 T^{8} + 62144 T^{12} + T^{16} \)
$89$ \( ( 1144900 - 257316 T^{2} + 16786 T^{4} - 260 T^{6} + T^{8} )^{2} \)
$97$ \( 583506543376 + 8112583184 T^{4} + 22546936 T^{8} + 15908 T^{12} + T^{16} \)
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