Properties

Label 360.2.w.d
Level $360$
Weight $2$
Character orbit 360.w
Analytic conductor $2.875$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 5 x^{12} + 28 x^{8} + 80 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_{15} q^{2} -\beta_{2} q^{4} + ( \beta_{10} - \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{6} + \beta_{11} ) q^{7} + ( -\beta_{4} + \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{15} q^{2} -\beta_{2} q^{4} + ( \beta_{10} - \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{6} + \beta_{11} ) q^{7} + ( -\beta_{4} + \beta_{5} ) q^{8} + ( \beta_{2} + 2 \beta_{8} + \beta_{9} ) q^{10} + ( -\beta_{1} + \beta_{14} + \beta_{15} ) q^{11} + ( -\beta_{3} - \beta_{7} ) q^{13} + ( -2 \beta_{4} + \beta_{5} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{14} + ( -1 - \beta_{2} - \beta_{3} - \beta_{9} + \beta_{11} ) q^{16} + ( \beta_{4} + \beta_{10} - 2 \beta_{13} - \beta_{14} ) q^{17} + 4 \beta_{8} q^{19} + ( \beta_{1} - \beta_{4} - \beta_{5} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{20} + ( 2 - \beta_{2} - \beta_{6} + 2 \beta_{8} ) q^{22} + ( 2 \beta_{4} + 2 \beta_{10} - 2 \beta_{14} ) q^{23} + ( -2 \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{25} + ( -\beta_{1} - \beta_{5} - 4 \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{26} + ( 4 - \beta_{2} - 2 \beta_{3} + \beta_{6} - 2 \beta_{8} ) q^{28} + ( -\beta_{1} + 2 \beta_{4} + 2 \beta_{12} - \beta_{14} - \beta_{15} ) q^{29} + ( 1 + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{31} + ( \beta_{4} - 2 \beta_{10} + \beta_{13} - \beta_{14} ) q^{32} + ( -2 + 2 \beta_{3} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{34} + ( \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{35} + ( -1 - \beta_{2} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{37} + 4 \beta_{1} q^{38} + ( 1 + 2 \beta_{2} + \beta_{3} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{40} + ( -2 \beta_{5} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{41} + ( -4 + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{6} - 2 \beta_{7} ) q^{43} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{44} + ( 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{46} + ( 2 \beta_{12} + 2 \beta_{14} - 4 \beta_{15} ) q^{47} + ( 1 - 3 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - 4 \beta_{8} - \beta_{11} ) q^{49} + ( -2 \beta_{1} + \beta_{5} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{50} + ( -2 - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{52} + ( 2 \beta_{1} + \beta_{4} + \beta_{10} - \beta_{12} ) q^{53} + ( 2 + \beta_{2} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{55} + ( -4 \beta_{1} - \beta_{5} - 4 \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{14} + 4 \beta_{15} ) q^{56} + ( -2 + \beta_{2} - \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{58} + ( -\beta_{1} - \beta_{12} - \beta_{15} ) q^{59} + ( -3 + \beta_{2} + \beta_{3} + \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{61} + ( -2 \beta_{4} + 4 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{62} + ( 1 - \beta_{3} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - \beta_{9} - \beta_{11} ) q^{64} + ( -4 \beta_{1} + \beta_{4} + \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{65} + ( 2 + 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{11} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{10} - 4 \beta_{12} + 2 \beta_{14} ) q^{68} + ( -6 + 3 \beta_{2} + 4 \beta_{3} - \beta_{6} - 2 \beta_{7} - 4 \beta_{8} ) q^{70} + ( 5 - 5 \beta_{8} ) q^{73} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{74} + 4 \beta_{6} q^{76} + ( -2 \beta_{4} + 2 \beta_{10} - 3 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{77} + ( -1 - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{79} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{80} + ( 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{9} - 4 \beta_{11} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{10} - \beta_{12} - \beta_{14} ) q^{83} + ( 2 + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{11} ) q^{85} + ( 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{13} + 6 \beta_{14} - 2 \beta_{15} ) q^{86} + ( -3 \beta_{2} - 2 \beta_{3} + 3 \beta_{6} + 2 \beta_{8} ) q^{88} + ( 4 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{12} + 2 \beta_{13} + \beta_{14} + 4 \beta_{15} ) q^{89} + ( -6 + 2 \beta_{2} + 2 \beta_{3} - 6 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{11} ) q^{91} + ( -4 \beta_{4} + 4 \beta_{10} - 4 \beta_{12} - 4 \beta_{15} ) q^{92} + ( 4 \beta_{2} + 8 \beta_{8} ) q^{94} + ( -4 \beta_{1} - 4 \beta_{4} ) q^{95} + ( -3 + 2 \beta_{2} + 2 \beta_{6} - 3 \beta_{8} + 2 \beta_{11} ) q^{97} + ( -5 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{12} - 2 \beta_{14} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 8q^{10} - 20q^{16} + 36q^{22} - 24q^{25} + 44q^{28} + 32q^{40} - 16q^{43} - 32q^{46} - 8q^{52} - 44q^{58} + 16q^{67} - 44q^{70} + 80q^{73} - 16q^{76} - 8q^{82} - 28q^{88} - 32q^{91} - 64q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 5 x^{12} + 28 x^{8} + 80 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{14} - 4 \nu^{12} + 29 \nu^{10} - 36 \nu^{8} + 84 \nu^{6} - 256 \nu^{4} + 176 \nu^{2} - 384 \)\()/320\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{15} - 4 \nu^{13} - 11 \nu^{11} - 36 \nu^{9} + 44 \nu^{7} + 64 \nu^{5} - 144 \nu^{3} - 384 \nu \)\()/640\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{15} + 12 \nu^{13} - 11 \nu^{11} + 108 \nu^{9} + 44 \nu^{7} + 448 \nu^{5} - 144 \nu^{3} + 1152 \nu \)\()/640\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{12} - 9 \nu^{8} - 24 \nu^{4} - 96 \)\()/40\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{14} - 8 \nu^{12} + 11 \nu^{10} + 8 \nu^{8} - 44 \nu^{6} - 112 \nu^{4} + 144 \nu^{2} - 128 \)\()/320\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{14} - 7 \nu^{10} - 12 \nu^{6} - 48 \nu^{2} \)\()/320\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{14} - 2 \nu^{12} - \nu^{10} + 2 \nu^{8} + 14 \nu^{6} - 28 \nu^{4} - 24 \nu^{2} - 32 \)\()/80\)
\(\beta_{10}\)\(=\)\((\)\( \nu^{15} + 2 \nu^{13} + 9 \nu^{11} - 2 \nu^{9} + 24 \nu^{7} + 28 \nu^{5} + 136 \nu^{3} - 48 \nu \)\()/160\)
\(\beta_{11}\)\(=\)\((\)\( 5 \nu^{14} - 12 \nu^{12} + 25 \nu^{10} - 28 \nu^{8} + 140 \nu^{6} - 48 \nu^{4} + 400 \nu^{2} - 192 \)\()/320\)
\(\beta_{12}\)\(=\)\((\)\( 5 \nu^{15} + 12 \nu^{13} + 25 \nu^{11} + 28 \nu^{9} + 140 \nu^{7} + 48 \nu^{5} + 400 \nu^{3} + 192 \nu \)\()/640\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{15} + 2 \nu^{13} - 9 \nu^{11} - 2 \nu^{9} - 24 \nu^{7} + 28 \nu^{5} + 24 \nu^{3} - 48 \nu \)\()/160\)
\(\beta_{14}\)\(=\)\((\)\( 5 \nu^{15} - 12 \nu^{13} + 25 \nu^{11} - 28 \nu^{9} + 140 \nu^{7} - 48 \nu^{5} + 400 \nu^{3} - 192 \nu \)\()/640\)
\(\beta_{15}\)\(=\)\((\)\( 3 \nu^{15} + 7 \nu^{11} + 12 \nu^{7} + 48 \nu^{3} \)\()/320\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{14} + \beta_{13} - \beta_{4}\)
\(\nu^{4}\)\(=\)\(\beta_{11} - \beta_{9} - \beta_{3} - \beta_{2} - 1\)
\(\nu^{5}\)\(=\)\(-2 \beta_{12} + 2 \beta_{10} + \beta_{5} + \beta_{4}\)
\(\nu^{6}\)\(=\)\(\beta_{11} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{3} - 1\)
\(\nu^{7}\)\(=\)\(-4 \beta_{15} + \beta_{14} - \beta_{13} + 4 \beta_{12} - 2 \beta_{10} + 3 \beta_{4}\)
\(\nu^{8}\)\(=\)\(-\beta_{11} + 3 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{6} + \beta_{3} + \beta_{2} - 7\)
\(\nu^{9}\)\(=\)\(4 \beta_{14} + 2 \beta_{12} - 6 \beta_{10} + 3 \beta_{5} - 9 \beta_{4} - 12 \beta_{1}\)
\(\nu^{10}\)\(=\)\(3 \beta_{11} - 9 \beta_{9} - 8 \beta_{8} + 6 \beta_{7} - 3 \beta_{6} + 3 \beta_{3} - 12 \beta_{2} - 3\)
\(\nu^{11}\)\(=\)\(-4 \beta_{15} - 9 \beta_{14} - 15 \beta_{13} - 12 \beta_{12} + 18 \beta_{10} - 3 \beta_{4}\)
\(\nu^{12}\)\(=\)\(-15 \beta_{11} - 3 \beta_{9} - 18 \beta_{8} - 18 \beta_{7} - 4 \beta_{6} + 15 \beta_{3} + 15 \beta_{2} - 9\)
\(\nu^{13}\)\(=\)\(-36 \beta_{14} + 30 \beta_{12} + 6 \beta_{10} - 11 \beta_{5} + 17 \beta_{4} + 12 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-11 \beta_{11} + 17 \beta_{9} - 104 \beta_{8} - 6 \beta_{7} + 11 \beta_{6} - 11 \beta_{3} + 12 \beta_{2} + 11\)
\(\nu^{15}\)\(=\)\(132 \beta_{15} + \beta_{14} + 23 \beta_{13} + 12 \beta_{12} - 34 \beta_{10} + 11 \beta_{4}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(\beta_{8}\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−0.512386 + 1.31813i
0.788026 + 1.17431i
1.17431 + 0.788026i
−1.31813 + 0.512386i
1.31813 0.512386i
−1.17431 0.788026i
−0.788026 1.17431i
0.512386 1.31813i
−0.512386 1.31813i
0.788026 1.17431i
1.17431 0.788026i
−1.31813 0.512386i
1.31813 + 0.512386i
−1.17431 + 0.788026i
−0.788026 + 1.17431i
0.512386 + 1.31813i
−1.31813 0.512386i 0 1.47492 + 1.35078i 1.83051 + 1.28422i 0 2.94984 + 2.94984i −1.25201 2.53623i 0 −1.75483 2.63069i
163.2 −1.17431 + 0.788026i 0 0.758030 1.85078i 0.386289 2.20245i 0 1.51606 + 1.51606i 0.568298 + 2.77075i 0 1.28196 + 2.89077i
163.3 −0.788026 + 1.17431i 0 −0.758030 1.85078i −0.386289 + 2.20245i 0 −1.51606 1.51606i 2.77075 + 0.568298i 0 −2.28196 2.18921i
163.4 −0.512386 1.31813i 0 −1.47492 + 1.35078i 1.83051 + 1.28422i 0 −2.94984 2.94984i 2.53623 + 1.25201i 0 0.754834 3.07087i
163.5 0.512386 + 1.31813i 0 −1.47492 + 1.35078i −1.83051 1.28422i 0 −2.94984 2.94984i −2.53623 1.25201i 0 0.754834 3.07087i
163.6 0.788026 1.17431i 0 −0.758030 1.85078i 0.386289 2.20245i 0 −1.51606 1.51606i −2.77075 0.568298i 0 −2.28196 2.18921i
163.7 1.17431 0.788026i 0 0.758030 1.85078i −0.386289 + 2.20245i 0 1.51606 + 1.51606i −0.568298 2.77075i 0 1.28196 + 2.89077i
163.8 1.31813 + 0.512386i 0 1.47492 + 1.35078i −1.83051 1.28422i 0 2.94984 + 2.94984i 1.25201 + 2.53623i 0 −1.75483 2.63069i
307.1 −1.31813 + 0.512386i 0 1.47492 1.35078i 1.83051 1.28422i 0 2.94984 2.94984i −1.25201 + 2.53623i 0 −1.75483 + 2.63069i
307.2 −1.17431 0.788026i 0 0.758030 + 1.85078i 0.386289 + 2.20245i 0 1.51606 1.51606i 0.568298 2.77075i 0 1.28196 2.89077i
307.3 −0.788026 1.17431i 0 −0.758030 + 1.85078i −0.386289 2.20245i 0 −1.51606 + 1.51606i 2.77075 0.568298i 0 −2.28196 + 2.18921i
307.4 −0.512386 + 1.31813i 0 −1.47492 1.35078i 1.83051 1.28422i 0 −2.94984 + 2.94984i 2.53623 1.25201i 0 0.754834 + 3.07087i
307.5 0.512386 1.31813i 0 −1.47492 1.35078i −1.83051 + 1.28422i 0 −2.94984 + 2.94984i −2.53623 + 1.25201i 0 0.754834 + 3.07087i
307.6 0.788026 + 1.17431i 0 −0.758030 + 1.85078i 0.386289 + 2.20245i 0 −1.51606 + 1.51606i −2.77075 + 0.568298i 0 −2.28196 + 2.18921i
307.7 1.17431 + 0.788026i 0 0.758030 + 1.85078i −0.386289 2.20245i 0 1.51606 1.51606i −0.568298 + 2.77075i 0 1.28196 2.89077i
307.8 1.31813 0.512386i 0 1.47492 1.35078i −1.83051 + 1.28422i 0 2.94984 2.94984i 1.25201 2.53623i 0 −1.75483 + 2.63069i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
8.d odd 2 1 inner
15.e even 4 1 inner
24.f even 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.w.d 16
3.b odd 2 1 inner 360.2.w.d 16
4.b odd 2 1 1440.2.bi.d 16
5.c odd 4 1 inner 360.2.w.d 16
8.b even 2 1 1440.2.bi.d 16
8.d odd 2 1 inner 360.2.w.d 16
12.b even 2 1 1440.2.bi.d 16
15.e even 4 1 inner 360.2.w.d 16
20.e even 4 1 1440.2.bi.d 16
24.f even 2 1 inner 360.2.w.d 16
24.h odd 2 1 1440.2.bi.d 16
40.i odd 4 1 1440.2.bi.d 16
40.k even 4 1 inner 360.2.w.d 16
60.l odd 4 1 1440.2.bi.d 16
120.q odd 4 1 inner 360.2.w.d 16
120.w even 4 1 1440.2.bi.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.w.d 16 1.a even 1 1 trivial
360.2.w.d 16 3.b odd 2 1 inner
360.2.w.d 16 5.c odd 4 1 inner
360.2.w.d 16 8.d odd 2 1 inner
360.2.w.d 16 15.e even 4 1 inner
360.2.w.d 16 24.f even 2 1 inner
360.2.w.d 16 40.k even 4 1 inner
360.2.w.d 16 120.q odd 4 1 inner
1440.2.bi.d 16 4.b odd 2 1
1440.2.bi.d 16 8.b even 2 1
1440.2.bi.d 16 12.b even 2 1
1440.2.bi.d 16 20.e even 4 1
1440.2.bi.d 16 24.h odd 2 1
1440.2.bi.d 16 40.i odd 4 1
1440.2.bi.d 16 60.l odd 4 1
1440.2.bi.d 16 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 324 T_{7}^{4} + 6400 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 80 T^{4} + 28 T^{8} + 5 T^{12} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 625 + 150 T^{2} + 18 T^{4} + 6 T^{6} + T^{8} )^{2} \)
$7$ \( ( 6400 + 324 T^{4} + T^{8} )^{2} \)
$11$ \( ( 40 - 18 T^{2} + T^{4} )^{4} \)
$13$ \( ( 102400 + 804 T^{4} + T^{8} )^{2} \)
$17$ \( ( 25600 + 1796 T^{4} + T^{8} )^{2} \)
$19$ \( ( 16 + T^{2} )^{8} \)
$23$ \( ( 16384 + 2880 T^{4} + T^{8} )^{2} \)
$29$ \( ( 800 - 58 T^{2} + T^{4} )^{4} \)
$31$ \( ( 1280 + 76 T^{2} + T^{4} )^{4} \)
$37$ \( ( 102400 + 804 T^{4} + T^{8} )^{2} \)
$41$ \( ( 640 - 92 T^{2} + T^{4} )^{4} \)
$43$ \( ( 6400 - 320 T + 8 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$47$ \( ( 262144 + 11520 T^{4} + T^{8} )^{2} \)
$53$ \( ( 16384 + 420 T^{4} + T^{8} )^{2} \)
$59$ \( ( 40 + 18 T^{2} + T^{4} )^{4} \)
$61$ \( ( 8000 + 220 T^{2} + T^{4} )^{4} \)
$67$ \( ( 6400 + 320 T + 8 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$71$ \( T^{16} \)
$73$ \( ( 50 - 10 T + T^{2} )^{8} \)
$79$ \( ( 1280 - 76 T^{2} + T^{4} )^{4} \)
$83$ \( ( 104857600 + 21136 T^{4} + T^{8} )^{2} \)
$89$ \( ( 2560 + 348 T^{2} + T^{4} )^{4} \)
$97$ \( ( 2500 - 800 T + 128 T^{2} + 16 T^{3} + T^{4} )^{4} \)
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