Properties

Label 605.2.j.e
Level $605$
Weight $2$
Character orbit 605.j
Analytic conductor $4.831$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.j (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.6879707136000000000000.7
Defining polynomial: \(x^{16} - 4 x^{14} + 15 x^{12} - 56 x^{10} + 209 x^{8} - 56 x^{6} + 15 x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + \beta_{3} q^{3} -\beta_{10} q^{4} + ( -\beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{14} ) q^{5} + \beta_{13} q^{6} + ( \beta_{1} - 2 \beta_{3} - \beta_{5} - 2 \beta_{7} + \beta_{9} + 2 \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{7} -\beta_{3} q^{8} + ( -\beta_{4} - \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{12} q^{2} + \beta_{3} q^{3} -\beta_{10} q^{4} + ( -\beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{14} ) q^{5} + \beta_{13} q^{6} + ( \beta_{1} - 2 \beta_{3} - \beta_{5} - 2 \beta_{7} + \beta_{9} + 2 \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{7} -\beta_{3} q^{8} + ( -\beta_{4} - \beta_{6} ) q^{9} + ( 1 - \beta_{2} - \beta_{9} + \beta_{11} ) q^{10} + ( -\beta_{9} - 2 \beta_{11} ) q^{12} -3 \beta_{15} q^{13} -\beta_{14} q^{14} + ( \beta_{1} - 2 \beta_{3} - \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{15} + ( -2 \beta_{2} - 2 \beta_{6} + 2 \beta_{10} - \beta_{13} + 2 \beta_{14} ) q^{16} -2 \beta_{7} q^{17} + ( 3 \beta_{1} - \beta_{3} - 3 \beta_{5} - \beta_{7} - 2 \beta_{9} + \beta_{11} - 2 \beta_{12} + \beta_{15} ) q^{18} + ( -1 + \beta_{4} + \beta_{8} - \beta_{13} + 3 \beta_{14} ) q^{19} + ( -3 \beta_{4} + 2 \beta_{12} + \beta_{15} ) q^{20} + ( -3 - 2 \beta_{2} ) q^{21} + ( 4 \beta_{9} + \beta_{11} ) q^{23} + ( -2 \beta_{4} + \beta_{6} ) q^{24} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{8} - \beta_{13} ) q^{25} + ( 3 \beta_{8} - 3 \beta_{10} ) q^{26} + ( -\beta_{5} + 2 \beta_{7} ) q^{27} + 3 \beta_{7} q^{28} -4 \beta_{10} q^{29} + ( \beta_{1} - \beta_{3} - \beta_{14} ) q^{30} + ( 7 \beta_{4} - \beta_{6} ) q^{31} + ( -\beta_{9} + 4 \beta_{11} ) q^{32} -2 q^{34} + ( -3 \beta_{4} + \beta_{6} - 3 \beta_{12} - 3 \beta_{15} ) q^{35} + ( 3 - 3 \beta_{4} - 3 \beta_{8} + 3 \beta_{13} - \beta_{14} ) q^{36} + ( 4 \beta_{1} - 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{15} ) q^{37} + ( 7 \beta_{5} + 3 \beta_{7} ) q^{38} + ( 3 \beta_{2} + 3 \beta_{6} - 3 \beta_{10} - 3 \beta_{13} - 3 \beta_{14} ) q^{39} + ( -\beta_{1} + 2 \beta_{3} + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{15} ) q^{40} + \beta_{14} q^{41} + ( -\beta_{12} - 2 \beta_{15} ) q^{42} + ( 7 \beta_{9} + 2 \beta_{11} ) q^{43} + ( -3 + \beta_{2} - 2 \beta_{9} ) q^{45} + ( 7 \beta_{4} + 3 \beta_{6} ) q^{46} + ( -4 \beta_{1} + 7 \beta_{3} ) q^{47} + ( 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} + \beta_{9} + 3 \beta_{11} + \beta_{12} + 3 \beta_{15} ) q^{48} + ( 3 \beta_{2} + 3 \beta_{6} - 3 \beta_{10} + \beta_{13} - 3 \beta_{14} ) q^{49} + ( -2 \beta_{2} + \beta_{5} - 2 \beta_{6} + 2 \beta_{10} - 6 \beta_{13} + 2 \beta_{14} ) q^{50} + ( -4 \beta_{8} - 2 \beta_{10} ) q^{51} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{52} + ( -4 \beta_{12} + 3 \beta_{15} ) q^{53} + ( 4 - \beta_{2} ) q^{54} + ( 3 + 2 \beta_{2} ) q^{56} + ( 2 \beta_{12} + 5 \beta_{15} ) q^{57} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{58} + ( -3 \beta_{8} + \beta_{10} ) q^{59} + ( -\beta_{2} - \beta_{6} + 3 \beta_{7} + \beta_{10} + 3 \beta_{13} + \beta_{14} ) q^{60} + ( -\beta_{2} - \beta_{6} + \beta_{10} + 10 \beta_{13} + \beta_{14} ) q^{61} + ( -5 \beta_{1} - \beta_{3} + 5 \beta_{5} - \beta_{7} + 6 \beta_{9} + \beta_{11} + 6 \beta_{12} + \beta_{15} ) q^{62} + ( -\beta_{1} - \beta_{3} ) q^{63} + ( -4 \beta_{4} - \beta_{6} ) q^{64} + ( 6 - 6 \beta_{9} - 3 \beta_{11} ) q^{65} + ( -7 \beta_{9} + \beta_{11} ) q^{67} + ( 2 \beta_{12} + 4 \beta_{15} ) q^{68} + ( -5 + 5 \beta_{4} + 5 \beta_{8} - 5 \beta_{13} - \beta_{14} ) q^{69} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{70} + ( 3 \beta_{2} + 3 \beta_{6} - 3 \beta_{10} - 3 \beta_{13} - 3 \beta_{14} ) q^{71} + ( \beta_{5} + \beta_{7} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - 4 \beta_{9} - 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{15} ) q^{73} + ( 6 - 6 \beta_{4} - 6 \beta_{8} + 6 \beta_{13} - 4 \beta_{14} ) q^{74} + ( -6 \beta_{4} + 2 \beta_{6} - \beta_{12} - \beta_{15} ) q^{75} + ( -9 + \beta_{2} ) q^{76} -3 \beta_{11} q^{78} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{79} + ( -6 - 3 \beta_{1} - \beta_{3} + 6 \beta_{4} + 6 \beta_{8} - 6 \beta_{13} + \beta_{14} ) q^{80} + ( 2 \beta_{8} + 5 \beta_{10} ) q^{81} + ( 2 \beta_{5} + \beta_{7} ) q^{82} + ( -7 \beta_{5} - 7 \beta_{7} ) q^{83} + ( 6 \beta_{8} + 3 \beta_{10} ) q^{84} + ( 2 + 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{8} + 2 \beta_{13} + 2 \beta_{14} ) q^{85} + ( 12 \beta_{4} + 5 \beta_{6} ) q^{86} + ( -4 \beta_{9} - 8 \beta_{11} ) q^{87} + ( 3 - 2 \beta_{2} ) q^{89} + ( -4 \beta_{4} - 2 \beta_{6} - 4 \beta_{12} + \beta_{15} ) q^{90} + ( 9 - 9 \beta_{4} - 9 \beta_{8} + 9 \beta_{13} + 3 \beta_{14} ) q^{91} + ( -7 \beta_{1} + 5 \beta_{3} + 7 \beta_{5} + 5 \beta_{7} + 2 \beta_{9} - 5 \beta_{11} + 2 \beta_{12} - 5 \beta_{15} ) q^{92} + ( -\beta_{5} - 9 \beta_{7} ) q^{93} + ( -4 \beta_{2} - 4 \beta_{6} + 4 \beta_{10} - \beta_{13} + 4 \beta_{14} ) q^{94} + ( -4 \beta_{1} - 2 \beta_{3} + 4 \beta_{5} - 2 \beta_{7} + 9 \beta_{8} + 6 \beta_{9} - \beta_{10} + 2 \beta_{11} + 6 \beta_{12} + 2 \beta_{15} ) q^{95} + ( -3 + 3 \beta_{4} + 3 \beta_{8} - 3 \beta_{13} - 4 \beta_{14} ) q^{96} + ( 4 \beta_{12} + 5 \beta_{15} ) q^{97} + ( 4 \beta_{9} - 3 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 4q^{6} - 4q^{9} + O(q^{10}) \) \( 16q - 4q^{6} - 4q^{9} + 16q^{10} - 4q^{15} + 4q^{16} - 4q^{19} - 12q^{20} - 48q^{21} - 8q^{24} - 4q^{25} + 12q^{26} + 28q^{31} - 32q^{34} - 12q^{35} + 12q^{36} + 12q^{39} + 4q^{40} - 48q^{45} + 28q^{46} - 4q^{49} + 24q^{50} - 16q^{51} + 64q^{54} + 48q^{56} - 12q^{59} - 12q^{60} - 40q^{61} - 16q^{64} + 96q^{65} - 20q^{69} - 12q^{70} + 12q^{71} + 24q^{74} - 24q^{75} - 144q^{76} + 8q^{79} - 24q^{80} + 8q^{81} + 24q^{84} + 8q^{85} + 48q^{86} + 48q^{89} - 16q^{90} + 36q^{91} + 4q^{94} + 36q^{95} - 12q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{14} + 15 x^{12} - 56 x^{10} + 209 x^{8} - 56 x^{6} + 15 x^{4} - 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{10} + 362 \)\()/209\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} + 780 \nu \)\()/209\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{12} + 780 \nu^{2} \)\()/209\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{13} + 780 \nu^{3} \)\()/209\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{12} - 1351 \nu^{2} \)\()/209\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{13} - 2911 \nu^{3} \)\()/209\)
\(\beta_{8}\)\(=\)\((\)\( -4 \nu^{14} - 2911 \nu^{4} \)\()/209\)
\(\beta_{9}\)\(=\)\((\)\( -4 \nu^{15} - 2911 \nu^{5} \)\()/209\)
\(\beta_{10}\)\(=\)\((\)\( -7 \nu^{14} - 5042 \nu^{4} \)\()/209\)
\(\beta_{11}\)\(=\)\((\)\( -\nu^{15} - 723 \nu^{5} \)\()/19\)
\(\beta_{12}\)\(=\)\((\)\( 60 \nu^{15} - 225 \nu^{13} + 840 \nu^{11} - 3135 \nu^{9} + 11704 \nu^{7} - 225 \nu^{5} + 60 \nu^{3} - 15 \nu \)\()/209\)
\(\beta_{13}\)\(=\)\((\)\( 56 \nu^{14} - 224 \nu^{12} + 840 \nu^{10} - 3135 \nu^{8} + 11704 \nu^{6} - 3136 \nu^{4} + 840 \nu^{2} - 224 \)\()/209\)
\(\beta_{14}\)\(=\)\((\)\( 104 \nu^{14} - 390 \nu^{12} + 1456 \nu^{10} - 5434 \nu^{8} + 20273 \nu^{6} - 390 \nu^{4} + 104 \nu^{2} - 26 \)\()/209\)
\(\beta_{15}\)\(=\)\((\)\( 164 \nu^{15} - 615 \nu^{13} + 2296 \nu^{11} - 8569 \nu^{9} + 31977 \nu^{7} - 615 \nu^{5} + 164 \nu^{3} - 41 \nu \)\()/209\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + 2 \beta_{4}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 4 \beta_{5}\)
\(\nu^{4}\)\(=\)\(4 \beta_{10} - 7 \beta_{8}\)
\(\nu^{5}\)\(=\)\(4 \beta_{11} - 11 \beta_{9}\)
\(\nu^{6}\)\(=\)\(-15 \beta_{14} + 26 \beta_{13} - 26 \beta_{8} - 26 \beta_{4} + 26\)
\(\nu^{7}\)\(=\)\(-15 \beta_{15} + 41 \beta_{12}\)
\(\nu^{8}\)\(=\)\(-56 \beta_{14} + 97 \beta_{13} - 56 \beta_{10} + 56 \beta_{6} + 56 \beta_{2}\)
\(\nu^{9}\)\(=\)\(-56 \beta_{15} + 153 \beta_{12} - 56 \beta_{11} + 153 \beta_{9} + 56 \beta_{7} + 209 \beta_{5} + 56 \beta_{3} - 209 \beta_{1}\)
\(\nu^{10}\)\(=\)\(209 \beta_{2} - 362\)
\(\nu^{11}\)\(=\)\(209 \beta_{3} - 780 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-780 \beta_{6} - 1351 \beta_{4}\)
\(\nu^{13}\)\(=\)\(-780 \beta_{7} - 2911 \beta_{5}\)
\(\nu^{14}\)\(=\)\(-2911 \beta_{10} + 5042 \beta_{8}\)
\(\nu^{15}\)\(=\)\(-2911 \beta_{11} + 7953 \beta_{9}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(-\beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−1.13551 1.56290i
−0.304260 0.418778i
0.304260 + 0.418778i
1.13551 + 1.56290i
1.83730 0.596975i
0.492303 0.159959i
−0.492303 + 0.159959i
−1.83730 + 0.596975i
−1.13551 + 1.56290i
−0.304260 + 0.418778i
0.304260 0.418778i
1.13551 1.56290i
1.83730 + 0.596975i
0.492303 + 0.159959i
−0.492303 0.159959i
−1.83730 0.596975i
−1.83730 0.596975i −0.304260 0.418778i 1.40126 + 1.01807i −1.88023 1.21026i 0.309017 + 0.951057i −0.526994 + 0.725345i 0.304260 + 0.418778i 0.844250 2.59833i 2.73205 + 3.34607i
9.2 −0.492303 0.159959i −1.13551 1.56290i −1.40126 1.01807i 1.88023 + 1.21026i 0.309017 + 0.951057i 1.96677 2.70702i 1.13551 + 1.56290i −0.226216 + 0.696222i −0.732051 0.896575i
9.3 0.492303 + 0.159959i 1.13551 + 1.56290i −1.40126 1.01807i −0.809764 + 2.08429i 0.309017 + 0.951057i −1.96677 + 2.70702i −1.13551 1.56290i −0.226216 + 0.696222i −0.732051 + 0.896575i
9.4 1.83730 + 0.596975i 0.304260 + 0.418778i 1.40126 + 1.01807i 0.809764 2.08429i 0.309017 + 0.951057i 0.526994 0.725345i −0.304260 0.418778i 0.844250 2.59833i 2.73205 3.34607i
124.1 −1.13551 1.56290i 0.492303 0.159959i −0.535233 + 1.64728i 0.570005 + 2.16220i −0.809017 0.587785i 0.852694 + 0.277057i −0.492303 + 0.159959i −2.21028 + 1.60586i 2.73205 3.34607i
124.2 −0.304260 0.418778i 1.83730 0.596975i 0.535233 1.64728i −0.570005 2.16220i −0.809017 0.587785i −3.18230 1.03399i −1.83730 + 0.596975i 0.592242 0.430289i −0.732051 + 0.896575i
124.3 0.304260 + 0.418778i −1.83730 + 0.596975i 0.535233 1.64728i −2.23251 + 0.126049i −0.809017 0.587785i 3.18230 + 1.03399i 1.83730 0.596975i 0.592242 0.430289i −0.732051 0.896575i
124.4 1.13551 + 1.56290i −0.492303 + 0.159959i −0.535233 + 1.64728i 2.23251 0.126049i −0.809017 0.587785i −0.852694 0.277057i 0.492303 0.159959i −2.21028 + 1.60586i 2.73205 + 3.34607i
269.1 −1.83730 + 0.596975i −0.304260 + 0.418778i 1.40126 1.01807i −1.88023 + 1.21026i 0.309017 0.951057i −0.526994 0.725345i 0.304260 0.418778i 0.844250 + 2.59833i 2.73205 3.34607i
269.2 −0.492303 + 0.159959i −1.13551 + 1.56290i −1.40126 + 1.01807i 1.88023 1.21026i 0.309017 0.951057i 1.96677 + 2.70702i 1.13551 1.56290i −0.226216 0.696222i −0.732051 + 0.896575i
269.3 0.492303 0.159959i 1.13551 1.56290i −1.40126 + 1.01807i −0.809764 2.08429i 0.309017 0.951057i −1.96677 2.70702i −1.13551 + 1.56290i −0.226216 0.696222i −0.732051 0.896575i
269.4 1.83730 0.596975i 0.304260 0.418778i 1.40126 1.01807i 0.809764 + 2.08429i 0.309017 0.951057i 0.526994 + 0.725345i −0.304260 + 0.418778i 0.844250 + 2.59833i 2.73205 + 3.34607i
444.1 −1.13551 + 1.56290i 0.492303 + 0.159959i −0.535233 1.64728i 0.570005 2.16220i −0.809017 + 0.587785i 0.852694 0.277057i −0.492303 0.159959i −2.21028 1.60586i 2.73205 + 3.34607i
444.2 −0.304260 + 0.418778i 1.83730 + 0.596975i 0.535233 + 1.64728i −0.570005 + 2.16220i −0.809017 + 0.587785i −3.18230 + 1.03399i −1.83730 0.596975i 0.592242 + 0.430289i −0.732051 0.896575i
444.3 0.304260 0.418778i −1.83730 0.596975i 0.535233 + 1.64728i −2.23251 0.126049i −0.809017 + 0.587785i 3.18230 1.03399i 1.83730 + 0.596975i 0.592242 + 0.430289i −0.732051 + 0.896575i
444.4 1.13551 1.56290i −0.492303 0.159959i −0.535233 1.64728i 2.23251 + 0.126049i −0.809017 + 0.587785i −0.852694 + 0.277057i 0.492303 + 0.159959i −2.21028 1.60586i 2.73205 3.34607i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 444.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 3 inner
55.j even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.e 16
5.b even 2 1 inner 605.2.j.e 16
11.b odd 2 1 605.2.j.f 16
11.c even 5 1 605.2.b.e yes 4
11.c even 5 3 inner 605.2.j.e 16
11.d odd 10 1 605.2.b.d 4
11.d odd 10 3 605.2.j.f 16
55.d odd 2 1 605.2.j.f 16
55.h odd 10 1 605.2.b.d 4
55.h odd 10 3 605.2.j.f 16
55.j even 10 1 605.2.b.e yes 4
55.j even 10 3 inner 605.2.j.e 16
55.k odd 20 2 3025.2.a.z 4
55.l even 20 2 3025.2.a.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.d 4 11.d odd 10 1
605.2.b.d 4 55.h odd 10 1
605.2.b.e yes 4 11.c even 5 1
605.2.b.e yes 4 55.j even 10 1
605.2.j.e 16 1.a even 1 1 trivial
605.2.j.e 16 5.b even 2 1 inner
605.2.j.e 16 11.c even 5 3 inner
605.2.j.e 16 55.j even 10 3 inner
605.2.j.f 16 11.b odd 2 1
605.2.j.f 16 11.d odd 10 3
605.2.j.f 16 55.d odd 2 1
605.2.j.f 16 55.h odd 10 3
3025.2.a.y 4 55.l even 20 2
3025.2.a.z 4 55.k odd 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\(T_{2}^{16} - \cdots\)
\(T_{19}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + 15 T^{4} - 56 T^{6} + 209 T^{8} - 56 T^{10} + 15 T^{12} - 4 T^{14} + T^{16} \)
$3$ \( 1 - 4 T^{2} + 15 T^{4} - 56 T^{6} + 209 T^{8} - 56 T^{10} + 15 T^{12} - 4 T^{14} + T^{16} \)
$5$ \( 390625 + 31250 T^{2} - 13125 T^{4} - 2300 T^{6} + 341 T^{8} - 92 T^{10} - 21 T^{12} + 2 T^{14} + T^{16} \)
$7$ \( 6561 - 8748 T^{2} + 10935 T^{4} - 13608 T^{6} + 16929 T^{8} - 1512 T^{10} + 135 T^{12} - 12 T^{14} + T^{16} \)
$11$ \( T^{16} \)
$13$ \( ( 104976 - 5832 T^{2} + 324 T^{4} - 18 T^{6} + T^{8} )^{2} \)
$17$ \( 65536 - 65536 T^{2} + 61440 T^{4} - 57344 T^{6} + 53504 T^{8} - 3584 T^{10} + 240 T^{12} - 16 T^{14} + T^{16} \)
$19$ \( ( 456976 - 35152 T + 20280 T^{2} - 2912 T^{3} + 1004 T^{4} + 112 T^{5} + 30 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$23$ \( ( 484 + 52 T^{2} + T^{4} )^{4} \)
$29$ \( ( 5308416 + 110592 T^{2} + 2304 T^{4} + 48 T^{6} + T^{8} )^{2} \)
$31$ \( ( 4477456 - 1362704 T + 317400 T^{2} - 66976 T^{3} + 13484 T^{4} - 1456 T^{5} + 150 T^{6} - 14 T^{7} + T^{8} )^{2} \)
$37$ \( 429981696 - 143327232 T^{2} + 44789760 T^{4} - 13934592 T^{6} + 4333824 T^{8} - 96768 T^{10} + 2160 T^{12} - 48 T^{14} + T^{16} \)
$41$ \( ( 81 + 27 T^{2} + 9 T^{4} + 3 T^{6} + T^{8} )^{2} \)
$43$ \( ( 4761 + 156 T^{2} + T^{4} )^{4} \)
$47$ \( 23811286661761 - 1595323868692 T^{2} + 96105317295 T^{4} - 5716732952 T^{6} + 339507089 T^{8} - 2587928 T^{10} + 19695 T^{12} - 148 T^{14} + T^{16} \)
$53$ \( 208827064576 - 45719534848 T^{2} + 9700686528 T^{4} - 2056186496 T^{6} + 435820880 T^{8} - 3041696 T^{10} + 21228 T^{12} - 148 T^{14} + T^{16} \)
$59$ \( ( 1296 + 1296 T + 1080 T^{2} + 864 T^{3} + 684 T^{4} + 144 T^{5} + 30 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$61$ \( ( 88529281 + 18253460 T + 2850927 T^{2} + 399640 T^{3} + 53009 T^{4} + 4120 T^{5} + 303 T^{6} + 20 T^{7} + T^{8} )^{2} \)
$67$ \( ( 1089 + 228 T^{2} + T^{4} )^{4} \)
$71$ \( ( 104976 + 34992 T + 17496 T^{2} + 7776 T^{3} + 3564 T^{4} - 432 T^{5} + 54 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$73$ \( ( 331776 - 13824 T^{2} + 576 T^{4} - 24 T^{6} + T^{8} )^{2} \)
$79$ \( ( 4096 + 2048 T + 1536 T^{2} + 1024 T^{3} + 704 T^{4} - 128 T^{5} + 24 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$83$ \( ( 92236816 - 941192 T^{2} + 9604 T^{4} - 98 T^{6} + T^{8} )^{2} \)
$89$ \( ( -3 - 6 T + T^{2} )^{8} \)
$97$ \( 1679616 - 3919104 T^{2} + 9097920 T^{4} - 21119616 T^{6} + 49026384 T^{8} - 586656 T^{10} + 7020 T^{12} - 84 T^{14} + T^{16} \)
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