Properties

Label 1500.2.o.b
Level $1500$
Weight $2$
Character orbit 1500.o
Analytic conductor $11.978$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1500.o (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.9775603032\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 5 x^{14} + 6 x^{12} - 20 x^{10} - 79 x^{8} - 80 x^{6} + 96 x^{4} + 320 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 300)
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.

\(f(q)\) \(=\) \( q -\beta_{6} q^{3} + ( \beta_{1} + \beta_{3} + \beta_{6} + \beta_{10} + 2 \beta_{13} - \beta_{14} ) q^{7} + ( 1 - \beta_{2} - \beta_{4} + \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{6} q^{3} + ( \beta_{1} + \beta_{3} + \beta_{6} + \beta_{10} + 2 \beta_{13} - \beta_{14} ) q^{7} + ( 1 - \beta_{2} - \beta_{4} + \beta_{9} ) q^{9} + ( 2 - 2 \beta_{2} - 2 \beta_{4} - \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{10} + \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{13} + ( \beta_{3} - \beta_{5} + \beta_{12} - \beta_{13} ) q^{17} + ( -1 - \beta_{8} - \beta_{11} - \beta_{15} ) q^{19} + ( 2 \beta_{2} + 2 \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{15} ) q^{21} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{23} + \beta_{1} q^{27} + ( \beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{15} ) q^{29} + ( -1 - 2 \beta_{4} - 3 \beta_{8} + \beta_{9} - \beta_{11} - 3 \beta_{15} ) q^{31} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{33} + ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{10} + 2 \beta_{13} ) q^{37} + ( 1 - \beta_{2} - 3 \beta_{4} - \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{39} + ( -1 + 3 \beta_{2} + 3 \beta_{4} + 4 \beta_{7} - \beta_{9} ) q^{41} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{6} + 2 \beta_{10} + 3 \beta_{13} - 2 \beta_{14} ) q^{43} + ( -4 \beta_{1} + 2 \beta_{3} - 6 \beta_{6} - 4 \beta_{13} + \beta_{14} ) q^{47} + ( 2 - 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{11} - 3 \beta_{15} ) q^{49} + ( -1 + \beta_{2} + \beta_{11} + \beta_{15} ) q^{51} + ( 5 \beta_{1} + 2 \beta_{3} + \beta_{6} + 5 \beta_{13} - \beta_{14} ) q^{53} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{57} + ( 5 + 3 \beta_{2} + 3 \beta_{4} - \beta_{7} + 5 \beta_{9} + \beta_{15} ) q^{59} + ( 6 - 6 \beta_{2} + 3 \beta_{4} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{61} + ( -\beta_{1} - \beta_{6} + \beta_{10} ) q^{63} + ( \beta_{1} + 2 \beta_{3} - \beta_{5} + 2 \beta_{10} + 2 \beta_{12} - 8 \beta_{13} - 2 \beta_{14} ) q^{67} + ( -\beta_{4} - \beta_{8} + 3 \beta_{9} + 2 \beta_{11} - \beta_{15} ) q^{69} + ( -7 \beta_{2} - 4 \beta_{4} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{15} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} ) q^{73} + ( 2 \beta_{1} + \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + \beta_{10} + \beta_{13} ) q^{77} + ( -\beta_{2} - 4 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} - \beta_{11} - 4 \beta_{15} ) q^{79} + \beta_{9} q^{81} + ( 10 \beta_{1} + 6 \beta_{3} + 4 \beta_{5} + 2 \beta_{10} + 6 \beta_{12} + 9 \beta_{13} - 2 \beta_{14} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} - \beta_{10} - 2 \beta_{13} ) q^{87} + ( 9 - 9 \beta_{2} - 11 \beta_{4} - 3 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + 3 \beta_{11} ) q^{89} + ( 4 + \beta_{7} + 4 \beta_{9} ) q^{91} + ( \beta_{1} + \beta_{3} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{10} + 3 \beta_{12} - \beta_{13} - \beta_{14} ) q^{93} + ( -6 \beta_{1} - 4 \beta_{3} + 3 \beta_{6} - 6 \beta_{13} + 4 \beta_{14} ) q^{97} + ( 2 - 2 \beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{11} + \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{9} + O(q^{10}) \) \( 16q + 4q^{9} + 16q^{11} - 10q^{19} + 14q^{21} + 6q^{29} - 6q^{31} + 20q^{41} + 16q^{49} - 16q^{51} + 76q^{59} + 92q^{61} - 4q^{69} - 50q^{71} + 32q^{79} - 4q^{81} + 60q^{89} + 50q^{91} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 5 x^{14} + 6 x^{12} - 20 x^{10} - 79 x^{8} - 80 x^{6} + 96 x^{4} + 320 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{15} - \nu^{13} + 14 \nu^{11} + 44 \nu^{9} - \nu^{7} - 108 \nu^{5} - 288 \nu^{3} - 192 \nu \)\()/384\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{14} - 9 \nu^{12} - 10 \nu^{10} + 12 \nu^{8} + 63 \nu^{6} + 76 \nu^{4} + 48 \nu^{2} + 64 \)\()/192\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} + \nu^{13} - 46 \nu^{11} - 76 \nu^{9} + 65 \nu^{7} + 364 \nu^{5} + 256 \nu^{3} - 192 \nu \)\()/384\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{14} - 9 \nu^{12} - 26 \nu^{10} - 4 \nu^{8} + 95 \nu^{6} + 204 \nu^{4} + 32 \nu^{2} - 128 \)\()/192\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} - 5 \nu^{13} + 10 \nu^{11} + 100 \nu^{9} + 175 \nu^{7} - 112 \nu^{5} - 720 \nu^{3} - 960 \nu \)\()/384\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{13} - 3 \nu^{11} - 4 \nu^{9} + 8 \nu^{7} + 23 \nu^{5} + 18 \nu^{3} + 8 \nu \)\()/48\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{14} - 11 \nu^{12} - 30 \nu^{10} - 12 \nu^{8} + 93 \nu^{6} + 180 \nu^{4} + 320 \nu^{2} + 192 \)\()/192\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{14} + 7 \nu^{12} + 26 \nu^{10} + 68 \nu^{8} + 83 \nu^{6} - 184 \nu^{4} - 752 \nu^{2} - 960 \)\()/192\)
\(\beta_{9}\)\(=\)\((\)\( 5 \nu^{14} + 9 \nu^{12} - 18 \nu^{10} - 100 \nu^{8} - 139 \nu^{6} + 96 \nu^{4} + 512 \nu^{2} + 512 \)\()/192\)
\(\beta_{10}\)\(=\)\((\)\( 2 \nu^{15} + 5 \nu^{13} + 3 \nu^{11} - 22 \nu^{9} - 58 \nu^{7} - 21 \nu^{5} + 96 \nu^{3} + 224 \nu \)\()/96\)
\(\beta_{11}\)\(=\)\((\)\( -\nu^{14} - 3 \nu^{12} + 2 \nu^{10} + 30 \nu^{8} + 51 \nu^{6} - 14 \nu^{4} - 162 \nu^{2} - 200 \)\()/24\)
\(\beta_{12}\)\(=\)\((\)\( -5 \nu^{15} - 33 \nu^{13} - 38 \nu^{11} + 84 \nu^{9} + 363 \nu^{7} + 392 \nu^{5} - 192 \nu^{3} - 832 \nu \)\()/384\)
\(\beta_{13}\)\(=\)\((\)\( -5 \nu^{15} - 15 \nu^{13} - 4 \nu^{11} + 72 \nu^{9} + 147 \nu^{7} + 26 \nu^{5} - 216 \nu^{3} - 224 \nu \)\()/192\)
\(\beta_{14}\)\(=\)\((\)\( -3 \nu^{15} - 6 \nu^{13} + 15 \nu^{11} + 86 \nu^{9} + 113 \nu^{7} - 103 \nu^{5} - 508 \nu^{3} - 448 \nu \)\()/96\)
\(\beta_{15}\)\(=\)\((\)\( 13 \nu^{14} + 49 \nu^{12} - 2 \nu^{10} - 356 \nu^{8} - 707 \nu^{6} + 96 \nu^{4} + 2016 \nu^{2} + 2304 \)\()/192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14} + 2 \beta_{13} - \beta_{12} + 3 \beta_{10} + 2 \beta_{6} - \beta_{5} + \beta_{3} - \beta_{1}\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{15} + 3 \beta_{11} + \beta_{9} + \beta_{8} + 3 \beta_{7} - 3 \beta_{4} + 4 \beta_{2} - 3\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{14} + \beta_{13} + 2 \beta_{12} - \beta_{10} - 9 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - 8 \beta_{1}\)\()/5\)
\(\nu^{4}\)\(=\)\(\beta_{15} + \beta_{11} - \beta_{9} - \beta_{7} + 2 \beta_{4} + \beta_{2} + 1\)
\(\nu^{5}\)\(=\)\((\)\(6 \beta_{14} - 8 \beta_{13} + 9 \beta_{12} + 3 \beta_{10} - 8 \beta_{6} - 16 \beta_{5} + 6 \beta_{3} + 9 \beta_{1}\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{15} - 7 \beta_{11} - 14 \beta_{9} + 16 \beta_{8} + 18 \beta_{7} - 3 \beta_{4} + 9 \beta_{2} + 12\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(8 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} - \beta_{10} + 6 \beta_{6} + 2 \beta_{5} - 7 \beta_{3} - 58 \beta_{1}\)\()/5\)
\(\nu^{8}\)\(=\)\(3 \beta_{15} + 10 \beta_{11} + 9 \beta_{9} - 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{4} + 4\)
\(\nu^{9}\)\(=\)\((\)\(-19 \beta_{14} + 42 \beta_{13} + 14 \beta_{12} + 38 \beta_{10} - 78 \beta_{6} + 19 \beta_{5} + 26 \beta_{3} + 49 \beta_{1}\)\()/5\)
\(\nu^{10}\)\(=\)\((\)\(27 \beta_{15} - 27 \beta_{11} - 114 \beta_{9} + 51 \beta_{8} + 3 \beta_{7} + 27 \beta_{4} + 114 \beta_{2} - 13\)\()/5\)
\(\nu^{11}\)\(=\)\((\)\(73 \beta_{14} - 139 \beta_{13} + 62 \beta_{12} - 51 \beta_{10} + 11 \beta_{6} - 133 \beta_{5} - 62 \beta_{3} - 133 \beta_{1}\)\()/5\)
\(\nu^{12}\)\(=\)\(10 \beta_{15} + 18 \beta_{11} + 5 \beta_{9} + 5 \beta_{8} + 18 \beta_{7} + 3 \beta_{4} - 31 \beta_{2} + 36\)
\(\nu^{13}\)\(=\)\((\)\(31 \beta_{14} + 67 \beta_{13} - 31 \beta_{12} + 68 \beta_{10} - 243 \beta_{6} - \beta_{5} + 136 \beta_{3} - 206 \beta_{1}\)\()/5\)
\(\nu^{14}\)\(=\)\((\)\(62 \beta_{15} + 143 \beta_{11} + 241 \beta_{9} + 81 \beta_{8} - 62 \beta_{7} - 98 \beta_{4} + 434 \beta_{2} + 62\)\()/5\)
\(\nu^{15}\)\(=\)\((\)\(-117 \beta_{14} + 31 \beta_{13} + 162 \beta_{12} + 279 \beta_{10} + 31 \beta_{6} + 317 \beta_{5} - 117 \beta_{3} + 162 \beta_{1}\)\()/5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1500\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(877\) \(1001\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.720348 1.21700i
−0.132563 + 1.40799i
0.132563 1.40799i
−0.720348 + 1.21700i
0.462894 + 1.33631i
−1.41395 0.0272949i
1.41395 + 0.0272949i
−0.462894 1.33631i
−1.41395 + 0.0272949i
0.462894 1.33631i
−0.462894 + 1.33631i
1.41395 0.0272949i
−0.132563 1.40799i
0.720348 + 1.21700i
−0.720348 1.21700i
0.132563 + 1.40799i
0 −0.951057 + 0.309017i 0 0 0 1.74037i 0 0.809017 0.587785i 0
49.2 0 −0.951057 + 0.309017i 0 0 0 1.50430i 0 0.809017 0.587785i 0
49.3 0 0.951057 0.309017i 0 0 0 1.50430i 0 0.809017 0.587785i 0
49.4 0 0.951057 0.309017i 0 0 0 1.74037i 0 0.809017 0.587785i 0
349.1 0 −0.587785 0.809017i 0 0 0 0.0883282i 0 −0.309017 + 0.951057i 0
349.2 0 −0.587785 0.809017i 0 0 0 4.32440i 0 −0.309017 + 0.951057i 0
349.3 0 0.587785 + 0.809017i 0 0 0 4.32440i 0 −0.309017 + 0.951057i 0
349.4 0 0.587785 + 0.809017i 0 0 0 0.0883282i 0 −0.309017 + 0.951057i 0
649.1 0 −0.587785 + 0.809017i 0 0 0 4.32440i 0 −0.309017 0.951057i 0
649.2 0 −0.587785 + 0.809017i 0 0 0 0.0883282i 0 −0.309017 0.951057i 0
649.3 0 0.587785 0.809017i 0 0 0 0.0883282i 0 −0.309017 0.951057i 0
649.4 0 0.587785 0.809017i 0 0 0 4.32440i 0 −0.309017 0.951057i 0
949.1 0 −0.951057 0.309017i 0 0 0 1.50430i 0 0.809017 + 0.587785i 0
949.2 0 −0.951057 0.309017i 0 0 0 1.74037i 0 0.809017 + 0.587785i 0
949.3 0 0.951057 + 0.309017i 0 0 0 1.74037i 0 0.809017 + 0.587785i 0
949.4 0 0.951057 + 0.309017i 0 0 0 1.50430i 0 0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 949.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1500.2.o.b 16
5.b even 2 1 inner 1500.2.o.b 16
5.c odd 4 1 300.2.m.b 8
5.c odd 4 1 1500.2.m.a 8
15.e even 4 1 900.2.n.b 8
25.d even 5 1 inner 1500.2.o.b 16
25.d even 5 1 7500.2.d.c 8
25.e even 10 1 inner 1500.2.o.b 16
25.e even 10 1 7500.2.d.c 8
25.f odd 20 1 300.2.m.b 8
25.f odd 20 1 1500.2.m.a 8
25.f odd 20 1 7500.2.a.e 4
25.f odd 20 1 7500.2.a.f 4
75.l even 20 1 900.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.b 8 5.c odd 4 1
300.2.m.b 8 25.f odd 20 1
900.2.n.b 8 15.e even 4 1
900.2.n.b 8 75.l even 20 1
1500.2.m.a 8 5.c odd 4 1
1500.2.m.a 8 25.f odd 20 1
1500.2.o.b 16 1.a even 1 1 trivial
1500.2.o.b 16 5.b even 2 1 inner
1500.2.o.b 16 25.d even 5 1 inner
1500.2.o.b 16 25.e even 10 1 inner
7500.2.a.e 4 25.f odd 20 1
7500.2.a.f 4 25.f odd 20 1
7500.2.d.c 8 25.d even 5 1
7500.2.d.c 8 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 24 T_{7}^{6} + 106 T_{7}^{4} + 129 T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1500, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$5$ \( T^{16} \)
$7$ \( ( 1 + 129 T^{2} + 106 T^{4} + 24 T^{6} + T^{8} )^{2} \)
$11$ \( ( 361 - 323 T + 1033 T^{2} + 149 T^{3} + 100 T^{4} + 39 T^{5} + 23 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$13$ \( 4100625 - 7745625 T^{2} + 5862375 T^{4} - 822375 T^{6} + 61150 T^{8} - 2725 T^{10} + 645 T^{12} + 10 T^{14} + T^{16} \)
$17$ \( 6561 - 42282 T^{2} + 103113 T^{4} + 11871 T^{6} + 8530 T^{8} - 19 T^{10} + 123 T^{12} - 17 T^{14} + T^{16} \)
$19$ \( ( 25 + 25 T + 225 T^{2} - 225 T^{3} + 160 T^{4} - 45 T^{5} + 5 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$23$ \( 855036081 - 748189467 T^{2} + 257169168 T^{4} - 6550029 T^{6} + 2680555 T^{8} - 169649 T^{10} + 4588 T^{12} + 13 T^{14} + T^{16} \)
$29$ \( ( 1 - 3 T + 58 T^{2} + 129 T^{3} + 105 T^{4} - 21 T^{5} + 8 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$31$ \( ( 962361 - 55917 T + 78228 T^{2} - 21699 T^{3} + 3625 T^{4} + 511 T^{5} + 58 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$37$ \( 1086683238481 - 23119256498 T^{2} + 3352544013 T^{4} - 170879101 T^{6} + 6122150 T^{8} - 189931 T^{10} + 6123 T^{12} - 113 T^{14} + T^{16} \)
$41$ \( ( 7317025 + 1704150 T + 228475 T^{2} - 5800 T^{3} + 4085 T^{4} - 520 T^{5} + 155 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$43$ \( ( 17161 + 9074 T^{2} + 1391 T^{4} + 74 T^{6} + T^{8} )^{2} \)
$47$ \( 282300516843601 - 18921633826767 T^{2} + 560665017528 T^{4} - 6824110249 T^{6} + 111178775 T^{8} - 1382449 T^{10} + 13208 T^{12} - 47 T^{14} + T^{16} \)
$53$ \( 96059601 - 23110758 T^{2} + 84654153 T^{4} - 82166121 T^{6} + 35560330 T^{8} - 498671 T^{10} + 6063 T^{12} - 73 T^{14} + T^{16} \)
$59$ \( ( 13697401 - 8874998 T + 2975713 T^{2} - 622126 T^{3} + 94675 T^{4} - 10026 T^{5} + 773 T^{6} - 38 T^{7} + T^{8} )^{2} \)
$61$ \( ( 62552281 - 11594594 T + 3590217 T^{2} - 782922 T^{3} + 126055 T^{4} - 13818 T^{5} + 1037 T^{6} - 46 T^{7} + T^{8} )^{2} \)
$67$ \( 166726039041 + 724658711688 T^{2} + 8322937870428 T^{4} - 208496255454 T^{6} + 2351516830 T^{8} - 12833564 T^{10} + 64093 T^{12} - 302 T^{14} + T^{16} \)
$71$ \( ( 25 - 650 T + 44875 T^{2} + 10175 T^{3} + 4060 T^{4} + 1395 T^{5} + 275 T^{6} + 25 T^{7} + T^{8} )^{2} \)
$73$ \( 14641 - 514008 T^{2} + 6850908 T^{4} + 2612174 T^{6} + 5935550 T^{8} - 996916 T^{10} + 62633 T^{12} + 142 T^{14} + T^{16} \)
$79$ \( ( 408321 - 300969 T + 629937 T^{2} + 22443 T^{3} + 2350 T^{4} + 247 T^{5} + 117 T^{6} - 16 T^{7} + T^{8} )^{2} \)
$83$ \( 2200435611257601 - 650847511404342 T^{2} + 74718431099703 T^{4} - 562589074404 T^{6} + 2583491605 T^{8} - 7485604 T^{10} + 104863 T^{12} + 98 T^{14} + T^{16} \)
$89$ \( ( 97515625 + 11109375 T + 359375 T^{2} - 106875 T^{3} + 57750 T^{4} - 7125 T^{5} + 625 T^{6} - 30 T^{7} + T^{8} )^{2} \)
$97$ \( 4135050843618321 - 583254572620698 T^{2} + 32339334356883 T^{4} - 198542073996 T^{6} + 1406165005 T^{8} - 8689916 T^{10} + 66403 T^{12} + 22 T^{14} + T^{16} \)
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