Properties

Label 1050.2.b.f
Level 1050
Weight 2
Character orbit 1050.b
Analytic conductor 8.384
Analytic rank 0
Dimension 16
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.101415451701035401216.7
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{10} q^{2} -\beta_{7} q^{3} - q^{4} -\beta_{2} q^{6} + \beta_{9} q^{7} + \beta_{10} q^{8} + ( -\beta_{6} + \beta_{15} ) q^{9} +O(q^{10})\) \( q -\beta_{10} q^{2} -\beta_{7} q^{3} - q^{4} -\beta_{2} q^{6} + \beta_{9} q^{7} + \beta_{10} q^{8} + ( -\beta_{6} + \beta_{15} ) q^{9} + \beta_{15} q^{11} + \beta_{7} q^{12} + ( -\beta_{4} + \beta_{7} ) q^{13} + \beta_{11} q^{14} + q^{16} + ( \beta_{4} + \beta_{7} + \beta_{12} ) q^{17} -\beta_{5} q^{18} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{14} ) q^{19} + ( -1 + \beta_{3} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{21} + \beta_{1} q^{22} + ( \beta_{1} + 2 \beta_{5} - 2 \beta_{10} ) q^{23} + \beta_{2} q^{24} + ( \beta_{2} + \beta_{3} ) q^{26} + ( -2 \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} ) q^{27} -\beta_{9} q^{28} + ( -\beta_{11} - \beta_{13} + 2 \beta_{15} ) q^{29} + ( \beta_{2} - \beta_{3} - \beta_{14} ) q^{31} -\beta_{10} q^{32} + ( \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} ) q^{33} + ( \beta_{2} - \beta_{3} + \beta_{14} ) q^{34} + ( \beta_{6} - \beta_{15} ) q^{36} + ( 3 \beta_{1} - \beta_{8} + \beta_{9} ) q^{37} + ( 2 \beta_{4} + 2 \beta_{7} + \beta_{12} ) q^{38} + ( 3 + \beta_{6} - \beta_{15} ) q^{39} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{11} + 2 \beta_{13} ) q^{41} + ( \beta_{1} + \beta_{4} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{42} + ( -2 \beta_{8} + 2 \beta_{9} ) q^{43} -\beta_{15} q^{44} + ( -2 - 2 \beta_{6} + \beta_{15} ) q^{46} + 2 \beta_{12} q^{47} -\beta_{7} q^{48} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{6} - \beta_{14} - \beta_{15} ) q^{49} + ( -2 - \beta_{11} - \beta_{13} - \beta_{15} ) q^{51} + ( \beta_{4} - \beta_{7} ) q^{52} + ( -2 \beta_{1} - 4 \beta_{5} - 2 \beta_{10} ) q^{53} + ( \beta_{2} + 2 \beta_{3} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{54} -\beta_{11} q^{56} + ( \beta_{1} - \beta_{5} + \beta_{8} - \beta_{9} - 5 \beta_{10} ) q^{57} + ( 2 \beta_{1} - \beta_{8} + \beta_{9} ) q^{58} + ( -\beta_{2} - \beta_{3} - \beta_{11} + \beta_{13} ) q^{59} + ( -3 \beta_{2} + 3 \beta_{3} ) q^{61} + ( -\beta_{4} - \beta_{7} + \beta_{12} ) q^{62} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{63} - q^{64} + ( \beta_{2} - \beta_{3} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{66} + ( -2 \beta_{8} + 2 \beta_{9} ) q^{67} + ( -\beta_{4} - \beta_{7} - \beta_{12} ) q^{68} + ( -3 \beta_{2} - 5 \beta_{3} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{69} + ( 3 \beta_{11} + 3 \beta_{13} - 2 \beta_{15} ) q^{71} + \beta_{5} q^{72} + ( -2 \beta_{4} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} + ( \beta_{11} + \beta_{13} - 3 \beta_{15} ) q^{74} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{14} ) q^{76} + ( \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{10} ) q^{77} + ( \beta_{5} - 3 \beta_{10} ) q^{78} + ( 2 - 2 \beta_{6} + \beta_{15} ) q^{79} + ( 5 - 2 \beta_{11} - 2 \beta_{13} + \beta_{15} ) q^{81} + ( -2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{82} + ( -\beta_{4} - \beta_{7} + 2 \beta_{12} ) q^{83} + ( 1 - \beta_{3} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{84} + ( 2 \beta_{11} + 2 \beta_{13} ) q^{86} + ( 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{87} -\beta_{1} q^{88} + ( 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{11} + 2 \beta_{13} ) q^{89} + ( 2 + \beta_{2} - \beta_{3} + 2 \beta_{14} ) q^{91} + ( -\beta_{1} - 2 \beta_{5} + 2 \beta_{10} ) q^{92} + ( \beta_{1} + 2 \beta_{5} + \beta_{8} - \beta_{9} + 4 \beta_{10} ) q^{93} + 2 \beta_{14} q^{94} -\beta_{2} q^{96} + ( -4 \beta_{4} + 4 \beta_{7} - \beta_{8} - \beta_{9} ) q^{97} + ( \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{10} + \beta_{12} ) q^{98} + ( -4 - 2 \beta_{11} - 2 \beta_{13} + \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{4} + O(q^{10}) \) \( 16q - 16q^{4} + 16q^{16} - 16q^{21} + 48q^{39} - 32q^{46} + 16q^{49} - 32q^{51} - 16q^{64} + 32q^{79} + 80q^{81} + 16q^{84} + 32q^{91} - 64q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 18 x^{12} + 145 x^{8} - 72 x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{12} + 10 \nu^{8} - 5 \nu^{4} - 508 \)\()/180\)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{13} - 130 \nu^{9} + 1055 \nu^{5} - 764 \nu \)\()/240\)
\(\beta_{3}\)\(=\)\((\)\( 49 \nu^{15} - 910 \nu^{11} + 7625 \nu^{7} - 7268 \nu^{3} \)\()/1440\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{13} + 194 \nu^{9} - 1531 \nu^{5} + 412 \nu \)\()/144\)
\(\beta_{5}\)\(=\)\((\)\( 11 \nu^{12} - 190 \nu^{8} + 1495 \nu^{4} - 212 \)\()/120\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{14} - 16 \nu^{10} + 113 \nu^{6} + 154 \nu^{2} \)\()/36\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{15} - 158 \nu^{11} + 1233 \nu^{7} - 52 \nu^{3} \)\()/96\)
\(\beta_{8}\)\(=\)\((\)\( 35 \nu^{15} + 51 \nu^{14} - 30 \nu^{13} - 2 \nu^{12} - 650 \nu^{11} - 930 \nu^{10} + 540 \nu^{9} + 20 \nu^{8} + 5395 \nu^{7} + 7635 \nu^{6} - 4470 \nu^{5} - 10 \nu^{4} - 5140 \nu^{3} - 6252 \nu^{2} + 3240 \nu - 1016 \)\()/1440\)
\(\beta_{9}\)\(=\)\((\)\( 35 \nu^{15} - 51 \nu^{14} - 30 \nu^{13} + 2 \nu^{12} - 650 \nu^{11} + 930 \nu^{10} + 540 \nu^{9} - 20 \nu^{8} + 5395 \nu^{7} - 7635 \nu^{6} - 4470 \nu^{5} + 10 \nu^{4} - 5140 \nu^{3} + 6252 \nu^{2} + 3240 \nu + 1016 \)\()/1440\)
\(\beta_{10}\)\(=\)\((\)\( -13 \nu^{14} + 230 \nu^{10} - 1805 \nu^{6} + 316 \nu^{2} \)\()/240\)
\(\beta_{11}\)\(=\)\((\)\( 96 \nu^{15} + 55 \nu^{14} - 76 \nu^{13} + 90 \nu^{12} - 1680 \nu^{11} - 970 \nu^{10} + 1360 \nu^{9} - 1620 \nu^{8} + 13080 \nu^{7} + 7655 \nu^{6} - 10820 \nu^{5} + 12690 \nu^{4} - 552 \nu^{3} - 1340 \nu^{2} + 2912 \nu - 3240 \)\()/1440\)
\(\beta_{12}\)\(=\)\((\)\( -41 \nu^{15} + 10 \nu^{13} + 734 \nu^{11} - 172 \nu^{9} - 5857 \nu^{7} + 1274 \nu^{5} + 2212 \nu^{3} + 472 \nu \)\()/288\)
\(\beta_{13}\)\(=\)\((\)\( -96 \nu^{15} + 55 \nu^{14} + 76 \nu^{13} + 90 \nu^{12} + 1680 \nu^{11} - 970 \nu^{10} - 1360 \nu^{9} - 1620 \nu^{8} - 13080 \nu^{7} + 7655 \nu^{6} + 10820 \nu^{5} + 12690 \nu^{4} + 552 \nu^{3} - 1340 \nu^{2} - 2912 \nu - 3240 \)\()/1440\)
\(\beta_{14}\)\(=\)\((\)\( 241 \nu^{15} + 110 \nu^{13} - 4270 \nu^{11} - 1940 \nu^{9} + 33785 \nu^{7} + 15310 \nu^{5} - 8372 \nu^{3} - 1240 \nu \)\()/1440\)
\(\beta_{15}\)\(=\)\((\)\( 11 \nu^{14} - 194 \nu^{10} + 1531 \nu^{6} - 268 \nu^{2} \)\()/72\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14} + \beta_{13} - \beta_{11} + 2 \beta_{4} - \beta_{3} - \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{15} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{14} + 2 \beta_{13} + \beta_{12} - 2 \beta_{11} - \beta_{9} - \beta_{8} + 7 \beta_{7} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{15} - 4 \beta_{13} - 4 \beta_{11} + 6 \beta_{5} + 9 \beta_{1} + 18\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(8 \beta_{14} + 8 \beta_{13} - 9 \beta_{12} - 8 \beta_{11} - 9 \beta_{9} - 9 \beta_{8} - 9 \beta_{7} + 13 \beta_{4} - 8 \beta_{3} - 18 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(32 \beta_{15} + 70 \beta_{10} + 10 \beta_{9} - 10 \beta_{8} - 14 \beta_{6} + 5 \beta_{1}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-2 \beta_{14} + 2 \beta_{13} + 29 \beta_{12} - 2 \beta_{11} - 29 \beta_{9} - 29 \beta_{8} + 35 \beta_{7} + 29 \beta_{4} + 84 \beta_{3} - 2 \beta_{2}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(36 \beta_{15} - 72 \beta_{13} - 72 \beta_{11} + 102 \beta_{5} + 63 \beta_{1} + 34\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(22 \beta_{14} + 22 \beta_{13} - 111 \beta_{12} - 22 \beta_{11} - 111 \beta_{9} - 111 \beta_{8} - 111 \beta_{7} - 49 \beta_{4} - 22 \beta_{3} - 292 \beta_{2}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(278 \beta_{15} + 902 \beta_{10} - 58 \beta_{9} + 58 \beta_{8} + 82 \beta_{6} - 29 \beta_{1}\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(244 \beta_{14} - 244 \beta_{13} + 391 \beta_{12} + 244 \beta_{11} - 391 \beta_{9} - 391 \beta_{8} - 299 \beta_{7} + 391 \beta_{4} + 862 \beta_{3} + 244 \beta_{2}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(350 \beta_{15} - 700 \beta_{13} - 700 \beta_{11} + 990 \beta_{5} - 135 \beta_{1} - 1782\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-688 \beta_{14} - 688 \beta_{13} - 705 \beta_{12} + 688 \beta_{11} - 705 \beta_{9} - 705 \beta_{8} - 705 \beta_{7} - 2651 \beta_{4} + 688 \beta_{3} - 2682 \beta_{2}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(524 \beta_{15} + 6214 \beta_{10} - 2366 \beta_{9} + 2366 \beta_{8} + 3346 \beta_{6} - 1183 \beta_{1}\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(4546 \beta_{14} - 4546 \beta_{13} + 2897 \beta_{12} + 4546 \beta_{11} - 2897 \beta_{9} - 2897 \beta_{8} - 9961 \beta_{7} + 2897 \beta_{4} + 3648 \beta_{3} + 4546 \beta_{2}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.81768 + 0.332046i
−0.137538 + 0.752908i
−0.332046 1.81768i
0.752908 0.137538i
−0.752908 + 0.137538i
0.332046 + 1.81768i
0.137538 0.752908i
−1.81768 0.332046i
−0.137538 0.752908i
1.81768 0.332046i
0.752908 + 0.137538i
−0.332046 + 1.81768i
0.332046 1.81768i
−0.752908 0.137538i
−1.81768 + 0.332046i
0.137538 + 0.752908i
1.00000i −1.68014 0.420861i −1.00000 0 −0.420861 + 1.68014i 1.16372 2.37608i 1.00000i 2.64575 + 1.41421i 0
251.2 1.00000i −1.68014 + 0.420861i −1.00000 0 0.420861 + 1.68014i −1.16372 + 2.37608i 1.00000i 2.64575 1.41421i 0
251.3 1.00000i −0.420861 1.68014i −1.00000 0 −1.68014 + 0.420861i −2.57794 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.4 1.00000i −0.420861 + 1.68014i −1.00000 0 1.68014 + 0.420861i 2.57794 + 0.595188i 1.00000i −2.64575 1.41421i 0
251.5 1.00000i 0.420861 1.68014i −1.00000 0 −1.68014 0.420861i 2.57794 0.595188i 1.00000i −2.64575 1.41421i 0
251.6 1.00000i 0.420861 + 1.68014i −1.00000 0 1.68014 0.420861i −2.57794 + 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.7 1.00000i 1.68014 0.420861i −1.00000 0 −0.420861 1.68014i −1.16372 2.37608i 1.00000i 2.64575 1.41421i 0
251.8 1.00000i 1.68014 + 0.420861i −1.00000 0 0.420861 1.68014i 1.16372 + 2.37608i 1.00000i 2.64575 + 1.41421i 0
251.9 1.00000i −1.68014 0.420861i −1.00000 0 0.420861 1.68014i −1.16372 2.37608i 1.00000i 2.64575 + 1.41421i 0
251.10 1.00000i −1.68014 + 0.420861i −1.00000 0 −0.420861 1.68014i 1.16372 + 2.37608i 1.00000i 2.64575 1.41421i 0
251.11 1.00000i −0.420861 1.68014i −1.00000 0 1.68014 0.420861i 2.57794 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.12 1.00000i −0.420861 + 1.68014i −1.00000 0 −1.68014 0.420861i −2.57794 + 0.595188i 1.00000i −2.64575 1.41421i 0
251.13 1.00000i 0.420861 1.68014i −1.00000 0 1.68014 + 0.420861i −2.57794 0.595188i 1.00000i −2.64575 1.41421i 0
251.14 1.00000i 0.420861 + 1.68014i −1.00000 0 −1.68014 + 0.420861i 2.57794 + 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.15 1.00000i 1.68014 0.420861i −1.00000 0 0.420861 + 1.68014i 1.16372 2.37608i 1.00000i 2.64575 1.41421i 0
251.16 1.00000i 1.68014 + 0.420861i −1.00000 0 −0.420861 + 1.68014i −1.16372 + 2.37608i 1.00000i 2.64575 + 1.41421i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.b.f 16
3.b odd 2 1 inner 1050.2.b.f 16
5.b even 2 1 inner 1050.2.b.f 16
5.c odd 4 1 210.2.d.a 8
5.c odd 4 1 210.2.d.b yes 8
7.b odd 2 1 inner 1050.2.b.f 16
15.d odd 2 1 inner 1050.2.b.f 16
15.e even 4 1 210.2.d.a 8
15.e even 4 1 210.2.d.b yes 8
20.e even 4 1 1680.2.k.e 8
20.e even 4 1 1680.2.k.f 8
21.c even 2 1 inner 1050.2.b.f 16
35.c odd 2 1 inner 1050.2.b.f 16
35.f even 4 1 210.2.d.a 8
35.f even 4 1 210.2.d.b yes 8
60.l odd 4 1 1680.2.k.e 8
60.l odd 4 1 1680.2.k.f 8
105.g even 2 1 inner 1050.2.b.f 16
105.k odd 4 1 210.2.d.a 8
105.k odd 4 1 210.2.d.b yes 8
140.j odd 4 1 1680.2.k.e 8
140.j odd 4 1 1680.2.k.f 8
420.w even 4 1 1680.2.k.e 8
420.w even 4 1 1680.2.k.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.d.a 8 5.c odd 4 1
210.2.d.a 8 15.e even 4 1
210.2.d.a 8 35.f even 4 1
210.2.d.a 8 105.k odd 4 1
210.2.d.b yes 8 5.c odd 4 1
210.2.d.b yes 8 15.e even 4 1
210.2.d.b yes 8 35.f even 4 1
210.2.d.b yes 8 105.k odd 4 1
1050.2.b.f 16 1.a even 1 1 trivial
1050.2.b.f 16 3.b odd 2 1 inner
1050.2.b.f 16 5.b even 2 1 inner
1050.2.b.f 16 7.b odd 2 1 inner
1050.2.b.f 16 15.d odd 2 1 inner
1050.2.b.f 16 21.c even 2 1 inner
1050.2.b.f 16 35.c odd 2 1 inner
1050.2.b.f 16 105.g even 2 1 inner
1680.2.k.e 8 20.e even 4 1
1680.2.k.e 8 60.l odd 4 1
1680.2.k.e 8 140.j odd 4 1
1680.2.k.e 8 420.w even 4 1
1680.2.k.f 8 20.e even 4 1
1680.2.k.f 8 60.l odd 4 1
1680.2.k.f 8 140.j odd 4 1
1680.2.k.f 8 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{2} + 8 \)
\( T_{17}^{4} - 24 T_{17}^{2} + 32 \)
\( T_{37}^{4} - 128 T_{37}^{2} + 1296 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( ( 1 - 10 T^{4} + 81 T^{8} )^{2} \)
$5$ 1
$7$ \( ( 1 - 4 T^{2} - 10 T^{4} - 196 T^{6} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{8}( 1 + 6 T + 11 T^{2} )^{8} \)
$13$ \( ( 1 - 40 T^{2} + 710 T^{4} - 6760 T^{6} + 28561 T^{8} )^{4} \)
$17$ \( ( 1 + 44 T^{2} + 950 T^{4} + 12716 T^{6} + 83521 T^{8} )^{4} \)
$19$ \( ( 1 - 24 T^{2} + 838 T^{4} - 8664 T^{6} + 130321 T^{8} )^{4} \)
$23$ \( ( 1 - 28 T^{2} + 806 T^{4} - 14812 T^{6} + 279841 T^{8} )^{4} \)
$29$ \( ( 1 - 52 T^{2} + 1350 T^{4} - 43732 T^{6} + 707281 T^{8} )^{4} \)
$31$ \( ( 1 - 84 T^{2} + 3574 T^{4} - 80724 T^{6} + 923521 T^{8} )^{4} \)
$37$ \( ( 1 + 20 T^{2} + 38 T^{4} + 27380 T^{6} + 1874161 T^{8} )^{4} \)
$41$ \( ( 1 + 84 T^{2} + 4678 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{4} \)
$43$ \( ( 1 + 44 T^{2} + 2390 T^{4} + 81356 T^{6} + 3418801 T^{8} )^{4} \)
$47$ \( ( 1 + 108 T^{2} + 6886 T^{4} + 238572 T^{6} + 4879681 T^{8} )^{4} \)
$53$ \( ( 1 + 20 T^{2} + 3926 T^{4} + 56180 T^{6} + 7890481 T^{8} )^{4} \)
$59$ \( ( 1 + 216 T^{2} + 18598 T^{4} + 751896 T^{6} + 12117361 T^{8} )^{4} \)
$61$ \( ( 1 - 136 T^{2} + 9798 T^{4} - 506056 T^{6} + 13845841 T^{8} )^{4} \)
$67$ \( ( 1 + 140 T^{2} + 12086 T^{4} + 628460 T^{6} + 20151121 T^{8} )^{4} \)
$71$ \( ( 1 - 28 T^{2} + 9270 T^{4} - 141148 T^{6} + 25411681 T^{8} )^{4} \)
$73$ \( ( 1 - 252 T^{2} + 26422 T^{4} - 1342908 T^{6} + 28398241 T^{8} )^{4} \)
$79$ \( ( 1 - 4 T + 134 T^{2} - 316 T^{3} + 6241 T^{4} )^{8} \)
$83$ \( ( 1 + 224 T^{2} + 26294 T^{4} + 1543136 T^{6} + 47458321 T^{8} )^{4} \)
$89$ \( ( 1 - 60 T^{2} + 14950 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{4} \)
$97$ \( ( 1 - 108 T^{2} + 16246 T^{4} - 1016172 T^{6} + 88529281 T^{8} )^{4} \)
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