Properties

Label 896.2.q.d
Level $896$
Weight $2$
Character orbit 896.q
Analytic conductor $7.155$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 24 x^{14} + 226 x^{12} - 972 x^{10} + 1575 x^{8} + 252 x^{6} + 550 x^{4} + 156 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - \beta_{2} + \beta_{3} - \beta_{7} ) q^{3} + ( \beta_{5} + \beta_{10} - \beta_{13} ) q^{5} + ( \beta_{4} - \beta_{8} - \beta_{10} ) q^{7} + ( 2 - \beta_{2} + 3 \beta_{3} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{9} +O(q^{10})\) \( q + ( 2 - \beta_{2} + \beta_{3} - \beta_{7} ) q^{3} + ( \beta_{5} + \beta_{10} - \beta_{13} ) q^{5} + ( \beta_{4} - \beta_{8} - \beta_{10} ) q^{7} + ( 2 - \beta_{2} + 3 \beta_{3} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{9} + ( -\beta_{2} + \beta_{9} + \beta_{11} ) q^{11} + ( -\beta_{1} + \beta_{5} - \beta_{13} + \beta_{15} ) q^{13} + ( \beta_{1} + 3 \beta_{4} + 2 \beta_{5} - \beta_{8} + \beta_{10} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{15} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{7} - \beta_{12} ) q^{17} + ( -\beta_{3} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{12} ) q^{19} + ( -3 \beta_{1} - \beta_{4} + 3 \beta_{5} + 2 \beta_{8} - \beta_{10} ) q^{21} + ( -\beta_{4} + 2 \beta_{5} + \beta_{10} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{23} + ( 2 \beta_{3} + 2 \beta_{9} + 2 \beta_{11} ) q^{25} + ( 2 - \beta_{2} + 5 \beta_{3} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{11} + 3 \beta_{12} ) q^{27} + ( -\beta_{1} - 2 \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{13} - \beta_{15} ) q^{29} + ( -2 \beta_{4} - 4 \beta_{5} - \beta_{8} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{31} + ( 1 - 2 \beta_{6} + \beta_{7} + 2 \beta_{11} ) q^{33} + ( -1 + \beta_{2} - 3 \beta_{3} + 2 \beta_{6} + 3 \beta_{7} + \beta_{9} - 3 \beta_{11} - 2 \beta_{12} ) q^{35} + ( -\beta_{1} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{10} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{37} + ( -2 \beta_{1} - \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{10} - \beta_{13} + \beta_{15} ) q^{39} + ( 2 - \beta_{2} + 3 \beta_{3} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{41} + ( -2 + 2 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{11} + 2 \beta_{12} ) q^{43} + ( 6 \beta_{4} + 9 \beta_{5} + 2 \beta_{8} - \beta_{10} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{45} + ( -\beta_{1} - \beta_{4} + 2 \beta_{5} + 6 \beta_{8} - \beta_{10} + \beta_{13} ) q^{47} + ( -1 - \beta_{2} + \beta_{3} - \beta_{6} + 3 \beta_{7} - \beta_{9} + 3 \beta_{11} + 2 \beta_{12} ) q^{49} + ( -6 - 3 \beta_{3} + \beta_{6} + 6 \beta_{7} + 3 \beta_{9} - 2 \beta_{11} - 3 \beta_{12} ) q^{51} + ( 2 \beta_{1} + \beta_{4} + 4 \beta_{8} + \beta_{10} - \beta_{13} + \beta_{15} ) q^{53} + ( 4 \beta_{1} + 2 \beta_{4} - 6 \beta_{5} + 2 \beta_{8} - 2 \beta_{10} + \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{55} + ( 5 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} ) q^{57} + ( 2 + 3 \beta_{2} - 3 \beta_{3} - \beta_{7} ) q^{59} + ( -5 \beta_{1} - 4 \beta_{4} + 3 \beta_{5} + 4 \beta_{8} - \beta_{14} + \beta_{15} ) q^{61} + ( -3 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{8} - 2 \beta_{10} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{63} + ( 4 - \beta_{2} - \beta_{3} + 2 \beta_{6} - 3 \beta_{7} + 3 \beta_{9} - \beta_{11} - 3 \beta_{12} ) q^{65} + ( 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{6} - \beta_{7} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} ) q^{67} + ( -\beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{10} + \beta_{14} + \beta_{15} ) q^{69} + ( -2 \beta_{1} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{13} - 2 \beta_{15} ) q^{71} + ( -6 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} ) q^{73} + ( -6 + 2 \beta_{3} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} ) q^{75} + ( -\beta_{1} + \beta_{4} - 4 \beta_{8} + 2 \beta_{10} - 2 \beta_{13} + \beta_{14} ) q^{77} + ( 2 \beta_{1} - 5 \beta_{4} + 2 \beta_{5} + \beta_{10} + \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{79} + ( 4 \beta_{2} + 4 \beta_{3} + \beta_{6} - 3 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{81} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{83} + ( -3 \beta_{1} - 5 \beta_{4} - \beta_{5} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{85} + ( \beta_{4} - 8 \beta_{5} - 5 \beta_{8} + \beta_{10} + \beta_{13} + \beta_{15} ) q^{87} + ( 1 + 4 \beta_{6} + \beta_{7} - 4 \beta_{11} ) q^{89} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{6} + 7 \beta_{7} + \beta_{11} + 3 \beta_{12} ) q^{91} + ( 3 \beta_{1} - 6 \beta_{4} - 9 \beta_{5} + 2 \beta_{10} + 2 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{93} + ( 4 \beta_{1} + \beta_{4} - 2 \beta_{5} + 2 \beta_{8} + \beta_{10} + \beta_{14} ) q^{95} + ( 2 - \beta_{2} + 5 \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{97} + ( 7 - 3 \beta_{2} + 2 \beta_{3} - \beta_{6} - 3 \beta_{7} - 2 \beta_{9} + \beta_{11} - \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 12q^{3} + 8q^{9} + O(q^{10}) \) \( 16q + 12q^{3} + 8q^{9} - 4q^{11} - 12q^{19} - 16q^{25} + 24q^{33} + 20q^{35} + 16q^{49} - 52q^{51} + 48q^{57} + 60q^{59} + 24q^{65} + 12q^{67} - 24q^{73} - 120q^{75} - 32q^{81} + 24q^{89} + 72q^{91} + 64q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 24 x^{14} + 226 x^{12} - 972 x^{10} + 1575 x^{8} + 252 x^{6} + 550 x^{4} + 156 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -13543 \nu^{14} + 429249 \nu^{12} - 5601614 \nu^{10} + 36966313 \nu^{8} - 121375292 \nu^{6} + 152255823 \nu^{4} + 12537019 \nu^{2} + 28089012 \)\()/39788476\)
\(\beta_{3}\)\(=\)\((\)\( 2742 \nu^{14} - 63413 \nu^{12} + 569287 \nu^{10} - 2259390 \nu^{8} + 2967075 \nu^{6} + 1844696 \nu^{4} + 3667011 \nu^{2} - 889595 \)\()/3060652\)
\(\beta_{4}\)\(=\)\((\)\( 31987 \nu^{15} - 825179 \nu^{13} + 8757356 \nu^{11} - 47162623 \nu^{9} + 130204306 \nu^{7} - 161319441 \nu^{5} + 83577427 \nu^{3} - 7068028 \nu \)\()/39788476\)
\(\beta_{5}\)\(=\)\((\)\( -45956 \nu^{15} + 1339385 \nu^{13} - 16099727 \nu^{11} + 98925010 \nu^{9} - 309182623 \nu^{7} + 388373668 \nu^{5} - 2026449 \nu^{3} + 77606203 \nu \)\()/39788476\)
\(\beta_{6}\)\(=\)\((\)\( 60741 \nu^{14} - 1736125 \nu^{12} + 20612094 \nu^{10} - 126911873 \nu^{8} + 409643968 \nu^{6} - 582737775 \nu^{4} + 134047179 \nu^{2} - 52088512 \)\()/39788476\)
\(\beta_{7}\)\(=\)\((\)\( 8928 \nu^{14} - 219390 \nu^{12} + 2137764 \nu^{10} - 9777437 \nu^{8} + 18618924 \nu^{6} - 4607374 \nu^{4} + 2579940 \nu^{2} + 498261 \)\()/1421017\)
\(\beta_{8}\)\(=\)\((\)\( 281971 \nu^{15} - 6968099 \nu^{13} + 68614748 \nu^{11} - 320930859 \nu^{9} + 651534178 \nu^{7} - 290325913 \nu^{5} + 155815747 \nu^{3} + 6883280 \nu \)\()/39788476\)
\(\beta_{9}\)\(=\)\((\)\( 394087 \nu^{14} - 9092864 \nu^{12} + 80172959 \nu^{10} - 297924065 \nu^{8} + 245316181 \nu^{6} + 745788619 \nu^{4} + 221974720 \nu^{2} + 174076245 \)\()/39788476\)
\(\beta_{10}\)\(=\)\((\)\( 317762 \nu^{15} - 7371063 \nu^{13} + 65516719 \nu^{11} - 246474572 \nu^{9} + 203226999 \nu^{7} + 713348290 \nu^{5} - 168783393 \nu^{3} + 132348297 \nu \)\()/39788476\)
\(\beta_{11}\)\(=\)\((\)\( 432725 \nu^{14} - 10433869 \nu^{12} + 98985936 \nu^{10} - 432430181 \nu^{8} + 737763466 \nu^{6} - 13186027 \nu^{4} + 335602213 \nu^{2} - 13901972 \)\()/39788476\)
\(\beta_{12}\)\(=\)\((\)\( 582531 \nu^{14} - 13910723 \nu^{12} + 129886478 \nu^{10} - 548229671 \nu^{8} + 825018216 \nu^{6} + 389977711 \nu^{4} + 35475657 \nu^{2} + 132028820 \)\()/39788476\)
\(\beta_{13}\)\(=\)\((\)\( -528296 \nu^{15} + 13111829 \nu^{13} - 129828765 \nu^{11} + 612489648 \nu^{9} - 1264496381 \nu^{7} + 604632874 \nu^{5} - 303748827 \nu^{3} + 213399561 \nu \)\()/39788476\)
\(\beta_{14}\)\(=\)\((\)\( -682235 \nu^{15} + 16582331 \nu^{13} - 159108005 \nu^{11} + 708348501 \nu^{9} - 1259606069 \nu^{7} + 79180573 \nu^{5} - 167884022 \nu^{3} - 83407358 \nu \)\()/19894238\)
\(\beta_{15}\)\(=\)\((\)\( 718355 \nu^{15} - 17401158 \nu^{13} + 166244059 \nu^{11} - 735570487 \nu^{9} + 1297665313 \nu^{7} - 118211451 \nu^{5} + 435617438 \nu^{3} + 48167315 \nu \)\()/19894238\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{11} - \beta_{9} - \beta_{6} + \beta_{3} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{15} - 2 \beta_{13} + 2 \beta_{8} + 4 \beta_{5} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(4 \beta_{11} - 6 \beta_{9} + 3 \beta_{7} - 6 \beta_{6} + 10 \beta_{3} + 8 \beta_{2} + 16\)
\(\nu^{5}\)\(=\)\(-14 \beta_{15} - 2 \beta_{14} - 18 \beta_{13} - 2 \beta_{10} + 39 \beta_{8} + 52 \beta_{5} - 3 \beta_{4} + 20 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-7 \beta_{12} + 6 \beta_{11} - 24 \beta_{9} + 42 \beta_{7} - 29 \beta_{6} + 80 \beta_{3} + 48 \beta_{2} + 67\)
\(\nu^{7}\)\(=\)\(-122 \beta_{15} - 12 \beta_{14} - 128 \beta_{13} - 19 \beta_{10} + 427 \beta_{8} + 457 \beta_{5} - 65 \beta_{4} + 18 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-96 \beta_{12} - 104 \beta_{11} - 24 \beta_{9} + 397 \beta_{7} - 108 \beta_{6} + 568 \beta_{3} + 188 \beta_{2} + 139\)
\(\nu^{9}\)\(=\)\(-860 \beta_{15} + 16 \beta_{14} - 756 \beta_{13} - 80 \beta_{10} + 3757 \beta_{8} + 3268 \beta_{5} - 805 \beta_{4} - 697 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-869 \beta_{12} - 1557 \beta_{11} + 801 \beta_{9} + 2930 \beta_{7} - 84 \beta_{6} + 3743 \beta_{3} - 261 \beta_{2} - 1030\)
\(\nu^{11}\)\(=\)\(-5039 \beta_{15} + 1214 \beta_{14} - 3482 \beta_{13} + 345 \beta_{10} + 28675 \beta_{8} + 19529 \beta_{5} - 7539 \beta_{4} - 9573 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-5980 \beta_{12} - 14612 \beta_{11} + 11130 \beta_{9} + 16606 \beta_{7} + 3936 \beta_{6} + 23202 \beta_{3} - 14986 \beta_{2} - 17303\)
\(\nu^{13}\)\(=\)\(-22828 \beta_{15} + 17030 \beta_{14} - 8216 \beta_{13} + 11050 \beta_{10} + 192318 \beta_{8} + 91542 \beta_{5} - 57486 \beta_{4} - 88053 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-29406 \beta_{12} - 110881 \beta_{11} + 102665 \beta_{9} + 55380 \beta_{7} + 56229 \beta_{6} + 133983 \beta_{3} - 196135 \beta_{2} - 158729\)
\(\nu^{15}\)\(=\)\(-48729 \beta_{15} + 169312 \beta_{14} + 62152 \beta_{13} + 139906 \beta_{10} + 1103468 \beta_{8} + 224920 \beta_{5} - 355186 \beta_{4} - 663575 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(1 - \beta_{7}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
2.65282 0.500000i
−2.65282 + 0.500000i
−0.245327 + 0.500000i
0.245327 0.500000i
0.526379 + 0.500000i
−0.526379 0.500000i
−2.37177 0.500000i
2.37177 + 0.500000i
2.65282 + 0.500000i
−2.65282 0.500000i
−0.245327 0.500000i
0.245327 + 0.500000i
0.526379 0.500000i
−0.526379 + 0.500000i
−2.37177 + 0.500000i
2.37177 0.500000i
0 −1.54741 0.893397i 0 −0.581841 1.00778i 0 2.23781 + 1.41146i 0 0.0963180 + 0.166828i 0
703.2 0 −1.54741 0.893397i 0 0.581841 + 1.00778i 0 −2.23781 1.41146i 0 0.0963180 + 0.166828i 0
703.3 0 0.537541 + 0.310349i 0 −0.0337794 0.0585076i 0 1.98532 1.74886i 0 −1.30737 2.26443i 0
703.4 0 0.537541 + 0.310349i 0 0.0337794 + 0.0585076i 0 −1.98532 + 1.74886i 0 −1.30737 2.26443i 0
703.5 0 1.20586 + 0.696202i 0 −2.07821 3.59957i 0 −0.632797 + 2.56896i 0 −0.530605 0.919035i 0
703.6 0 1.20586 + 0.696202i 0 2.07821 + 3.59957i 0 0.632797 2.56896i 0 −0.530605 0.919035i 0
703.7 0 2.80401 + 1.61890i 0 −1.53015 2.65030i 0 2.57882 0.591357i 0 3.74165 + 6.48073i 0
703.8 0 2.80401 + 1.61890i 0 1.53015 + 2.65030i 0 −2.57882 + 0.591357i 0 3.74165 + 6.48073i 0
831.1 0 −1.54741 + 0.893397i 0 −0.581841 + 1.00778i 0 2.23781 1.41146i 0 0.0963180 0.166828i 0
831.2 0 −1.54741 + 0.893397i 0 0.581841 1.00778i 0 −2.23781 + 1.41146i 0 0.0963180 0.166828i 0
831.3 0 0.537541 0.310349i 0 −0.0337794 + 0.0585076i 0 1.98532 + 1.74886i 0 −1.30737 + 2.26443i 0
831.4 0 0.537541 0.310349i 0 0.0337794 0.0585076i 0 −1.98532 1.74886i 0 −1.30737 + 2.26443i 0
831.5 0 1.20586 0.696202i 0 −2.07821 + 3.59957i 0 −0.632797 2.56896i 0 −0.530605 + 0.919035i 0
831.6 0 1.20586 0.696202i 0 2.07821 3.59957i 0 0.632797 + 2.56896i 0 −0.530605 + 0.919035i 0
831.7 0 2.80401 1.61890i 0 −1.53015 + 2.65030i 0 2.57882 + 0.591357i 0 3.74165 6.48073i 0
831.8 0 2.80401 1.61890i 0 1.53015 2.65030i 0 −2.57882 0.591357i 0 3.74165 6.48073i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 831.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.d odd 2 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.q.d yes 16
4.b odd 2 1 896.2.q.a 16
7.d odd 6 1 inner 896.2.q.d yes 16
8.b even 2 1 896.2.q.a 16
8.d odd 2 1 inner 896.2.q.d yes 16
28.f even 6 1 896.2.q.a 16
56.j odd 6 1 896.2.q.a 16
56.m even 6 1 inner 896.2.q.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.q.a 16 4.b odd 2 1
896.2.q.a 16 8.b even 2 1
896.2.q.a 16 28.f even 6 1
896.2.q.a 16 56.j odd 6 1
896.2.q.d yes 16 1.a even 1 1 trivial
896.2.q.d yes 16 7.d odd 6 1 inner
896.2.q.d yes 16 8.d odd 2 1 inner
896.2.q.d yes 16 56.m even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 25 - 90 T + 118 T^{2} - 36 T^{3} - 27 T^{4} + 12 T^{5} + 10 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$5$ \( 1 + 220 T^{2} + 48202 T^{4} + 43504 T^{6} + 33043 T^{8} + 5104 T^{10} + 586 T^{12} + 28 T^{14} + T^{16} \)
$7$ \( 5764801 - 941192 T^{2} + 124852 T^{4} + 1960 T^{6} - 1370 T^{8} + 40 T^{10} + 52 T^{12} - 8 T^{14} + T^{16} \)
$11$ \( ( 4489 - 670 T + 1306 T^{2} - 88 T^{3} + 277 T^{4} - 16 T^{5} + 22 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$13$ \( ( 144 - 1152 T^{2} + 552 T^{4} - 48 T^{6} + T^{8} )^{2} \)
$17$ \( ( 167281 - 29448 T - 18722 T^{2} + 3600 T^{3} + 2091 T^{4} - 50 T^{6} + T^{8} )^{2} \)
$19$ \( ( 441 + 1890 T + 3330 T^{2} + 2700 T^{3} + 741 T^{4} - 180 T^{5} - 18 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$23$ \( 625 - 40600 T^{2} + 2610526 T^{4} - 1740976 T^{6} + 1049515 T^{8} - 65488 T^{10} + 3022 T^{12} - 64 T^{14} + T^{16} \)
$29$ \( ( 24336 + 13248 T^{2} + 1992 T^{4} + 96 T^{6} + T^{8} )^{2} \)
$31$ \( 194481 + 381024 T^{2} + 526878 T^{4} + 366768 T^{6} + 185355 T^{8} + 34128 T^{10} + 4686 T^{12} + 72 T^{14} + T^{16} \)
$37$ \( 64013554081 - 48499801228 T^{2} + 34254696250 T^{4} - 1800364336 T^{6} + 63719683 T^{8} - 1310128 T^{10} + 19738 T^{12} - 172 T^{14} + T^{16} \)
$41$ \( ( 150544 + 52480 T^{2} + 4104 T^{4} + 112 T^{6} + T^{8} )^{2} \)
$43$ \( ( 960 + 192 T - 96 T^{2} + T^{4} )^{4} \)
$47$ \( 3154956561 + 2887535952 T^{2} + 2396425230 T^{4} + 211994928 T^{6} + 13011867 T^{8} + 423504 T^{10} + 10014 T^{12} + 120 T^{14} + T^{16} \)
$53$ \( 151807041 - 1152802044 T^{2} + 8575542954 T^{4} - 1350655344 T^{6} + 186717555 T^{8} - 3467376 T^{10} + 49002 T^{12} - 252 T^{14} + T^{16} \)
$59$ \( ( 227529 - 111618 T - 7506 T^{2} + 12636 T^{3} + 1053 T^{4} - 1620 T^{5} + 354 T^{6} - 30 T^{7} + T^{8} )^{2} \)
$61$ \( 1534548635361 + 639546704244 T^{2} + 240192291546 T^{4} + 10297390032 T^{6} + 308681955 T^{8} + 4837968 T^{10} + 54906 T^{12} + 276 T^{14} + T^{16} \)
$67$ \( ( 4149369 - 1918854 T + 679590 T^{2} - 120528 T^{3} + 18093 T^{4} - 1272 T^{5} + 138 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$71$ \( ( 114233344 + 4833280 T^{2} + 71808 T^{4} + 448 T^{6} + T^{8} )^{2} \)
$73$ \( ( 378225 + 863460 T + 734562 T^{2} + 176904 T^{3} + 10875 T^{4} - 1512 T^{5} - 78 T^{6} + 12 T^{7} + T^{8} )^{2} \)
$79$ \( 863609933592081 - 165853359776520 T^{2} + 29115917173710 T^{4} - 494340968304 T^{6} + 5656476699 T^{8} - 37864080 T^{10} + 185694 T^{12} - 528 T^{14} + T^{16} \)
$83$ \( ( 36864 + 73728 T^{2} + 8832 T^{4} + 192 T^{6} + T^{8} )^{2} \)
$89$ \( ( 1274641 - 257412 T - 215246 T^{2} + 46968 T^{3} + 42219 T^{4} + 2472 T^{5} - 158 T^{6} - 12 T^{7} + T^{8} )^{2} \)
$97$ \( ( 1296 + 12096 T^{2} + 4104 T^{4} + 192 T^{6} + T^{8} )^{2} \)
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