Properties

Label 2646.2.t.a
Level $2646$
Weight $2$
Character orbit 2646.t
Analytic conductor $21.128$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} + 9 x^{12} + 54 x^{10} - 288 x^{8} + 486 x^{6} + 729 x^{4} - 4374 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{5} + \beta_{7} ) q^{2} + ( 1 - \beta_{8} ) q^{4} + ( \beta_{3} - \beta_{6} ) q^{5} -\beta_{5} q^{8} +O(q^{10})\) \( q + ( -\beta_{5} + \beta_{7} ) q^{2} + ( 1 - \beta_{8} ) q^{4} + ( \beta_{3} - \beta_{6} ) q^{5} -\beta_{5} q^{8} + ( -\beta_{3} + \beta_{6} - \beta_{15} ) q^{10} + ( 2 - \beta_{1} + \beta_{5} - 2 \beta_{8} - \beta_{9} + \beta_{14} ) q^{11} + ( \beta_{3} - \beta_{6} - \beta_{11} ) q^{13} -\beta_{8} q^{16} + ( -\beta_{2} - \beta_{12} ) q^{17} + ( -\beta_{2} + \beta_{3} + \beta_{6} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{19} + ( \beta_{3} - \beta_{6} - \beta_{11} + \beta_{15} ) q^{20} + ( -2 + \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{9} - \beta_{14} ) q^{22} + ( -2 - \beta_{4} - 2 \beta_{5} + \beta_{7} + 4 \beta_{8} - \beta_{9} ) q^{23} + ( 2 + \beta_{4} + 3 \beta_{5} - 5 \beta_{7} - \beta_{9} ) q^{25} + ( -\beta_{3} - \beta_{15} ) q^{26} + ( 2 + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} + \beta_{14} ) q^{29} + ( -\beta_{2} - \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{31} -\beta_{7} q^{32} + ( -\beta_{2} + \beta_{13} ) q^{34} + ( 2 - 2 \beta_{1} + \beta_{4} + 5 \beta_{5} - 3 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{14} ) q^{37} + ( -\beta_{2} + \beta_{3} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{38} + ( -\beta_{3} + \beta_{11} - \beta_{15} ) q^{40} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{41} + ( 1 - \beta_{1} + \beta_{4} + 3 \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{43} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{44} + ( -1 - \beta_{4} + \beta_{5} + 3 \beta_{7} - \beta_{8} - \beta_{14} ) q^{46} + ( -2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{47} + ( -5 - \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{14} ) q^{50} + ( -\beta_{6} - \beta_{11} + \beta_{15} ) q^{52} + ( -3 \beta_{2} - 2 \beta_{3} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - 3 \beta_{13} - 2 \beta_{15} ) q^{55} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{14} ) q^{58} + ( -\beta_{3} - 2 \beta_{6} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{59} + ( 2 \beta_{2} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{61} + ( -\beta_{2} + 2 \beta_{10} - \beta_{12} + \beta_{13} ) q^{62} - q^{64} + ( 7 + \beta_{1} + \beta_{5} - \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{65} + ( -3 - \beta_{1} + \beta_{4} - 9 \beta_{5} + 4 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{67} + ( -\beta_{2} + \beta_{10} + \beta_{13} ) q^{68} + ( 1 - \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{71} + ( -\beta_{2} - 2 \beta_{11} + \beta_{12} - 2 \beta_{15} ) q^{73} + ( -3 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - \beta_{14} ) q^{74} + ( -\beta_{2} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{76} + ( -3 + 2 \beta_{1} - \beta_{4} - 5 \beta_{5} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{14} ) q^{79} + ( -\beta_{11} + \beta_{15} ) q^{80} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{10} + 2 \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{82} + ( -2 \beta_{2} + 3 \beta_{3} - 2 \beta_{6} + 4 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} + 3 \beta_{15} ) q^{83} + ( 1 - \beta_{1} - 2 \beta_{8} - \beta_{9} ) q^{85} + ( -1 + \beta_{4} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{86} + ( -1 - \beta_{4} - \beta_{7} + \beta_{9} ) q^{88} + ( -2 \beta_{3} + 2 \beta_{6} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{89} + ( 2 - \beta_{1} - \beta_{7} + 2 \beta_{8} - \beta_{14} ) q^{92} + ( -\beta_{2} + 2 \beta_{3} - \beta_{10} + \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{94} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} + 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{95} + ( -4 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + 3 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + \beta_{13} - 2 \beta_{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + O(q^{10}) \) \( 16 q + 8 q^{4} - 8 q^{16} + 16 q^{25} + 12 q^{29} + 4 q^{37} + 4 q^{43} - 12 q^{44} - 12 q^{46} - 60 q^{50} + 24 q^{58} - 16 q^{64} + 84 q^{65} - 28 q^{67} - 4 q^{79} - 12 q^{85} + 48 q^{92} + 12 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{14} + 9 x^{12} + 54 x^{10} - 288 x^{8} + 486 x^{6} + 729 x^{4} - 4374 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{14} - \nu^{12} + 6 \nu^{10} - 36 \nu^{8} + 72 \nu^{6} + 234 \nu^{4} + 729 \nu^{2} - 243 \)\()/1944\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} + 3 \nu^{13} - 45 \nu^{11} + 135 \nu^{9} + 198 \nu^{7} - 1377 \nu^{5} + 2916 \nu^{3} + 2187 \nu \)\()/4374\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{15} + 9 \nu^{13} - 18 \nu^{11} + 396 \nu^{7} - 216 \nu^{5} + 324 \nu^{3} + 9477 \nu \)\()/5832\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{14} - \nu^{12} + 6 \nu^{10} - 36 \nu^{8} + 180 \nu^{6} + 396 \nu^{4} - 972 \nu^{2} + 4131 \)\()/1944\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{12} - 9 \nu^{10} + 81 \nu^{8} - 126 \nu^{6} - 135 \nu^{4} + 1458 \nu^{2} - 2187 \)\()/1458\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{15} - 24 \nu^{13} + 36 \nu^{11} + 540 \nu^{9} - 1044 \nu^{7} + 1134 \nu^{5} + 8019 \nu^{3} - 13122 \nu \)\()/17496\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{14} + 18 \nu^{12} - 216 \nu^{8} + 792 \nu^{6} + 54 \nu^{4} - 4617 \nu^{2} + 8748 \)\()/5832\)
\(\beta_{8}\)\(=\)\((\)\( -2 \nu^{14} + 21 \nu^{12} - 18 \nu^{10} - 108 \nu^{8} + 576 \nu^{6} - 648 \nu^{4} - 972 \nu^{2} + 9477 \)\()/5832\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{14} - 21 \nu^{12} + 126 \nu^{10} - 612 \nu^{6} + 2862 \nu^{4} - 2673 \nu^{2} + 729 \)\()/5832\)
\(\beta_{10}\)\(=\)\((\)\( 7 \nu^{15} - 15 \nu^{13} - 18 \nu^{11} + 378 \nu^{9} - 558 \nu^{7} - 1458 \nu^{5} + 8019 \nu^{3} - 2187 \nu \)\()/8748\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{15} - 6 \nu^{11} + 36 \nu^{9} - 18 \nu^{7} - 108 \nu^{5} + 513 \nu^{3} \)\()/972\)
\(\beta_{12}\)\(=\)\((\)\( -\nu^{15} + 5 \nu^{13} - 54 \nu^{9} + 180 \nu^{7} - 36 \nu^{5} - 621 \nu^{3} + 1215 \nu \)\()/972\)
\(\beta_{13}\)\(=\)\((\)\( 5 \nu^{15} - 21 \nu^{13} + 216 \nu^{9} - 630 \nu^{7} + 324 \nu^{5} + 2511 \nu^{3} - 6561 \nu \)\()/2916\)
\(\beta_{14}\)\(=\)\((\)\( -3 \nu^{14} + 13 \nu^{12} + 12 \nu^{10} - 180 \nu^{8} + 594 \nu^{6} - 180 \nu^{4} - 3321 \nu^{2} + 8505 \)\()/972\)
\(\beta_{15}\)\(=\)\((\)\( 35 \nu^{15} - 138 \nu^{13} - 36 \nu^{11} + 2052 \nu^{9} - 6192 \nu^{7} + 2106 \nu^{5} + 37179 \nu^{3} - 87480 \nu \)\()/17496\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{3}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{7} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{12} - 2 \beta_{11} + \beta_{10} - \beta_{6} + 2 \beta_{3}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{14} + \beta_{9} + 3 \beta_{8} + 6 \beta_{7} - \beta_{5} + \beta_{4} + \beta_{1}\)
\(\nu^{5}\)\(=\)\(2 \beta_{15} - 3 \beta_{11} - 5 \beta_{10} + 2 \beta_{6} + 6 \beta_{3} + \beta_{2}\)
\(\nu^{6}\)\(=\)\(3 \beta_{14} - 6 \beta_{8} + 3 \beta_{7} + 12 \beta_{5} + 3 \beta_{4} + 3 \beta_{1} - 9\)
\(\nu^{7}\)\(=\)\(-3 \beta_{15} + 15 \beta_{13} + 18 \beta_{12} - 6 \beta_{11} + 9 \beta_{6} + 6 \beta_{2}\)
\(\nu^{8}\)\(=\)\(3 \beta_{14} + 12 \beta_{9} - 3 \beta_{8} + 30 \beta_{7} + 42 \beta_{5} - 3 \beta_{4} - 12 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-18 \beta_{15} - 6 \beta_{13} - 21 \beta_{12} + 21 \beta_{11} - 27 \beta_{10} + 72 \beta_{6} - 24 \beta_{3} + 18 \beta_{2}\)
\(\nu^{10}\)\(=\)\(45 \beta_{14} + 45 \beta_{9} + 27 \beta_{8} - 135 \beta_{7} + 63 \beta_{5} - 18 \beta_{4} - 108\)
\(\nu^{11}\)\(=\)\(-63 \beta_{15} + 18 \beta_{13} + 54 \beta_{12} + 36 \beta_{11} + 153 \beta_{10} + 117 \beta_{6} - 108 \beta_{3} - 81 \beta_{2}\)
\(\nu^{12}\)\(=\)\(-90 \beta_{14} + 126 \beta_{9} + 648 \beta_{8} - 126 \beta_{5} - 36 \beta_{4} - 90 \beta_{1} - 405\)
\(\nu^{13}\)\(=\)\(36 \beta_{15} - 405 \beta_{13} - 216 \beta_{12} + 459 \beta_{11} - 36 \beta_{10} + 144 \beta_{6} - 162 \beta_{3} - 198 \beta_{2}\)
\(\nu^{14}\)\(=\)\(-54 \beta_{9} + 621 \beta_{8} - 999 \beta_{7} - 378 \beta_{5} + 486 \beta_{4} - 243 \beta_{1} - 1134\)
\(\nu^{15}\)\(=\)\(-81 \beta_{15} + 594 \beta_{13} + 891 \beta_{12} + 1026 \beta_{11} + 837 \beta_{10} - 999 \beta_{6} - 162 \beta_{3} - 918 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(\beta_{8}\) \(1 - \beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1979.1
−1.62181 + 0.608059i
1.40917 + 1.00709i
−1.40917 1.00709i
1.62181 0.608059i
−1.69547 0.354107i
0.0967785 1.72934i
−0.0967785 + 1.72934i
1.69547 + 0.354107i
−1.62181 0.608059i
1.40917 1.00709i
−1.40917 + 1.00709i
1.62181 + 0.608059i
−1.69547 + 0.354107i
0.0967785 + 1.72934i
−0.0967785 1.72934i
1.69547 0.354107i
−0.866025 + 0.500000i 0 0.500000 0.866025i −3.89111 0 0 1.00000i 0 3.36980 1.94556i
1979.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −2.34936 0 0 1.00000i 0 2.03460 1.17468i
1979.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 2.34936 0 0 1.00000i 0 −2.03460 + 1.17468i
1979.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 3.89111 0 0 1.00000i 0 −3.36980 + 1.94556i
1979.5 0.866025 0.500000i 0 0.500000 0.866025i −1.79035 0 0 1.00000i 0 −1.55049 + 0.895175i
1979.6 0.866025 0.500000i 0 0.500000 0.866025i −0.366598 0 0 1.00000i 0 −0.317483 + 0.183299i
1979.7 0.866025 0.500000i 0 0.500000 0.866025i 0.366598 0 0 1.00000i 0 0.317483 0.183299i
1979.8 0.866025 0.500000i 0 0.500000 0.866025i 1.79035 0 0 1.00000i 0 1.55049 0.895175i
2285.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −3.89111 0 0 1.00000i 0 3.36980 + 1.94556i
2285.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.34936 0 0 1.00000i 0 2.03460 + 1.17468i
2285.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.34936 0 0 1.00000i 0 −2.03460 1.17468i
2285.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 3.89111 0 0 1.00000i 0 −3.36980 1.94556i
2285.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.79035 0 0 1.00000i 0 −1.55049 0.895175i
2285.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.366598 0 0 1.00000i 0 −0.317483 0.183299i
2285.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.366598 0 0 1.00000i 0 0.317483 + 0.183299i
2285.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.79035 0 0 1.00000i 0 1.55049 + 0.895175i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2285.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.t.a 16
3.b odd 2 1 882.2.t.b 16
7.b odd 2 1 inner 2646.2.t.a 16
7.c even 3 1 378.2.m.a 16
7.c even 3 1 2646.2.l.b 16
7.d odd 6 1 378.2.m.a 16
7.d odd 6 1 2646.2.l.b 16
9.c even 3 1 882.2.l.a 16
9.d odd 6 1 2646.2.l.b 16
21.c even 2 1 882.2.t.b 16
21.g even 6 1 126.2.m.a 16
21.g even 6 1 882.2.l.a 16
21.h odd 6 1 126.2.m.a 16
21.h odd 6 1 882.2.l.a 16
28.f even 6 1 3024.2.cc.b 16
28.g odd 6 1 3024.2.cc.b 16
63.g even 3 1 882.2.t.b 16
63.g even 3 1 1134.2.d.a 16
63.h even 3 1 126.2.m.a 16
63.i even 6 1 378.2.m.a 16
63.j odd 6 1 378.2.m.a 16
63.k odd 6 1 882.2.t.b 16
63.k odd 6 1 1134.2.d.a 16
63.l odd 6 1 882.2.l.a 16
63.n odd 6 1 1134.2.d.a 16
63.n odd 6 1 inner 2646.2.t.a 16
63.o even 6 1 2646.2.l.b 16
63.s even 6 1 1134.2.d.a 16
63.s even 6 1 inner 2646.2.t.a 16
63.t odd 6 1 126.2.m.a 16
84.j odd 6 1 1008.2.cc.b 16
84.n even 6 1 1008.2.cc.b 16
252.r odd 6 1 3024.2.cc.b 16
252.u odd 6 1 1008.2.cc.b 16
252.bb even 6 1 3024.2.cc.b 16
252.bj even 6 1 1008.2.cc.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.m.a 16 21.g even 6 1
126.2.m.a 16 21.h odd 6 1
126.2.m.a 16 63.h even 3 1
126.2.m.a 16 63.t odd 6 1
378.2.m.a 16 7.c even 3 1
378.2.m.a 16 7.d odd 6 1
378.2.m.a 16 63.i even 6 1
378.2.m.a 16 63.j odd 6 1
882.2.l.a 16 9.c even 3 1
882.2.l.a 16 21.g even 6 1
882.2.l.a 16 21.h odd 6 1
882.2.l.a 16 63.l odd 6 1
882.2.t.b 16 3.b odd 2 1
882.2.t.b 16 21.c even 2 1
882.2.t.b 16 63.g even 3 1
882.2.t.b 16 63.k odd 6 1
1008.2.cc.b 16 84.j odd 6 1
1008.2.cc.b 16 84.n even 6 1
1008.2.cc.b 16 252.u odd 6 1
1008.2.cc.b 16 252.bj even 6 1
1134.2.d.a 16 63.g even 3 1
1134.2.d.a 16 63.k odd 6 1
1134.2.d.a 16 63.n odd 6 1
1134.2.d.a 16 63.s even 6 1
2646.2.l.b 16 7.c even 3 1
2646.2.l.b 16 7.d odd 6 1
2646.2.l.b 16 9.d odd 6 1
2646.2.l.b 16 63.o even 6 1
2646.2.t.a 16 1.a even 1 1 trivial
2646.2.t.a 16 7.b odd 2 1 inner
2646.2.t.a 16 63.n odd 6 1 inner
2646.2.t.a 16 63.s even 6 1 inner
3024.2.cc.b 16 28.f even 6 1
3024.2.cc.b 16 28.g odd 6 1
3024.2.cc.b 16 252.r odd 6 1
3024.2.cc.b 16 252.bb even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24 T_{5}^{6} + 153 T_{5}^{4} - 288 T_{5}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( T^{16} \)
$5$ \( ( 36 - 288 T^{2} + 153 T^{4} - 24 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 1296 + 3240 T^{2} + 801 T^{4} + 54 T^{6} + T^{8} )^{2} \)
$13$ \( 331776 - 497664 T^{2} + 559872 T^{4} - 238464 T^{6} + 73296 T^{8} - 9936 T^{10} + 972 T^{12} - 36 T^{14} + T^{16} \)
$17$ \( 331776 + 746496 T^{2} + 1404864 T^{4} + 569808 T^{6} + 172521 T^{8} + 17442 T^{10} + 1287 T^{12} + 42 T^{14} + T^{16} \)
$19$ \( 2313441 - 2819934 T^{2} + 2533842 T^{4} - 937008 T^{6} + 251199 T^{8} - 28368 T^{10} + 2322 T^{12} - 54 T^{14} + T^{16} \)
$23$ \( ( 443556 + 82944 T^{2} + 5229 T^{4} + 126 T^{6} + T^{8} )^{2} \)
$29$ \( ( 20736 + 10368 T - 2592 T^{2} - 2160 T^{3} + 612 T^{4} + 180 T^{5} - 18 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$31$ \( 557256278016 - 92876046336 T^{2} + 10400182272 T^{4} - 631535616 T^{6} + 27632016 T^{8} - 730944 T^{10} + 13932 T^{12} - 144 T^{14} + T^{16} \)
$37$ \( ( 1784896 + 245824 T + 170128 T^{2} - 13424 T^{3} + 9436 T^{4} - 164 T^{5} + 106 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$41$ \( 73499483897856 + 6883786653696 T^{2} + 448658922240 T^{4} + 13938763392 T^{6} + 307258425 T^{8} + 4294314 T^{10} + 43695 T^{12} + 258 T^{14} + T^{16} \)
$43$ \( ( 10816 - 15392 T + 17848 T^{2} - 6188 T^{3} + 1921 T^{4} - 218 T^{5} + 43 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$47$ \( 1485512441856 + 399459631104 T^{2} + 87539678208 T^{4} + 4759817472 T^{6} + 186073488 T^{8} + 3258432 T^{10} + 41292 T^{12} + 240 T^{14} + T^{16} \)
$53$ \( T^{16} \)
$59$ \( 1296 + 2901744 T^{2} + 6496369452 T^{4} + 1422558828 T^{6} + 287789589 T^{8} + 5027598 T^{10} + 68787 T^{12} + 294 T^{14} + T^{16} \)
$61$ \( 2425818710016 - 545450360832 T^{2} + 96222587904 T^{4} - 5193676800 T^{6} + 202203801 T^{8} - 3371184 T^{10} + 40635 T^{12} - 240 T^{14} + T^{16} \)
$67$ \( ( 824464 + 1507280 T + 2654812 T^{2} + 209684 T^{3} + 36469 T^{4} + 1766 T^{5} + 307 T^{6} + 14 T^{7} + T^{8} )^{2} \)
$71$ \( ( 82944 + 31104 T^{2} + 2745 T^{4} + 90 T^{6} + T^{8} )^{2} \)
$73$ \( 2927055626496 - 422268609024 T^{2} + 40269720240 T^{4} - 2219198688 T^{6} + 89156745 T^{8} - 2185686 T^{10} + 37215 T^{12} - 222 T^{14} + T^{16} \)
$79$ \( ( 1444804 + 935156 T + 450226 T^{2} + 105170 T^{3} + 19399 T^{4} + 1298 T^{5} + 133 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$83$ \( 337116351515590656 + 10214329542377472 T^{2} + 207879529033728 T^{4} + 2256409253376 T^{6} + 17587710864 T^{8} + 88712784 T^{10} + 326268 T^{12} + 708 T^{14} + T^{16} \)
$89$ \( 34828517376 + 29023764480 T^{2} + 21767823360 T^{4} + 1934917632 T^{6} + 134182656 T^{8} + 2488320 T^{10} + 33696 T^{12} + 216 T^{14} + T^{16} \)
$97$ \( 4512402164941056 - 390697151362560 T^{2} + 24485300891568 T^{4} - 714581204256 T^{6} + 15192293193 T^{8} - 85999734 T^{10} + 353727 T^{12} - 702 T^{14} + T^{16} \)
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