Properties

Label 1134.2.d.a
Level $1134$
Weight $2$
Character orbit 1134.d
Analytic conductor $9.055$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(16\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} - q^{4} + \beta_{2} q^{5} -\beta_{8} q^{7} -\beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} - q^{4} + \beta_{2} q^{5} -\beta_{8} q^{7} -\beta_{1} q^{8} -\beta_{13} q^{10} + \beta_{5} q^{11} -\beta_{9} q^{13} -\beta_{6} q^{14} + q^{16} + \beta_{14} q^{17} + ( -\beta_{4} + \beta_{9} + \beta_{13} ) q^{19} -\beta_{2} q^{20} + \beta_{3} q^{22} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{23} + ( 1 + \beta_{3} - \beta_{12} ) q^{25} + \beta_{15} q^{26} + \beta_{8} q^{28} + ( -2 \beta_{1} + \beta_{6} + \beta_{7} ) q^{29} + ( -2 \beta_{4} - \beta_{8} - \beta_{10} ) q^{31} + \beta_{1} q^{32} -\beta_{4} q^{34} + ( -\beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{14} - \beta_{15} ) q^{35} + ( 2 \beta_{3} - \beta_{8} + \beta_{10} ) q^{37} + ( \beta_{2} - \beta_{14} - \beta_{15} ) q^{38} + \beta_{13} q^{40} + ( 2 \beta_{6} - 2 \beta_{7} + \beta_{14} - 2 \beta_{15} ) q^{41} + ( \beta_{3} - \beta_{8} + \beta_{10} ) q^{43} -\beta_{5} q^{44} + ( 2 + \beta_{3} - \beta_{8} + \beta_{10} - \beta_{12} ) q^{46} + ( \beta_{6} - \beta_{7} + 2 \beta_{14} + 2 \beta_{15} ) q^{47} + ( 1 + \beta_{3} - \beta_{4} + 2 \beta_{9} ) q^{49} + ( \beta_{1} - \beta_{5} + \beta_{11} ) q^{50} + \beta_{9} q^{52} + ( -2 \beta_{4} - 3 \beta_{8} - 3 \beta_{10} + 2 \beta_{13} ) q^{55} + \beta_{6} q^{56} + ( 2 - \beta_{8} + \beta_{10} ) q^{58} + ( -2 \beta_{2} + \beta_{14} + 3 \beta_{15} ) q^{59} + ( -2 \beta_{4} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} ) q^{61} + ( -\beta_{6} + \beta_{7} - 2 \beta_{14} ) q^{62} - q^{64} + ( -\beta_{1} - \beta_{6} - \beta_{7} - \beta_{11} ) q^{65} + ( 4 + \beta_{3} - \beta_{8} + \beta_{10} - 2 \beta_{12} ) q^{67} -\beta_{14} q^{68} + ( \beta_{3} - \beta_{4} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} ) q^{70} + ( \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{71} + ( \beta_{4} + \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{13} ) q^{73} + ( -2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{74} + ( \beta_{4} - \beta_{9} - \beta_{13} ) q^{76} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{6} + 2 \beta_{7} + \beta_{11} - \beta_{14} ) q^{77} + ( \beta_{3} + \beta_{8} - \beta_{10} + \beta_{12} ) q^{79} + \beta_{2} q^{80} + ( -\beta_{4} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{82} + ( -2 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{14} - \beta_{15} ) q^{83} + ( 2 - \beta_{8} + \beta_{10} ) q^{85} + ( -\beta_{5} - \beta_{6} - \beta_{7} ) q^{86} -\beta_{3} q^{88} + ( 2 \beta_{2} - 2 \beta_{14} ) q^{89} + ( 2 - \beta_{9} + 2 \beta_{10} - 2 \beta_{13} ) q^{91} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{11} ) q^{92} + ( -2 \beta_{4} - \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{94} + ( -3 \beta_{1} + \beta_{5} ) q^{95} + ( -\beta_{4} + 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - 2 \beta_{13} ) q^{97} + ( \beta_{1} - \beta_{5} - \beta_{14} - 2 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 4 q^{7} + O(q^{10}) \) \( 16 q - 16 q^{4} - 4 q^{7} + 16 q^{16} + 16 q^{25} + 4 q^{28} - 8 q^{37} - 8 q^{43} + 24 q^{46} + 16 q^{49} + 24 q^{58} - 16 q^{64} + 56 q^{67} + 8 q^{79} + 24 q^{85} + 24 q^{91} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{12} - 9 \nu^{10} + 81 \nu^{8} - 126 \nu^{6} - 135 \nu^{4} + 1458 \nu^{2} - 2187 \)\()/1458\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - 51 \nu^{13} + 90 \nu^{11} + 540 \nu^{9} - 2232 \nu^{7} + 1782 \nu^{5} + 7047 \nu^{3} - 41553 \nu \)\()/17496\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{12} - 3 \nu^{10} - 9 \nu^{8} + 54 \nu^{6} - 45 \nu^{4} - 135 \nu^{2} + 729 \)\()/162\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} + 15 \nu^{13} - 9 \nu^{11} - 54 \nu^{9} + 531 \nu^{7} - 891 \nu^{5} + 1215 \nu^{3} + 4374 \nu \)\()/4374\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{14} + 6 \nu^{12} - 9 \nu^{10} - 54 \nu^{8} + 288 \nu^{6} - 486 \nu^{4} - 1458 \nu^{2} + 4374 \)\()/729\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{15} + 12 \nu^{13} - 36 \nu^{12} - 45 \nu^{11} - 27 \nu^{10} + 54 \nu^{9} + 405 \nu^{8} + 333 \nu^{7} - 1944 \nu^{6} - 1701 \nu^{5} + 891 \nu^{4} + 1944 \nu^{3} + 10935 \nu^{2} + 6561 \nu - 39366 \)\()/8748\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{15} - 12 \nu^{13} - 36 \nu^{12} + 45 \nu^{11} - 27 \nu^{10} - 54 \nu^{9} + 405 \nu^{8} - 333 \nu^{7} - 1944 \nu^{6} + 1701 \nu^{5} + 891 \nu^{4} - 1944 \nu^{3} + 10935 \nu^{2} - 6561 \nu - 39366 \)\()/8748\)
\(\beta_{8}\)\(=\)\((\)\( -4 \nu^{15} + 27 \nu^{14} + 6 \nu^{13} - 81 \nu^{12} + 45 \nu^{11} - 81 \nu^{10} - 216 \nu^{9} + 1215 \nu^{8} - 63 \nu^{7} - 3402 \nu^{6} + 1053 \nu^{5} + 729 \nu^{4} - 3888 \nu^{3} + 18954 \nu^{2} + 2187 \nu - 37179 \)\()/8748\)
\(\beta_{9}\)\(=\)\((\)\( -5 \nu^{15} + 57 \nu^{13} - 18 \nu^{11} - 432 \nu^{9} + 2412 \nu^{7} - 1458 \nu^{5} - 9963 \nu^{3} + 37179 \nu \)\()/8748\)
\(\beta_{10}\)\(=\)\((\)\( -4 \nu^{15} - 27 \nu^{14} + 6 \nu^{13} + 81 \nu^{12} + 45 \nu^{11} + 81 \nu^{10} - 216 \nu^{9} - 1215 \nu^{8} - 63 \nu^{7} + 3402 \nu^{6} + 1053 \nu^{5} - 729 \nu^{4} - 3888 \nu^{3} - 18954 \nu^{2} + 2187 \nu + 37179 \)\()/8748\)
\(\beta_{11}\)\(=\)\((\)\( 2 \nu^{14} - 21 \nu^{12} + 18 \nu^{10} + 108 \nu^{8} - 576 \nu^{6} + 648 \nu^{4} + 972 \nu^{2} - 6561 \)\()/972\)
\(\beta_{12}\)\(=\)\((\)\( -7 \nu^{14} + 24 \nu^{12} + 18 \nu^{10} - 378 \nu^{8} + 1044 \nu^{6} + 324 \nu^{4} - 7533 \nu^{2} + 13122 \)\()/972\)
\(\beta_{13}\)\(=\)\((\)\( 23 \nu^{15} - 111 \nu^{13} + 18 \nu^{11} + 1404 \nu^{9} - 4680 \nu^{7} + 3402 \nu^{5} + 28917 \nu^{3} - 59049 \nu \)\()/17496\)
\(\beta_{14}\)\(=\)\((\)\( 5 \nu^{15} - 21 \nu^{13} + 18 \nu^{11} + 189 \nu^{9} - 711 \nu^{7} + 567 \nu^{5} + 2430 \nu^{3} - 6561 \nu \)\()/2187\)
\(\beta_{15}\)\(=\)\((\)\( -23 \nu^{15} + 57 \nu^{13} + 90 \nu^{11} - 1080 \nu^{9} + 2736 \nu^{7} + 486 \nu^{5} - 19197 \nu^{3} + 37179 \nu \)\()/8748\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} + \beta_{14} + 2 \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - \beta_{4} - 2 \beta_{2}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{12} + 3 \beta_{10} - 3 \beta_{8} - 3 \beta_{5} + 3 \beta_{3} + 6\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} + 2 \beta_{13} - \beta_{9} + 2 \beta_{4} - 2 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{12} - \beta_{10} + \beta_{8} - 3 \beta_{5} + \beta_{3} + 6 \beta_{1} + 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{15} - 5 \beta_{14} + 10 \beta_{13} - \beta_{10} + 5 \beta_{9} - \beta_{8} + 4 \beta_{7} - 4 \beta_{6} - 5 \beta_{4} - 2 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(2 \beta_{11} + 3 \beta_{10} - 3 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{3} + 15 \beta_{1} - 12\)
\(\nu^{7}\)\(=\)\((\)\(9 \beta_{15} - 3 \beta_{14} + 6 \beta_{13} - 21 \beta_{10} + 3 \beta_{9} - 21 \beta_{8} + 12 \beta_{7} - 12 \beta_{6} + 3 \beta_{4} + 6 \beta_{2}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(18 \beta_{12} - 9 \beta_{10} + 9 \beta_{8} - 12 \beta_{7} - 12 \beta_{6} + 3 \beta_{5} - 27 \beta_{3} + 114 \beta_{1}\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-3 \beta_{15} - 12 \beta_{14} + 30 \beta_{13} - 12 \beta_{10} + 69 \beta_{9} - 12 \beta_{8} - 12 \beta_{7} + 12 \beta_{6} - 42 \beta_{4} + 78 \beta_{2}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-30 \beta_{12} + 45 \beta_{10} - 45 \beta_{8} - 72 \beta_{7} - 72 \beta_{6} - 27 \beta_{5} - 63 \beta_{3} + 36 \beta_{1} - 126\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(27 \beta_{15} + 117 \beta_{14} - 54 \beta_{13} + 63 \beta_{10} + 45 \beta_{9} + 63 \beta_{8} - 18 \beta_{7} + 18 \beta_{6} + 189 \beta_{4} + 162 \beta_{2}\)\()/2\)
\(\nu^{12}\)\(=\)\(36 \beta_{12} - 108 \beta_{11} - 90 \beta_{10} + 90 \beta_{8} - 126 \beta_{3} - 108 \beta_{1} - 45\)
\(\nu^{13}\)\(=\)\((\)\(-495 \beta_{15} - 225 \beta_{14} + 18 \beta_{13} + 603 \beta_{10} + 441 \beta_{9} + 603 \beta_{8} + 18 \beta_{7} - 18 \beta_{6} + 153 \beta_{4} + 342 \beta_{2}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-270 \beta_{12} - 144 \beta_{11} - 243 \beta_{10} + 243 \beta_{8} - 432 \beta_{7} - 432 \beta_{6} - 189 \beta_{5} + 297 \beta_{3} - 2052 \beta_{1} - 2430\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-945 \beta_{15} + 540 \beta_{14} - 1242 \beta_{13} + 324 \beta_{10} - 135 \beta_{9} + 324 \beta_{8} + 972 \beta_{7} - 972 \beta_{6} + 1134 \beta_{4} - 918 \beta_{2}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1133.1
1.62181 0.608059i
−1.40917 1.00709i
1.69547 0.354107i
−0.0967785 1.72934i
0.0967785 + 1.72934i
−1.69547 + 0.354107i
1.40917 + 1.00709i
−1.62181 + 0.608059i
1.62181 + 0.608059i
−1.40917 + 1.00709i
1.69547 + 0.354107i
−0.0967785 + 1.72934i
0.0967785 1.72934i
−1.69547 0.354107i
1.40917 1.00709i
−1.62181 0.608059i
1.00000i 0 −1.00000 −3.89111 0 −2.44383 1.01375i 1.00000i 0 3.89111i
1133.2 1.00000i 0 −1.00000 −2.34936 0 1.07781 + 2.41626i 1.00000i 0 2.34936i
1133.3 1.00000i 0 −1.00000 −1.79035 0 2.28052 1.34136i 1.00000i 0 1.79035i
1133.4 1.00000i 0 −1.00000 −0.366598 0 −1.91449 + 1.82612i 1.00000i 0 0.366598i
1133.5 1.00000i 0 −1.00000 0.366598 0 −1.91449 1.82612i 1.00000i 0 0.366598i
1133.6 1.00000i 0 −1.00000 1.79035 0 2.28052 + 1.34136i 1.00000i 0 1.79035i
1133.7 1.00000i 0 −1.00000 2.34936 0 1.07781 2.41626i 1.00000i 0 2.34936i
1133.8 1.00000i 0 −1.00000 3.89111 0 −2.44383 + 1.01375i 1.00000i 0 3.89111i
1133.9 1.00000i 0 −1.00000 −3.89111 0 −2.44383 + 1.01375i 1.00000i 0 3.89111i
1133.10 1.00000i 0 −1.00000 −2.34936 0 1.07781 2.41626i 1.00000i 0 2.34936i
1133.11 1.00000i 0 −1.00000 −1.79035 0 2.28052 + 1.34136i 1.00000i 0 1.79035i
1133.12 1.00000i 0 −1.00000 −0.366598 0 −1.91449 1.82612i 1.00000i 0 0.366598i
1133.13 1.00000i 0 −1.00000 0.366598 0 −1.91449 + 1.82612i 1.00000i 0 0.366598i
1133.14 1.00000i 0 −1.00000 1.79035 0 2.28052 1.34136i 1.00000i 0 1.79035i
1133.15 1.00000i 0 −1.00000 2.34936 0 1.07781 + 2.41626i 1.00000i 0 2.34936i
1133.16 1.00000i 0 −1.00000 3.89111 0 −2.44383 1.01375i 1.00000i 0 3.89111i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1133.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.d.a 16
3.b odd 2 1 inner 1134.2.d.a 16
7.b odd 2 1 inner 1134.2.d.a 16
9.c even 3 1 126.2.m.a 16
9.c even 3 1 378.2.m.a 16
9.d odd 6 1 126.2.m.a 16
9.d odd 6 1 378.2.m.a 16
21.c even 2 1 inner 1134.2.d.a 16
36.f odd 6 1 1008.2.cc.b 16
36.f odd 6 1 3024.2.cc.b 16
36.h even 6 1 1008.2.cc.b 16
36.h even 6 1 3024.2.cc.b 16
63.g even 3 1 882.2.t.b 16
63.g even 3 1 2646.2.t.a 16
63.h even 3 1 882.2.l.a 16
63.h even 3 1 2646.2.l.b 16
63.i even 6 1 882.2.l.a 16
63.i even 6 1 2646.2.l.b 16
63.j odd 6 1 882.2.l.a 16
63.j odd 6 1 2646.2.l.b 16
63.k odd 6 1 882.2.t.b 16
63.k odd 6 1 2646.2.t.a 16
63.l odd 6 1 126.2.m.a 16
63.l odd 6 1 378.2.m.a 16
63.n odd 6 1 882.2.t.b 16
63.n odd 6 1 2646.2.t.a 16
63.o even 6 1 126.2.m.a 16
63.o even 6 1 378.2.m.a 16
63.s even 6 1 882.2.t.b 16
63.s even 6 1 2646.2.t.a 16
63.t odd 6 1 882.2.l.a 16
63.t odd 6 1 2646.2.l.b 16
252.s odd 6 1 1008.2.cc.b 16
252.s odd 6 1 3024.2.cc.b 16
252.bi even 6 1 1008.2.cc.b 16
252.bi even 6 1 3024.2.cc.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.m.a 16 9.c even 3 1
126.2.m.a 16 9.d odd 6 1
126.2.m.a 16 63.l odd 6 1
126.2.m.a 16 63.o even 6 1
378.2.m.a 16 9.c even 3 1
378.2.m.a 16 9.d odd 6 1
378.2.m.a 16 63.l odd 6 1
378.2.m.a 16 63.o even 6 1
882.2.l.a 16 63.h even 3 1
882.2.l.a 16 63.i even 6 1
882.2.l.a 16 63.j odd 6 1
882.2.l.a 16 63.t odd 6 1
882.2.t.b 16 63.g even 3 1
882.2.t.b 16 63.k odd 6 1
882.2.t.b 16 63.n odd 6 1
882.2.t.b 16 63.s even 6 1
1008.2.cc.b 16 36.f odd 6 1
1008.2.cc.b 16 36.h even 6 1
1008.2.cc.b 16 252.s odd 6 1
1008.2.cc.b 16 252.bi even 6 1
1134.2.d.a 16 1.a even 1 1 trivial
1134.2.d.a 16 3.b odd 2 1 inner
1134.2.d.a 16 7.b odd 2 1 inner
1134.2.d.a 16 21.c even 2 1 inner
2646.2.l.b 16 63.h even 3 1
2646.2.l.b 16 63.i even 6 1
2646.2.l.b 16 63.j odd 6 1
2646.2.l.b 16 63.t odd 6 1
2646.2.t.a 16 63.g even 3 1
2646.2.t.a 16 63.k odd 6 1
2646.2.t.a 16 63.n odd 6 1
2646.2.t.a 16 63.s even 6 1
3024.2.cc.b 16 36.f odd 6 1
3024.2.cc.b 16 36.h even 6 1
3024.2.cc.b 16 252.s odd 6 1
3024.2.cc.b 16 252.bi 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 \(S_{2}^{\mathrm{new}}(1134, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( T^{16} \)
$5$ \( ( 36 - 288 T^{2} + 153 T^{4} - 24 T^{6} + T^{8} )^{2} \)
$7$ \( ( 2401 + 686 T - 98 T^{2} + 14 T^{3} + 58 T^{4} + 2 T^{5} - 2 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$11$ \( ( 1296 + 3240 T^{2} + 801 T^{4} + 54 T^{6} + T^{8} )^{2} \)
$13$ \( ( 576 + 864 T^{2} + 324 T^{4} + 36 T^{6} + T^{8} )^{2} \)
$17$ \( ( 576 - 1296 T^{2} + 477 T^{4} - 42 T^{6} + T^{8} )^{2} \)
$19$ \( ( 1521 + 1854 T^{2} + 594 T^{4} + 54 T^{6} + T^{8} )^{2} \)
$23$ \( ( 443556 + 82944 T^{2} + 5229 T^{4} + 126 T^{6} + T^{8} )^{2} \)
$29$ \( ( 20736 + 10368 T^{2} + 1476 T^{4} + 72 T^{6} + T^{8} )^{2} \)
$31$ \( ( 746496 + 124416 T^{2} + 6804 T^{4} + 144 T^{6} + T^{8} )^{2} \)
$37$ \( ( 1336 - 184 T - 102 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$41$ \( ( 8573184 - 802944 T^{2} + 22869 T^{4} - 258 T^{6} + T^{8} )^{2} \)
$43$ \( ( -104 - 148 T - 39 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$47$ \( ( 1218816 - 327744 T^{2} + 16308 T^{4} - 240 T^{6} + T^{8} )^{2} \)
$53$ \( T^{16} \)
$59$ \( ( 36 - 80604 T^{2} + 17649 T^{4} - 294 T^{6} + T^{8} )^{2} \)
$61$ \( ( 1557504 + 350208 T^{2} + 16965 T^{4} + 240 T^{6} + T^{8} )^{2} \)
$67$ \( ( -908 + 1660 T - 111 T^{2} - 14 T^{3} + T^{4} )^{4} \)
$71$ \( ( 82944 + 31104 T^{2} + 2745 T^{4} + 90 T^{6} + T^{8} )^{2} \)
$73$ \( ( 1710864 + 246816 T^{2} + 12069 T^{4} + 222 T^{6} + T^{8} )^{2} \)
$79$ \( ( -1202 + 778 T - 129 T^{2} - 2 T^{3} + T^{4} )^{4} \)
$83$ \( ( 580617216 - 17592192 T^{2} + 174996 T^{4} - 708 T^{6} + T^{8} )^{2} \)
$89$ \( ( 186624 - 155520 T^{2} + 12960 T^{4} - 216 T^{6} + T^{8} )^{2} \)
$97$ \( ( 67174416 + 5816160 T^{2} + 139077 T^{4} + 702 T^{6} + T^{8} )^{2} \)
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