Properties

Label 216.3.b.a
Level $216$
Weight $3$
Character orbit 216.b
Analytic conductor $5.886$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + x^{14} - 8 x^{12} + 4 x^{10} + 160 x^{8} + 64 x^{6} - 2048 x^{4} + 4096 x^{2} + 65536\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{10} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{7} q^{5} -\beta_{6} q^{7} + \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{7} q^{5} -\beta_{6} q^{7} + \beta_{3} q^{8} + ( -1 - \beta_{10} ) q^{10} + ( \beta_{3} - \beta_{13} ) q^{11} + ( \beta_{2} - \beta_{9} ) q^{13} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{11} + \beta_{15} ) q^{14} + ( 2 + \beta_{5} - \beta_{9} ) q^{16} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{12} ) q^{17} + ( 2 + \beta_{5} + \beta_{8} - \beta_{10} ) q^{19} + ( -\beta_{1} - \beta_{4} + \beta_{7} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{20} + ( -2 - \beta_{6} - \beta_{9} - \beta_{14} ) q^{22} + ( \beta_{3} - \beta_{4} + 2 \beta_{11} + \beta_{13} - \beta_{15} ) q^{23} + ( -5 + \beta_{2} + \beta_{6} - \beta_{8} - 2 \beta_{10} - \beta_{14} ) q^{25} + ( -2 \beta_{4} + 2 \beta_{13} - \beta_{15} ) q^{26} + ( 7 + \beta_{2} - 3 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{14} ) q^{28} + ( 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{11} + \beta_{12} - \beta_{15} ) q^{29} + ( 1 + 4 \beta_{2} + \beta_{5} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{14} ) q^{31} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 4 \beta_{7} + 2 \beta_{13} - \beta_{15} ) q^{32} + ( 5 + 2 \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{14} ) q^{34} + ( -6 \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{12} - 2 \beta_{15} ) q^{35} + ( -\beta_{2} - 2 \beta_{5} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{4} + 4 \beta_{7} + 2 \beta_{12} + \beta_{15} ) q^{38} + ( -1 + \beta_{5} - 5 \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{14} ) q^{40} + ( -7 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{41} + ( 8 + 6 \beta_{2} - 4 \beta_{6} + 4 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} ) q^{43} + ( -\beta_{1} - 2 \beta_{3} - 3 \beta_{4} + \beta_{7} - \beta_{11} - \beta_{12} + 3 \beta_{13} + 2 \beta_{15} ) q^{44} + ( 3 - 4 \beta_{2} + 7 \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{14} ) q^{46} + ( -2 \beta_{1} - 4 \beta_{3} - 4 \beta_{7} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{47} + ( -5 - 7 \beta_{2} - 2 \beta_{5} + 3 \beta_{6} - 5 \beta_{8} - 2 \beta_{9} + \beta_{14} ) q^{49} + ( -2 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} + 6 \beta_{7} - 2 \beta_{11} - 2 \beta_{13} + 3 \beta_{15} ) q^{50} + ( -10 + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{14} ) q^{52} + ( -11 \beta_{1} + 3 \beta_{3} - \beta_{4} - \beta_{7} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{53} + ( -2 \beta_{2} + 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{55} + ( 2 \beta_{1} - \beta_{3} + 6 \beta_{4} - 6 \beta_{7} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{56} + ( -11 - 2 \beta_{2} - 2 \beta_{5} + 5 \beta_{6} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{14} ) q^{58} + ( 10 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{12} + 2 \beta_{15} ) q^{59} + ( -4 - 12 \beta_{2} + 4 \beta_{6} + 4 \beta_{8} - 4 \beta_{14} ) q^{61} + ( -\beta_{1} + 3 \beta_{3} + 7 \beta_{4} + 3 \beta_{7} - \beta_{11} - 2 \beta_{12} - 4 \beta_{15} ) q^{62} + ( -4 + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{9} + 4 \beta_{10} + 2 \beta_{14} ) q^{64} + ( -\beta_{1} + 5 \beta_{3} + 3 \beta_{4} + \beta_{12} - 4 \beta_{13} ) q^{65} + ( 8 - 10 \beta_{2} + 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{14} ) q^{67} + ( 4 \beta_{1} + 2 \beta_{3} - 8 \beta_{4} - 8 \beta_{7} - 4 \beta_{15} ) q^{68} + ( -23 - 4 \beta_{2} - 4 \beta_{5} + 2 \beta_{6} - 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{14} ) q^{70} + ( 18 \beta_{1} + 5 \beta_{3} - \beta_{4} - 4 \beta_{7} + 2 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} - 7 \beta_{15} ) q^{71} + ( -10 + \beta_{2} - 4 \beta_{5} + \beta_{6} - 5 \beta_{8} + 2 \beta_{10} - \beta_{14} ) q^{73} + ( -10 \beta_{4} - 8 \beta_{7} + 4 \beta_{12} - 2 \beta_{13} + 3 \beta_{15} ) q^{74} + ( 10 + 2 \beta_{5} - 2 \beta_{6} + 6 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} - 2 \beta_{14} ) q^{76} + ( 16 \beta_{1} + \beta_{7} - 4 \beta_{15} ) q^{77} + ( -1 - 10 \beta_{2} + \beta_{5} - 5 \beta_{6} + 7 \beta_{9} + \beta_{10} - \beta_{14} ) q^{79} + ( -4 \beta_{1} - 3 \beta_{3} + 12 \beta_{4} - 2 \beta_{7} - 2 \beta_{11} - 2 \beta_{12} + 7 \beta_{15} ) q^{80} + ( -33 - 6 \beta_{2} - 2 \beta_{5} - \beta_{6} - 3 \beta_{8} - 3 \beta_{9} - \beta_{14} ) q^{82} + ( 6 \beta_{1} + 3 \beta_{3} + 2 \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{83} + ( 4 + 13 \beta_{2} - 4 \beta_{5} + 12 \beta_{6} - \beta_{9} - 4 \beta_{10} + 4 \beta_{14} ) q^{85} + ( 4 \beta_{1} + 4 \beta_{3} + 8 \beta_{4} - 12 \beta_{7} + 4 \beta_{11} + 4 \beta_{13} + 2 \beta_{15} ) q^{86} + ( 37 + 2 \beta_{2} + \beta_{5} + \beta_{6} - 5 \beta_{8} + 2 \beta_{10} + 5 \beta_{14} ) q^{88} + ( 24 \beta_{1} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{13} + 6 \beta_{15} ) q^{89} + ( -6 + 6 \beta_{2} + \beta_{5} - 4 \beta_{6} + 5 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{91} + ( 12 \beta_{1} + 2 \beta_{3} - 12 \beta_{4} + 12 \beta_{7} + 4 \beta_{11} - 4 \beta_{15} ) q^{92} + ( -4 \beta_{2} - 4 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} + 8 \beta_{9} - 4 \beta_{10} - 4 \beta_{14} ) q^{94} + ( -32 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} - 6 \beta_{11} - 3 \beta_{13} + 11 \beta_{15} ) q^{95} + ( -13 + 10 \beta_{2} - 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{14} ) q^{97} + ( -4 \beta_{1} - 6 \beta_{3} - 8 \beta_{4} - 2 \beta_{7} - 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} - 7 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 2q^{4} + O(q^{10}) \) \( 16q - 2q^{4} - 18q^{10} + 34q^{16} + 32q^{19} - 22q^{22} - 80q^{25} + 102q^{28} + 68q^{34} - 6q^{40} + 128q^{43} + 60q^{46} - 80q^{49} - 180q^{52} - 156q^{58} - 74q^{64} + 128q^{67} - 378q^{70} - 160q^{73} + 188q^{76} - 508q^{82} + 542q^{88} - 96q^{91} + 24q^{94} - 208q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + x^{14} - 8 x^{12} + 4 x^{10} + 160 x^{8} + 64 x^{6} - 2048 x^{4} + 4096 x^{2} + 65536\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} - 9 \nu^{13} - 196 \nu^{9} + 576 \nu^{7} + 1728 \nu^{5} + 512 \nu^{3} \)\()/32768\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{12} + 3 \nu^{10} - 4 \nu^{8} + 12 \nu^{6} + 304 \nu^{4} + 256 \nu^{2} \)\()/512\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{14} + \nu^{12} + 24 \nu^{10} + 164 \nu^{8} + 544 \nu^{6} - 320 \nu^{4} + 3584 \nu^{2} + 16384 \)\()/8192\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{15} - 7 \nu^{13} + 48 \nu^{11} + 132 \nu^{9} - 384 \nu^{7} + 3136 \nu^{5} + 3584 \nu^{3} \)\()/32768\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{14} - \nu^{12} - 24 \nu^{10} + 92 \nu^{8} + 224 \nu^{6} - 192 \nu^{4} + 512 \nu^{2} + 4096 \)\()/4096\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{12} + 3 \nu^{10} - 4 \nu^{8} + 12 \nu^{6} - 208 \nu^{4} + 256 \nu^{2} + 1024 \)\()/512\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{14} + 7 \nu^{12} + 16 \nu^{10} - 68 \nu^{8} + 384 \nu^{6} + 704 \nu^{4} - 512 \nu^{2} - 12288 \)\()/4096\)
\(\beta_{11}\)\(=\)\((\)\( -3 \nu^{15} - 3 \nu^{13} + 56 \nu^{11} + 148 \nu^{9} + 928 \nu^{7} - 1600 \nu^{5} - 2560 \nu^{3} \)\()/16384\)
\(\beta_{12}\)\(=\)\((\)\( 3 \nu^{15} - 37 \nu^{13} - 192 \nu^{11} + 460 \nu^{9} + 576 \nu^{7} - 1600 \nu^{5} - 3584 \nu^{3} + 81920 \nu \)\()/32768\)
\(\beta_{13}\)\(=\)\((\)\( -5 \nu^{15} + 19 \nu^{13} - 64 \nu^{11} - 84 \nu^{9} - 448 \nu^{7} + 8128 \nu^{5} + 16896 \nu^{3} - 49152 \nu \)\()/32768\)
\(\beta_{14}\)\(=\)\((\)\( 5 \nu^{14} - 11 \nu^{12} - 88 \nu^{10} - 12 \nu^{8} + 1376 \nu^{6} - 2880 \nu^{4} - 14848 \nu^{2} + 32768 \)\()/8192\)
\(\beta_{15}\)\(=\)\((\)\( -\nu^{15} - \nu^{13} + 8 \nu^{11} - 4 \nu^{9} - 160 \nu^{7} - 64 \nu^{5} + 2048 \nu^{3} - 4096 \nu \)\()/4096\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{5} + 2\)
\(\nu^{5}\)\(=\)\(-\beta_{15} + 2 \beta_{13} + 4 \beta_{7} + 2 \beta_{4} - \beta_{3} + 2 \beta_{1}\)
\(\nu^{6}\)\(=\)\(2 \beta_{14} + 4 \beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{6} + \beta_{5} + 2 \beta_{2} - 4\)
\(\nu^{7}\)\(=\)\(-9 \beta_{15} + 2 \beta_{13} + 8 \beta_{11} - 4 \beta_{7} + 10 \beta_{4} + 5 \beta_{3} - 6 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-6 \beta_{14} - 12 \beta_{10} - 5 \beta_{9} + 10 \beta_{8} + 26 \beta_{6} - \beta_{5} - 22 \beta_{2} - 64\)
\(\nu^{9}\)\(=\)\(-7 \beta_{15} + 14 \beta_{13} + 16 \beta_{12} + 24 \beta_{11} + 36 \beta_{7} - 74 \beta_{4} - \beta_{3} - 26 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-26 \beta_{14} + 12 \beta_{10} + 33 \beta_{9} - 42 \beta_{8} + 70 \beta_{6} + 5 \beta_{5} - 90 \beta_{2} - 24\)
\(\nu^{11}\)\(=\)\(3 \beta_{15} - 70 \beta_{13} - 80 \beta_{12} + 40 \beta_{11} + 172 \beta_{7} + 18 \beta_{4} + 13 \beta_{3} + 98 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-30 \beta_{14} + 132 \beta_{10} - 173 \beta_{9} - 142 \beta_{8} + 130 \beta_{6} - 177 \beta_{5} + 98 \beta_{2} + 744\)
\(\nu^{13}\)\(=\)\(-375 \beta_{15} + 366 \beta_{13} - 304 \beta_{12} + 120 \beta_{11} - 508 \beta_{7} - 970 \beta_{4} + 55 \beta_{3} + 934 \beta_{1}\)
\(\nu^{14}\)\(=\)\(550 \beta_{14} - 628 \beta_{10} - 663 \beta_{9} - 1578 \beta_{8} + 1030 \beta_{6} - 3 \beta_{5} + 998 \beta_{2} - 3240\)
\(\nu^{15}\)\(=\)\(-2165 \beta_{15} - 1430 \beta_{13} - 400 \beta_{12} - 1176 \beta_{11} + 2124 \beta_{7} - 318 \beta_{4} + 1365 \beta_{3} - 3310 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−1.95589 0.417734i
−1.95589 + 0.417734i
−1.71413 1.03041i
−1.71413 + 1.03041i
−0.941181 1.76470i
−0.941181 + 1.76470i
−0.316912 1.97473i
−0.316912 + 1.97473i
0.316912 1.97473i
0.316912 + 1.97473i
0.941181 1.76470i
0.941181 + 1.76470i
1.71413 1.03041i
1.71413 + 1.03041i
1.95589 0.417734i
1.95589 + 0.417734i
−1.95589 0.417734i 0 3.65100 + 1.63408i 5.14075i 0 12.6525i −6.45833 4.72122i 0 2.14746 10.0547i
163.2 −1.95589 + 0.417734i 0 3.65100 1.63408i 5.14075i 0 12.6525i −6.45833 + 4.72122i 0 2.14746 + 10.0547i
163.3 −1.71413 1.03041i 0 1.87649 + 3.53253i 0.169019i 0 4.44165i 0.423412 7.98879i 0 −0.174160 + 0.289721i
163.4 −1.71413 + 1.03041i 0 1.87649 3.53253i 0.169019i 0 4.44165i 0.423412 + 7.98879i 0 −0.174160 0.289721i
163.5 −0.941181 1.76470i 0 −2.22836 + 3.32181i 8.60639i 0 4.81948i 7.95930 + 0.805966i 0 −15.1877 + 8.10017i
163.6 −0.941181 + 1.76470i 0 −2.22836 3.32181i 8.60639i 0 4.81948i 7.95930 0.805966i 0 −15.1877 8.10017i
163.7 −0.316912 1.97473i 0 −3.79913 + 1.25163i 4.41296i 0 3.59985i 3.67563 + 7.10561i 0 8.71442 1.39852i
163.8 −0.316912 + 1.97473i 0 −3.79913 1.25163i 4.41296i 0 3.59985i 3.67563 7.10561i 0 8.71442 + 1.39852i
163.9 0.316912 1.97473i 0 −3.79913 1.25163i 4.41296i 0 3.59985i −3.67563 + 7.10561i 0 8.71442 + 1.39852i
163.10 0.316912 + 1.97473i 0 −3.79913 + 1.25163i 4.41296i 0 3.59985i −3.67563 7.10561i 0 8.71442 1.39852i
163.11 0.941181 1.76470i 0 −2.22836 3.32181i 8.60639i 0 4.81948i −7.95930 + 0.805966i 0 −15.1877 8.10017i
163.12 0.941181 + 1.76470i 0 −2.22836 + 3.32181i 8.60639i 0 4.81948i −7.95930 0.805966i 0 −15.1877 + 8.10017i
163.13 1.71413 1.03041i 0 1.87649 3.53253i 0.169019i 0 4.44165i −0.423412 7.98879i 0 −0.174160 0.289721i
163.14 1.71413 + 1.03041i 0 1.87649 + 3.53253i 0.169019i 0 4.44165i −0.423412 + 7.98879i 0 −0.174160 + 0.289721i
163.15 1.95589 0.417734i 0 3.65100 1.63408i 5.14075i 0 12.6525i 6.45833 4.72122i 0 2.14746 + 10.0547i
163.16 1.95589 + 0.417734i 0 3.65100 + 1.63408i 5.14075i 0 12.6525i 6.45833 + 4.72122i 0 2.14746 10.0547i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.b.a 16
3.b odd 2 1 inner 216.3.b.a 16
4.b odd 2 1 864.3.b.a 16
8.b even 2 1 864.3.b.a 16
8.d odd 2 1 inner 216.3.b.a 16
12.b even 2 1 864.3.b.a 16
24.f even 2 1 inner 216.3.b.a 16
24.h odd 2 1 864.3.b.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.b.a 16 1.a even 1 1 trivial
216.3.b.a 16 3.b odd 2 1 inner
216.3.b.a 16 8.d odd 2 1 inner
216.3.b.a 16 24.f even 2 1 inner
864.3.b.a 16 4.b odd 2 1
864.3.b.a 16 8.b even 2 1
864.3.b.a 16 12.b even 2 1
864.3.b.a 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 120 T_{5}^{6} + 3918 T_{5}^{4} + 38232 T_{5}^{2} + 1089 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 65536 + 4096 T^{2} - 2048 T^{4} + 64 T^{6} + 160 T^{8} + 4 T^{10} - 8 T^{12} + T^{14} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 1089 + 38232 T^{2} + 3918 T^{4} + 120 T^{6} + T^{8} )^{2} \)
$7$ \( ( 950625 + 168408 T^{2} + 9966 T^{4} + 216 T^{6} + T^{8} )^{2} \)
$11$ \( ( 4060225 - 2024572 T^{2} + 60486 T^{4} - 508 T^{6} + T^{8} )^{2} \)
$13$ \( ( 58982400 + 4165632 T^{2} + 90768 T^{4} + 600 T^{6} + T^{8} )^{2} \)
$17$ \( ( 1435500544 - 43393024 T^{2} + 409104 T^{4} - 1336 T^{6} + T^{8} )^{2} \)
$19$ \( ( 129088 + 2080 T - 756 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$23$ \( ( 49861103616 + 499069440 T^{2} + 1728912 T^{4} + 2376 T^{6} + T^{8} )^{2} \)
$29$ \( ( 197136000000 + 1506675456 T^{2} + 3887184 T^{4} + 3768 T^{6} + T^{8} )^{2} \)
$31$ \( ( 69463346481 + 806493672 T^{2} + 2821278 T^{4} + 3144 T^{6} + T^{8} )^{2} \)
$37$ \( ( 4052652134400 + 13575403008 T^{2} + 15643152 T^{4} + 7128 T^{6} + T^{8} )^{2} \)
$41$ \( ( 84239257600 - 1652173312 T^{2} + 7058832 T^{4} - 5368 T^{6} + T^{8} )^{2} \)
$43$ \( ( 5478400 + 174208 T - 6096 T^{2} - 32 T^{3} + T^{4} )^{4} \)
$47$ \( ( 7614420811776 + 28040675328 T^{2} + 31084032 T^{4} + 10176 T^{6} + T^{8} )^{2} \)
$53$ \( ( 269102600001 + 4463394840 T^{2} + 17537742 T^{4} + 10296 T^{6} + T^{8} )^{2} \)
$59$ \( ( 12687769600 - 1801781248 T^{2} + 15159552 T^{4} - 8416 T^{6} + T^{8} )^{2} \)
$61$ \( ( 872632396283904 + 798744379392 T^{2} + 228200448 T^{4} + 25728 T^{6} + T^{8} )^{2} \)
$67$ \( ( 8139520 + 654592 T - 13440 T^{2} - 32 T^{3} + T^{4} )^{4} \)
$71$ \( ( 90326016000000 + 505103579136 T^{2} + 199530000 T^{4} + 25416 T^{6} + T^{8} )^{2} \)
$73$ \( ( 15407905 - 288728 T - 10962 T^{2} + 40 T^{3} + T^{4} )^{4} \)
$79$ \( ( 6249517207916544 + 3096211060224 T^{2} + 539267088 T^{4} + 39000 T^{6} + T^{8} )^{2} \)
$83$ \( ( 5715511201 - 752382172 T^{2} + 10611558 T^{4} - 6940 T^{6} + T^{8} )^{2} \)
$89$ \( ( 338984068710400 - 457230696448 T^{2} + 165122304 T^{4} - 22240 T^{6} + T^{8} )^{2} \)
$97$ \( ( 4535185 - 786092 T - 12810 T^{2} + 52 T^{3} + T^{4} )^{4} \)
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