Properties

Label 1380.2.i.a
Level $1380$
Weight $2$
Character orbit 1380.i
Analytic conductor $11.019$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{14} - 4 x^{13} - 2 x^{12} + 12 x^{11} + 17 x^{10} + 16 x^{9} - 110 x^{8} + 48 x^{7} + 153 x^{6} + 324 x^{5} - 162 x^{4} - 972 x^{3} - 729 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q -\beta_{7} q^{3} - q^{5} -\beta_{8} q^{7} -\beta_{6} q^{9} +O(q^{10})\) \( q -\beta_{7} q^{3} - q^{5} -\beta_{8} q^{7} -\beta_{6} q^{9} -\beta_{9} q^{11} -\beta_{12} q^{13} + \beta_{7} q^{15} -\beta_{5} q^{17} + \beta_{14} q^{19} + ( -\beta_{1} - \beta_{13} ) q^{21} + ( 1 + \beta_{6} + \beta_{11} - \beta_{13} ) q^{23} + q^{25} + ( -1 + \beta_{4} ) q^{27} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{29} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{12} ) q^{31} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{11} ) q^{33} + \beta_{8} q^{35} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{37} + ( -1 + \beta_{2} + \beta_{5} - \beta_{9} - \beta_{10} - \beta_{14} ) q^{39} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{41} + ( -\beta_{2} - \beta_{6} - \beta_{8} - \beta_{10} ) q^{43} + \beta_{6} q^{45} + ( \beta_{1} - \beta_{7} + \beta_{8} - \beta_{11} - \beta_{14} ) q^{47} + ( 1 - \beta_{1} + \beta_{5} - \beta_{7} - \beta_{9} - \beta_{13} + \beta_{15} ) q^{49} + ( \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{51} + ( \beta_{2} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{12} ) q^{53} + \beta_{9} q^{55} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{57} + ( \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{59} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{8} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{63} + \beta_{12} q^{65} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{13} + \beta_{15} ) q^{67} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{69} + ( \beta_{8} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{71} + ( -\beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{73} -\beta_{7} q^{75} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{77} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{15} ) q^{79} + ( -\beta_{1} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{81} + ( \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{9} - \beta_{13} + \beta_{15} ) q^{83} + \beta_{5} q^{85} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{87} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{89} + ( -\beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{91} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} + 2 \beta_{14} ) q^{93} -\beta_{14} q^{95} + ( -2 \beta_{1} - 3 \beta_{2} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{97} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{5} + 2q^{9} + O(q^{10}) \) \( 16q - 16q^{5} + 2q^{9} - 4q^{21} + 10q^{23} + 16q^{25} - 12q^{27} + 4q^{31} + 2q^{33} - 14q^{39} - 2q^{45} + 8q^{49} - 2q^{51} + 4q^{53} - 2q^{57} + 30q^{63} - 4q^{73} + 10q^{81} - 4q^{83} - 10q^{87} - 28q^{89} - 10q^{93} - 26q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{14} - 4 x^{13} - 2 x^{12} + 12 x^{11} + 17 x^{10} + 16 x^{9} - 110 x^{8} + 48 x^{7} + 153 x^{6} + 324 x^{5} - 162 x^{4} - 972 x^{3} - 729 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 1 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{15} + 8 \nu^{13} - 4 \nu^{12} - 11 \nu^{11} - 24 \nu^{10} - \nu^{9} + 124 \nu^{8} + 43 \nu^{7} + 192 \nu^{6} - 837 \nu^{5} + 756 \nu^{4} + 1215 \nu^{3} + 1944 \nu^{2} - 2187 \nu - 6561 \)\()/2187\)
\(\beta_{5}\)\(=\)\((\)\( 8 \nu^{15} + 9 \nu^{14} + 64 \nu^{13} + 13 \nu^{12} - 124 \nu^{11} - 21 \nu^{10} - 116 \nu^{9} + 794 \nu^{8} + 164 \nu^{7} - 1209 \nu^{6} - 2484 \nu^{5} + 3429 \nu^{4} + 10368 \nu^{3} - 2187 \nu^{2} - 5832 \nu - 43740 \)\()/17496\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{14} - \nu^{12} - 4 \nu^{11} - 2 \nu^{10} + 12 \nu^{9} + 17 \nu^{8} + 16 \nu^{7} - 110 \nu^{6} + 48 \nu^{5} + 153 \nu^{4} + 324 \nu^{3} - 162 \nu^{2} - 972 \nu - 729 \)\()/729\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} + \nu^{13} + 4 \nu^{12} + 2 \nu^{11} - 12 \nu^{10} - 17 \nu^{9} - 16 \nu^{8} + 110 \nu^{7} - 48 \nu^{6} - 153 \nu^{5} - 324 \nu^{4} + 162 \nu^{3} + 972 \nu^{2} + 729 \nu \)\()/2187\)
\(\beta_{8}\)\(=\)\((\)\( 8 \nu^{15} - 27 \nu^{14} - 8 \nu^{13} - 5 \nu^{12} + 92 \nu^{11} - 93 \nu^{10} - 188 \nu^{9} - 88 \nu^{8} - 340 \nu^{7} + 1653 \nu^{6} - 2988 \nu^{5} - 3483 \nu^{4} - 5184 \nu^{3} + 8019 \nu^{2} - 17496 \)\()/17496\)
\(\beta_{9}\)\(=\)\((\)\( -2 \nu^{15} - 12 \nu^{14} + 11 \nu^{13} + 20 \nu^{12} - 38 \nu^{11} - 36 \nu^{10} - 115 \nu^{9} + 196 \nu^{8} + 343 \nu^{7} + 396 \nu^{6} - 1062 \nu^{5} - 3348 \nu^{4} + 2349 \nu^{3} + 2916 \nu^{2} + 1458 \nu - 8748 \)\()/8748\)
\(\beta_{10}\)\(=\)\((\)\( 8 \nu^{15} + 24 \nu^{14} + 64 \nu^{13} - 191 \nu^{12} - 184 \nu^{11} - 105 \nu^{10} + 820 \nu^{9} - 517 \nu^{8} - 892 \nu^{7} - 1212 \nu^{6} + 1044 \nu^{5} + 18009 \nu^{4} + 2268 \nu^{3} - 17739 \nu^{2} - 75816 \nu + 54675 \)\()/17496\)
\(\beta_{11}\)\(=\)\((\)\( 16 \nu^{15} - 27 \nu^{14} - 16 \nu^{13} + 71 \nu^{12} + 76 \nu^{11} - 105 \nu^{10} - 484 \nu^{9} - 176 \nu^{8} + 76 \nu^{7} + 3873 \nu^{6} - 36 \nu^{5} - 4023 \nu^{4} - 1296 \nu^{3} + 8019 \nu^{2} + 11664 \nu - 34992 \)\()/17496\)
\(\beta_{12}\)\(=\)\((\)\( -7 \nu^{15} + 70 \nu^{13} + 136 \nu^{12} - 130 \nu^{11} - 444 \nu^{10} + 133 \nu^{9} + 752 \nu^{8} + 383 \nu^{7} - 1380 \nu^{6} - 4734 \nu^{5} - 3672 \nu^{4} + 24138 \nu^{3} + 25272 \nu^{2} - 38637 \nu - 52488 \)\()/17496\)
\(\beta_{13}\)\(=\)\((\)\( -8 \nu^{15} - 24 \nu^{14} + 89 \nu^{13} + 56 \nu^{12} + 31 \nu^{11} - 372 \nu^{10} + 143 \nu^{9} + 436 \nu^{8} + 1144 \nu^{7} + 636 \nu^{6} - 6183 \nu^{5} + 6372 \nu^{4} + 11745 \nu^{3} + 23328 \nu^{2} - 35721 \nu \)\()/17496\)
\(\beta_{14}\)\(=\)\((\)\( -10 \nu^{15} + 15 \nu^{14} + \nu^{13} + 25 \nu^{12} - 112 \nu^{11} + 129 \nu^{10} + 109 \nu^{9} + 68 \nu^{8} + 377 \nu^{7} - 1545 \nu^{6} - 468 \nu^{5} - 729 \nu^{4} + 9153 \nu^{3} - 10935 \nu^{2} + 4374 \nu + 8748 \)\()/8748\)
\(\beta_{15}\)\(=\)\((\)\( 19 \nu^{15} - 19 \nu^{13} - 76 \nu^{12} + 205 \nu^{11} + 228 \nu^{10} + 80 \nu^{9} - 668 \nu^{8} - 389 \nu^{7} + 3828 \nu^{6} + 4851 \nu^{5} + 1296 \nu^{4} - 14499 \nu^{3} + 1944 \nu^{2} + 23328 \nu + 52488 \)\()/17496\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{15} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{2} + \beta_{1}\)
\(\nu^{5}\)\(=\)\(-2 \beta_{14} + 2 \beta_{9} - 4 \beta_{8} - \beta_{7} - \beta_{4} - \beta_{3} - 2\)
\(\nu^{6}\)\(=\)\(\beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} - 5 \beta_{8} - \beta_{7} - 3 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} - 3 \beta_{2} - 3 \beta_{1} - 3\)
\(\nu^{7}\)\(=\)\(4 \beta_{15} + 4 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} + 4 \beta_{11} + 4 \beta_{9} + 4 \beta_{8} + 8 \beta_{7} + 12 \beta_{6} - \beta_{4} + 3 \beta_{3} + \beta_{1} - 6\)
\(\nu^{8}\)\(=\)\(-\beta_{15} - 8 \beta_{14} + \beta_{13} - \beta_{12} - 7 \beta_{11} - 8 \beta_{10} + 7 \beta_{9} - 13 \beta_{8} - 17 \beta_{7} + 5 \beta_{6} - 4 \beta_{5} + 8 \beta_{4} - 10 \beta_{2} - 7 \beta_{1} + 32\)
\(\nu^{9}\)\(=\)\(20 \beta_{15} + 18 \beta_{14} + 12 \beta_{13} + 12 \beta_{12} - 4 \beta_{11} + 6 \beta_{9} + 24 \beta_{8} - 28 \beta_{7} + 24 \beta_{6} - 16 \beta_{5} - 5 \beta_{4} + 18 \beta_{3} - 4 \beta_{2} + 48 \beta_{1} - 69\)
\(\nu^{10}\)\(=\)\(16 \beta_{15} + 32 \beta_{14} + 16 \beta_{13} - 16 \beta_{12} - 6 \beta_{11} - 8 \beta_{10} + 22 \beta_{8} - 30 \beta_{7} - 7 \beta_{6} + 24 \beta_{5} - 12 \beta_{4} + 4 \beta_{3} + 50 \beta_{2} - 74 \beta_{1} - 80\)
\(\nu^{11}\)\(=\)\(28 \beta_{15} - 26 \beta_{14} + 20 \beta_{13} + 4 \beta_{12} - 52 \beta_{11} - 32 \beta_{10} - 42 \beta_{9} + 16 \beta_{8} + 19 \beta_{7} - 4 \beta_{6} + 16 \beta_{5} - 6 \beta_{4} - 42 \beta_{3} - 72 \beta_{2} - 84 \beta_{1} - 16\)
\(\nu^{12}\)\(=\)\(72 \beta_{14} + 64 \beta_{13} + 32 \beta_{12} + 90 \beta_{11} - 24 \beta_{10} - 88 \beta_{9} + 2 \beta_{8} - 106 \beta_{7} + 6 \beta_{6} + 20 \beta_{5} - 56 \beta_{4} + 104 \beta_{3} + 22 \beta_{2} + 10 \beta_{1} - 23\)
\(\nu^{13}\)\(=\)\(108 \beta_{15} - 54 \beta_{14} - 4 \beta_{13} + 76 \beta_{12} - 132 \beta_{11} + 32 \beta_{10} + 70 \beta_{9} - 72 \beta_{8} + 108 \beta_{7} - 200 \beta_{6} + 192 \beta_{5} - 74 \beta_{4} + 2 \beta_{3} - 84 \beta_{2} - 55 \beta_{1} - 352\)
\(\nu^{14}\)\(=\)\(120 \beta_{15} - 16 \beta_{14} - 312 \beta_{13} + 216 \beta_{12} - 70 \beta_{11} - 32 \beta_{10} - 344 \beta_{9} - 226 \beta_{8} + 498 \beta_{7} - 190 \beta_{6} - 48 \beta_{5} + 324 \beta_{4} + 52 \beta_{3} - 269 \beta_{2} - 458 \beta_{1} + 304\)
\(\nu^{15}\)\(=\)\(364 \beta_{15} + 366 \beta_{14} + 44 \beta_{13} + 52 \beta_{12} + 444 \beta_{11} + 96 \beta_{10} - 170 \beta_{9} + 1120 \beta_{8} + 48 \beta_{7} + 876 \beta_{6} + 544 \beta_{5} - 358 \beta_{4} + 301 \beta_{3} + 280 \beta_{2} + 660 \beta_{1} + 1097\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1241.1
1.68259 0.410981i
1.68259 + 0.410981i
1.58090 0.707640i
1.58090 + 0.707640i
1.12976 1.31287i
1.12976 + 1.31287i
−0.0831550 1.73005i
−0.0831550 + 1.73005i
−0.208374 1.71947i
−0.208374 + 1.71947i
−1.04974 1.37769i
−1.04974 + 1.37769i
−1.32140 1.11978i
−1.32140 + 1.11978i
−1.73058 0.0713366i
−1.73058 + 0.0713366i
0 −1.68259 0.410981i 0 −1.00000 0 4.25920i 0 2.66219 + 1.38302i 0
1241.2 0 −1.68259 + 0.410981i 0 −1.00000 0 4.25920i 0 2.66219 1.38302i 0
1241.3 0 −1.58090 0.707640i 0 −1.00000 0 1.33803i 0 1.99849 + 2.23742i 0
1241.4 0 −1.58090 + 0.707640i 0 −1.00000 0 1.33803i 0 1.99849 2.23742i 0
1241.5 0 −1.12976 1.31287i 0 −1.00000 0 1.67773i 0 −0.447268 + 2.96647i 0
1241.6 0 −1.12976 + 1.31287i 0 −1.00000 0 1.67773i 0 −0.447268 2.96647i 0
1241.7 0 0.0831550 1.73005i 0 −1.00000 0 4.02092i 0 −2.98617 0.287725i 0
1241.8 0 0.0831550 + 1.73005i 0 −1.00000 0 4.02092i 0 −2.98617 + 0.287725i 0
1241.9 0 0.208374 1.71947i 0 −1.00000 0 0.187545i 0 −2.91316 0.716587i 0
1241.10 0 0.208374 + 1.71947i 0 −1.00000 0 0.187545i 0 −2.91316 + 0.716587i 0
1241.11 0 1.04974 1.37769i 0 −1.00000 0 2.10004i 0 −0.796081 2.89245i 0
1241.12 0 1.04974 + 1.37769i 0 −1.00000 0 2.10004i 0 −0.796081 + 2.89245i 0
1241.13 0 1.32140 1.11978i 0 −1.00000 0 1.85076i 0 0.492177 2.95935i 0
1241.14 0 1.32140 + 1.11978i 0 −1.00000 0 1.85076i 0 0.492177 + 2.95935i 0
1241.15 0 1.73058 0.0713366i 0 −1.00000 0 2.28380i 0 2.98982 0.246908i 0
1241.16 0 1.73058 + 0.0713366i 0 −1.00000 0 2.28380i 0 2.98982 + 0.246908i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1241.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.i.a 16
3.b odd 2 1 1380.2.i.b yes 16
23.b odd 2 1 1380.2.i.b yes 16
69.c even 2 1 inner 1380.2.i.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.i.a 16 1.a even 1 1 trivial
1380.2.i.a 16 69.c even 2 1 inner
1380.2.i.b yes 16 3.b odd 2 1
1380.2.i.b yes 16 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} - 44 T_{11}^{6} + 50 T_{11}^{5} + 486 T_{11}^{4} - 924 T_{11}^{3} - 816 T_{11}^{2} + 2368 T_{11} - 1024 \) acting on \(S_{2}^{\mathrm{new}}(1380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 6561 - 729 T^{2} + 972 T^{3} - 162 T^{4} - 324 T^{5} + 153 T^{6} - 48 T^{7} - 110 T^{8} - 16 T^{9} + 17 T^{10} - 12 T^{11} - 2 T^{12} + 4 T^{13} - T^{14} + T^{16} \)
$5$ \( ( 1 + T )^{16} \)
$7$ \( 4096 + 123584 T^{2} + 207712 T^{4} + 142280 T^{6} + 50313 T^{8} + 9772 T^{10} + 1022 T^{12} + 52 T^{14} + T^{16} \)
$11$ \( ( -1024 + 2368 T - 816 T^{2} - 924 T^{3} + 486 T^{4} + 50 T^{5} - 44 T^{6} + T^{8} )^{2} \)
$13$ \( ( 10936 - 1544 T - 7342 T^{2} + 566 T^{3} + 1256 T^{4} - 30 T^{5} - 67 T^{6} + T^{8} )^{2} \)
$17$ \( ( -4592 - 3872 T + 4840 T^{2} + 5312 T^{3} + 1245 T^{4} - 150 T^{5} - 70 T^{6} + T^{8} )^{2} \)
$19$ \( 2166784 + 27048704 T^{2} + 31794560 T^{4} + 12909600 T^{6} + 2061156 T^{8} + 159708 T^{10} + 6416 T^{12} + 128 T^{14} + T^{16} \)
$23$ \( 78310985281 - 34048254470 T + 11250727564 T^{2} - 3771696998 T^{3} + 1098096084 T^{4} - 282882750 T^{5} + 71023540 T^{6} - 16562622 T^{7} + 3468694 T^{8} - 720114 T^{9} + 134260 T^{10} - 23250 T^{11} + 3924 T^{12} - 586 T^{13} + 76 T^{14} - 10 T^{15} + T^{16} \)
$29$ \( 28283166976 + 16075421600 T^{2} + 3572280249 T^{4} + 411594054 T^{6} + 27012503 T^{8} + 1031408 T^{10} + 22183 T^{12} + 242 T^{14} + T^{16} \)
$31$ \( ( 89776 + 70912 T - 51887 T^{2} - 8950 T^{3} + 4799 T^{4} + 304 T^{5} - 137 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$37$ \( 223781979136 + 114700283584 T^{2} + 21618975840 T^{4} + 1974237208 T^{6} + 96864353 T^{8} + 2647812 T^{10} + 40198 T^{12} + 316 T^{14} + T^{16} \)
$41$ \( 1587317772544 + 489708504880 T^{2} + 62282339689 T^{4} + 4239516602 T^{6} + 167857263 T^{8} + 3928944 T^{10} + 52639 T^{12} + 366 T^{14} + T^{16} \)
$43$ \( 15320498176 + 16691293184 T^{2} + 5162366464 T^{4} + 721565312 T^{6} + 51564304 T^{8} + 1918704 T^{10} + 36216 T^{12} + 320 T^{14} + T^{16} \)
$47$ \( 2124103744 + 56956236656 T^{2} + 13350047764 T^{4} + 1265229052 T^{6} + 63348984 T^{8} + 1822404 T^{10} + 30277 T^{12} + 270 T^{14} + T^{16} \)
$53$ \( ( -7168 + 35712 T - 23532 T^{2} - 14524 T^{3} + 4025 T^{4} + 548 T^{5} - 146 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$59$ \( 111851776 + 20874722624 T^{2} + 77397481920 T^{4} + 12004641172 T^{6} + 582308225 T^{8} + 12345448 T^{10} + 123686 T^{12} + 576 T^{14} + T^{16} \)
$61$ \( 507510784 + 1161625600 T^{2} + 857996288 T^{4} + 255257264 T^{6} + 33576020 T^{8} + 1960092 T^{10} + 44912 T^{12} + 372 T^{14} + T^{16} \)
$67$ \( 3255387136 + 750110523584 T^{2} + 540187885856 T^{4} + 50971279720 T^{6} + 1619039529 T^{8} + 23883212 T^{10} + 180222 T^{12} + 676 T^{14} + T^{16} \)
$71$ \( 108601884304 + 169629257388 T^{2} + 81010225761 T^{4} + 13283343182 T^{6} + 735840911 T^{8} + 15934244 T^{10} + 152343 T^{12} + 650 T^{14} + T^{16} \)
$73$ \( ( -3042376 - 2268512 T - 21562 T^{2} + 198974 T^{3} + 26204 T^{4} - 1784 T^{5} - 339 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$79$ \( 12647639547904 + 4845286666240 T^{2} + 688396376064 T^{4} + 45788382720 T^{6} + 1504567104 T^{8} + 24276480 T^{10} + 193600 T^{12} + 724 T^{14} + T^{16} \)
$83$ \( ( -27536 + 118408 T - 116168 T^{2} - 2534 T^{3} + 10613 T^{4} - 158 T^{5} - 266 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$89$ \( ( 275968 - 407424 T + 51776 T^{2} + 63776 T^{3} - 1776 T^{4} - 2440 T^{5} - 132 T^{6} + 14 T^{7} + T^{8} )^{2} \)
$97$ \( 1201694948982784 + 312670350527488 T^{2} + 25874450907648 T^{4} + 770699168896 T^{6} + 11292823440 T^{8} + 91406512 T^{10} + 417432 T^{12} + 1008 T^{14} + T^{16} \)
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