Properties

Label 1078.2.i.b
Level $1078$
Weight $2$
Character orbit 1078.i
Analytic conductor $8.608$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.162447943996702457856.1
Defining polynomial: \(x^{16} - x^{12} - 15 x^{8} - 16 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q + ( \beta_{2} - \beta_{3} ) q^{2} + ( -\beta_{4} + \beta_{9} ) q^{3} + \beta_{1} q^{4} + \beta_{11} q^{5} + \beta_{12} q^{6} + \beta_{2} q^{8} + ( 3 - 3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} - \beta_{3} ) q^{2} + ( -\beta_{4} + \beta_{9} ) q^{3} + \beta_{1} q^{4} + \beta_{11} q^{5} + \beta_{12} q^{6} + \beta_{2} q^{8} + ( 3 - 3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{9} + ( \beta_{13} - \beta_{14} ) q^{10} + ( -\beta_{1} - \beta_{10} ) q^{11} -\beta_{4} q^{12} + \beta_{12} q^{13} -2 q^{15} + ( -1 + \beta_{1} ) q^{16} + ( -\beta_{5} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{17} + ( -3 \beta_{3} - \beta_{8} - \beta_{10} ) q^{18} + ( -2 \beta_{5} + \beta_{13} ) q^{19} -\beta_{15} q^{20} + ( -\beta_{2} - \beta_{7} - \beta_{8} ) q^{22} + ( -4 + 4 \beta_{1} ) q^{23} + ( -\beta_{5} + \beta_{12} ) q^{24} + ( -\beta_{1} + \beta_{8} - \beta_{10} ) q^{25} -\beta_{4} q^{26} + ( 4 \beta_{9} + 2 \beta_{15} ) q^{27} + ( -4 \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{29} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{30} + ( -\beta_{4} + \beta_{9} + 3 \beta_{11} + 3 \beta_{15} ) q^{31} + \beta_{3} q^{32} + ( 3 \beta_{4} + 2 \beta_{5} - \beta_{11} - \beta_{13} ) q^{33} + ( -\beta_{9} + \beta_{15} ) q^{34} + ( 3 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{36} + ( 2 - 2 \beta_{1} - \beta_{6} - \beta_{7} ) q^{37} + ( 2 \beta_{4} - 2 \beta_{9} - \beta_{11} - \beta_{15} ) q^{38} + ( -6 \beta_{3} - \beta_{8} - \beta_{10} ) q^{39} + \beta_{13} q^{40} + ( -\beta_{12} - 3 \beta_{14} ) q^{41} + ( 4 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} ) q^{43} + ( 1 - \beta_{1} - \beta_{6} ) q^{44} + ( 2 \beta_{4} - 2 \beta_{9} - 3 \beta_{11} - 3 \beta_{15} ) q^{45} + 4 \beta_{3} q^{46} + ( -\beta_{4} - 3 \beta_{11} ) q^{47} -\beta_{9} q^{48} + ( -\beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{50} + ( 8 \beta_{2} - 8 \beta_{3} - \beta_{6} + \beta_{7} ) q^{51} + ( -\beta_{5} + \beta_{12} ) q^{52} + ( 6 \beta_{1} + \beta_{8} - \beta_{10} ) q^{53} + ( 4 \beta_{5} - 2 \beta_{13} ) q^{54} + ( -\beta_{9} - \beta_{12} + 3 \beta_{14} - 2 \beta_{15} ) q^{55} + ( 10 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} ) q^{57} + ( 4 - 4 \beta_{1} - \beta_{6} - \beta_{7} ) q^{58} + ( \beta_{4} - \beta_{9} ) q^{59} -2 \beta_{1} q^{60} -3 \beta_{5} q^{61} + ( \beta_{12} - 3 \beta_{14} ) q^{62} - q^{64} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -2 \beta_{4} + 3 \beta_{5} + 2 \beta_{9} + \beta_{11} - 3 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{66} + ( 2 \beta_{1} + \beta_{8} - \beta_{10} ) q^{67} + ( -\beta_{5} - \beta_{13} ) q^{68} -4 \beta_{9} q^{69} + 2 q^{71} + ( 3 \beta_{2} - 3 \beta_{3} - \beta_{6} + \beta_{7} ) q^{72} + ( \beta_{5} - \beta_{12} + 5 \beta_{13} - 5 \beta_{14} ) q^{73} + ( -2 \beta_{3} - \beta_{8} - \beta_{10} ) q^{74} + ( 5 \beta_{4} - 2 \beta_{11} ) q^{75} + ( -2 \beta_{12} + \beta_{14} ) q^{76} + ( 6 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{78} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{79} + ( -\beta_{11} - \beta_{15} ) q^{80} + ( -11 \beta_{1} + \beta_{8} - \beta_{10} ) q^{81} + ( \beta_{4} + 3 \beta_{11} ) q^{82} + ( -4 \beta_{12} + 5 \beta_{14} ) q^{83} + ( -6 \beta_{2} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{85} + ( -4 + 4 \beta_{1} + 2 \beta_{6} + 2 \beta_{7} ) q^{86} + ( 8 \beta_{5} - 8 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{87} + ( -\beta_{3} - \beta_{8} ) q^{88} + ( -4 \beta_{4} + 4 \beta_{11} ) q^{89} + ( -2 \beta_{12} + 3 \beta_{14} ) q^{90} -4 q^{92} + ( -\beta_{6} - \beta_{7} ) q^{93} + ( -\beta_{5} + \beta_{12} - 3 \beta_{13} + 3 \beta_{14} ) q^{94} + ( -\beta_{8} - \beta_{10} ) q^{95} -\beta_{5} q^{96} + ( -6 \beta_{9} - 4 \beta_{15} ) q^{97} + ( -13 - 10 \beta_{2} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 16 q^{9} + O(q^{10}) \) \( 16 q + 8 q^{4} + 16 q^{9} - 4 q^{11} - 32 q^{15} - 8 q^{16} - 8 q^{22} - 32 q^{23} + 32 q^{36} + 8 q^{37} + 4 q^{44} + 56 q^{53} + 24 q^{58} - 16 q^{60} - 16 q^{64} + 24 q^{67} + 32 q^{71} + 80 q^{78} - 80 q^{81} - 16 q^{86} - 4 q^{88} - 64 q^{92} - 8 q^{93} - 184 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{12} - 15 x^{8} - 16 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{12} + 15 \nu^{8} - 15 \nu^{4} - 16 \)\()/240\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{14} - 15 \nu^{10} - 33 \nu^{6} + 256 \nu^{2} \)\()/576\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{14} + 91 \nu^{2} \)\()/180\)
\(\beta_{4}\)\(=\)\((\)\( -13 \nu^{15} + 4 \nu^{13} + 45 \nu^{11} + 60 \nu^{9} + 675 \nu^{7} - 1020 \nu^{5} - 2192 \nu^{3} + 4736 \nu \)\()/5760\)
\(\beta_{5}\)\(=\)\((\)\( 13 \nu^{15} + 4 \nu^{13} - 45 \nu^{11} + 60 \nu^{9} - 675 \nu^{7} - 1020 \nu^{5} + 2192 \nu^{3} + 4736 \nu \)\()/5760\)
\(\beta_{6}\)\(=\)\((\)\( -7 \nu^{14} - 66 \nu^{12} + 15 \nu^{10} - 30 \nu^{8} - 255 \nu^{6} + 510 \nu^{4} - 968 \nu^{2} + 2016 \)\()/1440\)
\(\beta_{7}\)\(=\)\((\)\( 7 \nu^{14} - 66 \nu^{12} - 15 \nu^{10} - 30 \nu^{8} + 255 \nu^{6} + 510 \nu^{4} + 968 \nu^{2} + 2016 \)\()/1440\)
\(\beta_{8}\)\(=\)\((\)\( 41 \nu^{14} + 72 \nu^{12} + 135 \nu^{10} + 120 \nu^{8} - 855 \nu^{6} + 840 \nu^{4} - 2336 \nu^{2} - 2112 \)\()/2880\)
\(\beta_{9}\)\(=\)\((\)\( 5 \nu^{15} - 8 \nu^{13} - 21 \nu^{11} + 72 \nu^{9} + 69 \nu^{7} - 72 \nu^{5} - 224 \nu^{3} - 64 \nu \)\()/1152\)
\(\beta_{10}\)\(=\)\((\)\( 41 \nu^{14} - 72 \nu^{12} + 135 \nu^{10} - 120 \nu^{8} - 855 \nu^{6} - 840 \nu^{4} - 2336 \nu^{2} + 2112 \)\()/2880\)
\(\beta_{11}\)\(=\)\((\)\( 13 \nu^{15} - 34 \nu^{13} + 75 \nu^{11} - 30 \nu^{9} + 165 \nu^{7} + 510 \nu^{5} - 568 \nu^{3} + 3424 \nu \)\()/2880\)
\(\beta_{12}\)\(=\)\((\)\( -5 \nu^{15} - 8 \nu^{13} + 21 \nu^{11} + 72 \nu^{9} - 69 \nu^{7} - 72 \nu^{5} + 224 \nu^{3} - 64 \nu \)\()/1152\)
\(\beta_{13}\)\(=\)\((\)\( -13 \nu^{15} - 34 \nu^{13} - 75 \nu^{11} - 30 \nu^{9} - 165 \nu^{7} + 510 \nu^{5} + 568 \nu^{3} + 3424 \nu \)\()/2880\)
\(\beta_{14}\)\(=\)\((\)\( -11 \nu^{15} + 4 \nu^{13} - 21 \nu^{11} + 60 \nu^{9} + 69 \nu^{7} + 132 \nu^{5} + 656 \nu^{3} + 128 \nu \)\()/1152\)
\(\beta_{15}\)\(=\)\((\)\( -11 \nu^{15} - 4 \nu^{13} - 21 \nu^{11} - 60 \nu^{9} + 69 \nu^{7} - 132 \nu^{5} + 656 \nu^{3} - 128 \nu \)\()/1152\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{9} + 2 \beta_{5} + 2 \beta_{4}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 8 \beta_{3} + 2 \beta_{2}\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{15} + 4 \beta_{14} - 5 \beta_{13} - \beta_{12} + 5 \beta_{11} + \beta_{9} + 5 \beta_{5} - 5 \beta_{4}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{10} + 2 \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-7 \beta_{15} + 7 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{9} - 7 \beta_{5} - 7 \beta_{4}\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{10} - 5 \beta_{8} + 5 \beta_{7} - 5 \beta_{6} - 10 \beta_{3} - 22 \beta_{2}\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{15} + 13 \beta_{14} - 17 \beta_{13} - 13 \beta_{12} + 17 \beta_{11} + 13 \beta_{9} - 4 \beta_{5} + 4 \beta_{4}\)\()/6\)
\(\nu^{8}\)\(=\)\((\)\(-\beta_{10} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 32 \beta_{1} - 2\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-28 \beta_{15} + 28 \beta_{14} - 5 \beta_{13} + 23 \beta_{12} - 5 \beta_{11} + 23 \beta_{9} + 5 \beta_{5} + 5 \beta_{4}\)\()/6\)
\(\nu^{10}\)\(=\)\((\)\(22 \beta_{10} + 22 \beta_{8} + 11 \beta_{7} - 11 \beta_{6} + 182 \beta_{3} - 160 \beta_{2}\)\()/6\)
\(\nu^{11}\)\(=\)\((\)\(\beta_{15} + \beta_{14} - 68 \beta_{13} + 68 \beta_{12} + 68 \beta_{11} - 68 \beta_{9} - \beta_{5} + \beta_{4}\)\()/6\)
\(\nu^{12}\)\(=\)\((\)\(-15 \beta_{10} + 15 \beta_{8} - 15 \beta_{7} - 15 \beta_{6} - 30 \beta_{1} + 62\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-181 \beta_{15} + 181 \beta_{14} - 89 \beta_{13} - 181 \beta_{12} - 89 \beta_{11} - 181 \beta_{9} + 92 \beta_{5} + 92 \beta_{4}\)\()/6\)
\(\nu^{14}\)\(=\)\((\)\(91 \beta_{10} + 91 \beta_{8} + 182 \beta_{7} - 182 \beta_{6} - 352 \beta_{3} + 182 \beta_{2}\)\()/6\)
\(\nu^{15}\)\(=\)\((\)\(4 \beta_{15} + 4 \beta_{14} - 275 \beta_{13} - 271 \beta_{12} + 275 \beta_{11} + 271 \beta_{9} + 275 \beta_{5} - 275 \beta_{4}\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(981\)
\(\chi(n)\) \(1 - \beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
1.14839 0.825348i
1.40721 + 0.140577i
−1.40721 0.140577i
−1.14839 + 0.825348i
0.825348 + 1.14839i
−0.140577 + 1.40721i
0.140577 1.40721i
−0.825348 1.14839i
1.14839 + 0.825348i
1.40721 0.140577i
−1.40721 + 0.140577i
−1.14839 0.825348i
0.825348 1.14839i
−0.140577 1.40721i
0.140577 + 1.40721i
−0.825348 + 1.14839i
−0.866025 0.500000i −2.68085 + 1.54779i 0.500000 + 0.866025i 0.559525 + 0.323042i 3.09557 0 1.00000i 3.29129 5.70068i −0.323042 0.559525i
901.2 −0.866025 0.500000i −0.559525 + 0.323042i 0.500000 + 0.866025i 2.68085 + 1.54779i 0.646084 0 1.00000i −1.29129 + 2.23658i −1.54779 2.68085i
901.3 −0.866025 0.500000i 0.559525 0.323042i 0.500000 + 0.866025i −2.68085 1.54779i −0.646084 0 1.00000i −1.29129 + 2.23658i 1.54779 + 2.68085i
901.4 −0.866025 0.500000i 2.68085 1.54779i 0.500000 + 0.866025i −0.559525 0.323042i −3.09557 0 1.00000i 3.29129 5.70068i 0.323042 + 0.559525i
901.5 0.866025 + 0.500000i −2.68085 + 1.54779i 0.500000 + 0.866025i 0.559525 + 0.323042i −3.09557 0 1.00000i 3.29129 5.70068i 0.323042 + 0.559525i
901.6 0.866025 + 0.500000i −0.559525 + 0.323042i 0.500000 + 0.866025i 2.68085 + 1.54779i −0.646084 0 1.00000i −1.29129 + 2.23658i 1.54779 + 2.68085i
901.7 0.866025 + 0.500000i 0.559525 0.323042i 0.500000 + 0.866025i −2.68085 1.54779i 0.646084 0 1.00000i −1.29129 + 2.23658i −1.54779 2.68085i
901.8 0.866025 + 0.500000i 2.68085 1.54779i 0.500000 + 0.866025i −0.559525 0.323042i 3.09557 0 1.00000i 3.29129 5.70068i −0.323042 0.559525i
1011.1 −0.866025 + 0.500000i −2.68085 1.54779i 0.500000 0.866025i 0.559525 0.323042i 3.09557 0 1.00000i 3.29129 + 5.70068i −0.323042 + 0.559525i
1011.2 −0.866025 + 0.500000i −0.559525 0.323042i 0.500000 0.866025i 2.68085 1.54779i 0.646084 0 1.00000i −1.29129 2.23658i −1.54779 + 2.68085i
1011.3 −0.866025 + 0.500000i 0.559525 + 0.323042i 0.500000 0.866025i −2.68085 + 1.54779i −0.646084 0 1.00000i −1.29129 2.23658i 1.54779 2.68085i
1011.4 −0.866025 + 0.500000i 2.68085 + 1.54779i 0.500000 0.866025i −0.559525 + 0.323042i −3.09557 0 1.00000i 3.29129 + 5.70068i 0.323042 0.559525i
1011.5 0.866025 0.500000i −2.68085 1.54779i 0.500000 0.866025i 0.559525 0.323042i −3.09557 0 1.00000i 3.29129 + 5.70068i 0.323042 0.559525i
1011.6 0.866025 0.500000i −0.559525 0.323042i 0.500000 0.866025i 2.68085 1.54779i −0.646084 0 1.00000i −1.29129 2.23658i 1.54779 2.68085i
1011.7 0.866025 0.500000i 0.559525 + 0.323042i 0.500000 0.866025i −2.68085 + 1.54779i 0.646084 0 1.00000i −1.29129 2.23658i −1.54779 + 2.68085i
1011.8 0.866025 0.500000i 2.68085 + 1.54779i 0.500000 0.866025i −0.559525 + 0.323042i 3.09557 0 1.00000i 3.29129 + 5.70068i −0.323042 + 0.559525i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1011.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner
77.h odd 6 1 inner
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.i.b 16
7.b odd 2 1 inner 1078.2.i.b 16
7.c even 3 1 154.2.c.a 8
7.c even 3 1 inner 1078.2.i.b 16
7.d odd 6 1 154.2.c.a 8
7.d odd 6 1 inner 1078.2.i.b 16
11.b odd 2 1 inner 1078.2.i.b 16
21.g even 6 1 1386.2.e.b 8
21.h odd 6 1 1386.2.e.b 8
28.f even 6 1 1232.2.e.e 8
28.g odd 6 1 1232.2.e.e 8
77.b even 2 1 inner 1078.2.i.b 16
77.h odd 6 1 154.2.c.a 8
77.h odd 6 1 inner 1078.2.i.b 16
77.i even 6 1 154.2.c.a 8
77.i even 6 1 inner 1078.2.i.b 16
231.k odd 6 1 1386.2.e.b 8
231.l even 6 1 1386.2.e.b 8
308.m odd 6 1 1232.2.e.e 8
308.n even 6 1 1232.2.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.c.a 8 7.c even 3 1
154.2.c.a 8 7.d odd 6 1
154.2.c.a 8 77.h odd 6 1
154.2.c.a 8 77.i even 6 1
1078.2.i.b 16 1.a even 1 1 trivial
1078.2.i.b 16 7.b odd 2 1 inner
1078.2.i.b 16 7.c even 3 1 inner
1078.2.i.b 16 7.d odd 6 1 inner
1078.2.i.b 16 11.b odd 2 1 inner
1078.2.i.b 16 77.b even 2 1 inner
1078.2.i.b 16 77.h odd 6 1 inner
1078.2.i.b 16 77.i even 6 1 inner
1232.2.e.e 8 28.f even 6 1
1232.2.e.e 8 28.g odd 6 1
1232.2.e.e 8 308.m odd 6 1
1232.2.e.e 8 308.n even 6 1
1386.2.e.b 8 21.g even 6 1
1386.2.e.b 8 21.h odd 6 1
1386.2.e.b 8 231.k odd 6 1
1386.2.e.b 8 231.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 10 T_{3}^{6} + 96 T_{3}^{4} - 40 T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( ( 16 - 40 T^{2} + 96 T^{4} - 10 T^{6} + T^{8} )^{2} \)
$5$ \( ( 16 - 40 T^{2} + 96 T^{4} - 10 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 14641 + 2662 T + 242 T^{2} - 440 T^{3} - 161 T^{4} - 40 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$13$ \( ( 4 - 10 T^{2} + T^{4} )^{4} \)
$17$ \( ( 196 + 14 T^{2} + T^{4} )^{4} \)
$19$ \( ( 10000 + 3400 T^{2} + 1056 T^{4} + 34 T^{6} + T^{8} )^{2} \)
$23$ \( ( 16 + 4 T + T^{2} )^{8} \)
$29$ \( ( 144 + 60 T^{2} + T^{4} )^{4} \)
$31$ \( ( 10000 - 7600 T^{2} + 5676 T^{4} - 76 T^{6} + T^{8} )^{2} \)
$37$ \( ( 400 + 40 T + 24 T^{2} - 2 T^{3} + T^{4} )^{4} \)
$41$ \( ( 2500 - 124 T^{2} + T^{4} )^{4} \)
$43$ \( ( 6400 + 176 T^{2} + T^{4} )^{4} \)
$47$ \( ( 6250000 - 310000 T^{2} + 12876 T^{4} - 124 T^{6} + T^{8} )^{2} \)
$53$ \( ( 784 - 392 T + 168 T^{2} - 14 T^{3} + T^{4} )^{4} \)
$59$ \( ( 16 - 40 T^{2} + 96 T^{4} - 10 T^{6} + T^{8} )^{2} \)
$61$ \( ( 104976 + 29160 T^{2} + 7776 T^{4} + 90 T^{6} + T^{8} )^{2} \)
$67$ \( ( 144 + 72 T + 48 T^{2} - 6 T^{3} + T^{4} )^{4} \)
$71$ \( ( -2 + T )^{16} \)
$73$ \( ( 108243216 + 3121200 T^{2} + 79596 T^{4} + 300 T^{6} + T^{8} )^{2} \)
$79$ \( ( 256 - 16 T^{2} + T^{4} )^{4} \)
$83$ \( ( 13924 - 250 T^{2} + T^{4} )^{4} \)
$89$ \( ( 9216 - 96 T^{2} + T^{4} )^{4} \)
$97$ \( ( 18496 + 328 T^{2} + T^{4} )^{4} \)
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