Properties

Label 1120.2.h.b
Level $1120$
Weight $2$
Character orbit 1120.h
Analytic conductor $8.943$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} - 2 x^{12} + 6 x^{11} - 12 x^{9} + 8 x^{8} - 24 x^{7} + 48 x^{5} - 32 x^{4} - 128 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 280)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + q^{5} + \beta_{8} q^{7} + ( -1 - \beta_{12} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + q^{5} + \beta_{8} q^{7} + ( -1 - \beta_{12} ) q^{9} + \beta_{10} q^{11} + \beta_{5} q^{13} + \beta_{2} q^{15} + ( -\beta_{2} + \beta_{13} ) q^{17} + ( -\beta_{7} + \beta_{11} + \beta_{13} ) q^{19} + ( -\beta_{7} + \beta_{15} ) q^{21} + ( \beta_{2} + \beta_{14} ) q^{23} + q^{25} + ( -\beta_{2} + \beta_{4} + \beta_{7} + \beta_{9} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{27} + ( \beta_{2} - \beta_{11} ) q^{29} + ( 2 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{10} + \beta_{15} ) q^{31} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{14} ) q^{33} + \beta_{8} q^{35} + ( -\beta_{9} - \beta_{11} - \beta_{13} + \beta_{15} ) q^{37} + ( \beta_{1} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{13} + \beta_{15} ) q^{39} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{41} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{12} ) q^{43} + ( -1 - \beta_{12} ) q^{45} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{47} + ( -1 + \beta_{3} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{49} + ( 2 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{12} - \beta_{15} ) q^{51} + ( -\beta_{1} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{53} + \beta_{10} q^{55} + ( -2 - 2 \beta_{1} + \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{57} + ( \beta_{1} + \beta_{2} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{5} - \beta_{9} - \beta_{12} + \beta_{15} ) q^{61} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{12} + \beta_{13} ) q^{63} + \beta_{5} q^{65} + ( -2 + \beta_{3} - \beta_{5} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{15} ) q^{67} + ( -2 + \beta_{5} - 2 \beta_{10} - \beta_{12} ) q^{69} + ( -\beta_{2} - \beta_{4} - \beta_{9} - \beta_{11} + \beta_{15} ) q^{71} + ( -\beta_{1} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{13} - \beta_{14} ) q^{73} + \beta_{2} q^{75} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{77} + ( -\beta_{1} - \beta_{2} + \beta_{7} + \beta_{8} - \beta_{11} + \beta_{13} - 2 \beta_{14} ) q^{79} + ( 1 + 2 \beta_{3} + 2 \beta_{8} + \beta_{9} + 2 \beta_{12} - \beta_{15} ) q^{81} + ( -2 \beta_{2} + \beta_{4} - \beta_{7} - \beta_{9} - 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{83} + ( -\beta_{2} + \beta_{13} ) q^{85} + ( -4 + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{12} ) q^{87} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} - \beta_{14} ) q^{89} + ( 2 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{8} + 2 \beta_{9} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{91} + ( \beta_{1} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{93} + ( -\beta_{7} + \beta_{11} + \beta_{13} ) q^{95} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{8} + 2 \beta_{11} - \beta_{13} ) q^{97} + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{12} - \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{5} - 16q^{9} + O(q^{10}) \) \( 16q + 16q^{5} - 16q^{9} + 4q^{11} - 4q^{21} + 16q^{25} + 16q^{31} + 4q^{43} - 16q^{45} - 8q^{49} + 40q^{51} + 4q^{55} - 16q^{57} - 8q^{61} - 28q^{63} - 20q^{67} - 40q^{69} - 4q^{77} + 24q^{81} - 72q^{87} + 32q^{91} - 20q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} - 2 x^{12} + 6 x^{11} - 12 x^{9} + 8 x^{8} - 24 x^{7} + 48 x^{5} - 32 x^{4} - 128 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{11} + \nu^{10} + 2 \nu^{8} - 2 \nu^{7} - 2 \nu^{6} - 8 \nu^{2} - 32 \nu - 16 \)\()/16\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} + \nu^{14} + 2 \nu^{13} + 4 \nu^{12} - 2 \nu^{11} - 6 \nu^{10} + 4 \nu^{9} - 4 \nu^{8} + 16 \nu^{7} - 8 \nu^{6} - 48 \nu^{5} - 16 \nu^{4} - 64 \nu^{3} - 64 \nu^{2} + 128 \nu \)\()/128\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{14} + 3 \nu^{12} - 6 \nu^{8} + 4 \nu^{7} + 8 \nu^{6} - 16 \nu^{5} + 8 \nu^{4} - 16 \nu^{3} - 96 \nu^{2} + 32 \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} - \nu^{14} - 6 \nu^{13} + 2 \nu^{11} - 10 \nu^{10} + 20 \nu^{9} - 20 \nu^{8} - 16 \nu^{7} + 24 \nu^{6} - 16 \nu^{5} + 112 \nu^{4} + 128 \nu^{3} \)\()/128\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} + 3 \nu^{14} - 2 \nu^{13} + 10 \nu^{11} - 2 \nu^{10} - 4 \nu^{9} + 12 \nu^{8} - 48 \nu^{7} - 8 \nu^{6} + 48 \nu^{5} - 80 \nu^{4} - 128 \nu^{2} - 384 \nu + 256 \)\()/128\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{15} + \nu^{14} + 2 \nu^{11} - 6 \nu^{10} + 12 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 48 \nu^{4} + 32 \nu^{3} + 128 \nu + 128 \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} + \nu^{14} + 2 \nu^{11} - 6 \nu^{10} + 12 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 48 \nu^{4} + 32 \nu^{3} - 128 \nu + 128 \)\()/64\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{15} + \nu^{14} + 6 \nu^{11} - 10 \nu^{10} - 8 \nu^{9} + 4 \nu^{8} - 16 \nu^{7} + 32 \nu^{6} + 16 \nu^{5} - 80 \nu^{4} - 32 \nu^{2} + 320 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( -3 \nu^{15} + \nu^{14} + 2 \nu^{13} + 14 \nu^{11} + 10 \nu^{10} - 12 \nu^{9} + 4 \nu^{8} - 16 \nu^{7} + 40 \nu^{6} + 112 \nu^{5} + 16 \nu^{4} - 64 \nu^{3} - 128 \nu^{2} - 128 \nu \)\()/128\)
\(\beta_{10}\)\(=\)\((\)\( -3 \nu^{15} + \nu^{14} - 2 \nu^{13} + 4 \nu^{12} + 14 \nu^{11} - 6 \nu^{10} - 20 \nu^{9} + 12 \nu^{8} - 16 \nu^{7} + 56 \nu^{6} + 112 \nu^{5} - 80 \nu^{4} - 64 \nu^{2} - 256 \nu + 512 \)\()/128\)
\(\beta_{11}\)\(=\)\((\)\( -\nu^{15} - \nu^{14} - \nu^{13} - \nu^{12} + 2 \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 2 \nu^{8} + 12 \nu^{6} + 32 \nu^{5} + 8 \nu^{4} + 16 \nu^{3} - 16 \nu^{2} - 32 \nu + 64 \)\()/32\)
\(\beta_{12}\)\(=\)\((\)\( 3 \nu^{15} + 3 \nu^{14} - 6 \nu^{13} - 8 \nu^{12} - 6 \nu^{11} - 2 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} - 16 \nu^{7} - 40 \nu^{6} - 80 \nu^{5} + 48 \nu^{4} + 256 \nu^{3} + 128 \nu^{2} - 256 \)\()/128\)
\(\beta_{13}\)\(=\)\((\)\( \nu^{15} + \nu^{12} + 4 \nu^{10} + 8 \nu^{9} - 2 \nu^{8} - 12 \nu^{6} - 16 \nu^{5} + 8 \nu^{4} + 16 \nu^{3} - 48 \nu^{2} - 64 \nu - 128 \)\()/32\)
\(\beta_{14}\)\(=\)\((\)\( 3 \nu^{15} - \nu^{14} - 6 \nu^{13} + 4 \nu^{12} + 2 \nu^{11} + 22 \nu^{10} + 36 \nu^{9} - 28 \nu^{8} - 16 \nu^{7} + 8 \nu^{6} - 48 \nu^{5} + 144 \nu^{4} - 192 \nu^{2} - 256 \nu - 512 \)\()/128\)
\(\beta_{15}\)\(=\)\((\)\( -5 \nu^{15} - \nu^{14} - 2 \nu^{13} + 8 \nu^{12} + 18 \nu^{11} - 26 \nu^{10} - 20 \nu^{9} + 12 \nu^{8} - 48 \nu^{7} + 152 \nu^{6} + 144 \nu^{5} - 80 \nu^{4} + 64 \nu^{3} - 384 \nu^{2} - 128 \nu + 1024 \)\()/128\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + \beta_{6}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{14} + 2 \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{15} - 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 3 \beta_{1} + 4\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{15} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_{1} - 4\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_{1} - 4\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-6 \beta_{15} + 2 \beta_{13} + 5 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + 5 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 11 \beta_{2} - 5 \beta_{1} + 12\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(6 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} - 7 \beta_{9} - 10 \beta_{8} + 6 \beta_{6} - \beta_{5} - 7 \beta_{4} - 5 \beta_{3} + 11 \beta_{2} + 15 \beta_{1} + 4\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-2 \beta_{15} + 4 \beta_{14} + 10 \beta_{13} - 11 \beta_{12} + 7 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 10 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} - 9 \beta_{3} + 7 \beta_{2} - 5 \beta_{1} + 36\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-10 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + \beta_{12} - 5 \beta_{11} + 7 \beta_{10} + 9 \beta_{9} - 2 \beta_{8} + 14 \beta_{6} - \beta_{5} - 7 \beta_{4} + 11 \beta_{3} - 29 \beta_{2} - 9 \beta_{1} + 52\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-10 \beta_{15} - 4 \beta_{14} + 10 \beta_{13} + \beta_{12} + 3 \beta_{11} + 15 \beta_{10} + 9 \beta_{9} + 30 \beta_{8} - 16 \beta_{7} - 2 \beta_{6} - 25 \beta_{5} + 17 \beta_{4} - 5 \beta_{3} + 19 \beta_{2} + 39 \beta_{1} + 20\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(14 \beta_{15} - 12 \beta_{14} + 10 \beta_{13} - 23 \beta_{12} - 37 \beta_{11} + 47 \beta_{10} - 31 \beta_{9} - 42 \beta_{8} - 16 \beta_{7} + 22 \beta_{6} - \beta_{5} + 25 \beta_{4} - 5 \beta_{3} - 13 \beta_{2} + 15 \beta_{1} - 60\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(30 \beta_{15} - 44 \beta_{14} + 42 \beta_{13} - 23 \beta_{12} - 37 \beta_{11} - 49 \beta_{10} + 33 \beta_{9} + 6 \beta_{8} + 24 \beta_{7} - 34 \beta_{6} - 33 \beta_{5} + 25 \beta_{4} - 21 \beta_{3} - 29 \beta_{2} + 15 \beta_{1} + 68\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-18 \beta_{15} + 4 \beta_{14} - 22 \beta_{13} + 57 \beta_{12} - 21 \beta_{11} + 31 \beta_{10} + 81 \beta_{9} - 26 \beta_{8} + 32 \beta_{7} - 10 \beta_{6} + 31 \beta_{5} - 7 \beta_{4} + 11 \beta_{3} + 83 \beta_{2} - 81 \beta_{1} + 36\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(94 \beta_{15} + 52 \beta_{14} + 10 \beta_{13} + 33 \beta_{12} - 13 \beta_{11} + 7 \beta_{10} - 71 \beta_{9} + 54 \beta_{8} - 64 \beta_{7} - 90 \beta_{6} - 89 \beta_{5} - 79 \beta_{4} - 93 \beta_{3} + 75 \beta_{2} + 71 \beta_{1} - 60\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
−0.275585 1.38710i
1.07046 + 0.924187i
1.41214 0.0765298i
0.244064 + 1.39299i
−1.24098 + 0.678208i
−1.38133 0.303194i
1.14218 0.833926i
−0.470943 1.33350i
−0.470943 + 1.33350i
1.14218 + 0.833926i
−1.38133 + 0.303194i
−1.24098 0.678208i
0.244064 1.39299i
1.41214 + 0.0765298i
1.07046 0.924187i
−0.275585 + 1.38710i
0 3.19977i 0 1.00000 0 2.59303 + 0.525543i 0 −7.23851 0
111.2 0 2.99734i 0 1.00000 0 −0.183359 2.63939i 0 −5.98405 0
111.3 0 2.21915i 0 1.00000 0 −1.20923 + 2.35325i 0 −1.92464 0
111.4 0 1.68420i 0 1.00000 0 −0.695780 + 2.55262i 0 0.163484 0
111.5 0 1.61069i 0 1.00000 0 −2.13463 1.56312i 0 0.405694 0
111.6 0 1.34113i 0 1.00000 0 1.28003 2.31550i 0 1.20136 0
111.7 0 0.586834i 0 1.00000 0 2.52442 0.792014i 0 2.65563 0
111.8 0 0.528177i 0 1.00000 0 −2.17448 + 1.50719i 0 2.72103 0
111.9 0 0.528177i 0 1.00000 0 −2.17448 1.50719i 0 2.72103 0
111.10 0 0.586834i 0 1.00000 0 2.52442 + 0.792014i 0 2.65563 0
111.11 0 1.34113i 0 1.00000 0 1.28003 + 2.31550i 0 1.20136 0
111.12 0 1.61069i 0 1.00000 0 −2.13463 + 1.56312i 0 0.405694 0
111.13 0 1.68420i 0 1.00000 0 −0.695780 2.55262i 0 0.163484 0
111.14 0 2.21915i 0 1.00000 0 −1.20923 2.35325i 0 −1.92464 0
111.15 0 2.99734i 0 1.00000 0 −0.183359 + 2.63939i 0 −5.98405 0
111.16 0 3.19977i 0 1.00000 0 2.59303 0.525543i 0 −7.23851 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.h.b 16
4.b odd 2 1 280.2.h.b yes 16
7.b odd 2 1 1120.2.h.a 16
8.b even 2 1 280.2.h.a 16
8.d odd 2 1 1120.2.h.a 16
28.d even 2 1 280.2.h.a 16
56.e even 2 1 inner 1120.2.h.b 16
56.h odd 2 1 280.2.h.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.h.a 16 8.b even 2 1
280.2.h.a 16 28.d even 2 1
280.2.h.b yes 16 4.b odd 2 1
280.2.h.b yes 16 56.h odd 2 1
1120.2.h.a 16 7.b odd 2 1
1120.2.h.a 16 8.d odd 2 1
1120.2.h.b 16 1.a even 1 1 trivial
1120.2.h.b 16 56.e even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{8} - 62 T_{13}^{6} + 12 T_{13}^{5} + 1213 T_{13}^{4} - 228 T_{13}^{3} - 7792 T_{13}^{2} - 1232 T_{13} + 10032 \) acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 576 + 4720 T^{2} + 13024 T^{4} + 14676 T^{6} + 8217 T^{8} + 2468 T^{10} + 398 T^{12} + 32 T^{14} + T^{16} \)
$5$ \( ( -1 + T )^{16} \)
$7$ \( 5764801 + 470596 T^{2} - 605052 T^{3} + 9604 T^{4} - 69972 T^{5} + 34300 T^{6} - 6328 T^{7} + 5814 T^{8} - 904 T^{9} + 700 T^{10} - 204 T^{11} + 4 T^{12} - 36 T^{13} + 4 T^{14} + T^{16} \)
$11$ \( ( 2400 - 1960 T - 3932 T^{2} - 138 T^{3} + 761 T^{4} + 52 T^{5} - 50 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$13$ \( ( 10032 - 1232 T - 7792 T^{2} - 228 T^{3} + 1213 T^{4} + 12 T^{5} - 62 T^{6} + T^{8} )^{2} \)
$17$ \( 2359296 + 27167680 T^{2} + 34001456 T^{4} + 13596164 T^{6} + 2255465 T^{8} + 178476 T^{10} + 7086 T^{12} + 136 T^{14} + T^{16} \)
$19$ \( 1327104 + 26099712 T^{2} + 81127424 T^{4} + 49375232 T^{6} + 6949696 T^{8} + 423168 T^{10} + 12736 T^{12} + 184 T^{14} + T^{16} \)
$23$ \( 7573504 + 585268224 T^{2} + 289346560 T^{4} + 57190400 T^{6} + 5807360 T^{8} + 325872 T^{10} + 10096 T^{12} + 160 T^{14} + T^{16} \)
$29$ \( 65536 + 2408448 T^{2} + 15058432 T^{4} + 9795008 T^{6} + 2200529 T^{8} + 217548 T^{10} + 9718 T^{12} + 172 T^{14} + T^{16} \)
$31$ \( ( 98304 + 2816 T - 33920 T^{2} - 3040 T^{3} + 3496 T^{4} + 440 T^{5} - 92 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$37$ \( 4063297536 + 30390333440 T^{2} + 7832304640 T^{4} + 825741568 T^{6} + 45832768 T^{8} + 1448768 T^{10} + 26192 T^{12} + 252 T^{14} + T^{16} \)
$41$ \( 6423183360000 + 1819708211200 T^{2} + 201056874496 T^{4} + 11370832896 T^{6} + 364052032 T^{8} + 6823040 T^{10} + 73760 T^{12} + 424 T^{14} + T^{16} \)
$43$ \( ( -69504 + 91360 T - 17664 T^{2} - 14912 T^{3} + 4232 T^{4} + 332 T^{5} - 128 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$47$ \( ( -10368 - 5820 T + 36188 T^{2} + 28292 T^{3} + 5461 T^{4} - 424 T^{5} - 166 T^{6} + T^{8} )^{2} \)
$53$ \( 262144 + 27312128 T^{2} + 158445568 T^{4} + 251149312 T^{6} + 69431040 T^{8} + 3445312 T^{10} + 58128 T^{12} + 404 T^{14} + T^{16} \)
$59$ \( 1090584576 + 2931687424 T^{2} + 2362048512 T^{4} + 734793728 T^{6} + 82300416 T^{8} + 3287040 T^{10} + 55552 T^{12} + 400 T^{14} + T^{16} \)
$61$ \( ( 10368 - 51712 T + 26432 T^{2} + 85216 T^{3} + 15176 T^{4} - 1256 T^{5} - 268 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$67$ \( ( 165632 - 197184 T - 55616 T^{2} + 42240 T^{3} + 7712 T^{4} - 1204 T^{5} - 180 T^{6} + 10 T^{7} + T^{8} )^{2} \)
$71$ \( 2621440000 + 5845811200 T^{2} + 4596301824 T^{4} + 1494106112 T^{6} + 180376576 T^{8} + 6017280 T^{10} + 82752 T^{12} + 492 T^{14} + T^{16} \)
$73$ \( 28179280429056 + 12310919774208 T^{2} + 1248988233728 T^{4} + 57276760064 T^{6} + 1406521344 T^{8} + 19483584 T^{10} + 151344 T^{12} + 612 T^{14} + T^{16} \)
$79$ \( 39806206534656 + 12666930453504 T^{2} + 1373008608512 T^{4} + 61858049132 T^{6} + 1444670401 T^{8} + 19139020 T^{10} + 145478 T^{12} + 592 T^{14} + T^{16} \)
$83$ \( 88794464256 + 165451399168 T^{2} + 51045990400 T^{4} + 6096578560 T^{6} + 337953344 T^{8} + 8985456 T^{10} + 109632 T^{12} + 564 T^{14} + T^{16} \)
$89$ \( 347892350976 + 245081571328 T^{2} + 61230546944 T^{4} + 6628880384 T^{6} + 331338304 T^{8} + 8073856 T^{10} + 95808 T^{12} + 520 T^{14} + T^{16} \)
$97$ \( 17227945230336 + 27670537460160 T^{2} + 4314469148336 T^{4} + 212781536324 T^{6} + 4708474873 T^{8} + 52726332 T^{10} + 307550 T^{12} + 888 T^{14} + T^{16} \)
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