Properties

Label 1764.2.e.h
Level $1764$
Weight $2$
Character orbit 1764.e
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + x^{12} - 10 x^{10} + 4 x^{8} - 40 x^{6} + 16 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{5} q^{5} + ( \beta_{3} - \beta_{5} + \beta_{7} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{5} q^{5} + ( \beta_{3} - \beta_{5} + \beta_{7} ) q^{8} + ( \beta_{9} + \beta_{10} ) q^{10} + \beta_{15} q^{11} + ( -1 - \beta_{2} - \beta_{6} - \beta_{8} ) q^{13} + ( \beta_{6} + \beta_{8} + \beta_{9} ) q^{16} + ( -\beta_{13} - \beta_{14} ) q^{17} + ( -\beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{19} + ( -\beta_{3} - \beta_{7} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{20} + ( -1 - \beta_{4} - \beta_{9} - \beta_{11} ) q^{22} + ( -\beta_{13} + \beta_{14} ) q^{23} + ( -1 - \beta_{2} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{25} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} - \beta_{12} - \beta_{13} ) q^{26} + ( \beta_{5} - \beta_{7} - \beta_{13} - \beta_{14} ) q^{29} + ( \beta_{2} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{31} + ( -\beta_{3} - \beta_{7} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{32} + ( 1 - \beta_{4} + \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{34} + ( -1 + \beta_{2} - \beta_{8} - \beta_{10} ) q^{37} + ( -\beta_{1} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{13} - 2 \beta_{14} ) q^{38} + ( -3 + \beta_{4} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{40} + ( 2 \beta_{1} + \beta_{7} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{41} + ( \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{43} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{12} + 2 \beta_{14} ) q^{44} + ( 1 - \beta_{4} - \beta_{6} + \beta_{11} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{47} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} - 3 \beta_{7} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{50} + ( -3 - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{52} + ( -2 \beta_{1} + \beta_{5} + \beta_{7} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{53} + ( \beta_{2} - 2 \beta_{6} + 3 \beta_{8} - \beta_{10} ) q^{55} + ( 1 - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{58} + ( -2 \beta_{3} + \beta_{5} - \beta_{7} + 2 \beta_{13} - 2 \beta_{14} ) q^{59} + ( -2 + \beta_{4} - \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{61} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} + 2 \beta_{7} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{62} + ( 4 + 2 \beta_{4} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{64} + ( 2 \beta_{1} + \beta_{7} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{65} + ( \beta_{6} - 3 \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{67} + ( -2 \beta_{1} - \beta_{5} - 4 \beta_{7} - \beta_{14} + \beta_{15} ) q^{68} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{7} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{71} + ( -1 - 3 \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{73} + ( \beta_{1} + \beta_{3} - \beta_{5} + 3 \beta_{7} + \beta_{12} - \beta_{13} ) q^{74} + ( -1 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{76} + ( -3 \beta_{2} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{79} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - 3 \beta_{7} - \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{80} + ( 5 - 2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{82} + ( -6 \beta_{1} + 3 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{83} + ( -2 - 2 \beta_{2} - \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{85} + ( \beta_{1} + \beta_{3} + \beta_{5} + 3 \beta_{7} + \beta_{13} + 2 \beta_{14} ) q^{86} + ( -4 - 2 \beta_{2} - 3 \beta_{6} - \beta_{8} + \beta_{9} ) q^{88} + ( 4 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{12} ) q^{89} + ( -2 \beta_{1} - \beta_{5} + 4 \beta_{7} - \beta_{14} + \beta_{15} ) q^{92} + ( 4 + 2 \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{8} + \beta_{10} ) q^{94} + ( 6 \beta_{1} - 2 \beta_{3} + \beta_{5} - \beta_{7} - 3 \beta_{12} + \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{95} + ( 4 + \beta_{2} + \beta_{4} - \beta_{6} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 8q^{13} - 4q^{16} - 16q^{22} - 24q^{25} + 8q^{34} - 8q^{37} - 52q^{40} + 24q^{46} - 52q^{52} + 12q^{58} - 16q^{61} + 60q^{64} - 8q^{73} - 36q^{76} + 68q^{82} - 16q^{85} - 44q^{88} + 60q^{94} + 88q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + x^{12} - 10 x^{10} + 4 x^{8} - 40 x^{6} + 16 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{13} + \nu^{9} - 10 \nu^{7} + 4 \nu^{5} - 40 \nu^{3} + 16 \nu \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{12} - \nu^{8} + 10 \nu^{6} - 4 \nu^{4} + 24 \nu^{2} - 16 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{15} + 4 \nu^{13} + 17 \nu^{11} - 6 \nu^{9} - 20 \nu^{7} - 56 \nu^{5} + 48 \nu^{3} - 320 \nu \)\()/256\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{14} - 4 \nu^{12} + 15 \nu^{10} + 6 \nu^{8} + 52 \nu^{6} - 8 \nu^{4} + 80 \nu^{2} - 448 \)\()/128\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{15} - 4 \nu^{13} + 17 \nu^{11} - 14 \nu^{9} + 60 \nu^{7} - 88 \nu^{5} + 112 \nu^{3} - 448 \nu \)\()/256\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{14} - 4 \nu^{12} + \nu^{10} - 14 \nu^{8} + 12 \nu^{6} - 24 \nu^{4} + 112 \nu^{2} + 64 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{14} + 12 \nu^{12} - 17 \nu^{10} + 22 \nu^{8} - 76 \nu^{6} + 184 \nu^{4} - 304 \nu^{2} + 320 \)\()/128\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{14} - 4 \nu^{12} + 19 \nu^{10} - 34 \nu^{8} + 68 \nu^{6} - 168 \nu^{4} + 144 \nu^{2} - 448 \)\()/128\)
\(\beta_{11}\)\(=\)\((\)\( 5 \nu^{14} - 4 \nu^{12} - 11 \nu^{10} + 10 \nu^{8} - 20 \nu^{6} - 56 \nu^{4} - 80 \nu^{2} + 704 \)\()/128\)
\(\beta_{12}\)\(=\)\((\)\( \nu^{15} + \nu^{11} - 10 \nu^{9} + 4 \nu^{7} - 40 \nu^{5} + 16 \nu^{3} \)\()/64\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{15} + 4 \nu^{13} - \nu^{11} + 14 \nu^{9} - 12 \nu^{7} + 24 \nu^{5} - 112 \nu^{3} - 64 \nu \)\()/64\)
\(\beta_{14}\)\(=\)\((\)\( 7 \nu^{15} + 4 \nu^{13} - 9 \nu^{11} - 2 \nu^{9} - 28 \nu^{7} - 40 \nu^{5} - 240 \nu^{3} + 448 \nu \)\()/256\)
\(\beta_{15}\)\(=\)\((\)\( -\nu^{15} + \nu^{13} - \nu^{11} + 3 \nu^{9} + 2 \nu^{7} + 36 \nu^{5} + 8 \nu^{3} - 48 \nu \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{5} - \beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{9} + \beta_{8} + \beta_{6}\)
\(\nu^{5}\)\(=\)\(\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{7} + \beta_{3}\)
\(\nu^{6}\)\(=\)\(\beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{4} + 4\)
\(\nu^{7}\)\(=\)\(\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + 3 \beta_{7} - 2 \beta_{5} - \beta_{3} + 4 \beta_{1}\)
\(\nu^{8}\)\(=\)\(2 \beta_{11} - \beta_{10} - \beta_{9} - 3 \beta_{8} + 2 \beta_{6} + 4 \beta_{2} - 2\)
\(\nu^{9}\)\(=\)\(-3 \beta_{15} + \beta_{14} + 3 \beta_{13} - 5 \beta_{12} - 3 \beta_{7} + 4 \beta_{5} - 7 \beta_{3}\)
\(\nu^{10}\)\(=\)\(\beta_{10} + \beta_{9} + 3 \beta_{8} + 8 \beta_{6} - 6 \beta_{4} + 20\)
\(\nu^{11}\)\(=\)\(\beta_{15} + \beta_{14} - 3 \beta_{13} - 7 \beta_{12} + 11 \beta_{7} + 6 \beta_{5} + 7 \beta_{3} + 20 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-2 \beta_{11} + 11 \beta_{10} + 7 \beta_{9} - 11 \beta_{8} - 6 \beta_{6} + 4 \beta_{4} + 20 \beta_{2} + 26\)
\(\nu^{13}\)\(=\)\(9 \beta_{15} + 5 \beta_{14} + 11 \beta_{13} + 19 \beta_{12} - 11 \beta_{7} + 16 \beta_{5} - 15 \beta_{3} + 24 \beta_{1}\)
\(\nu^{14}\)\(=\)\(20 \beta_{11} + 17 \beta_{10} + 25 \beta_{9} + 11 \beta_{8} + 20 \beta_{6} - 2 \beta_{4} + 24 \beta_{2} - 56\)
\(\nu^{15}\)\(=\)\(5 \beta_{15} + 45 \beta_{14} - 11 \beta_{13} - 23 \beta_{12} + 3 \beta_{7} + 26 \beta_{5} - 17 \beta_{3} - 36 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1079.1
1.41135 + 0.0900240i
1.41135 0.0900240i
1.13008 + 0.850247i
1.13008 0.850247i
0.658334 + 1.25164i
0.658334 1.25164i
0.545545 + 1.30475i
0.545545 1.30475i
−0.545545 + 1.30475i
−0.545545 1.30475i
−0.658334 + 1.25164i
−0.658334 1.25164i
−1.13008 + 0.850247i
−1.13008 0.850247i
−1.41135 + 0.0900240i
−1.41135 0.0900240i
−1.41135 0.0900240i 0 1.98379 + 0.254110i 2.48866i 0 0 −2.77694 0.537226i 0 −0.224040 + 3.51237i
1079.2 −1.41135 + 0.0900240i 0 1.98379 0.254110i 2.48866i 0 0 −2.77694 + 0.537226i 0 −0.224040 3.51237i
1079.3 −1.13008 0.850247i 0 0.554161 + 1.92169i 3.87147i 0 0 1.00767 2.64284i 0 −3.29170 + 4.37507i
1079.4 −1.13008 + 0.850247i 0 0.554161 1.92169i 3.87147i 0 0 1.00767 + 2.64284i 0 −3.29170 4.37507i
1079.5 −0.658334 1.25164i 0 −1.13319 + 1.64799i 2.08104i 0 0 2.80871 + 0.333415i 0 2.60471 1.37002i
1079.6 −0.658334 + 1.25164i 0 −1.13319 1.64799i 2.08104i 0 0 2.80871 0.333415i 0 2.60471 + 1.37002i
1079.7 −0.545545 1.30475i 0 −1.40476 + 1.42360i 0.698240i 0 0 2.62381 + 1.05623i 0 0.911031 0.380921i
1079.8 −0.545545 + 1.30475i 0 −1.40476 1.42360i 0.698240i 0 0 2.62381 1.05623i 0 0.911031 + 0.380921i
1079.9 0.545545 1.30475i 0 −1.40476 1.42360i 0.698240i 0 0 −2.62381 + 1.05623i 0 0.911031 + 0.380921i
1079.10 0.545545 + 1.30475i 0 −1.40476 + 1.42360i 0.698240i 0 0 −2.62381 1.05623i 0 0.911031 0.380921i
1079.11 0.658334 1.25164i 0 −1.13319 1.64799i 2.08104i 0 0 −2.80871 + 0.333415i 0 2.60471 + 1.37002i
1079.12 0.658334 + 1.25164i 0 −1.13319 + 1.64799i 2.08104i 0 0 −2.80871 0.333415i 0 2.60471 1.37002i
1079.13 1.13008 0.850247i 0 0.554161 1.92169i 3.87147i 0 0 −1.00767 2.64284i 0 −3.29170 4.37507i
1079.14 1.13008 + 0.850247i 0 0.554161 + 1.92169i 3.87147i 0 0 −1.00767 + 2.64284i 0 −3.29170 + 4.37507i
1079.15 1.41135 0.0900240i 0 1.98379 0.254110i 2.48866i 0 0 2.77694 0.537226i 0 −0.224040 3.51237i
1079.16 1.41135 + 0.0900240i 0 1.98379 + 0.254110i 2.48866i 0 0 2.77694 + 0.537226i 0 −0.224040 + 3.51237i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1079.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.e.h 16
3.b odd 2 1 inner 1764.2.e.h 16
4.b odd 2 1 inner 1764.2.e.h 16
7.b odd 2 1 1764.2.e.i 16
7.d odd 6 2 252.2.be.a 32
12.b even 2 1 inner 1764.2.e.h 16
21.c even 2 1 1764.2.e.i 16
21.g even 6 2 252.2.be.a 32
28.d even 2 1 1764.2.e.i 16
28.f even 6 2 252.2.be.a 32
84.h odd 2 1 1764.2.e.i 16
84.j odd 6 2 252.2.be.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.be.a 32 7.d odd 6 2
252.2.be.a 32 21.g even 6 2
252.2.be.a 32 28.f even 6 2
252.2.be.a 32 84.j odd 6 2
1764.2.e.h 16 1.a even 1 1 trivial
1764.2.e.h 16 3.b odd 2 1 inner
1764.2.e.h 16 4.b odd 2 1 inner
1764.2.e.h 16 12.b even 2 1 inner
1764.2.e.i 16 7.b odd 2 1
1764.2.e.i 16 21.c even 2 1
1764.2.e.i 16 28.d even 2 1
1764.2.e.i 16 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{8} + 26 T_{5}^{6} + 197 T_{5}^{4} + 492 T_{5}^{2} + 196 \)
\( T_{13}^{4} + 2 T_{13}^{3} - 25 T_{13}^{2} - 40 T_{13} + 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} - 10 T^{6} + 4 T^{8} - 40 T^{10} + 16 T^{12} + 256 T^{16} \)
$3$ 1
$5$ \( ( 1 - 14 T^{2} + 117 T^{4} - 698 T^{6} + 3576 T^{8} - 17450 T^{10} + 73125 T^{12} - 218750 T^{14} + 390625 T^{16} )^{2} \)
$7$ 1
$11$ \( ( 1 + 30 T^{2} + 757 T^{4} + 11666 T^{6} + 154360 T^{8} + 1411586 T^{10} + 11083237 T^{12} + 53146830 T^{14} + 214358881 T^{16} )^{2} \)
$13$ \( ( 1 + 2 T + 27 T^{2} + 38 T^{3} + 378 T^{4} + 494 T^{5} + 4563 T^{6} + 4394 T^{7} + 28561 T^{8} )^{4} \)
$17$ \( ( 1 - 76 T^{2} + 3096 T^{4} - 83940 T^{6} + 1656750 T^{8} - 24258660 T^{10} + 258581016 T^{12} - 1834455244 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 - 34 T^{2} + 1041 T^{4} - 25110 T^{6} + 610032 T^{8} - 9064710 T^{10} + 135664161 T^{12} - 1599559954 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 + 116 T^{2} + 6936 T^{4} + 269340 T^{6} + 7319790 T^{8} + 142480860 T^{10} + 1940977176 T^{12} + 17172163124 T^{14} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 - 146 T^{2} + 10849 T^{4} - 522290 T^{6} + 17802004 T^{8} - 439245890 T^{10} + 7673291569 T^{12} - 86844204866 T^{14} + 500246412961 T^{16} )^{2} \)
$31$ \( ( 1 - 132 T^{2} + 6430 T^{4} - 130264 T^{6} + 1736719 T^{8} - 125183704 T^{10} + 5938240030 T^{12} - 117150485892 T^{14} + 852891037441 T^{16} )^{2} \)
$37$ \( ( 1 + 2 T + 115 T^{2} + 282 T^{3} + 5758 T^{4} + 10434 T^{5} + 157435 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4} \)
$41$ \( ( 1 - 188 T^{2} + 15912 T^{4} - 846644 T^{6} + 36194574 T^{8} - 1423208564 T^{10} + 44963509032 T^{12} - 893019597308 T^{14} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 - 202 T^{2} + 17289 T^{4} - 881742 T^{6} + 37179072 T^{8} - 1630340958 T^{10} + 59107650489 T^{12} - 1276915335898 T^{14} + 11688200277601 T^{16} )^{2} \)
$47$ \( ( 1 + 200 T^{2} + 22212 T^{4} + 1666392 T^{6} + 91084806 T^{8} + 3681059928 T^{10} + 108387474372 T^{12} + 2155843065800 T^{14} + 23811286661761 T^{16} )^{2} \)
$53$ \( ( 1 - 282 T^{2} + 34177 T^{4} - 2497906 T^{6} + 140957380 T^{8} - 7016617954 T^{10} + 269672969137 T^{12} - 6250349838378 T^{14} + 62259690411361 T^{16} )^{2} \)
$59$ \( ( 1 + 186 T^{2} + 18161 T^{4} + 1259962 T^{6} + 77631844 T^{8} + 4385927722 T^{10} + 220063393121 T^{12} + 7845579257226 T^{14} + 146830437604321 T^{16} )^{2} \)
$61$ \( ( 1 + 4 T + 162 T^{2} + 588 T^{3} + 12546 T^{4} + 35868 T^{5} + 602802 T^{6} + 907924 T^{7} + 13845841 T^{8} )^{4} \)
$67$ \( ( 1 - 254 T^{2} + 32925 T^{4} - 3201102 T^{6} + 246095804 T^{8} - 14369746878 T^{10} + 663475658925 T^{12} - 22976429070926 T^{14} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 + 360 T^{2} + 62596 T^{4} + 7074488 T^{6} + 580384198 T^{8} + 35662494008 T^{10} + 1590669583876 T^{12} + 46116102211560 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 + 2 T + 141 T^{2} - 58 T^{3} + 12336 T^{4} - 4234 T^{5} + 751389 T^{6} + 778034 T^{7} + 28398241 T^{8} )^{4} \)
$79$ \( ( 1 - 332 T^{2} + 56998 T^{4} - 6691240 T^{6} + 595616087 T^{8} - 41760028840 T^{10} + 2220076716838 T^{12} - 80705035232972 T^{14} + 1517108809906561 T^{16} )^{2} \)
$83$ \( ( 1 + 110 T^{2} + 7989 T^{4} + 466194 T^{6} + 66686616 T^{8} + 3211610466 T^{10} + 379144526469 T^{12} + 35963441070590 T^{14} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 - 352 T^{2} + 72316 T^{4} - 10162848 T^{6} + 1047015622 T^{8} - 80499919008 T^{10} + 4537267900156 T^{12} - 174937414418272 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 - 22 T + 373 T^{2} - 3806 T^{3} + 41336 T^{4} - 369182 T^{5} + 3509557 T^{6} - 20078806 T^{7} + 88529281 T^{8} )^{4} \)
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