Properties

Label 8281.2.a.co
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 16 x^{10} + 88 x^{8} - 197 x^{6} + 172 x^{4} - 36 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( 1 - \beta_{8} + \beta_{9} ) q^{4} + ( -\beta_{1} - \beta_{11} ) q^{5} + \beta_{4} q^{6} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{8} + ( \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( 1 - \beta_{8} + \beta_{9} ) q^{4} + ( -\beta_{1} - \beta_{11} ) q^{5} + \beta_{4} q^{6} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{8} + ( \beta_{2} - \beta_{3} ) q^{9} + ( -2 + \beta_{8} ) q^{10} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{11} + ( \beta_{6} + \beta_{9} ) q^{12} + ( -\beta_{4} + \beta_{5} - \beta_{7} ) q^{15} + ( 2 - \beta_{2} + \beta_{3} - \beta_{6} ) q^{16} + ( -3 - \beta_{3} + \beta_{8} + \beta_{9} ) q^{17} + ( -\beta_{1} - \beta_{4} - \beta_{5} - 2 \beta_{10} + \beta_{11} ) q^{18} + ( -\beta_{1} + \beta_{4} + \beta_{5} + \beta_{11} ) q^{19} + ( -\beta_{1} - \beta_{5} - \beta_{10} + 2 \beta_{11} ) q^{20} + ( 2 - 2 \beta_{3} - \beta_{6} ) q^{22} + ( -\beta_{2} - \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{23} + ( \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{24} + ( -1 + \beta_{3} - \beta_{8} - 2 \beta_{9} ) q^{25} + ( -2 - 2 \beta_{2} - \beta_{9} ) q^{27} + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{8} - \beta_{9} ) q^{29} + ( 1 - \beta_{3} - \beta_{8} - \beta_{9} ) q^{30} + ( -2 \beta_{1} - \beta_{4} - \beta_{5} - 2 \beta_{7} + 2 \beta_{11} ) q^{31} + ( \beta_{1} - \beta_{4} + 2 \beta_{10} - \beta_{11} ) q^{32} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} + 2 \beta_{10} + \beta_{11} ) q^{33} + ( -3 \beta_{1} - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{34} + ( -3 + \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{36} + ( 3 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{11} ) q^{37} + ( -3 + 2 \beta_{3} + \beta_{6} - \beta_{9} ) q^{38} + ( -1 + \beta_{2} + \beta_{6} - 2 \beta_{9} ) q^{40} + ( -\beta_{1} - 2 \beta_{4} - \beta_{7} + 3 \beta_{10} - 4 \beta_{11} ) q^{41} + ( 1 - \beta_{3} + 2 \beta_{6} + 2 \beta_{9} ) q^{43} + ( 3 \beta_{5} + 2 \beta_{7} + 2 \beta_{10} - 2 \beta_{11} ) q^{44} + ( \beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{45} + ( -2 \beta_{1} - 2 \beta_{4} + \beta_{5} + 3 \beta_{10} - \beta_{11} ) q^{46} + ( \beta_{7} - \beta_{10} + \beta_{11} ) q^{47} + ( 3 + 2 \beta_{2} - \beta_{3} - \beta_{9} ) q^{48} + ( -2 \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{50} + ( -1 - \beta_{2} - \beta_{3} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{51} + ( 1 - 2 \beta_{3} - \beta_{8} + 2 \beta_{9} ) q^{53} + ( -\beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{7} + 4 \beta_{10} - \beta_{11} ) q^{54} + ( -3 - \beta_{2} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{55} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + \beta_{7} - 2 \beta_{10} - \beta_{11} ) q^{57} + ( -\beta_{1} - \beta_{5} - \beta_{7} - 3 \beta_{10} + 3 \beta_{11} ) q^{58} + ( -4 \beta_{1} + 3 \beta_{7} - 4 \beta_{10} - \beta_{11} ) q^{59} + ( \beta_{1} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{60} + ( -2 \beta_{2} + \beta_{3} - 2 \beta_{9} ) q^{61} + ( -9 + \beta_{6} + 3 \beta_{8} - 5 \beta_{9} ) q^{62} + ( -2 - 4 \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{64} + ( -5 - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{9} ) q^{66} + ( -\beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{10} + 5 \beta_{11} ) q^{67} + ( -2 + \beta_{2} + \beta_{3} + 2 \beta_{8} - 3 \beta_{9} ) q^{68} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{69} + ( 3 \beta_{1} - \beta_{7} - 3 \beta_{10} + 2 \beta_{11} ) q^{71} + ( -4 \beta_{1} + \beta_{4} - \beta_{5} - 3 \beta_{7} + \beta_{11} ) q^{72} + ( -\beta_{1} - 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{10} + 4 \beta_{11} ) q^{73} + ( 7 + \beta_{3} - 2 \beta_{8} + 3 \beta_{9} ) q^{74} + ( \beta_{2} - 3 \beta_{3} - 2 \beta_{6} + 3 \beta_{8} + \beta_{9} ) q^{75} + ( -2 \beta_{1} - \beta_{4} - 3 \beta_{5} - 3 \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{76} + ( 6 + \beta_{2} + \beta_{6} + 3 \beta_{8} + \beta_{9} ) q^{79} + ( -2 \beta_{1} - 2 \beta_{4} - 4 \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{80} + ( -3 - \beta_{2} - 3 \beta_{3} - \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{81} + ( -2 - 3 \beta_{2} - 3 \beta_{3} - 4 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{82} + ( -3 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{10} - 4 \beta_{11} ) q^{83} + ( 5 \beta_{1} - \beta_{4} - 3 \beta_{7} + \beta_{10} + 6 \beta_{11} ) q^{85} + ( 3 \beta_{1} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 4 \beta_{10} + 2 \beta_{11} ) q^{86} + ( 3 - \beta_{2} + 3 \beta_{3} - \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{87} + ( -1 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - 3 \beta_{8} ) q^{88} + ( 4 \beta_{1} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{10} + 3 \beta_{11} ) q^{89} + ( 1 - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{90} + ( -7 - \beta_{2} - 4 \beta_{3} - 3 \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{92} + ( -\beta_{1} - \beta_{4} - \beta_{5} + 4 \beta_{7} - \beta_{11} ) q^{93} + ( \beta_{2} - \beta_{3} ) q^{94} + ( 1 + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{95} + ( -2 \beta_{1} + \beta_{7} + \beta_{11} ) q^{96} + ( -2 \beta_{1} - \beta_{4} - 2 \beta_{5} - 4 \beta_{10} + 2 \beta_{11} ) q^{97} + ( -2 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} - 6 \beta_{10} + 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6q^{3} + 8q^{4} + 2q^{9} + O(q^{10}) \) \( 12q - 6q^{3} + 8q^{4} + 2q^{9} - 24q^{10} + 2q^{12} + 16q^{16} - 34q^{17} + 30q^{22} + 6q^{23} - 10q^{25} - 12q^{27} + 2q^{29} + 22q^{30} - 26q^{36} - 38q^{38} - 2q^{40} + 22q^{43} + 38q^{48} + 8q^{51} + 16q^{53} - 30q^{55} + 10q^{61} - 82q^{62} - 2q^{64} - 68q^{66} - 22q^{68} - 14q^{69} + 66q^{74} - 2q^{75} + 70q^{79} - 28q^{81} - 10q^{82} + 20q^{87} - 28q^{88} - 66q^{92} + 2q^{94} + 4q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 16 x^{10} + 88 x^{8} - 197 x^{6} + 172 x^{4} - 36 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{10} - 66 \nu^{8} + 242 \nu^{6} - 127 \nu^{4} - 248 \nu^{2} + 32 \)\()/11\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{10} + 44 \nu^{8} - 209 \nu^{6} + 360 \nu^{4} - 223 \nu^{2} + 27 \)\()/11\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{11} + 44 \nu^{9} - 209 \nu^{7} + 360 \nu^{5} - 223 \nu^{3} + 27 \nu \)\()/11\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{11} + 22 \nu^{9} - 33 \nu^{7} - 244 \nu^{5} + 559 \nu^{3} - 191 \nu \)\()/11\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{10} + 110 \nu^{8} - 451 \nu^{6} + 476 \nu^{4} + 91 \nu^{2} - 27 \)\()/11\)
\(\beta_{7}\)\(=\)\( \nu^{9} - 14 \nu^{7} + 60 \nu^{5} - 76 \nu^{3} + 12 \nu \)
\(\beta_{8}\)\(=\)\((\)\( 10 \nu^{10} - 143 \nu^{8} + 638 \nu^{6} - 903 \nu^{4} + 263 \nu^{2} + 9 \)\()/11\)
\(\beta_{9}\)\(=\)\((\)\( 10 \nu^{10} - 143 \nu^{8} + 638 \nu^{6} - 903 \nu^{4} + 274 \nu^{2} - 24 \)\()/11\)
\(\beta_{10}\)\(=\)\((\)\( -8 \nu^{11} + 121 \nu^{9} - 605 \nu^{7} + 1147 \nu^{5} - 822 \nu^{3} + 171 \nu \)\()/11\)
\(\beta_{11}\)\(=\)\((\)\( -13 \nu^{11} + 198 \nu^{9} - 1001 \nu^{7} + 1923 \nu^{5} - 1333 \nu^{3} + 194 \nu \)\()/11\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} - \beta_{8} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{4} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(6 \beta_{9} - 6 \beta_{8} - \beta_{6} + \beta_{3} - \beta_{2} + 16\)
\(\nu^{5}\)\(=\)\(-9 \beta_{11} + 10 \beta_{10} + 8 \beta_{7} + 8 \beta_{5} + 7 \beta_{4} + 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(35 \beta_{9} - 37 \beta_{8} - 11 \beta_{6} + 6 \beta_{3} - 10 \beta_{2} + 94\)
\(\nu^{7}\)\(=\)\(-67 \beta_{11} + 79 \beta_{10} + 57 \beta_{7} + 58 \beta_{5} + 41 \beta_{4} + 176 \beta_{1}\)
\(\nu^{8}\)\(=\)\(205 \beta_{9} - 234 \beta_{8} - 95 \beta_{6} + 25 \beta_{3} - 79 \beta_{2} + 574\)
\(\nu^{9}\)\(=\)\(-474 \beta_{11} + 582 \beta_{10} + 395 \beta_{7} + 408 \beta_{5} + 230 \beta_{4} + 1092 \beta_{1}\)
\(\nu^{10}\)\(=\)\(1214 \beta_{9} - 1500 \beta_{8} - 747 \beta_{6} + 65 \beta_{3} - 582 \beta_{2} + 3576\)
\(\nu^{11}\)\(=\)\(-3290 \beta_{11} + 4158 \beta_{10} + 2708 \beta_{7} + 2829 \beta_{5} + 1279 \beta_{4} + 6872 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.58860
−2.30327
−1.37905
−1.34523
−0.499987
−0.180824
0.180824
0.499987
1.34523
1.37905
2.30327
2.58860
−2.58860 −0.518466 4.70085 1.61205 1.34210 0 −6.99143 −2.73119 −4.17296
1.2 −2.30327 1.47336 3.30504 0.847292 −3.39354 0 −3.00585 −0.829208 −1.95154
1.3 −1.37905 −2.88120 −0.0982074 0.805948 3.97334 0 2.89354 5.30133 −1.11145
1.4 −1.34523 −2.05010 −0.190366 3.56778 2.75785 0 2.94654 1.20292 −4.79947
1.5 −0.499987 −0.849601 −1.75001 −1.04248 0.424789 0 1.87496 −2.27818 0.521224
1.6 −0.180824 1.82601 −1.96730 2.68664 −0.330186 0 0.717383 0.334323 −0.485809
1.7 0.180824 1.82601 −1.96730 −2.68664 0.330186 0 −0.717383 0.334323 −0.485809
1.8 0.499987 −0.849601 −1.75001 1.04248 −0.424789 0 −1.87496 −2.27818 0.521224
1.9 1.34523 −2.05010 −0.190366 −3.56778 −2.75785 0 −2.94654 1.20292 −4.79947
1.10 1.37905 −2.88120 −0.0982074 −0.805948 −3.97334 0 −2.89354 5.30133 −1.11145
1.11 2.30327 1.47336 3.30504 −0.847292 3.39354 0 3.00585 −0.829208 −1.95154
1.12 2.58860 −0.518466 4.70085 −1.61205 −1.34210 0 6.99143 −2.73119 −4.17296
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.co 12
7.b odd 2 1 8281.2.a.cp 12
7.d odd 6 2 1183.2.e.j 24
13.b even 2 1 inner 8281.2.a.co 12
13.f odd 12 2 637.2.q.i 12
91.b odd 2 1 8281.2.a.cp 12
91.s odd 6 2 1183.2.e.j 24
91.w even 12 2 91.2.u.b yes 12
91.x odd 12 2 637.2.k.i 12
91.ba even 12 2 91.2.k.b 12
91.bc even 12 2 637.2.q.g 12
91.bd odd 12 2 637.2.u.g 12
273.bs odd 12 2 819.2.bm.f 12
273.ch odd 12 2 819.2.do.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b 12 91.ba even 12 2
91.2.u.b yes 12 91.w even 12 2
637.2.k.i 12 91.x odd 12 2
637.2.q.g 12 91.bc even 12 2
637.2.q.i 12 13.f odd 12 2
637.2.u.g 12 91.bd odd 12 2
819.2.bm.f 12 273.bs odd 12 2
819.2.do.e 12 273.ch odd 12 2
1183.2.e.j 24 7.d odd 6 2
1183.2.e.j 24 91.s odd 6 2
8281.2.a.co 12 1.a even 1 1 trivial
8281.2.a.co 12 13.b even 2 1 inner
8281.2.a.cp 12 7.b odd 2 1
8281.2.a.cp 12 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{12} - 16 T_{2}^{10} + 88 T_{2}^{8} - 197 T_{2}^{6} + 172 T_{2}^{4} - 36 T_{2}^{2} + 1 \)
\( T_{3}^{6} + 3 T_{3}^{5} - 5 T_{3}^{4} - 16 T_{3}^{3} + 4 T_{3}^{2} + 19 T_{3} + 7 \)
\( T_{5}^{12} - 25 T_{5}^{10} + 201 T_{5}^{8} - 636 T_{5}^{6} + 878 T_{5}^{4} - 539 T_{5}^{2} + 121 \)
\( T_{11}^{12} - 62 T_{11}^{10} + 1355 T_{11}^{8} - 13284 T_{11}^{6} + 61227 T_{11}^{4} - 122593 T_{11}^{2} + 85849 \)
\( T_{17}^{6} + 17 T_{17}^{5} + 96 T_{17}^{4} + 198 T_{17}^{3} + 56 T_{17}^{2} - 146 T_{17} + 19 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 36 T^{2} + 172 T^{4} - 197 T^{6} + 88 T^{8} - 16 T^{10} + T^{12} \)
$3$ \( ( 7 + 19 T + 4 T^{2} - 16 T^{3} - 5 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$5$ \( 121 - 539 T^{2} + 878 T^{4} - 636 T^{6} + 201 T^{8} - 25 T^{10} + T^{12} \)
$7$ \( T^{12} \)
$11$ \( 85849 - 122593 T^{2} + 61227 T^{4} - 13284 T^{6} + 1355 T^{8} - 62 T^{10} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( ( 19 - 146 T + 56 T^{2} + 198 T^{3} + 96 T^{4} + 17 T^{5} + T^{6} )^{2} \)
$19$ \( 1 - 474 T^{2} + 13117 T^{4} - 15833 T^{6} + 1984 T^{8} - 79 T^{10} + T^{12} \)
$23$ \( ( 793 - 646 T - 185 T^{2} + 259 T^{3} - 50 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$29$ \( ( 4009 - 5040 T + 1438 T^{2} + 190 T^{3} - 86 T^{4} - T^{5} + T^{6} )^{2} \)
$31$ \( 241274089 - 82568820 T^{2} + 10607464 T^{4} - 639617 T^{6} + 18424 T^{8} - 232 T^{10} + T^{12} \)
$37$ \( 123201 - 371790 T^{2} + 363609 T^{4} - 123741 T^{6} + 7164 T^{8} - 147 T^{10} + T^{12} \)
$41$ \( 389707081 - 501255685 T^{2} + 58359667 T^{4} - 2449760 T^{6} + 44563 T^{8} - 354 T^{10} + T^{12} \)
$43$ \( ( 20467 - 7744 T - 1377 T^{2} + 816 T^{3} - 49 T^{4} - 11 T^{5} + T^{6} )^{2} \)
$47$ \( 121 - 1243 T^{2} + 2634 T^{4} - 1968 T^{6} + 509 T^{8} - 41 T^{10} + T^{12} \)
$53$ \( ( -17 - 264 T - 611 T^{2} + 404 T^{3} - 38 T^{4} - 8 T^{5} + T^{6} )^{2} \)
$59$ \( 35582408689 - 14985781851 T^{2} + 693359866 T^{4} - 13023632 T^{6} + 120841 T^{8} - 553 T^{10} + T^{12} \)
$61$ \( ( 1777 - 4825 T + 1100 T^{2} + 354 T^{3} - 75 T^{4} - 5 T^{5} + T^{6} )^{2} \)
$67$ \( 5708255809 - 1907282039 T^{2} + 147600062 T^{4} - 4680243 T^{6} + 68286 T^{8} - 439 T^{10} + T^{12} \)
$71$ \( 639230089 - 469455277 T^{2} + 72145275 T^{4} - 3845232 T^{6} + 66611 T^{8} - 446 T^{10} + T^{12} \)
$73$ \( 484396081 - 886799840 T^{2} + 87039596 T^{4} - 3205221 T^{6} + 53832 T^{8} - 400 T^{10} + T^{12} \)
$79$ \( ( 255121 + 36737 T - 20673 T^{2} + 504 T^{3} + 344 T^{4} - 35 T^{5} + T^{6} )^{2} \)
$83$ \( 402363481 - 194879694 T^{2} + 33361345 T^{4} - 2359793 T^{6} + 59836 T^{8} - 463 T^{10} + T^{12} \)
$89$ \( 145033849 - 107270948 T^{2} + 24138686 T^{4} - 1621338 T^{6} + 42861 T^{8} - 430 T^{10} + T^{12} \)
$97$ \( 1681 - 25106 T^{2} + 125675 T^{4} - 216876 T^{6} + 29979 T^{8} - 361 T^{10} + T^{12} \)
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