Properties

Label 1089.3.c.l
Level $1089$
Weight $3$
Character orbit 1089.c
Analytic conductor $29.673$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 21 x^{14} + 227 x^{12} - 1488 x^{10} + 24225 x^{8} - 62832 x^{6} + 64372 x^{4} + 7986 x^{2} + 14641\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 11^{4} \)
Twist minimal: no (minimal twist has level 99)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + ( -3 + \beta_{2} ) q^{4} + \beta_{8} q^{5} + ( -\beta_{1} - \beta_{7} ) q^{7} + ( 4 \beta_{3} - \beta_{10} ) q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} + ( -3 + \beta_{2} ) q^{4} + \beta_{8} q^{5} + ( -\beta_{1} - \beta_{7} ) q^{7} + ( 4 \beta_{3} - \beta_{10} ) q^{8} + ( \beta_{1} - \beta_{5} - \beta_{12} ) q^{10} + ( \beta_{1} + \beta_{5} + 4 \beta_{7} ) q^{13} + ( 2 \beta_{6} + \beta_{9} ) q^{14} + ( 17 - 3 \beta_{2} + 2 \beta_{4} + \beta_{11} ) q^{16} + ( 3 \beta_{3} + \beta_{10} - 2 \beta_{13} ) q^{17} + ( \beta_{1} - 3 \beta_{5} + \beta_{7} ) q^{19} + ( -2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - \beta_{14} ) q^{20} + ( -\beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{23} + ( 1 + 2 \beta_{2} + \beta_{4} + 2 \beta_{11} ) q^{25} + ( -5 \beta_{6} + \beta_{8} + \beta_{14} ) q^{26} + ( 10 \beta_{1} - \beta_{5} + 4 \beta_{7} - 4 \beta_{12} ) q^{28} + ( \beta_{3} + \beta_{10} - 3 \beta_{13} - \beta_{15} ) q^{29} + ( -4 - 2 \beta_{2} + \beta_{4} + 4 \beta_{11} ) q^{31} + ( -16 \beta_{3} + 3 \beta_{10} - \beta_{13} - 2 \beta_{15} ) q^{32} + ( 20 - 2 \beta_{2} - 2 \beta_{4} + \beta_{11} ) q^{34} + ( 3 \beta_{3} - 2 \beta_{10} + 2 \beta_{13} ) q^{35} + ( -18 + 2 \beta_{2} - \beta_{11} ) q^{37} + ( -2 \beta_{6} - 3 \beta_{8} - 4 \beta_{9} - 3 \beta_{14} ) q^{38} + ( -14 \beta_{1} + 4 \beta_{5} - 6 \beta_{7} - \beta_{12} ) q^{40} + ( 5 \beta_{3} + 3 \beta_{10} + \beta_{13} ) q^{41} + ( \beta_{1} - \beta_{5} - 6 \beta_{12} ) q^{43} + ( -11 \beta_{1} + 3 \beta_{5} - \beta_{7} ) q^{46} + ( 2 \beta_{6} - 3 \beta_{8} - 3 \beta_{14} ) q^{47} + ( 26 + 3 \beta_{2} - \beta_{4} - \beta_{11} ) q^{49} + ( 12 \beta_{3} + \beta_{13} - \beta_{15} ) q^{50} + ( -18 \beta_{1} - 8 \beta_{5} - 9 \beta_{7} + 8 \beta_{12} ) q^{52} + ( 3 \beta_{6} - 4 \beta_{8} - 5 \beta_{9} ) q^{53} + ( -10 \beta_{6} - 5 \beta_{8} - 7 \beta_{9} - \beta_{14} ) q^{56} + ( 5 - 3 \beta_{2} - 9 \beta_{4} + 2 \beta_{11} ) q^{58} + ( -6 \beta_{6} - 6 \beta_{8} - 5 \beta_{9} + 4 \beta_{14} ) q^{59} + ( 7 \beta_{1} + 5 \beta_{5} - 11 \beta_{7} - 2 \beta_{12} ) q^{61} + ( \beta_{3} + 4 \beta_{10} + 3 \beta_{13} - \beta_{15} ) q^{62} + ( -49 + 8 \beta_{2} - 12 \beta_{4} + 2 \beta_{11} ) q^{64} + ( 4 \beta_{10} - 9 \beta_{13} ) q^{65} + ( 16 + 7 \beta_{2} - 2 \beta_{4} + 2 \beta_{11} ) q^{67} + ( -14 \beta_{3} + 2 \beta_{10} - 5 \beta_{13} + 2 \beta_{15} ) q^{68} + ( 23 - 7 \beta_{2} + 4 \beta_{4} ) q^{70} + ( -\beta_{6} + \beta_{8} - 4 \beta_{9} + 5 \beta_{14} ) q^{71} + ( 5 \beta_{1} - \beta_{5} + 13 \beta_{7} - 11 \beta_{12} ) q^{73} + ( 26 \beta_{3} - 2 \beta_{10} - \beta_{13} ) q^{74} + ( -31 \beta_{1} + 19 \beta_{5} + 2 \beta_{7} + 10 \beta_{12} ) q^{76} + ( -16 \beta_{1} - 14 \beta_{5} - 13 \beta_{7} - 7 \beta_{12} ) q^{79} + ( 11 \beta_{6} + 11 \beta_{8} + 10 \beta_{9} ) q^{80} + ( 32 + 5 \beta_{2} - 6 \beta_{4} - 4 \beta_{11} ) q^{82} + ( -6 \beta_{3} + 4 \beta_{10} + 4 \beta_{13} - 3 \beta_{15} ) q^{83} + ( -5 \beta_{1} - \beta_{5} + 24 \beta_{7} + 16 \beta_{12} ) q^{85} + ( -7 \beta_{6} - 7 \beta_{8} - 2 \beta_{9} - \beta_{14} ) q^{86} + ( 11 \beta_{6} - 2 \beta_{8} + 6 \beta_{9} ) q^{89} + ( 48 - 4 \beta_{2} - 2 \beta_{4} + 3 \beta_{11} ) q^{91} + ( 8 \beta_{6} + 11 \beta_{8} + 6 \beta_{9} + 3 \beta_{14} ) q^{92} + ( \beta_{1} + 23 \beta_{5} + 10 \beta_{7} + 2 \beta_{12} ) q^{94} + ( -21 \beta_{3} + 5 \beta_{10} - 8 \beta_{13} ) q^{95} + ( -47 - 10 \beta_{2} - 9 \beta_{4} + 6 \beta_{11} ) q^{97} + ( -12 \beta_{3} - 5 \beta_{10} + \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 44q^{4} + O(q^{10}) \) \( 16q - 44q^{4} + 244q^{16} + 16q^{25} - 80q^{31} + 328q^{34} - 280q^{37} + 436q^{49} + 140q^{58} - 656q^{64} + 300q^{67} + 308q^{70} + 580q^{82} + 768q^{91} - 720q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 21 x^{14} + 227 x^{12} - 1488 x^{10} + 24225 x^{8} - 62832 x^{6} + 64372 x^{4} + 7986 x^{2} + 14641\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(45667752444 \nu^{14} - 597804978248 \nu^{12} + 3528390713769 \nu^{10} - 621669021475 \nu^{8} + 721124010273630 \nu^{6} + 5017459744442882 \nu^{4} - 2844369989518299 \nu^{2} + 332158447600837\)\()/ 3911331152625545 \)
\(\beta_{2}\)\(=\)\((\)\(5084684619 \nu^{14} - 102502036623 \nu^{12} + 1068420907817 \nu^{10} - 6488335560210 \nu^{8} + 114684022273170 \nu^{6} - 194838435402303 \nu^{4} - 38302095275829 \nu^{2} + 2293183755931321\)\()/ 355575559329595 \)
\(\beta_{3}\)\(=\)\((\)\(7530891822 \nu^{15} - 161922628410 \nu^{13} + 1773071437239 \nu^{11} - 11760003820535 \nu^{9} + 185084618505135 \nu^{7} - 549582649580084 \nu^{5} + 383464959034245 \nu^{3} + 243765923221832 \nu\)\()/ 355575559329595 \)
\(\beta_{4}\)\(=\)\((\)\(-6262229615 \nu^{14} + 123470462925 \nu^{12} - 1253712964279 \nu^{10} + 7604818484385 \nu^{8} - 143729875320340 \nu^{6} + 245252299046205 \nu^{4} + 48330735202860 \nu^{2} + 159603221202733\)\()/ 355575559329595 \)
\(\beta_{5}\)\(=\)\((\)\(-226078922543 \nu^{14} + 5080279956993 \nu^{12} - 57615885824889 \nu^{10} + 397910034267221 \nu^{8} - 5837952403045238 \nu^{6} + 21521280774481015 \nu^{4} - 22558715111757198 \nu^{2} + 619468120074969\)\()/ 3911331152625545 \)
\(\beta_{6}\)\(=\)\((\)\(445551515192 \nu^{15} - 9307380902916 \nu^{13} + 100246591559245 \nu^{11} - 655811636243113 \nu^{9} + 10767587725429269 \nu^{7} - 27112360148554791 \nu^{5} + 28933184524975093 \nu^{3} - 11359471726161328 \nu\)\()/ 3911331152625545 \)
\(\beta_{7}\)\(=\)\((\)\(373009540956 \nu^{14} - 8272667397206 \nu^{12} + 93216689770491 \nu^{10} - 641417043065372 \nu^{8} + 9554475942835496 \nu^{6} - 33184103725585560 \nu^{4} + 35403359064010586 \nu^{2} - 908176758810299\)\()/ 3911331152625545 \)
\(\beta_{8}\)\(=\)\((\)\(-598241835319 \nu^{15} + 13289199325505 \nu^{13} - 150985563543754 \nu^{11} + 1053101453448714 \nu^{9} - 15549320783909747 \nu^{7} + 54993468657667254 \nu^{5} - 81927088464326798 \nu^{3} + 26696886947325782 \nu\)\()/ 3911331152625545 \)
\(\beta_{9}\)\(=\)\((\)\(-675341589602 \nu^{15} + 13868921723197 \nu^{13} - 147009169987501 \nu^{11} + 940994096476483 \nu^{9} - 15981967603157154 \nu^{7} + 35444318312643801 \nu^{5} - 30785562544956175 \nu^{3} + 13743078230669150 \nu\)\()/ 3911331152625545 \)
\(\beta_{10}\)\(=\)\((\)\(87882372397 \nu^{15} - 1883160071640 \nu^{13} + 20577181071988 \nu^{11} - 136163991479220 \nu^{9} + 2153687201361460 \nu^{7} - 6248425886583264 \nu^{5} + 4466286022482460 \nu^{3} + 2839478527701024 \nu\)\()/ 355575559329595 \)
\(\beta_{11}\)\(=\)\((\)\(76482029149 \nu^{14} - 1530855816173 \nu^{12} + 15742685210216 \nu^{10} - 96069570646780 \nu^{8} + 1734861504036050 \nu^{6} - 2951600294906813 \nu^{4} - 580703621221449 \nu^{2} + 4020757524848768\)\()/ 355575559329595 \)
\(\beta_{12}\)\(=\)\((\)\(914548814197 \nu^{14} - 18896431939389 \nu^{12} + 202148975636349 \nu^{10} - 1310700954966809 \nu^{8} + 21893640153645292 \nu^{6} - 51225043168201705 \nu^{4} + 62937332947605996 \nu^{2} - 757739325471612\)\()/ 3911331152625545 \)
\(\beta_{13}\)\(=\)\((\)\(-125817353665 \nu^{15} + 2586721454850 \nu^{13} - 27473712050138 \nu^{11} + 176068897871470 \nu^{9} - 2979228810240980 \nu^{7} + 6639244523357525 \nu^{5} - 6254386540011170 \nu^{3} - 3981574876275964 \nu\)\()/ 355575559329595 \)
\(\beta_{14}\)\(=\)\((\)\(1590497768908 \nu^{15} - 33743007445253 \nu^{13} + 368452116183311 \nu^{11} - 2450024395518921 \nu^{9} + 39102063088234488 \nu^{7} - 108717170977956420 \nu^{5} + 131071031074296203 \nu^{3} - 47917933122741968 \nu\)\()/ 3911331152625545 \)
\(\beta_{15}\)\(=\)\((\)\(26847260804 \nu^{15} - 566324648655 \nu^{13} + 6124941929985 \nu^{11} - 40072005667775 \nu^{9} + 649352121603960 \nu^{7} - 1717349454316168 \nu^{5} + 1352701250986305 \nu^{3} + 860414508504650 \nu\)\()/ 32325050848145 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} - \beta_{14} - 2 \beta_{13} + 2 \beta_{10} + 4 \beta_{9} + 6 \beta_{6} + 2 \beta_{3}\)\()/22\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{2} + \beta_{1} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{15} + 2 \beta_{14} - \beta_{13} + 3 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 12 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-15 \beta_{12} + \beta_{11} + 27 \beta_{7} - 6 \beta_{5} + 11 \beta_{4} - 3 \beta_{2} + 52 \beta_{1} + 3\)\()/2\)
\(\nu^{5}\)\(=\)\(-2 \beta_{15} - \beta_{13} + 19 \beta_{10} - 160 \beta_{3}\)
\(\nu^{6}\)\(=\)\((\)\(-215 \beta_{12} - 15 \beta_{11} + 354 \beta_{7} - 137 \beta_{5} - 127 \beta_{4} + 70 \beta_{2} + 735 \beta_{1} - 225\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-179 \beta_{15} - 272 \beta_{14} - 146 \beta_{13} + 640 \beta_{10} - 589 \beta_{9} - 354 \beta_{8} - 397 \beta_{6} - 2889 \beta_{3}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-1420 \beta_{12} - 494 \beta_{11} + 807 \beta_{7} - 3356 \beta_{5} - 2533 \beta_{4} + 4305 \beta_{2} + 5239 \beta_{1} - 21042\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-1537 \beta_{15} - 2486 \beta_{14} - 1325 \beta_{13} + 4352 \beta_{10} - 13120 \beta_{9} - 5583 \beta_{8} - 18507 \beta_{6} - 12658 \beta_{3}\)\()/2\)
\(\nu^{10}\)\(=\)\(-4305 \beta_{11} - 17010 \beta_{4} + 43805 \beta_{2} - 226193\)
\(\nu^{11}\)\(=\)\((\)\(19463 \beta_{15} - 41352 \beta_{14} + 16436 \beta_{13} - 60241 \beta_{10} - 196922 \beta_{9} - 86435 \beta_{8} - 266948 \beta_{6} + 214479 \beta_{3}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(362330 \beta_{12} - 108925 \beta_{11} - 379152 \beta_{7} + 578566 \beta_{5} - 531443 \beta_{4} + 980208 \beta_{2} - 1290483 \beta_{1} - 4851218\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(531443 \beta_{15} - 578566 \beta_{14} + 422518 \beta_{13} - 2043094 \beta_{10} - 1869049 \beta_{9} - 940896 \beta_{8} - 2031965 \beta_{6} + 10065851 \beta_{3}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(9766705 \beta_{12} - 640368 \beta_{11} - 13570866 \beta_{7} + 10225888 \beta_{5} - 3503633 \beta_{4} + 5282864 \beta_{2} - 33907236 \beta_{1} - 25255599\)\()/2\)
\(\nu^{15}\)\(=\)\(4302656 \beta_{15} + 3322448 \beta_{13} - 18032177 \beta_{10} + 97256460 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
604.1
−3.67414 1.19380i
3.67414 1.19380i
1.83190 + 2.52140i
−1.83190 + 2.52140i
1.32111 0.429256i
−1.32111 0.429256i
−0.386583 + 0.532086i
0.386583 + 0.532086i
−0.386583 0.532086i
0.386583 0.532086i
1.32111 + 0.429256i
−1.32111 + 0.429256i
1.83190 2.52140i
−1.83190 2.52140i
−3.67414 + 1.19380i
3.67414 + 1.19380i
3.86322i 0 −10.9245 −4.15040 0 7.44465i 26.7509i 0 16.0339i
604.2 3.86322i 0 −10.9245 4.15040 0 7.44465i 26.7509i 0 16.0339i
604.3 3.11662i 0 −5.71334 −1.42146 0 3.80859i 5.33982i 0 4.43016i
604.4 3.11662i 0 −5.71334 1.42146 0 3.80859i 5.33982i 0 4.43016i
604.5 1.38910i 0 2.07040 −4.37284 0 0.612830i 8.43240i 0 6.07432i
604.6 1.38910i 0 2.07040 4.37284 0 0.612830i 8.43240i 0 6.07432i
604.7 0.657695i 0 3.56744 −8.10135 0 4.08611i 4.97707i 0 5.32822i
604.8 0.657695i 0 3.56744 8.10135 0 4.08611i 4.97707i 0 5.32822i
604.9 0.657695i 0 3.56744 −8.10135 0 4.08611i 4.97707i 0 5.32822i
604.10 0.657695i 0 3.56744 8.10135 0 4.08611i 4.97707i 0 5.32822i
604.11 1.38910i 0 2.07040 −4.37284 0 0.612830i 8.43240i 0 6.07432i
604.12 1.38910i 0 2.07040 4.37284 0 0.612830i 8.43240i 0 6.07432i
604.13 3.11662i 0 −5.71334 −1.42146 0 3.80859i 5.33982i 0 4.43016i
604.14 3.11662i 0 −5.71334 1.42146 0 3.80859i 5.33982i 0 4.43016i
604.15 3.86322i 0 −10.9245 −4.15040 0 7.44465i 26.7509i 0 16.0339i
604.16 3.86322i 0 −10.9245 4.15040 0 7.44465i 26.7509i 0 16.0339i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 604.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.c.l 16
3.b odd 2 1 inner 1089.3.c.l 16
11.b odd 2 1 inner 1089.3.c.l 16
11.c even 5 1 99.3.k.b 16
11.d odd 10 1 99.3.k.b 16
33.d even 2 1 inner 1089.3.c.l 16
33.f even 10 1 99.3.k.b 16
33.h odd 10 1 99.3.k.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.k.b 16 11.c even 5 1
99.3.k.b 16 11.d odd 10 1
99.3.k.b 16 33.f even 10 1
99.3.k.b 16 33.h odd 10 1
1089.3.c.l 16 1.a even 1 1 trivial
1089.3.c.l 16 3.b odd 2 1 inner
1089.3.c.l 16 11.b odd 2 1 inner
1089.3.c.l 16 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 27 T_{2}^{6} + 204 T_{2}^{4} + 363 T_{2}^{2} + 121 \) acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 121 + 363 T^{2} + 204 T^{4} + 27 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( ( 43681 - 27104 T^{2} + 2921 T^{4} - 104 T^{6} + T^{8} )^{2} \)
$7$ \( ( 5041 + 14163 T^{2} + 2004 T^{4} + 87 T^{6} + T^{8} )^{2} \)
$11$ \( T^{16} \)
$13$ \( ( 441168016 + 16689588 T^{2} + 199029 T^{4} + 822 T^{6} + T^{8} )^{2} \)
$17$ \( ( 28344976 + 5547608 T^{2} + 275249 T^{4} + 1177 T^{6} + T^{8} )^{2} \)
$19$ \( ( 5628000400 + 111594900 T^{2} + 697785 T^{4} + 1545 T^{6} + T^{8} )^{2} \)
$23$ \( ( 1301766400 - 68631200 T^{2} + 581765 T^{4} - 1430 T^{6} + T^{8} )^{2} \)
$29$ \( ( 136766832400 + 1403466900 T^{2} + 4217685 T^{4} + 3795 T^{6} + T^{8} )^{2} \)
$31$ \( ( 592295 - 14190 T - 2045 T^{2} + 20 T^{3} + T^{4} )^{4} \)
$37$ \( ( -21780 + 9570 T + 1535 T^{2} + 70 T^{3} + T^{4} )^{4} \)
$41$ \( ( 328443610000 + 2839083500 T^{2} + 7400225 T^{4} + 5410 T^{6} + T^{8} )^{2} \)
$43$ \( ( 431696305296 + 5069235348 T^{2} + 8632349 T^{4} + 5102 T^{6} + T^{8} )^{2} \)
$47$ \( ( 1065065280400 - 13123720500 T^{2} + 17652665 T^{4} - 7755 T^{6} + T^{8} )^{2} \)
$53$ \( ( 130872341603025 - 186513356700 T^{2} + 85663765 T^{4} - 15780 T^{6} + T^{8} )^{2} \)
$59$ \( ( 640510301973025 - 552445404500 T^{2} + 170933165 T^{4} - 22220 T^{6} + T^{8} )^{2} \)
$61$ \( ( 3114589632400 + 33068752500 T^{2} + 59688585 T^{4} + 14865 T^{6} + T^{8} )^{2} \)
$67$ \( ( -3482380 + 263450 T - 3655 T^{2} - 75 T^{3} + T^{4} )^{4} \)
$71$ \( ( 128324037120400 - 230194235700 T^{2} + 122338865 T^{4} - 21555 T^{6} + T^{8} )^{2} \)
$73$ \( ( 494466201667216 + 914205087428 T^{2} + 289934849 T^{4} + 30217 T^{6} + T^{8} )^{2} \)
$79$ \( ( 436852010010025 + 1461577192875 T^{2} + 468510660 T^{4} + 39735 T^{6} + T^{8} )^{2} \)
$83$ \( ( 1450273074167401 + 1766333903213 T^{2} + 418791164 T^{4} + 35437 T^{6} + T^{8} )^{2} \)
$89$ \( ( 33083847444736 - 131362519376 T^{2} + 132376541 T^{4} - 29246 T^{6} + T^{8} )^{2} \)
$97$ \( ( -45053405 - 1687290 T - 7305 T^{2} + 180 T^{3} + T^{4} )^{4} \)
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