Properties

Label 380.2.l.b
Level $380$
Weight $2$
Character orbit 380.l
Analytic conductor $3.034$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} - 1564 x^{3} + 2284 x^{2} - 1088 x + 1370\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( \beta_{3} - \beta_{8} ) q^{5} + \beta_{6} q^{7} + ( \beta_{3} + 4 \beta_{5} - \beta_{6} - \beta_{8} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( \beta_{3} - \beta_{8} ) q^{5} + \beta_{6} q^{7} + ( \beta_{3} + 4 \beta_{5} - \beta_{6} - \beta_{8} ) q^{9} + ( 1 - \beta_{1} ) q^{11} + ( \beta_{2} + \beta_{7} + \beta_{11} ) q^{13} + ( \beta_{4} + \beta_{9} - \beta_{11} ) q^{15} + ( \beta_{1} - 2 \beta_{6} - \beta_{8} ) q^{17} + ( \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( \beta_{10} - \beta_{11} ) q^{21} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{8} ) q^{23} + ( -1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{25} + ( 2 \beta_{4} + \beta_{9} - \beta_{10} ) q^{27} + ( \beta_{2} + \beta_{4} ) q^{29} + ( \beta_{10} - \beta_{11} ) q^{31} + ( \beta_{7} + \beta_{11} ) q^{33} + ( 3 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{8} ) q^{35} + ( -\beta_{4} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{37} + ( -\beta_{5} + 5 \beta_{8} ) q^{39} + ( \beta_{2} - \beta_{4} + \beta_{10} - \beta_{11} ) q^{41} + ( -2 - 2 \beta_{1} + \beta_{3} + 2 \beta_{5} - 2 \beta_{8} ) q^{43} + ( -4 - 3 \beta_{1} - 3 \beta_{5} + 5 \beta_{6} + \beta_{8} ) q^{45} + ( 2 + 2 \beta_{1} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{8} ) q^{47} + ( -3 \beta_{5} - 2 \beta_{8} ) q^{49} + ( -\beta_{2} + \beta_{4} - 2 \beta_{10} ) q^{51} + ( -\beta_{2} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{53} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{55} + ( 3 - 2 \beta_{1} + \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{57} + ( -3 \beta_{2} - 3 \beta_{4} - 3 \beta_{7} - \beta_{9} ) q^{59} + ( 1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{61} + ( 3 + \beta_{1} - 5 \beta_{3} - 3 \beta_{5} + \beta_{8} ) q^{63} + ( 2 \beta_{2} - \beta_{4} + 2 \beta_{7} - \beta_{10} + \beta_{11} ) q^{65} + ( -3 \beta_{4} - \beta_{7} + \beta_{11} ) q^{67} + ( -2 \beta_{2} - 2 \beta_{4} - 3 \beta_{7} - \beta_{9} ) q^{69} + ( -\beta_{10} - \beta_{11} ) q^{71} + ( -6 - \beta_{1} + 6 \beta_{5} - \beta_{8} ) q^{73} + ( -\beta_{2} - 2 \beta_{4} - \beta_{7} - \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{75} + ( -1 - \beta_{1} - \beta_{5} + \beta_{8} ) q^{77} + ( \beta_{2} + \beta_{4} - \beta_{7} - \beta_{9} ) q^{79} + ( -2 - 5 \beta_{1} + 6 \beta_{3} + 6 \beta_{6} ) q^{81} + ( 7 - \beta_{1} + \beta_{3} - 7 \beta_{5} - \beta_{8} ) q^{83} + ( -7 + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{85} + ( -7 - \beta_{1} - 7 \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{87} + ( \beta_{2} + \beta_{4} + 2 \beta_{9} ) q^{89} + ( 2 \beta_{2} - 2 \beta_{4} - \beta_{10} + 3 \beta_{11} ) q^{91} + ( 3 + \beta_{1} - 8 \beta_{3} - 3 \beta_{5} + \beta_{8} ) q^{93} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{95} + ( \beta_{4} - 2 \beta_{9} + 2 \beta_{10} ) q^{97} + ( \beta_{3} + 3 \beta_{5} - \beta_{6} + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{5} + 4q^{7} + O(q^{10}) \) \( 12q + 4q^{5} + 4q^{7} + 8q^{11} - 4q^{17} + 4q^{23} - 4q^{25} + 36q^{35} - 28q^{43} - 40q^{45} + 20q^{47} - 16q^{55} + 24q^{57} + 16q^{61} + 20q^{63} - 76q^{73} - 16q^{77} + 4q^{81} + 84q^{83} - 76q^{85} - 80q^{87} + 8q^{93} - 16q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} - 1564 x^{3} + 2284 x^{2} - 1088 x + 1370\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(43066472536 \nu^{11} - 449979675808 \nu^{10} + 2364419690616 \nu^{9} - 10578246250319 \nu^{8} + 27902022591136 \nu^{7} - 79408221084824 \nu^{6} + 141181287966024 \nu^{5} - 265116944629568 \nu^{4} + 312068843894400 \nu^{3} - 358597198759640 \nu^{2} + 218631797615712 \nu - 279575999635621\)\()/ 79197078563257 \)
\(\beta_{2}\)\(=\)\((\)\(-20904914650254 \nu^{11} + 151211473768106 \nu^{10} - 794968673642910 \nu^{9} + 2887911715476177 \nu^{8} - 7417029052839577 \nu^{7} + 19179607630692878 \nu^{6} - 33837760885089870 \nu^{5} + 55088809656985999 \nu^{4} - 59614149648195498 \nu^{3} + 73129468722892382 \nu^{2} - 13260382630146966 \nu + 61336751231783802\)\()/ 22254379076275217 \)
\(\beta_{3}\)\(=\)\((\)\(-1434891834382 \nu^{11} + 4349786383365 \nu^{10} - 30559554581954 \nu^{9} + 42878862093020 \nu^{8} - 159886688468036 \nu^{7} + 171437976711065 \nu^{6} - 135293109090940 \nu^{5} - 195667629153178 \nu^{4} + 1037776394961430 \nu^{3} - 1640243274648676 \nu^{2} + 1313836759023712 \nu - 614674855329394\)\()/ 1171283109277643 \)
\(\beta_{4}\)\(=\)\((\)\(-36029778947630 \nu^{11} + 211750342123647 \nu^{10} - 1238018225399786 \nu^{9} + 3905072695392607 \nu^{8} - 11494568432177163 \nu^{7} + 26121941475654550 \nu^{6} - 52601844594880850 \nu^{5} + 76369650536598931 \nu^{4} - 107621131739087582 \nu^{3} + 103021520567154984 \nu^{2} - 109128561548555516 \nu + 82760354423821606\)\()/ 22254379076275217 \)
\(\beta_{5}\)\(=\)\((\)\(109895826 \nu^{11} - 371224964 \nu^{10} + 2654024378 \nu^{9} - 4749330090 \nu^{8} + 18447551030 \nu^{7} - 26443913423 \nu^{6} + 55312820760 \nu^{5} - 45534731950 \nu^{4} + 73343454746 \nu^{3} + 3299301738 \nu^{2} + 68436366132 \nu + 32432735911\)\()/ 57697329013 \)
\(\beta_{6}\)\(=\)\((\)\(46589648822398 \nu^{11} - 179791490723729 \nu^{10} + 1294747135943166 \nu^{9} - 2848570771913886 \nu^{8} + 10820535063658264 \nu^{7} - 17972775160857487 \nu^{6} + 39813418259552772 \nu^{5} - 44865504993333180 \nu^{4} + 60390304202600358 \nu^{3} - 34043951404704188 \nu^{2} + 31530397540338688 \nu + 6809468628902868\)\()/ 22254379076275217 \)
\(\beta_{7}\)\(=\)\((\)\(176200650116 \nu^{11} - 1329132544436 \nu^{10} + 6501003581929 \nu^{9} - 24026915922758 \nu^{8} + 54318387857268 \nu^{7} - 145683835248313 \nu^{6} + 220874607342796 \nu^{5} - 357135080326986 \nu^{4} + 349453006431700 \nu^{3} - 471595750135346 \nu^{2} + 272502725402962 \nu - 625898181929096\)\()/ 79197078563257 \)
\(\beta_{8}\)\(=\)\((\)\(60614681294844 \nu^{11} - 176924960309206 \nu^{10} + 1343717589033390 \nu^{9} - 1909836933490731 \nu^{8} + 8609778550923730 \nu^{7} - 10682537136506275 \nu^{6} + 24968890071655488 \nu^{5} - 24287455010154182 \nu^{4} + 49085377800415758 \nu^{3} - 35882940036857220 \nu^{2} + 77615330419094180 \nu - 21606519557536053\)\()/ 22254379076275217 \)
\(\beta_{9}\)\(=\)\((\)\(256791075958 \nu^{11} - 2213799473497 \nu^{10} + 10545830020631 \nu^{9} - 43417051770468 \nu^{8} + 98549420880028 \nu^{7} - 293506877658653 \nu^{6} + 456518283579298 \nu^{5} - 854726461820000 \nu^{4} + 874080236420692 \nu^{3} - 1149195769701478 \nu^{2} + 642334628102864 \nu - 965449441292380\)\()/ 79197078563257 \)
\(\beta_{10}\)\(=\)\((\)\(-115790335988434 \nu^{11} + 395417420255441 \nu^{10} - 2966086386470279 \nu^{9} + 5628049459277240 \nu^{8} - 23005474619523732 \nu^{7} + 35317879974459683 \nu^{6} - 80577050400795482 \nu^{5} + 98786412752558234 \nu^{4} - 135626919464003418 \nu^{3} + 128200897738300130 \nu^{2} - 191498153537718348 \nu + 88642056768645630\)\()/ 22254379076275217 \)
\(\beta_{11}\)\(=\)\((\)\(-120425382922218 \nu^{11} + 468739736789324 \nu^{10} - 3221660843066217 \nu^{9} + 6816331515095650 \nu^{8} - 24346391496367094 \nu^{7} + 38423409192569915 \nu^{6} - 81116872823766392 \nu^{5} + 82309718088189014 \nu^{4} - 110128017822072330 \nu^{3} + 32522730416457886 \nu^{2} - 114045130802806098 \nu - 6834518991404602\)\()/ 22254379076275217 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{2} - \beta_{1} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 3 \beta_{6} + 12 \beta_{5} + 10 \beta_{4} + 15 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} - 18\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(8 \beta_{11} - 16 \beta_{10} - 4 \beta_{9} - 30 \beta_{8} + 4 \beta_{7} - 37 \beta_{6} + 84 \beta_{5} + 16 \beta_{4} + 29 \beta_{3} - 24 \beta_{2} + 4 \beta_{1} + 20\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{11} - 10 \beta_{10} - 55 \beta_{9} - 21 \beta_{8} + 5 \beta_{7} - 150 \beta_{6} + 37 \beta_{5} - 85 \beta_{4} - 116 \beta_{3} - 105 \beta_{2} + 89 \beta_{1} + 273\)\()/2\)
\(\nu^{6}\)\(=\)\(-32 \beta_{11} + 99 \beta_{10} - 51 \beta_{9} + 183 \beta_{8} + 8 \beta_{7} + 99 \beta_{6} - 498 \beta_{5} - 210 \beta_{4} - 325 \beta_{3} + 60 \beta_{2} + 93 \beta_{1} + 238\)
\(\nu^{7}\)\(=\)\((\)\(-100 \beta_{11} + 586 \beta_{10} + 532 \beta_{9} + 1074 \beta_{8} - 70 \beta_{7} + 2615 \beta_{6} - 2858 \beta_{5} - 20 \beta_{4} - 77 \beta_{3} + 1742 \beta_{2} - 934 \beta_{1} - 2648\)\()/2\)
\(\nu^{8}\)\(=\)\(176 \beta_{11} - 616 \beta_{10} + 1388 \beta_{9} - 1120 \beta_{8} - 292 \beta_{7} + 1768 \beta_{6} + 3024 \beta_{5} + 2876 \beta_{4} + 4400 \beta_{3} + 1164 \beta_{2} - 2527 \beta_{1} - 6723\)
\(\nu^{9}\)\(=\)\(993 \beta_{11} - 5524 \beta_{10} - 348 \beta_{9} - 10000 \beta_{8} - 13512 \beta_{6} + 26866 \beta_{5} + 7641 \beta_{4} + 11633 \beta_{3} - 8961 \beta_{2} + 584 \beta_{1} + 1838\)
\(\nu^{10}\)\(=\)\(877 \beta_{11} - 5469 \beta_{10} - 19989 \beta_{9} - 9929 \beta_{8} + 4117 \beta_{7} - 57259 \beta_{6} + 26452 \beta_{5} - 21000 \beta_{4} - 32160 \beta_{3} - 37564 \beta_{2} + 36399 \beta_{1} + 97170\)
\(\nu^{11}\)\(=\)\((\)\(-23101 \beta_{11} + 124939 \beta_{10} - 88121 \beta_{9} + 226350 \beta_{8} + 17347 \beta_{7} + 85400 \beta_{6} - 607538 \beta_{5} - 316704 \beta_{4} - 481316 \beta_{3} + 56994 \beta_{2} + 160388 \beta_{1} + 427540\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(\beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.24060 + 1.01288i
−0.585043 + 2.22350i
0.344446 + 1.15131i
0.344446 1.84020i
−0.585043 1.05342i
1.24060 3.49408i
1.24060 1.01288i
−0.585043 2.22350i
0.344446 1.15131i
0.344446 + 1.84020i
−0.585043 + 1.05342i
1.24060 + 3.49408i
0 −2.25348 2.25348i 0 −1.48119 + 1.67513i 0 −1.48119 1.48119i 0 7.15633i 0
37.2 0 −1.63846 1.63846i 0 2.17009 + 0.539189i 0 2.17009 + 2.17009i 0 2.36910i 0
37.3 0 −1.49576 1.49576i 0 0.311108 2.21432i 0 0.311108 + 0.311108i 0 1.47457i 0
37.4 0 1.49576 + 1.49576i 0 0.311108 2.21432i 0 0.311108 + 0.311108i 0 1.47457i 0
37.5 0 1.63846 + 1.63846i 0 2.17009 + 0.539189i 0 2.17009 + 2.17009i 0 2.36910i 0
37.6 0 2.25348 + 2.25348i 0 −1.48119 + 1.67513i 0 −1.48119 1.48119i 0 7.15633i 0
113.1 0 −2.25348 + 2.25348i 0 −1.48119 1.67513i 0 −1.48119 + 1.48119i 0 7.15633i 0
113.2 0 −1.63846 + 1.63846i 0 2.17009 0.539189i 0 2.17009 2.17009i 0 2.36910i 0
113.3 0 −1.49576 + 1.49576i 0 0.311108 + 2.21432i 0 0.311108 0.311108i 0 1.47457i 0
113.4 0 1.49576 1.49576i 0 0.311108 + 2.21432i 0 0.311108 0.311108i 0 1.47457i 0
113.5 0 1.63846 1.63846i 0 2.17009 0.539189i 0 2.17009 2.17009i 0 2.36910i 0
113.6 0 2.25348 2.25348i 0 −1.48119 1.67513i 0 −1.48119 + 1.48119i 0 7.15633i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.l.b 12
3.b odd 2 1 3420.2.bb.d 12
5.b even 2 1 1900.2.l.b 12
5.c odd 4 1 inner 380.2.l.b 12
5.c odd 4 1 1900.2.l.b 12
15.e even 4 1 3420.2.bb.d 12
19.b odd 2 1 inner 380.2.l.b 12
57.d even 2 1 3420.2.bb.d 12
95.d odd 2 1 1900.2.l.b 12
95.g even 4 1 inner 380.2.l.b 12
95.g even 4 1 1900.2.l.b 12
285.j odd 4 1 3420.2.bb.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.l.b 12 1.a even 1 1 trivial
380.2.l.b 12 5.c odd 4 1 inner
380.2.l.b 12 19.b odd 2 1 inner
380.2.l.b 12 95.g even 4 1 inner
1900.2.l.b 12 5.b even 2 1
1900.2.l.b 12 5.c odd 4 1
1900.2.l.b 12 95.d odd 2 1
1900.2.l.b 12 95.g even 4 1
3420.2.bb.d 12 3.b odd 2 1
3420.2.bb.d 12 15.e even 4 1
3420.2.bb.d 12 57.d even 2 1
3420.2.bb.d 12 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 152 T_{3}^{8} + 5616 T_{3}^{4} + 59536 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 59536 + 5616 T^{4} + 152 T^{8} + T^{12} \)
$5$ \( ( 125 - 50 T + 15 T^{2} - 12 T^{3} + 3 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$7$ \( ( 8 - 24 T + 36 T^{2} + 8 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$11$ \( ( 4 - 4 T - 2 T^{2} + T^{3} )^{4} \)
$13$ \( 59536 + 408272 T^{4} + 1816 T^{8} + T^{12} \)
$17$ \( ( 1352 + 936 T + 324 T^{2} + 16 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$19$ \( 47045881 - 781926 T^{2} + 25631 T^{4} - 10484 T^{6} + 71 T^{8} - 6 T^{10} + T^{12} \)
$23$ \( ( 5000 - 3000 T + 900 T^{2} - 40 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$29$ \( ( -1952 + 496 T^{2} - 40 T^{4} + T^{6} )^{2} \)
$31$ \( ( 1952 + 2336 T^{2} + 96 T^{4} + T^{6} )^{2} \)
$37$ \( 59536 + 54727472 T^{4} + 20536 T^{8} + T^{12} \)
$41$ \( ( 7808 + 2432 T^{2} + 144 T^{4} + T^{6} )^{2} \)
$43$ \( ( 14792 - 344 T + 4 T^{2} + 200 T^{3} + 98 T^{4} + 14 T^{5} + T^{6} )^{2} \)
$47$ \( ( 2312 - 2040 T + 900 T^{2} + 232 T^{3} + 50 T^{4} - 10 T^{5} + T^{6} )^{2} \)
$53$ \( 37210000 + 91250864 T^{4} + 29016 T^{8} + T^{12} \)
$59$ \( ( -2818688 + 63808 T^{2} - 448 T^{4} + T^{6} )^{2} \)
$61$ \( ( 20 - 4 T - 4 T^{2} + T^{3} )^{4} \)
$67$ \( 59536 + 33883600 T^{4} + 16568 T^{8} + T^{12} \)
$71$ \( ( 1952 + 1088 T^{2} + 104 T^{4} + T^{6} )^{2} \)
$73$ \( ( 390728 + 203320 T + 52900 T^{2} + 7856 T^{3} + 722 T^{4} + 38 T^{5} + T^{6} )^{2} \)
$79$ \( ( -7808 + 4032 T^{2} - 184 T^{4} + T^{6} )^{2} \)
$83$ \( ( 803912 - 362648 T + 81796 T^{2} - 10744 T^{3} + 882 T^{4} - 42 T^{5} + T^{6} )^{2} \)
$89$ \( ( -329888 + 18960 T^{2} - 288 T^{4} + T^{6} )^{2} \)
$97$ \( 46306881767056 + 5323303152 T^{4} + 140856 T^{8} + T^{12} \)
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