Properties

Label 1520.2.g.g
Level $1520$
Weight $2$
Character orbit 1520.g
Analytic conductor $12.137$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 20 x^{14} + 271 x^{12} - 2000 x^{10} + 10645 x^{8} - 29570 x^{6} + 58816 x^{4} - 56840 x^{2} + 38416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{15} q^{5} + \beta_{10} q^{7} + ( -2 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + \beta_{6} q^{3} + \beta_{15} q^{5} + \beta_{10} q^{7} + ( -2 + \beta_{1} ) q^{9} + ( \beta_{3} - \beta_{4} ) q^{11} + ( \beta_{8} - \beta_{10} ) q^{13} + ( 2 - \beta_{1} + \beta_{6} - \beta_{12} + \beta_{14} ) q^{15} + \beta_{12} q^{17} + ( -1 - \beta_{4} + \beta_{5} ) q^{19} + ( \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{21} + ( -\beta_{10} - 2 \beta_{11} ) q^{23} + ( -\beta_{5} + \beta_{12} ) q^{25} + ( -\beta_{6} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{27} + ( \beta_{2} - \beta_{3} - \beta_{7} - \beta_{9} ) q^{29} + ( 6 - \beta_{14} + \beta_{15} ) q^{31} + ( \beta_{7} - \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{33} + ( \beta_{2} - \beta_{3} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{35} + ( -\beta_{8} - \beta_{10} - \beta_{11} ) q^{37} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{39} + ( -4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{41} + ( \beta_{7} - \beta_{9} - 2 \beta_{11} ) q^{43} + ( -2 + \beta_{5} + 3 \beta_{6} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{45} + ( 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{47} + ( -\beta_{5} - 2 \beta_{14} + 2 \beta_{15} ) q^{49} + ( -1 - \beta_{5} - 2 \beta_{14} + 2 \beta_{15} ) q^{51} + ( \beta_{8} - 3 \beta_{11} ) q^{53} + ( \beta_{2} - \beta_{4} - 2 \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{55} + ( \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{57} + ( -5 - 2 \beta_{1} + \beta_{5} ) q^{59} + ( -2 - \beta_{1} + \beta_{5} + \beta_{14} - \beta_{15} ) q^{61} + ( -2 \beta_{8} - 2 \beta_{10} ) q^{63} + ( -3 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{11} ) q^{65} + ( 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{67} + ( \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{69} + ( -6 + \beta_{14} - \beta_{15} ) q^{71} + ( -2 \beta_{6} + \beta_{12} + 2 \beta_{13} ) q^{73} + ( -1 - \beta_{5} - \beta_{6} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{75} + ( 4 \beta_{6} - 2 \beta_{12} + \beta_{14} + \beta_{15} ) q^{77} + ( 2 + 2 \beta_{1} - 2 \beta_{14} + 2 \beta_{15} ) q^{79} + ( -1 - \beta_{1} + \beta_{5} ) q^{81} -2 \beta_{8} q^{83} + ( -\beta_{1} - \beta_{6} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{85} + ( -2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + 6 \beta_{11} ) q^{87} + ( -2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - \beta_{7} - \beta_{9} ) q^{89} + ( -3 - 2 \beta_{1} - \beta_{5} ) q^{91} + ( 8 \beta_{6} - 2 \beta_{12} + \beta_{14} + \beta_{15} ) q^{93} + ( \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} + \beta_{13} + 3 \beta_{14} ) q^{95} + ( -2 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{97} + ( 4 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + \beta_{7} + \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 32q^{9} + O(q^{10}) \) \( 16q - 32q^{9} + 32q^{15} - 8q^{19} - 8q^{25} + 96q^{31} - 24q^{45} - 8q^{49} - 24q^{51} - 72q^{59} - 24q^{61} - 96q^{71} - 24q^{75} + 32q^{79} - 8q^{81} - 56q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 20 x^{14} + 271 x^{12} - 2000 x^{10} + 10645 x^{8} - 29570 x^{6} + 58816 x^{4} - 56840 x^{2} + 38416\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4429 \nu^{14} - 77314 \nu^{12} + 1007855 \nu^{10} - 6294330 \nu^{8} + 31135885 \nu^{6} - 38435480 \nu^{4} + 37711184 \nu^{2} + 469265976 \)\()/ 125016960 \)
\(\beta_{2}\)\(=\)\((\)\( 397497 \nu^{14} - 7515898 \nu^{12} + 100144915 \nu^{10} - 696224210 \nu^{8} + 3614511225 \nu^{6} - 8702669560 \nu^{4} + 19612506512 \nu^{2} - 12772202408 \)\()/ 6125831040 \)
\(\beta_{3}\)\(=\)\((\)\(-2445217 \nu^{14} + 50395018 \nu^{12} - 682415115 \nu^{10} + 5182162850 \nu^{8} - 27198089825 \nu^{6} + 77259028920 \nu^{4} - 109267132432 \nu^{2} + 81350328808\)\()/ 18377493120 \)
\(\beta_{4}\)\(=\)\((\)\( -32348 \nu^{14} + 671313 \nu^{12} - 9132730 \nu^{10} + 68382760 \nu^{8} - 357807455 \nu^{6} + 968268310 \nu^{4} - 1598980868 \nu^{2} + 1133374508 \)\()/ 191432220 \)
\(\beta_{5}\)\(=\)\((\)\( 258 \nu^{14} - 4537 \nu^{12} + 58710 \nu^{10} - 366660 \nu^{8} + 1709600 \nu^{6} - 2238960 \nu^{4} + 2196768 \nu^{2} + 3806333 \)\()/976695\)
\(\beta_{6}\)\(=\)\((\)\( 397497 \nu^{15} - 7515898 \nu^{13} + 100144915 \nu^{11} - 696224210 \nu^{9} + 3614511225 \nu^{7} - 8702669560 \nu^{5} + 19612506512 \nu^{3} - 6646371368 \nu \)\()/ 12251662080 \)
\(\beta_{7}\)\(=\)\((\)\(-8087003 \nu^{15} + 15441888 \nu^{14} + 174764750 \nu^{13} - 258490848 \nu^{12} - 2399690025 \nu^{11} + 3340938720 \nu^{10} + 19149751990 \nu^{9} - 20673461760 \nu^{8} - 106365244795 \nu^{7} + 107038243200 \nu^{6} + 362008005480 \nu^{5} - 223553890560 \nu^{4} - 784321942448 \nu^{3} + 688708328448 \nu^{2} + 1260457378040 \nu - 419116013568\)\()/ 257284903680 \)
\(\beta_{8}\)\(=\)\((\)\(-3060399 \nu^{15} + 28801046 \nu^{13} - 243837925 \nu^{11} - 1681047170 \nu^{9} + 19983308945 \nu^{7} - 176800898360 \nu^{5} + 369343421456 \nu^{3} - 610375641064 \nu\)\()/ 85761634560 \)
\(\beta_{9}\)\(=\)\((\)\(8087003 \nu^{15} + 15441888 \nu^{14} - 174764750 \nu^{13} - 258490848 \nu^{12} + 2399690025 \nu^{11} + 3340938720 \nu^{10} - 19149751990 \nu^{9} - 20673461760 \nu^{8} + 106365244795 \nu^{7} + 107038243200 \nu^{6} - 362008005480 \nu^{5} - 223553890560 \nu^{4} + 784321942448 \nu^{3} + 688708328448 \nu^{2} - 1260457378040 \nu - 419116013568\)\()/ 257284903680 \)
\(\beta_{10}\)\(=\)\((\)\( 202691 \nu^{15} - 2601950 \nu^{13} + 26162145 \nu^{11} - 34220230 \nu^{9} - 388311005 \nu^{7} + 5948481240 \nu^{5} - 12690218704 \nu^{3} + 21347294920 \nu \)\()/ 5593150080 \)
\(\beta_{11}\)\(=\)\((\)\(-8503883 \nu^{15} + 139049390 \nu^{13} - 1713925305 \nu^{11} + 9317827030 \nu^{9} - 38437913515 \nu^{7} + 5428627560 \nu^{5} - 24375491888 \nu^{3} + 17822113400 \nu\)\()/ 128642451840 \)
\(\beta_{12}\)\(=\)\((\)\( 794819 \nu^{15} - 14201470 \nu^{13} + 189226225 \nu^{11} - 1281113910 \nu^{9} + 6829705875 \nu^{7} - 16443903400 \nu^{5} + 38758828144 \nu^{3} - 12558478520 \nu \)\()/ 10720204320 \)
\(\beta_{13}\)\(=\)\((\)\(-12433839 \nu^{15} + 245478550 \nu^{13} - 3270857125 \nu^{11} + 23498819390 \nu^{9} - 118054419375 \nu^{7} + 284239981000 \nu^{5} - 344266874864 \nu^{3} + 217078731800 \nu\)\()/ 85761634560 \)
\(\beta_{14}\)\(=\)\((\)\(6388160 \nu^{15} - 4527943 \nu^{14} - 125036448 \nu^{13} + 81780118 \nu^{12} + 1666037040 \nu^{11} - 1030370285 \nu^{10} - 11841223600 \nu^{9} + 6434944110 \nu^{8} + 60131955600 \nu^{7} - 28618680055 \nu^{6} - 144779890560 \nu^{5} + 39294121160 \nu^{4} + 227361858560 \nu^{3} - 38553644528 \nu^{2} - 110570775168 \nu - 16824899112\)\()/ 42880817280 \)
\(\beta_{15}\)\(=\)\((\)\(6388160 \nu^{15} + 4527943 \nu^{14} - 125036448 \nu^{13} - 81780118 \nu^{12} + 1666037040 \nu^{11} + 1030370285 \nu^{10} - 11841223600 \nu^{9} - 6434944110 \nu^{8} + 60131955600 \nu^{7} + 28618680055 \nu^{6} - 144779890560 \nu^{5} - 39294121160 \nu^{4} + 227361858560 \nu^{3} + 38553644528 \nu^{2} - 110570775168 \nu + 16824899112\)\()/ 42880817280 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{9} - \beta_{7} + \beta_{4} + 5 \beta_{2} - \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + 7 \beta_{6}\)
\(\nu^{4}\)\(=\)\((\)\(-10 \beta_{9} - 10 \beta_{7} - \beta_{5} + 10 \beta_{4} - 3 \beta_{3} + 37 \beta_{2} + 10 \beta_{1} - 35\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-10 \beta_{15} - 10 \beta_{14} - 12 \beta_{13} - 9 \beta_{12} + 100 \beta_{11} + 23 \beta_{10} + 57 \beta_{9} - 64 \beta_{8} - 57 \beta_{7} + 56 \beta_{6}\)\()/2\)
\(\nu^{6}\)\(=\)\(4 \beta_{15} - 4 \beta_{14} - 15 \beta_{5} + 88 \beta_{1} - 275\)
\(\nu^{7}\)\(=\)\((\)\(92 \beta_{15} + 92 \beta_{14} + 118 \beta_{13} + 65 \beta_{12} + 878 \beta_{11} + 237 \beta_{10} + 485 \beta_{9} - 524 \beta_{8} - 485 \beta_{7} - 458 \beta_{6}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(80 \beta_{15} - 80 \beta_{14} + 680 \beta_{9} + 680 \beta_{7} - 171 \beta_{5} - 920 \beta_{4} + 513 \beta_{3} - 2581 \beta_{2} + 760 \beta_{1} - 2239\)\()/2\)
\(\nu^{9}\)\(=\)\(840 \beta_{15} + 840 \beta_{14} + 1102 \beta_{13} + 429 \beta_{12} - 3770 \beta_{6}\)
\(\nu^{10}\)\(=\)\((\)\(-1084 \beta_{15} + 1084 \beta_{14} + 5468 \beta_{9} + 5468 \beta_{7} + 1775 \beta_{5} - 8720 \beta_{4} + 5325 \beta_{3} - 22025 \beta_{2} - 6552 \beta_{1} + 18475\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(7636 \beta_{15} + 7636 \beta_{14} + 10102 \beta_{13} + 2609 \beta_{12} - 66662 \beta_{11} - 22765 \beta_{10} - 36213 \beta_{9} + 36356 \beta_{8} + 36213 \beta_{7} - 31186 \beta_{6}\)\()/2\)
\(\nu^{12}\)\(=\)\(-12520 \beta_{15} + 12520 \beta_{14} + 17595 \beta_{5} - 56560 \beta_{1} + 153559\)
\(\nu^{13}\)\(=\)\((\)\(-69080 \beta_{15} - 69080 \beta_{14} - 91750 \beta_{13} - 13925 \beta_{12} - 580894 \beta_{11} - 215985 \beta_{10} - 314389 \beta_{9} + 305644 \beta_{8} + 314389 \beta_{7} + 259234 \beta_{6}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-132980 \beta_{15} + 132980 \beta_{14} - 356164 \beta_{9} - 356164 \beta_{7} + 169575 \beta_{5} + 755104 \beta_{4} - 508725 \beta_{3} + 1622465 \beta_{2} - 489144 \beta_{1} + 1283315\)\()/2\)
\(\nu^{15}\)\(=\)\(-622124 \beta_{15} - 622124 \beta_{14} - 828294 \beta_{13} - 53609 \beta_{12} + 2165218 \beta_{6}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1519.1
2.56586 1.48140i
−2.56586 1.48140i
−2.38641 1.37780i
2.38641 1.37780i
1.34352 0.775681i
−1.34352 0.775681i
−0.957255 0.552672i
0.957255 0.552672i
−0.957255 + 0.552672i
0.957255 + 0.552672i
1.34352 + 0.775681i
−1.34352 + 0.775681i
−2.38641 + 1.37780i
2.38641 + 1.37780i
2.56586 + 1.48140i
−2.56586 + 1.48140i
0 2.96281i 0 −0.602114 + 2.15348i 0 −0.562696 0 −5.77822 0
1519.2 0 2.96281i 0 −0.602114 + 2.15348i 0 0.562696 0 −5.77822 0
1519.3 0 2.75559i 0 2.03407 + 0.928731i 0 −4.29083 0 −4.59328 0
1519.4 0 2.75559i 0 2.03407 + 0.928731i 0 4.29083 0 −4.59328 0
1519.5 0 1.55136i 0 −2.03407 + 0.928731i 0 −1.46240 0 0.593276 0
1519.6 0 1.55136i 0 −2.03407 + 0.928731i 0 1.46240 0 0.593276 0
1519.7 0 1.10534i 0 0.602114 2.15348i 0 −2.26573 0 1.77822 0
1519.8 0 1.10534i 0 0.602114 2.15348i 0 2.26573 0 1.77822 0
1519.9 0 1.10534i 0 0.602114 + 2.15348i 0 −2.26573 0 1.77822 0
1519.10 0 1.10534i 0 0.602114 + 2.15348i 0 2.26573 0 1.77822 0
1519.11 0 1.55136i 0 −2.03407 0.928731i 0 −1.46240 0 0.593276 0
1519.12 0 1.55136i 0 −2.03407 0.928731i 0 1.46240 0 0.593276 0
1519.13 0 2.75559i 0 2.03407 0.928731i 0 −4.29083 0 −4.59328 0
1519.14 0 2.75559i 0 2.03407 0.928731i 0 4.29083 0 −4.59328 0
1519.15 0 2.96281i 0 −0.602114 2.15348i 0 −0.562696 0 −5.77822 0
1519.16 0 2.96281i 0 −0.602114 2.15348i 0 0.562696 0 −5.77822 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1519.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
76.d even 2 1 inner
380.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.g.g yes 16
4.b odd 2 1 1520.2.g.e 16
5.b even 2 1 inner 1520.2.g.g yes 16
19.b odd 2 1 1520.2.g.e 16
20.d odd 2 1 1520.2.g.e 16
76.d even 2 1 inner 1520.2.g.g yes 16
95.d odd 2 1 1520.2.g.e 16
380.d even 2 1 inner 1520.2.g.g yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.g.e 16 4.b odd 2 1
1520.2.g.e 16 19.b odd 2 1
1520.2.g.e 16 20.d odd 2 1
1520.2.g.e 16 95.d odd 2 1
1520.2.g.g yes 16 1.a even 1 1 trivial
1520.2.g.g yes 16 5.b even 2 1 inner
1520.2.g.g yes 16 76.d even 2 1 inner
1520.2.g.g yes 16 380.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3}^{8} + 20 T_{3}^{6} + 129 T_{3}^{4} + 290 T_{3}^{2} + 196 \)
\( T_{7}^{8} - 26 T_{7}^{6} + 153 T_{7}^{4} - 248 T_{7}^{2} + 64 \)
\( T_{31}^{4} - 24 T_{31}^{3} + 198 T_{31}^{2} - 648 T_{31} + 672 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 196 + 290 T^{2} + 129 T^{4} + 20 T^{6} + T^{8} )^{2} \)
$5$ \( ( 625 + 50 T^{2} - 6 T^{4} + 2 T^{6} + T^{8} )^{2} \)
$7$ \( ( 64 - 248 T^{2} + 153 T^{4} - 26 T^{6} + T^{8} )^{2} \)
$11$ \( ( 3136 + 2320 T^{2} + 516 T^{4} + 40 T^{6} + T^{8} )^{2} \)
$13$ \( ( 196 - 926 T^{2} + 585 T^{4} - 56 T^{6} + T^{8} )^{2} \)
$17$ \( ( 96 + 21 T^{2} + T^{4} )^{4} \)
$19$ \( ( 361 + 38 T - 18 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$23$ \( ( 28224 - 19224 T^{2} + 2361 T^{4} - 90 T^{6} + T^{8} )^{2} \)
$29$ \( ( 12544 + 33760 T^{2} + 3489 T^{4} + 106 T^{6} + T^{8} )^{2} \)
$31$ \( ( 672 - 648 T + 198 T^{2} - 24 T^{3} + T^{4} )^{4} \)
$37$ \( ( 50176 - 30080 T^{2} + 3204 T^{4} - 110 T^{6} + T^{8} )^{2} \)
$41$ \( ( 18113536 + 1219648 T^{2} + 29220 T^{4} + 292 T^{6} + T^{8} )^{2} \)
$43$ \( ( 12544 - 29408 T^{2} + 3588 T^{4} - 116 T^{6} + T^{8} )^{2} \)
$47$ \( ( 4064256 - 501120 T^{2} + 18576 T^{4} - 240 T^{6} + T^{8} )^{2} \)
$53$ \( ( 13972644 - 1074294 T^{2} + 28137 T^{4} - 288 T^{6} + T^{8} )^{2} \)
$59$ \( ( -2352 - 876 T + 9 T^{2} + 18 T^{3} + T^{4} )^{4} \)
$61$ \( ( -504 - 324 T - 42 T^{2} + 6 T^{3} + T^{4} )^{4} \)
$67$ \( ( 86436 + 360090 T^{2} + 23601 T^{4} + 300 T^{6} + T^{8} )^{2} \)
$71$ \( ( 672 + 648 T + 198 T^{2} + 24 T^{3} + T^{4} )^{4} \)
$73$ \( ( 9216 + 27936 T^{2} + 13881 T^{4} + 234 T^{6} + T^{8} )^{2} \)
$79$ \( ( 448 + 400 T - 84 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$83$ \( ( 589824 - 184320 T^{2} + 12864 T^{4} - 216 T^{6} + T^{8} )^{2} \)
$89$ \( ( 802816 + 1265920 T^{2} + 45636 T^{4} + 460 T^{6} + T^{8} )^{2} \)
$97$ \( ( 3136 - 9920 T^{2} + 5988 T^{4} - 230 T^{6} + T^{8} )^{2} \)
show more
show less