# Properties

 Label 1584.2.cd.c Level $1584$ Weight $2$ Character orbit 1584.cd Analytic conductor $12.648$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1584 = 2^{4} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1584.cd (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.6483036802$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 99) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{10} - \beta_{12} - 2 \beta_{13} ) q^{5} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{10} - \beta_{12} - 2 \beta_{13} ) q^{5} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} + ( -\beta_{1} - \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{11} + ( \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{9} ) q^{13} + ( -2 \beta_{1} + \beta_{8} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{14} ) q^{17} + ( -2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{19} + ( \beta_{1} + \beta_{8} - \beta_{11} - \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{23} + ( -4 \beta_{3} - 4 \beta_{4} - \beta_{6} - \beta_{7} ) q^{25} + ( 2 \beta_{8} + \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + 3 \beta_{15} ) q^{29} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{31} + ( 4 \beta_{8} - 2 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + 4 \beta_{15} ) q^{35} + ( -1 - 2 \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{37} + ( -\beta_{8} - 5 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - 4 \beta_{14} + 2 \beta_{15} ) q^{41} + ( 1 - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{9} ) q^{43} + ( -6 \beta_{1} + 2 \beta_{8} - 5 \beta_{10} - 3 \beta_{12} - 3 \beta_{14} + 3 \beta_{15} ) q^{47} + ( -2 - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{7} ) q^{49} + ( -\beta_{1} + \beta_{8} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{53} + ( -3 - 2 \beta_{2} + 3 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{9} ) q^{55} + ( \beta_{1} - \beta_{8} + \beta_{10} + 2 \beta_{11} + 3 \beta_{14} - \beta_{15} ) q^{59} + ( -2 - 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{9} ) q^{61} + ( -4 \beta_{1} + \beta_{8} - 2 \beta_{10} - 4 \beta_{11} - \beta_{13} - 2 \beta_{14} + 5 \beta_{15} ) q^{65} + ( -4 - 2 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} + \beta_{7} - \beta_{9} ) q^{67} + ( \beta_{1} + 4 \beta_{8} - 2 \beta_{10} - \beta_{12} - 2 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{71} + ( -2 + \beta_{2} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} + 4 \beta_{9} ) q^{73} + ( -\beta_{1} + 4 \beta_{8} - 2 \beta_{10} - 5 \beta_{11} + 3 \beta_{12} - \beta_{13} + 5 \beta_{15} ) q^{77} + ( -3 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{9} ) q^{79} + ( -\beta_{1} - 4 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + 3 \beta_{15} ) q^{83} + ( -10 + 2 \beta_{3} + 8 \beta_{4} + 5 \beta_{5} ) q^{85} + ( -3 \beta_{1} + 3 \beta_{11} - 4 \beta_{12} + \beta_{13} - 2 \beta_{15} ) q^{89} + ( \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} ) q^{91} + ( -5 \beta_{8} + 3 \beta_{12} + 3 \beta_{13} + 4 \beta_{14} - 5 \beta_{15} ) q^{95} + ( 3 + 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} - 6 \beta_{7} + 2 \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + O(q^{10})$$ $$16 q - 32 q^{25} - 16 q^{31} - 12 q^{37} - 24 q^{49} - 16 q^{55} - 96 q^{67} - 20 q^{73} - 100 q^{85} + 72 q^{91} + 60 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$155 \nu^{14} - 1408 \nu^{12} - 4611 \nu^{10} + 9188 \nu^{8} + 65146 \nu^{6} - 26953 \nu^{4} + 145025 \nu^{2} + 108500$$$$)/176275$$ $$\beta_{3}$$ $$=$$ $$($$$$5996 \nu^{14} + 4807 \nu^{12} - 95306 \nu^{10} - 277252 \nu^{8} + 500966 \nu^{6} + 1108500 \nu^{4} + 1748000 \nu^{2} + 2787000$$$$)/881375$$ $$\beta_{4}$$ $$=$$ $$($$$$8474 \nu^{14} + 2743 \nu^{12} - 149594 \nu^{10} - 389798 \nu^{8} + 955934 \nu^{6} + 2009635 \nu^{4} + 3037000 \nu^{2} + 3175500$$$$)/881375$$ $$\beta_{5}$$ $$=$$ $$($$$$-8672 \nu^{14} - 3649 \nu^{12} + 153342 \nu^{10} + 402539 \nu^{8} - 938462 \nu^{6} - 1990750 \nu^{4} - 3034300 \nu^{2} - 3730750$$$$)/881375$$ $$\beta_{6}$$ $$=$$ $$($$$$-12916 \nu^{14} - 10637 \nu^{12} + 225196 \nu^{10} + 690882 \nu^{8} - 1135131 \nu^{6} - 3696115 \nu^{4} - 6351875 \nu^{2} - 6749625$$$$)/881375$$ $$\beta_{7}$$ $$=$$ $$($$$$-14403 \nu^{14} - 14916 \nu^{12} + 224078 \nu^{10} + 748351 \nu^{8} - 1023533 \nu^{6} - 3087765 \nu^{4} - 6177175 \nu^{2} - 6732875$$$$)/881375$$ $$\beta_{8}$$ $$=$$ $$($$$$-1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} - 478015 \nu^{5} - 412275 \nu^{3} - 598500 \nu$$$$)/400625$$ $$\beta_{9}$$ $$=$$ $$($$$$-22098 \nu^{14} - 13706 \nu^{12} + 377023 \nu^{10} + 1116041 \nu^{8} - 1983428 \nu^{6} - 5540615 \nu^{4} - 10624800 \nu^{2} - 11238000$$$$)/881375$$ $$\beta_{10}$$ $$=$$ $$($$$$22194 \nu^{15} + 11523 \nu^{13} - 339409 \nu^{11} - 1024903 \nu^{9} + 1669899 \nu^{7} + 3769000 \nu^{5} + 11127550 \nu^{3} + 9814875 \nu$$$$)/4406875$$ $$\beta_{11}$$ $$=$$ $$($$$$25903 \nu^{15} - 3944 \nu^{13} - 422373 \nu^{11} - 826866 \nu^{9} + 3248903 \nu^{7} + 2357930 \nu^{5} + 4320250 \nu^{3} + 4286500 \nu$$$$)/4406875$$ $$\beta_{12}$$ $$=$$ $$($$$$8474 \nu^{15} + 2743 \nu^{13} - 149594 \nu^{11} - 389798 \nu^{9} + 955934 \nu^{7} + 2009635 \nu^{5} + 3037000 \nu^{3} + 2294125 \nu$$$$)/881375$$ $$\beta_{13}$$ $$=$$ $$($$$$-8672 \nu^{15} - 3649 \nu^{13} + 153342 \nu^{11} + 402539 \nu^{9} - 938462 \nu^{7} - 1990750 \nu^{5} - 3034300 \nu^{3} - 3730750 \nu$$$$)/881375$$ $$\beta_{14}$$ $$=$$ $$($$$$-50508 \nu^{15} - 41726 \nu^{13} + 817683 \nu^{11} + 2587261 \nu^{9} - 4065438 \nu^{7} - 11753915 \nu^{5} - 22984550 \nu^{3} - 22711875 \nu$$$$)/4406875$$ $$\beta_{15}$$ $$=$$ $$($$$$-11661 \nu^{15} - 8721 \nu^{13} + 202398 \nu^{11} + 611766 \nu^{9} - 1064278 \nu^{7} - 3191899 \nu^{5} - 5385345 \nu^{3} - 5486525 \nu$$$$)/881375$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 3 \beta_{10} + 2 \beta_{8} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 2 \beta_{6} + 8 \beta_{5} + 5 \beta_{4} + 5 \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$-13 \beta_{15} + 13 \beta_{14} + 12 \beta_{13} + 7 \beta_{11} + 7 \beta_{10} - 5 \beta_{8} + 7 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{9} - 11 \beta_{7} - 6 \beta_{6} + 11 \beta_{4} - 15 \beta_{3} + 5 \beta_{2} + 15$$ $$\nu^{7}$$ $$=$$ $$20 \beta_{15} - 43 \beta_{14} + 9 \beta_{12} - 4 \beta_{11} - 43 \beta_{10} + 20 \beta_{8} + 9 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$32 \beta_{9} - 52 \beta_{6} + 53 \beta_{5} + 18 \beta_{4} + 53 \beta_{3} + 32 \beta_{2} - 18$$ $$\nu^{9}$$ $$=$$ $$-209 \beta_{15} + 137 \beta_{14} + 137 \beta_{13} - 87 \beta_{12} + 137 \beta_{11} + 21 \beta_{10} - 21 \beta_{8}$$ $$\nu^{10}$$ $$=$$ $$21 \beta_{9} - 36 \beta_{7} + 275 \beta_{5} + 275 \beta_{4} + 166$$ $$\nu^{11}$$ $$=$$ $$78 \beta_{15} - 129 \beta_{14} + 239 \beta_{13} + 239 \beta_{12} - 78 \beta_{10} + 384 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$752 \beta_{9} - 462 \beta_{7} - 752 \beta_{6} - 110 \beta_{5} - 177 \beta_{3} + 462 \beta_{2} + 110$$ $$\nu^{13}$$ $$=$$ $$-1037 \beta_{15} - 639 \beta_{14} + 642 \beta_{13} - 1037 \beta_{12} + 1037 \beta_{11} - 1681 \beta_{10} + 639 \beta_{8} - 642 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$639 \beta_{7} - 639 \beta_{6} + 5431 \beta_{5} + 3360 \beta_{4} + 3360 \beta_{3} + 400 \beta_{2}$$ $$\nu^{15}$$ $$=$$ $$-6716 \beta_{15} + 6716 \beta_{14} + 7109 \beta_{13} + 4399 \beta_{11} + 4399 \beta_{10} - 2960 \beta_{8} + 4399 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$353$$ $$991$$ $$1189$$ $$\chi(n)$$ $$\beta_{4}$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 1.90184 + 0.0324487i −0.556839 + 1.81878i 0.556839 − 1.81878i −1.90184 − 0.0324487i −0.752864 − 0.902863i 0.0783900 − 1.17295i −0.0783900 + 1.17295i 0.752864 + 0.902863i −0.752864 + 0.902863i 0.0783900 + 1.17295i −0.0783900 − 1.17295i 0.752864 − 0.902863i 1.90184 − 0.0324487i −0.556839 − 1.81878i 0.556839 + 1.81878i −1.90184 + 0.0324487i
0 0 0 −2.13811 0.694712i 0 2.38116 + 3.27739i 0 0 0
17.2 0 0 0 −0.0381457 0.0123943i 0 −0.145094 0.199704i 0 0 0
17.3 0 0 0 0.0381457 + 0.0123943i 0 −0.145094 0.199704i 0 0 0
17.4 0 0 0 2.13811 + 0.694712i 0 2.38116 + 3.27739i 0 0 0
161.1 0 0 0 −2.23109 3.07083i 0 0.349790 0.113654i 0 0 0
161.2 0 0 0 −1.71735 2.36373i 0 −2.58586 + 0.840196i 0 0 0
161.3 0 0 0 1.71735 + 2.36373i 0 −2.58586 + 0.840196i 0 0 0
161.4 0 0 0 2.23109 + 3.07083i 0 0.349790 0.113654i 0 0 0
305.1 0 0 0 −2.23109 + 3.07083i 0 0.349790 + 0.113654i 0 0 0
305.2 0 0 0 −1.71735 + 2.36373i 0 −2.58586 0.840196i 0 0 0
305.3 0 0 0 1.71735 2.36373i 0 −2.58586 0.840196i 0 0 0
305.4 0 0 0 2.23109 3.07083i 0 0.349790 + 0.113654i 0 0 0
1025.1 0 0 0 −2.13811 + 0.694712i 0 2.38116 3.27739i 0 0 0
1025.2 0 0 0 −0.0381457 + 0.0123943i 0 −0.145094 + 0.199704i 0 0 0
1025.3 0 0 0 0.0381457 0.0123943i 0 −0.145094 + 0.199704i 0 0 0
1025.4 0 0 0 2.13811 0.694712i 0 2.38116 3.27739i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1025.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.2.cd.c 16
3.b odd 2 1 inner 1584.2.cd.c 16
4.b odd 2 1 99.2.j.a 16
11.d odd 10 1 inner 1584.2.cd.c 16
12.b even 2 1 99.2.j.a 16
33.f even 10 1 inner 1584.2.cd.c 16
36.f odd 6 2 891.2.u.c 32
36.h even 6 2 891.2.u.c 32
44.g even 10 1 99.2.j.a 16
44.g even 10 1 1089.2.d.g 16
44.h odd 10 1 1089.2.d.g 16
132.n odd 10 1 99.2.j.a 16
132.n odd 10 1 1089.2.d.g 16
132.o even 10 1 1089.2.d.g 16
396.bb odd 30 2 891.2.u.c 32
396.bf even 30 2 891.2.u.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.j.a 16 4.b odd 2 1
99.2.j.a 16 12.b even 2 1
99.2.j.a 16 44.g even 10 1
99.2.j.a 16 132.n odd 10 1
891.2.u.c 32 36.f odd 6 2
891.2.u.c 32 36.h even 6 2
891.2.u.c 32 396.bb odd 30 2
891.2.u.c 32 396.bf even 30 2
1089.2.d.g 16 44.g even 10 1
1089.2.d.g 16 44.h odd 10 1
1089.2.d.g 16 132.n odd 10 1
1089.2.d.g 16 132.o even 10 1
1584.2.cd.c 16 1.a even 1 1 trivial
1584.2.cd.c 16 3.b odd 2 1 inner
1584.2.cd.c 16 11.d odd 10 1 inner
1584.2.cd.c 16 33.f even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1584, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$1 - 1006 T^{2} + 386612 T^{4} - 79178 T^{6} + 9230 T^{8} - 572 T^{10} + 237 T^{12} + 6 T^{14} + T^{16}$$
$7$ $$( 1 - T^{2} - 50 T^{3} + 101 T^{4} + 50 T^{5} - T^{6} + T^{8} )^{2}$$
$11$ $$214358881 - 106293660 T^{2} + 26046339 T^{4} - 4066810 T^{6} + 439341 T^{8} - 33610 T^{10} + 1779 T^{12} - 60 T^{14} + T^{16}$$
$13$ $$( 841 + 1740 T + 1961 T^{2} + 950 T^{3} + 131 T^{4} - 50 T^{5} - 19 T^{6} + T^{8} )^{2}$$
$17$ $$625 + 173750 T^{2} + 18450125 T^{4} + 94500 T^{6} + 128150 T^{8} + 7350 T^{10} + 380 T^{12} + 10 T^{14} + T^{16}$$
$19$ $$( 14641 - 3630 T - 2539 T^{2} - 680 T^{3} + 566 T^{4} + 170 T^{5} - 4 T^{6} + T^{8} )^{2}$$
$23$ $$( 7921 + 13138 T^{2} + 1934 T^{4} + 82 T^{6} + T^{8} )^{2}$$
$29$ $$3544535296 + 275294464 T^{2} + 235287872 T^{4} + 64057472 T^{6} + 8205280 T^{8} + 38288 T^{10} + 7472 T^{12} + 136 T^{14} + T^{16}$$
$31$ $$( 28561 + 8788 T + 9633 T^{2} - 624 T^{3} + 100 T^{4} + 36 T^{5} + 38 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$37$ $$( 1565001 - 307746 T + 51687 T^{2} - 378 T^{3} + 1855 T^{4} + 178 T^{5} + 67 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$41$ $$11943113486161 + 2701787030514 T^{2} + 240395280827 T^{4} + 2225026622 T^{6} + 56799105 T^{8} + 791558 T^{10} + 9747 T^{12} + 26 T^{14} + T^{16}$$
$43$ $$( 9801 + 12564 T^{2} + 3326 T^{4} + 164 T^{6} + T^{8} )^{2}$$
$47$ $$14641 - 55782694 T^{2} + 81180810472 T^{4} - 501195932 T^{6} + 42758850 T^{8} - 960688 T^{10} + 12517 T^{12} + 4 T^{14} + T^{16}$$
$53$ $$166726039041 - 177666183594 T^{2} + 72224511597 T^{4} + 250175268 T^{6} + 67595950 T^{8} - 1017338 T^{10} + 9492 T^{12} - 46 T^{14} + T^{16}$$
$59$ $$1 - 184 T^{2} + 12857 T^{4} + 23128 T^{6} + 117530 T^{8} + 5272 T^{10} + 1632 T^{12} - 66 T^{14} + T^{16}$$
$61$ $$( 450241 - 67100 T + 6206 T^{2} + 31200 T^{3} + 3596 T^{4} - 800 T^{5} - 39 T^{6} + T^{8} )^{2}$$
$67$ $$( -649 + 184 T + 161 T^{2} + 24 T^{3} + T^{4} )^{4}$$
$71$ $$373301041 - 515754774 T^{2} + 283963772 T^{4} - 25333482 T^{6} + 1799350 T^{8} - 126588 T^{10} + 10757 T^{12} + 54 T^{14} + T^{16}$$
$73$ $$( 1488400 + 2269200 T + 1284400 T^{2} + 313800 T^{3} + 32960 T^{4} + 1440 T^{5} + 130 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$79$ $$( 24025 + 52700 T + 11225 T^{2} - 27450 T^{3} + 7835 T^{4} - 610 T^{5} + 45 T^{6} + T^{8} )^{2}$$
$83$ $$228745085711041 + 12776965883516 T^{2} + 335480370407 T^{4} + 4007514018 T^{6} + 127442525 T^{8} + 1172562 T^{10} + 12887 T^{12} + 124 T^{14} + T^{16}$$
$89$ $$( 126025 + 47650 T^{2} + 4790 T^{4} + 130 T^{6} + T^{8} )^{2}$$
$97$ $$( 1575025 + 1618950 T + 799150 T^{2} + 151350 T^{3} + 32460 T^{4} - 6060 T^{5} + 605 T^{6} - 30 T^{7} + T^{8} )^{2}$$