# Properties

 Label 1089.2.d.g Level $1089$ Weight $2$ Character orbit 1089.d Analytic conductor $8.696$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.69570878012$$ Analytic rank: $$0$$ Dimension: $$16$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{15} q^{2} + ( 1 + \beta_{4} ) q^{4} + ( -\beta_{2} + \beta_{5} - \beta_{6} + \beta_{12} ) q^{5} + ( -\beta_{8} + \beta_{9} ) q^{7} + ( -\beta_{7} + 2 \beta_{15} ) q^{8} +O(q^{10})$$ $$q + \beta_{15} q^{2} + ( 1 + \beta_{4} ) q^{4} + ( -\beta_{2} + \beta_{5} - \beta_{6} + \beta_{12} ) q^{5} + ( -\beta_{8} + \beta_{9} ) q^{7} + ( -\beta_{7} + 2 \beta_{15} ) q^{8} + ( -2 \beta_{8} + \beta_{11} + 2 \beta_{13} ) q^{10} + ( -\beta_{9} + \beta_{11} - \beta_{13} ) q^{13} + ( -\beta_{2} - \beta_{5} - \beta_{6} - \beta_{12} ) q^{14} + ( 3 + \beta_{1} ) q^{16} + ( -2 \beta_{10} + \beta_{14} + \beta_{15} ) q^{17} + ( \beta_{8} + 2 \beta_{9} ) q^{19} + ( -2 \beta_{2} + \beta_{5} - 5 \beta_{6} + 6 \beta_{12} ) q^{20} + ( \beta_{5} - 3 \beta_{6} + \beta_{12} ) q^{23} + ( -4 - 4 \beta_{1} - \beta_{4} ) q^{25} + ( \beta_{2} + \beta_{5} + \beta_{6} + 2 \beta_{12} ) q^{26} + ( -2 \beta_{8} + 2 \beta_{9} - 3 \beta_{11} + 2 \beta_{13} ) q^{28} + ( 3 \beta_{7} + 2 \beta_{10} + \beta_{14} ) q^{29} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{31} + ( 2 \beta_{7} + \beta_{14} - \beta_{15} ) q^{32} + ( 3 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{34} + ( 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{14} - \beta_{15} ) q^{35} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{37} + ( \beta_{2} - 2 \beta_{5} + \beta_{6} - 5 \beta_{12} ) q^{38} + ( -4 \beta_{8} + 3 \beta_{11} + 3 \beta_{13} ) q^{40} + ( -\beta_{7} + 4 \beta_{10} - 2 \beta_{14} + 2 \beta_{15} ) q^{41} + ( -\beta_{8} + 3 \beta_{11} - 2 \beta_{13} ) q^{43} + ( -2 \beta_{8} + \beta_{9} + 2 \beta_{11} + 3 \beta_{13} ) q^{46} + ( 3 \beta_{2} - 2 \beta_{6} ) q^{47} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{49} + ( \beta_{7} - 4 \beta_{14} - 7 \beta_{15} ) q^{50} + ( 4 \beta_{8} - 3 \beta_{9} + 2 \beta_{11} ) q^{52} + ( 3 \beta_{2} - 2 \beta_{5} - \beta_{6} - 2 \beta_{12} ) q^{53} + ( -2 \beta_{2} - \beta_{5} - \beta_{6} - 2 \beta_{12} ) q^{56} + ( 3 + 2 \beta_{1} - \beta_{3} ) q^{58} + ( \beta_{2} - \beta_{5} + \beta_{6} - 3 \beta_{12} ) q^{59} + ( -\beta_{9} - 2 \beta_{11} - 3 \beta_{13} ) q^{61} + ( -\beta_{10} + \beta_{14} + 3 \beta_{15} ) q^{62} + ( -7 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{64} + ( \beta_{7} + 3 \beta_{10} - 3 \beta_{14} ) q^{65} + ( 4 - 4 \beta_{1} + \beta_{4} ) q^{67} + ( 3 \beta_{10} + 2 \beta_{14} + 4 \beta_{15} ) q^{68} + ( -1 + 6 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{70} + ( \beta_{2} - \beta_{5} + 2 \beta_{6} - 5 \beta_{12} ) q^{71} + ( \beta_{8} - 4 \beta_{9} - \beta_{11} - 3 \beta_{13} ) q^{73} + ( -\beta_{7} + 2 \beta_{10} - 7 \beta_{14} - 3 \beta_{15} ) q^{74} + ( -\beta_{8} + \beta_{9} - 6 \beta_{11} - 2 \beta_{13} ) q^{76} + ( \beta_{9} - \beta_{11} - 3 \beta_{13} ) q^{79} + ( -3 \beta_{2} + 4 \beta_{5} - 3 \beta_{6} + 4 \beta_{12} ) q^{80} + ( 5 - \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{82} + ( 3 \beta_{7} + \beta_{10} + 4 \beta_{14} + \beta_{15} ) q^{83} + ( -3 \beta_{8} + 5 \beta_{11} ) q^{85} + ( \beta_{2} + \beta_{5} + 3 \beta_{12} ) q^{86} + ( 2 \beta_{2} + \beta_{5} - 2 \beta_{12} ) q^{89} + ( -6 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{91} + ( -5 \beta_{2} + 2 \beta_{5} - 4 \beta_{6} + 8 \beta_{12} ) q^{92} + ( 4 \beta_{8} - \beta_{9} + 3 \beta_{11} - \beta_{13} ) q^{94} + ( \beta_{7} + \beta_{10} + 4 \beta_{14} + 4 \beta_{15} ) q^{95} + ( -3 - 6 \beta_{1} + 3 \beta_{3} - \beta_{4} ) q^{97} + ( 2 \beta_{7} - 2 \beta_{10} + 8 \beta_{14} + 2 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{4} + O(q^{10})$$ $$16q + 16q^{4} + 40q^{16} - 32q^{25} + 16q^{31} + 40q^{34} + 8q^{37} + 16q^{49} + 32q^{58} - 104q^{64} + 96q^{67} - 64q^{70} + 88q^{82} + 48q^{91} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$198 \nu^{14} + 906 \nu^{12} - 3748 \nu^{10} - 12741 \nu^{8} - 17472 \nu^{6} - 18885 \nu^{4} - 2700 \nu^{2} + 555250$$$$)/881375$$ $$\beta_{2}$$ $$=$$ $$($$$$532 \nu^{15} - 12231 \nu^{13} - 61577 \nu^{11} - 12284 \nu^{9} + 989047 \nu^{7} + 1778925 \nu^{5} - 876975 \nu^{3} - 4371000 \nu$$$$)/4406875$$ $$\beta_{3}$$ $$=$$ $$($$$$-3734 \nu^{14} - 7568 \nu^{12} + 73369 \nu^{10} + 265723 \nu^{8} - 286834 \nu^{6} - 1851615 \nu^{4} - 2078950 \nu^{2} - 2261250$$$$)/881375$$ $$\beta_{4}$$ $$=$$ $$($$$$1229 \nu^{14} + 2574 \nu^{12} - 21367 \nu^{10} - 91914 \nu^{8} + 61687 \nu^{6} + 544476 \nu^{4} + 599475 \nu^{2} + 684025$$$$)/176275$$ $$\beta_{5}$$ $$=$$ $$($$$$-6796 \nu^{15} - 7982 \nu^{13} + 118381 \nu^{11} + 297652 \nu^{9} - 922291 \nu^{7} - 1867925 \nu^{5} + 2374050 \nu^{3} + 7469875 \nu$$$$)/4406875$$ $$\beta_{6}$$ $$=$$ $$($$$$-7797 \nu^{15} - 1879 \nu^{13} + 194307 \nu^{11} + 471569 \nu^{9} - 1255952 \nu^{7} - 4205580 \nu^{5} - 3942175 \nu^{3} - 4720750 \nu$$$$)/4406875$$ $$\beta_{7}$$ $$=$$ $$($$$$1932 \nu^{15} + 4062 \nu^{13} - 30006 \nu^{11} - 142852 \nu^{9} + 37491 \nu^{7} + 678048 \nu^{5} + 1381815 \nu^{3} + 1047925 \nu$$$$)/881375$$ $$\beta_{8}$$ $$=$$ $$($$$$11794 \nu^{14} + 8708 \nu^{12} - 186864 \nu^{10} - 541763 \nu^{8} + 1019404 \nu^{6} + 2235885 \nu^{4} + 3498700 \nu^{2} + 4137375$$$$)/881375$$ $$\beta_{9}$$ $$=$$ $$($$$$14403 \nu^{14} + 14916 \nu^{12} - 224078 \nu^{10} - 748351 \nu^{8} + 1023533 \nu^{6} + 3087765 \nu^{4} + 6177175 \nu^{2} + 6732875$$$$)/881375$$ $$\beta_{10}$$ $$=$$ $$($$$$2989 \nu^{15} + 5072 \nu^{13} - 49056 \nu^{11} - 209227 \nu^{9} + 125816 \nu^{7} + 1201149 \nu^{5} + 2351045 \nu^{3} + 1755775 \nu$$$$)/881375$$ $$\beta_{11}$$ $$=$$ $$($$$$-17146 \nu^{14} - 6392 \nu^{12} + 302936 \nu^{10} + 792337 \nu^{8} - 1894396 \nu^{6} - 4000385 \nu^{4} - 6071300 \nu^{2} - 6906250$$$$)/881375$$ $$\beta_{12}$$ $$=$$ $$($$$$1496 \nu^{15} + 2572 \nu^{13} - 24401 \nu^{11} - 91992 \nu^{9} + 100886 \nu^{7} + 478015 \nu^{5} + 412275 \nu^{3} + 598500 \nu$$$$)/400625$$ $$\beta_{13}$$ $$=$$ $$($$$$-22098 \nu^{14} - 13706 \nu^{12} + 377023 \nu^{10} + 1116041 \nu^{8} - 1983428 \nu^{6} - 5540615 \nu^{4} - 10624800 \nu^{2} - 11238000$$$$)/881375$$ $$\beta_{14}$$ $$=$$ $$($$$$58981 \nu^{15} + 32907 \nu^{13} - 991506 \nu^{11} - 2819027 \nu^{9} + 5518041 \nu^{7} + 13611505 \nu^{5} + 23823225 \nu^{3} + 25694375 \nu$$$$)/4406875$$ $$\beta_{15}$$ $$=$$ $$($$$$93868 \nu^{15} + 59971 \nu^{13} - 1584393 \nu^{11} - 4599956 \nu^{9} + 8757748 \nu^{7} + 21707665 \nu^{5} + 38156050 \nu^{3} + 41365625 \nu$$$$)/4406875$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{12} - \beta_{7} + \beta_{5}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} - \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{15} - 3 \beta_{14} - 2 \beta_{12} + 3 \beta_{10} - 3 \beta_{7} - \beta_{5} - \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{13} + 3 \beta_{11} - 4 \beta_{9} + 5 \beta_{8} - 2 \beta_{3} - 8 \beta_{1} + 5$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{15} + 7 \beta_{14} + 3 \beta_{12} + 5 \beta_{10} - 7 \beta_{7}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$11 \beta_{13} - 11 \beta_{11} + 17 \beta_{9} - 15 \beta_{8} - 5 \beta_{4} - 6 \beta_{3} - 26 \beta_{1} + 15$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$28 \beta_{15} - 43 \beta_{14} + 7 \beta_{12} + 27 \beta_{10} - 48 \beta_{7} + 16 \beta_{6} + 4 \beta_{5} + 9 \beta_{2}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-20 \beta_{13} + 35 \beta_{11} - 32 \beta_{9} + 53 \beta_{8} - 32 \beta_{4} - 52 \beta_{3} - 18 \beta_{1} + 17$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-15 \beta_{15} + 21 \beta_{14} - 29 \beta_{12} + 72 \beta_{10} - 116 \beta_{7} - 51 \beta_{6} - 50 \beta_{5} - 87 \beta_{2}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$21 \beta_{13} + 36 \beta_{9} - 275 \beta_{1} + 166$$ $$\nu^{11}$$ $$=$$ $$($$$$51 \beta_{15} - 78 \beta_{14} + 223 \beta_{12} + 290 \beta_{10} - 462 \beta_{7} + 368 \beta_{6} + 145 \beta_{5} + 239 \beta_{2}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$290 \beta_{13} - 110 \beta_{11} + 462 \beta_{9} - 177 \beta_{8} - 462 \beta_{4} - 752 \beta_{3} - 67 \beta_{1} + 43$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$1034 \beta_{15} - 1681 \beta_{14} - 239 \beta_{12} + 1281 \beta_{10} - 2076 \beta_{7} - 400 \beta_{6} - 642 \beta_{5} - 1037 \beta_{2}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-1039 \beta_{13} + 2071 \beta_{11} - 1678 \beta_{9} + 3360 \beta_{8} - 400 \beta_{4} - 639 \beta_{3} - 5431 \beta_{1} + 3360$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-2710 \beta_{15} + 4399 \beta_{14} + 1521 \beta_{12} + 2710 \beta_{10} - 4399 \beta_{7} + 2475 \beta_{6}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1088.1
 0.0783900 − 1.17295i 0.0783900 + 1.17295i 0.556839 + 1.81878i 0.556839 − 1.81878i 1.90184 + 0.0324487i 1.90184 − 0.0324487i 0.752864 − 0.902863i 0.752864 + 0.902863i −0.752864 − 0.902863i −0.752864 + 0.902863i −1.90184 + 0.0324487i −1.90184 − 0.0324487i −0.556839 + 1.81878i −0.556839 − 1.81878i −0.0783900 − 1.17295i −0.0783900 + 1.17295i
−2.43632 0 3.93565 3.79576i 0 0.367791i −4.71586 0 9.24768i
1088.2 −2.43632 0 3.93565 3.79576i 0 0.367791i −4.71586 0 9.24768i
1088.3 −2.35080 0 3.52626 2.24814i 0 4.05107i −3.58792 0 5.28492i
1088.4 −2.35080 0 3.52626 2.24814i 0 4.05107i −3.58792 0 5.28492i
1088.5 −0.688291 0 −1.52626 0.0401087i 0 0.246848i 2.42709 0 0.0276065i
1088.6 −0.688291 0 −1.52626 0.0401087i 0 0.246848i 2.42709 0 0.0276065i
1088.7 −0.253675 0 −1.93565 2.92173i 0 2.71893i 0.998377 0 0.741170i
1088.8 −0.253675 0 −1.93565 2.92173i 0 2.71893i 0.998377 0 0.741170i
1088.9 0.253675 0 −1.93565 2.92173i 0 2.71893i −0.998377 0 0.741170i
1088.10 0.253675 0 −1.93565 2.92173i 0 2.71893i −0.998377 0 0.741170i
1088.11 0.688291 0 −1.52626 0.0401087i 0 0.246848i −2.42709 0 0.0276065i
1088.12 0.688291 0 −1.52626 0.0401087i 0 0.246848i −2.42709 0 0.0276065i
1088.13 2.35080 0 3.52626 2.24814i 0 4.05107i 3.58792 0 5.28492i
1088.14 2.35080 0 3.52626 2.24814i 0 4.05107i 3.58792 0 5.28492i
1088.15 2.43632 0 3.93565 3.79576i 0 0.367791i 4.71586 0 9.24768i
1088.16 2.43632 0 3.93565 3.79576i 0 0.367791i 4.71586 0 9.24768i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1088.16 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.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 1089.2.d.g 16
3.b odd 2 1 inner 1089.2.d.g 16
11.b odd 2 1 inner 1089.2.d.g 16
11.c even 5 1 99.2.j.a 16
11.d odd 10 1 99.2.j.a 16
33.d even 2 1 inner 1089.2.d.g 16
33.f even 10 1 99.2.j.a 16
33.h odd 10 1 99.2.j.a 16
44.g even 10 1 1584.2.cd.c 16
44.h odd 10 1 1584.2.cd.c 16
99.m even 15 2 891.2.u.c 32
99.n odd 30 2 891.2.u.c 32
99.o odd 30 2 891.2.u.c 32
99.p even 30 2 891.2.u.c 32
132.n odd 10 1 1584.2.cd.c 16
132.o even 10 1 1584.2.cd.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.j.a 16 11.c even 5 1
99.2.j.a 16 11.d odd 10 1
99.2.j.a 16 33.f even 10 1
99.2.j.a 16 33.h odd 10 1
891.2.u.c 32 99.m even 15 2
891.2.u.c 32 99.n odd 30 2
891.2.u.c 32 99.o odd 30 2
891.2.u.c 32 99.p even 30 2
1089.2.d.g 16 1.a even 1 1 trivial
1089.2.d.g 16 3.b odd 2 1 inner
1089.2.d.g 16 11.b odd 2 1 inner
1089.2.d.g 16 33.d even 2 1 inner
1584.2.cd.c 16 44.g even 10 1
1584.2.cd.c 16 44.h odd 10 1
1584.2.cd.c 16 132.n odd 10 1
1584.2.cd.c 16 132.o even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1089, [\chi])$$:

 $$T_{2}^{8} - 12 T_{2}^{6} + 39 T_{2}^{4} - 18 T_{2}^{2} + 1$$ $$T_{17}^{8} - 50 T_{17}^{6} + 815 T_{17}^{4} - 4300 T_{17}^{2} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 18 T^{2} + 39 T^{4} - 12 T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$( 1 + 622 T^{2} + 239 T^{4} + 28 T^{6} + T^{8} )^{2}$$
$7$ $$( 1 + 24 T^{2} + 126 T^{4} + 24 T^{6} + T^{8} )^{2}$$
$11$ $$T^{16}$$
$13$ $$( 841 + 1236 T^{2} + 366 T^{4} + 36 T^{6} + T^{8} )^{2}$$
$17$ $$( 25 - 4300 T^{2} + 815 T^{4} - 50 T^{6} + T^{8} )^{2}$$
$19$ $$( 14641 + 7746 T^{2} + 1251 T^{4} + 66 T^{6} + T^{8} )^{2}$$
$23$ $$( 7921 + 13138 T^{2} + 1934 T^{4} + 82 T^{6} + T^{8} )^{2}$$
$29$ $$( 59536 - 29808 T^{2} + 4044 T^{4} - 132 T^{6} + T^{8} )^{2}$$
$31$ $$( -169 + 156 T - 29 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$37$ $$( 1251 + 102 T - 76 T^{2} - 2 T^{3} + T^{4} )^{4}$$
$41$ $$( 3455881 - 601978 T^{2} + 19934 T^{4} - 242 T^{6} + T^{8} )^{2}$$
$43$ $$( 9801 + 12564 T^{2} + 3326 T^{4} + 164 T^{6} + T^{8} )^{2}$$
$47$ $$( 121 + 284928 T^{2} + 14459 T^{4} + 222 T^{6} + T^{8} )^{2}$$
$53$ $$( 408321 + 292428 T^{2} + 14959 T^{4} + 222 T^{6} + T^{8} )^{2}$$
$59$ $$( 1 + 118 T^{2} + 359 T^{4} + 52 T^{6} + T^{8} )^{2}$$
$61$ $$( 450241 + 140406 T^{2} + 10071 T^{4} + 186 T^{6} + T^{8} )^{2}$$
$67$ $$( -649 - 184 T + 161 T^{2} - 24 T^{3} + T^{4} )^{4}$$
$71$ $$( 19321 + 18798 T^{2} + 2879 T^{4} + 132 T^{6} + T^{8} )^{2}$$
$73$ $$( 1488400 + 576800 T^{2} + 32060 T^{4} + 340 T^{6} + T^{8} )^{2}$$
$79$ $$( 24025 + 58500 T^{2} + 7710 T^{4} + 180 T^{6} + T^{8} )^{2}$$
$83$ $$( 15124321 - 1110502 T^{2} + 27374 T^{4} - 278 T^{6} + T^{8} )^{2}$$
$89$ $$( 126025 + 47650 T^{2} + 4790 T^{4} + 130 T^{6} + T^{8} )^{2}$$
$97$ $$( 1255 - 330 T - 255 T^{2} + T^{4} )^{4}$$