# Properties

 Label 108.4.h.a Level $108$ Weight $4$ Character orbit 108.h Analytic conductor $6.372$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 108.h (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.37220628062$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.553553856144.1 Defining polynomial: $$x^{8} + 11 x^{6} + 96 x^{4} + 704 x^{2} + 4096$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{4} + ( -10 + \beta_{1} + \beta_{3} - 5 \beta_{4} + \beta_{6} ) q^{5} + ( -3 \beta_{1} - 5 \beta_{3} - 7 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{7} + ( -10 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 20 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{7} q^{2} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{4} + ( -10 + \beta_{1} + \beta_{3} - 5 \beta_{4} + \beta_{6} ) q^{5} + ( -3 \beta_{1} - 5 \beta_{3} - 7 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{7} + ( -10 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 20 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} ) q^{8} + ( -10 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{10} + ( -\beta_{1} + 2 \beta_{2} - 10 \beta_{3} + \beta_{5} + \beta_{6} - 9 \beta_{7} ) q^{11} + ( -\beta_{1} + \beta_{3} - 55 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{13} + ( 4 - 8 \beta_{1} - 10 \beta_{2} - 18 \beta_{3} + 2 \beta_{4} - 20 \beta_{5} + 2 \beta_{6} - 10 \beta_{7} ) q^{14} + ( 26 - 2 \beta_{1} - 8 \beta_{3} + 26 \beta_{4} - 14 \beta_{5} + 24 \beta_{6} + 5 \beta_{7} ) q^{16} + ( -30 - 13 \beta_{1} - 60 \beta_{4} - 13 \beta_{6} - 13 \beta_{7} ) q^{17} + ( 5 \beta_{1} + 30 \beta_{2} + 40 \beta_{3} + 30 \beta_{5} - 5 \beta_{6} + 35 \beta_{7} ) q^{19} + ( -20 + 12 \beta_{1} - 24 \beta_{2} + 16 \beta_{3} + 20 \beta_{4} - 12 \beta_{5} - 12 \beta_{6} - 18 \beta_{7} ) q^{20} + ( -4 \beta_{1} - 16 \beta_{2} + 20 \beta_{3} - 60 \beta_{4} - 7 \beta_{6} + 2 \beta_{7} ) q^{22} + ( 21 \beta_{1} + 5 \beta_{2} + 26 \beta_{3} + 10 \beta_{5} - 21 \beta_{6} + 5 \beta_{7} ) q^{23} + ( -30 - 22 \beta_{1} - 11 \beta_{3} - 30 \beta_{4} - 22 \beta_{6} - 11 \beta_{7} ) q^{25} + ( 10 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 20 \beta_{4} + 2 \beta_{5} + 54 \beta_{6} + 56 \beta_{7} ) q^{26} + ( 60 + 4 \beta_{1} - 44 \beta_{2} - 36 \beta_{3} - 44 \beta_{5} - 14 \beta_{6} - 30 \beta_{7} ) q^{28} + ( 53 + 23 \beta_{3} - 53 \beta_{4} - 23 \beta_{7} ) q^{29} + ( -7 \beta_{1} + 43 \beta_{2} - 36 \beta_{3} + 7 \beta_{6} - 29 \beta_{7} ) q^{31} + ( -60 + 20 \beta_{1} - 38 \beta_{2} - 18 \beta_{3} - 30 \beta_{4} - 76 \beta_{5} - 31 \beta_{6} - 38 \beta_{7} ) q^{32} + ( 130 - 26 \beta_{1} + 130 \beta_{4} + 26 \beta_{5} + 60 \beta_{6} + 43 \beta_{7} ) q^{34} + ( 70 \beta_{1} - 55 \beta_{2} + 55 \beta_{3} + 55 \beta_{5} - 70 \beta_{6} - 15 \beta_{7} ) q^{35} + ( -160 - 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{37} + ( -90 - 70 \beta_{1} + 140 \beta_{2} - 20 \beta_{3} + 90 \beta_{4} + 70 \beta_{5} + 70 \beta_{6} + 65 \beta_{7} ) q^{38} + ( 48 \beta_{1} - 60 \beta_{2} + 12 \beta_{3} + 140 \beta_{4} - 62 \beta_{6} - 64 \beta_{7} ) q^{40} + ( 162 - 50 \beta_{1} - 50 \beta_{3} + 81 \beta_{4} - 50 \beta_{6} ) q^{41} + ( 51 \beta_{1} + 52 \beta_{3} + 53 \beta_{5} - 51 \beta_{6} + \beta_{7} ) q^{43} + ( 110 + 50 \beta_{1} - 14 \beta_{2} + 14 \beta_{3} + 220 \beta_{4} + 14 \beta_{5} + 37 \beta_{6} + 51 \beta_{7} ) q^{44} + ( -126 - 42 \beta_{1} + 62 \beta_{2} - 22 \beta_{3} + 62 \beta_{5} + 26 \beta_{6} + 36 \beta_{7} ) q^{46} + ( -87 \beta_{1} + 174 \beta_{2} - 77 \beta_{3} + 87 \beta_{5} + 87 \beta_{6} + 10 \beta_{7} ) q^{47} + ( 37 \beta_{1} - 37 \beta_{3} + 94 \beta_{4} + 37 \beta_{6} + 74 \beta_{7} ) q^{49} + ( 220 - 44 \beta_{1} + 22 \beta_{2} - 22 \beta_{3} + 110 \beta_{4} + 44 \beta_{5} + 41 \beta_{6} + 22 \beta_{7} ) q^{50} + ( -80 + 120 \beta_{1} + 8 \beta_{3} - 80 \beta_{4} - 104 \beta_{5} - 24 \beta_{6} - 64 \beta_{7} ) q^{52} + ( 10 + 74 \beta_{1} + 20 \beta_{4} + 74 \beta_{6} + 74 \beta_{7} ) q^{53} + ( 26 \beta_{1} + 17 \beta_{2} + 69 \beta_{3} + 17 \beta_{5} - 26 \beta_{6} + 43 \beta_{7} ) q^{55} + ( 84 + 60 \beta_{1} - 120 \beta_{2} - 56 \beta_{3} - 84 \beta_{4} - 60 \beta_{5} - 60 \beta_{6} - 14 \beta_{7} ) q^{56} + ( -46 \beta_{2} + 46 \beta_{3} + 230 \beta_{4} + 30 \beta_{6} + 106 \beta_{7} ) q^{58} + ( 70 \beta_{1} + 3 \beta_{2} + 73 \beta_{3} + 6 \beta_{5} - 70 \beta_{6} + 3 \beta_{7} ) q^{59} + ( -41 - 142 \beta_{1} - 71 \beta_{3} - 41 \beta_{4} - 142 \beta_{6} - 71 \beta_{7} ) q^{61} + ( -130 - 158 \beta_{1} + 14 \beta_{2} - 14 \beta_{3} - 260 \beta_{4} - 14 \beta_{5} + 50 \beta_{6} + 36 \beta_{7} ) q^{62} + ( -98 - 62 \beta_{1} - 90 \beta_{2} - 214 \beta_{3} - 90 \beta_{5} + 23 \beta_{6} - 113 \beta_{7} ) q^{64} + ( -255 + 37 \beta_{3} + 255 \beta_{4} - 37 \beta_{7} ) q^{65} + ( 76 \beta_{1} - 101 \beta_{2} + 25 \beta_{3} - 76 \beta_{6} - 51 \beta_{7} ) q^{67} + ( 140 + 172 \beta_{1} - 34 \beta_{2} + 138 \beta_{3} + 70 \beta_{4} - 68 \beta_{5} - 173 \beta_{6} - 34 \beta_{7} ) q^{68} + ( -90 + 190 \beta_{1} + 220 \beta_{3} - 90 \beta_{4} + 250 \beta_{5} - 110 \beta_{6} + 70 \beta_{7} ) q^{70} + ( -106 \beta_{1} + 26 \beta_{2} - 26 \beta_{3} - 26 \beta_{5} + 106 \beta_{6} + 80 \beta_{7} ) q^{71} + ( 620 + 85 \beta_{1} + 170 \beta_{3} + 85 \beta_{6} - 85 \beta_{7} ) q^{73} + ( 20 - 4 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - 158 \beta_{7} ) q^{74} + ( -280 \beta_{1} + 270 \beta_{2} + 10 \beta_{3} - 150 \beta_{4} + 115 \beta_{6} - 40 \beta_{7} ) q^{76} + ( -10 + 45 \beta_{1} + 45 \beta_{3} - 5 \beta_{4} + 45 \beta_{6} ) q^{77} + ( -341 \beta_{1} - 207 \beta_{3} - 73 \beta_{5} + 341 \beta_{6} + 134 \beta_{7} ) q^{79} + ( -20 + 116 \beta_{1} - 124 \beta_{2} + 124 \beta_{3} - 40 \beta_{4} + 124 \beta_{5} - 262 \beta_{6} - 138 \beta_{7} ) q^{80} + ( 500 - 100 \beta_{1} + 100 \beta_{2} - 100 \beta_{3} + 100 \beta_{5} - 31 \beta_{6} + 131 \beta_{7} ) q^{82} + ( 161 \beta_{1} - 322 \beta_{2} + 131 \beta_{3} - 161 \beta_{5} - 161 \beta_{6} - 30 \beta_{7} ) q^{83} + ( 48 \beta_{1} - 48 \beta_{3} + 190 \beta_{4} + 48 \beta_{6} + 96 \beta_{7} ) q^{85} + ( -200 + 4 \beta_{1} + 104 \beta_{2} + 108 \beta_{3} - 100 \beta_{4} + 208 \beta_{5} - \beta_{6} + 104 \beta_{7} ) q^{86} + ( -390 + 158 \beta_{1} + 56 \beta_{3} - 390 \beta_{4} - 46 \beta_{5} - 248 \beta_{6} - 147 \beta_{7} ) q^{88} + ( -102 - 46 \beta_{1} - 204 \beta_{4} - 46 \beta_{6} - 46 \beta_{7} ) q^{89} + ( 100 \beta_{1} - 325 \beta_{2} - 125 \beta_{3} - 325 \beta_{5} - 100 \beta_{6} - 225 \beta_{7} ) q^{91} + ( 160 - 72 \beta_{1} + 144 \beta_{2} + 104 \beta_{3} - 160 \beta_{4} + 72 \beta_{5} + 72 \beta_{6} - 28 \beta_{7} ) q^{92} + ( -348 \beta_{1} + 194 \beta_{2} + 154 \beta_{3} - 462 \beta_{4} + 184 \beta_{6} + 174 \beta_{7} ) q^{94} + ( -200 \beta_{1} - 290 \beta_{2} - 490 \beta_{3} - 580 \beta_{5} + 200 \beta_{6} - 290 \beta_{7} ) q^{95} + ( -5 + 184 \beta_{1} + 92 \beta_{3} - 5 \beta_{4} + 184 \beta_{6} + 92 \beta_{7} ) q^{97} + ( -370 + 74 \beta_{1} + 74 \beta_{2} - 74 \beta_{3} - 740 \beta_{4} - 74 \beta_{5} - 57 \beta_{6} - 131 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 3q^{2} + 11q^{4} - 66q^{5} + O(q^{10})$$ $$8q + 3q^{2} + 11q^{4} - 66q^{5} - 116q^{10} + 214q^{13} + 42q^{14} + 71q^{16} - 306q^{20} + 207q^{22} - 54q^{25} + 540q^{28} + 498q^{29} - 327q^{32} + 469q^{34} - 1256q^{37} - 1035q^{38} - 602q^{40} + 1272q^{41} - 912q^{46} - 154q^{49} + 1329q^{50} - 464q^{52} + 1314q^{56} - 830q^{58} + 262q^{61} - 550q^{64} - 3282q^{65} + 843q^{68} - 480q^{70} + 3940q^{73} - 222q^{74} + 105q^{76} - 330q^{77} + 4786q^{82} - 472q^{85} - 1209q^{86} - 1425q^{88} + 1308q^{92} + 1356q^{94} - 572q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 11 x^{6} + 96 x^{4} + 704 x^{2} + 4096$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 5 \nu^{5} - \nu^{4} - 15 \nu^{3} + 28 \nu^{2} - 40 \nu + 128$$$$)/320$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 19 \nu^{4} - 88 \nu^{2} - 160 \nu - 768$$$$)/320$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{7} - 17 \nu^{5} + 16 \nu^{3} - 704 \nu - 2560$$$$)/5120$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + 8 \nu^{6} + 25 \nu^{5} + 152 \nu^{4} - 240 \nu^{3} + 1984 \nu^{2} - 320 \nu + 8704$$$$)/5120$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} - 24 \nu^{6} + 55 \nu^{5} - 136 \nu^{4} + 480 \nu^{3} + 128 \nu^{2} + 3520 \nu - 5632$$$$)/5120$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 24 \nu^{6} + 55 \nu^{5} + 136 \nu^{4} + 480 \nu^{3} - 128 \nu^{2} + 3520 \nu + 5632$$$$)/5120$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{5} + 2 \beta_{2} - \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} + 20 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_{1} + 10$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{7} - 6 \beta_{6} - 14 \beta_{5} - 24 \beta_{3} - 14 \beta_{2} - 5 \beta_{1} - 26$$ $$\nu^{5}$$ $$=$$ $$20 \beta_{7} - 18 \beta_{6} + 38 \beta_{5} - 60 \beta_{4} + 38 \beta_{3} - 38 \beta_{2} + 7 \beta_{1} - 30$$ $$\nu^{6}$$ $$=$$ $$152 \beta_{7} - 62 \beta_{6} + 90 \beta_{5} + 136 \beta_{3} + 90 \beta_{2} + 23 \beta_{1} - 98$$ $$\nu^{7}$$ $$=$$ $$-92 \beta_{7} + 134 \beta_{6} - 226 \beta_{5} - 1260 \beta_{4} - 226 \beta_{3} + 226 \beta_{2} - 301 \beta_{1} - 630$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$-\beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1
 2.14417 − 1.84460i −2.14417 − 1.84460i 0.807795 + 2.71062i −0.807795 + 2.71062i 2.14417 + 1.84460i −2.14417 + 1.84460i 0.807795 − 2.71062i −0.807795 − 2.71062i
−2.66955 + 0.934608i 0 6.25302 4.98997i −4.30507 + 2.48553i 0 3.07102 + 1.77306i −12.0291 + 19.1651i 0 9.16960 10.6588i
35.2 −0.525382 2.77920i 0 −7.44795 + 2.92028i −4.30507 + 2.48553i 0 −3.07102 1.77306i 12.0291 + 19.1651i 0 9.16960 + 10.6588i
35.3 1.94357 + 2.05488i 0 −0.445079 + 7.98761i −12.1949 + 7.04075i 0 −21.1499 12.2109i −17.2786 + 14.6099i 0 −38.1696 11.3750i
35.4 2.75136 + 0.655739i 0 7.14001 + 3.60836i −12.1949 + 7.04075i 0 21.1499 + 12.2109i 17.2786 + 14.6099i 0 −38.1696 + 11.3750i
71.1 −2.66955 0.934608i 0 6.25302 + 4.98997i −4.30507 2.48553i 0 3.07102 1.77306i −12.0291 19.1651i 0 9.16960 + 10.6588i
71.2 −0.525382 + 2.77920i 0 −7.44795 2.92028i −4.30507 2.48553i 0 −3.07102 + 1.77306i 12.0291 19.1651i 0 9.16960 10.6588i
71.3 1.94357 2.05488i 0 −0.445079 7.98761i −12.1949 7.04075i 0 −21.1499 + 12.2109i −17.2786 14.6099i 0 −38.1696 + 11.3750i
71.4 2.75136 0.655739i 0 7.14001 3.60836i −12.1949 7.04075i 0 21.1499 12.2109i 17.2786 14.6099i 0 −38.1696 11.3750i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.h.a 8
3.b odd 2 1 36.4.h.a 8
4.b odd 2 1 inner 108.4.h.a 8
9.c even 3 1 36.4.h.a 8
9.c even 3 1 324.4.b.b 8
9.d odd 6 1 inner 108.4.h.a 8
9.d odd 6 1 324.4.b.b 8
12.b even 2 1 36.4.h.a 8
36.f odd 6 1 36.4.h.a 8
36.f odd 6 1 324.4.b.b 8
36.h even 6 1 inner 108.4.h.a 8
36.h even 6 1 324.4.b.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.h.a 8 3.b odd 2 1
36.4.h.a 8 9.c even 3 1
36.4.h.a 8 12.b even 2 1
36.4.h.a 8 36.f odd 6 1
108.4.h.a 8 1.a even 1 1 trivial
108.4.h.a 8 4.b odd 2 1 inner
108.4.h.a 8 9.d odd 6 1 inner
108.4.h.a 8 36.h even 6 1 inner
324.4.b.b 8 9.c even 3 1
324.4.b.b 8 9.d odd 6 1
324.4.b.b 8 36.f odd 6 1
324.4.b.b 8 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 33 T_{5}^{3} + 433 T_{5}^{2} + 2310 T_{5} + 4900$$ acting on $$S_{4}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4096 - 1536 T - 64 T^{2} + 96 T^{3} - 24 T^{4} + 12 T^{5} - T^{6} - 3 T^{7} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 4900 + 2310 T + 433 T^{2} + 33 T^{3} + T^{4} )^{2}$$
$7$ $$56250000 - 4567500 T^{2} + 363381 T^{4} - 609 T^{6} + T^{8}$$
$11$ $$44383955625 + 437361300 T^{2} + 4099101 T^{4} + 2076 T^{6} + T^{8}$$
$13$ $$( 7840000 - 299600 T + 8649 T^{2} - 107 T^{3} + T^{4} )^{2}$$
$17$ $$( 3422500 + 10327 T^{2} + T^{4} )^{2}$$
$19$ $$( 149107500 + 27675 T^{2} + T^{4} )^{2}$$
$23$ $$732895782914304 + 292134469968 T^{2} + 89373633 T^{4} + 10791 T^{6} + T^{8}$$
$29$ $$( 33756100 + 1446690 T + 14857 T^{2} - 249 T^{3} + T^{4} )^{2}$$
$31$ $$75175111088640000 - 12659750078400 T^{2} + 1857765129 T^{4} - 46173 T^{6} + T^{8}$$
$37$ $$( 24400 + 314 T + T^{2} )^{4}$$
$41$ $$( 330039889 + 11554212 T + 116665 T^{2} - 636 T^{3} + T^{4} )^{2}$$
$43$ $$94959303724355625 - 17413204374900 T^{2} + 2884999389 T^{4} - 56508 T^{6} + T^{8}$$
$47$ $$94\!\cdots\!24$$$$+ 10965873299346432 T^{2} + 96571490433 T^{4} + 356799 T^{6} + T^{8}$$
$53$ $$( 12418873600 + 231628 T^{2} + T^{4} )^{2}$$
$59$ $$2219693770084505625 + 155666916218700 T^{2} + 9427042581 T^{4} + 104484 T^{6} + T^{8}$$
$61$ $$( 95797678144 + 40546072 T + 326673 T^{2} - 131 T^{3} + T^{4} )^{2}$$
$67$ $$4470730707971529 - 8852462991108 T^{2} + 17461837293 T^{4} - 132396 T^{6} + T^{8}$$
$71$ $$( 7279642800 - 171216 T^{2} + T^{4} )^{2}$$
$73$ $$( -207200 - 985 T + T^{2} )^{4}$$
$79$ $$34\!\cdots\!00$$$$- 1004737992349797900 T^{2} + 2346599752629 T^{4} - 1712673 T^{6} + T^{8}$$
$83$ $$11\!\cdots\!64$$$$+ 393234541396216128 T^{2} + 1024903491873 T^{4} + 1166991 T^{6} + T^{8}$$
$89$ $$( 634233856 + 125260 T^{2} + T^{4} )^{2}$$
$97$ $$( 256476409225 - 144840410 T + 588231 T^{2} + 286 T^{3} + T^{4} )^{2}$$