# Properties

 Label 33.4.d.b Level $33$ Weight $4$ Character orbit 33.d Analytic conductor $1.947$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2 x^{7} - 35 x^{6} + 10 x^{5} + 2614 x^{4} + 16258 x^{3} + 120841 x^{2} + 205270 x + 821047$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( 5 + \beta_{1} + \beta_{3} ) q^{4} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{5} + ( -\beta_{2} + \beta_{6} ) q^{6} + \beta_{7} q^{7} + ( \beta_{2} - \beta_{4} - \beta_{6} ) q^{8} + ( -3 - 3 \beta_{1} + 3 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( 5 + \beta_{1} + \beta_{3} ) q^{4} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{5} + ( -\beta_{2} + \beta_{6} ) q^{6} + \beta_{7} q^{7} + ( \beta_{2} - \beta_{4} - \beta_{6} ) q^{8} + ( -3 - 3 \beta_{1} + 3 \beta_{5} ) q^{9} + ( \beta_{4} - \beta_{6} - \beta_{7} ) q^{10} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -17 + 3 \beta_{1} - 7 \beta_{3} - 3 \beta_{5} ) q^{12} -\beta_{7} q^{13} + ( 4 \beta_{1} - 4 \beta_{3} - 9 \beta_{5} ) q^{14} + ( 16 + 9 \beta_{1} + 2 \beta_{3} ) q^{15} + ( -35 + 3 \beta_{1} + 3 \beta_{3} ) q^{16} + ( 8 \beta_{2} - 4 \beta_{4} - 4 \beta_{6} ) q^{17} + ( -9 \beta_{2} + 3 \beta_{4} - 3 \beta_{7} ) q^{18} + ( -4 \beta_{4} + 4 \beta_{6} ) q^{19} + ( -14 \beta_{1} + 14 \beta_{3} + 7 \beta_{5} ) q^{20} + ( -2 \beta_{2} + 6 \beta_{4} + 2 \beta_{6} + 3 \beta_{7} ) q^{21} + ( -31 - 13 \beta_{1} - 13 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{22} + ( 3 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{23} + ( -17 \beta_{2} - 3 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{24} + ( 39 - 9 \beta_{1} - 9 \beta_{3} ) q^{25} + ( -4 \beta_{1} + 4 \beta_{3} + 9 \beta_{5} ) q^{26} + ( 33 + 18 \beta_{1} + 12 \beta_{3} + 9 \beta_{5} ) q^{27} + ( -4 \beta_{4} + 4 \beta_{6} + \beta_{7} ) q^{28} + ( 18 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} ) q^{29} + ( 38 \beta_{2} - 9 \beta_{4} - 2 \beta_{6} ) q^{30} + ( 104 - 3 \beta_{1} - 3 \beta_{3} ) q^{31} + ( -31 \beta_{2} + 5 \beta_{4} + 5 \beta_{6} ) q^{32} + ( 104 + 3 \beta_{1} + 19 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 12 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{33} + ( 72 + 48 \beta_{1} + 48 \beta_{3} ) q^{34} + ( 46 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} ) q^{35} + ( -81 - 39 \beta_{1} + 15 \beta_{3} + 12 \beta_{5} ) q^{36} + ( -98 - 25 \beta_{1} - 25 \beta_{3} ) q^{37} + ( 72 \beta_{1} - 72 \beta_{3} - 24 \beta_{5} ) q^{38} + ( 2 \beta_{2} - 6 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} ) q^{39} + ( 6 \beta_{4} - 6 \beta_{6} + \beta_{7} ) q^{40} + ( -58 \beta_{2} - 6 \beta_{4} - 6 \beta_{6} ) q^{41} + ( 6 - 66 \beta_{1} - 18 \beta_{3} - 15 \beta_{5} ) q^{42} + ( 8 \beta_{4} - 8 \beta_{6} - 2 \beta_{7} ) q^{43} + ( -26 \beta_{1} - 59 \beta_{2} + 26 \beta_{3} + 5 \beta_{4} + 17 \beta_{5} + 5 \beta_{6} ) q^{44} + ( -210 + 6 \beta_{1} - 27 \beta_{3} - 6 \beta_{5} ) q^{45} + ( -3 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{46} + ( -34 \beta_{1} + 34 \beta_{3} + 25 \beta_{5} ) q^{47} + ( -101 + 9 \beta_{1} + 29 \beta_{3} - 9 \beta_{5} ) q^{48} + ( -283 + 58 \beta_{1} + 58 \beta_{3} ) q^{49} + ( 3 \beta_{2} + 9 \beta_{4} + 9 \beta_{6} ) q^{50} + ( -72 \beta_{2} - 12 \beta_{4} + 12 \beta_{7} ) q^{51} + ( 4 \beta_{4} - 4 \beta_{6} - \beta_{7} ) q^{52} + ( -8 \beta_{1} + 8 \beta_{3} - 47 \beta_{5} ) q^{53} + ( 93 \beta_{2} - 18 \beta_{4} - 12 \beta_{6} - 9 \beta_{7} ) q^{54} + ( -42 - 35 \beta_{1} - 35 \beta_{3} - 8 \beta_{4} + 8 \beta_{6} + \beta_{7} ) q^{55} + ( 44 \beta_{1} - 44 \beta_{3} + 39 \beta_{5} ) q^{56} + ( -144 \beta_{2} + 12 \beta_{4} - 12 \beta_{7} ) q^{57} + ( 250 - 2 \beta_{1} - 2 \beta_{3} ) q^{58} + ( 23 \beta_{1} - 23 \beta_{3} - 26 \beta_{5} ) q^{59} + ( 322 + 84 \beta_{1} + 14 \beta_{3} - 21 \beta_{5} ) q^{60} + ( 4 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{61} + ( 92 \beta_{2} + 3 \beta_{4} + 3 \beta_{6} ) q^{62} + ( 132 \beta_{2} + 24 \beta_{4} + 12 \beta_{6} + 3 \beta_{7} ) q^{63} + ( -83 - 105 \beta_{1} - 105 \beta_{3} ) q^{64} + ( -46 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{65} + ( 255 - 39 \beta_{1} + 106 \beta_{2} + 57 \beta_{3} - 3 \beta_{4} + 39 \beta_{5} + 2 \beta_{6} + 12 \beta_{7} ) q^{66} + ( 8 + 27 \beta_{1} + 27 \beta_{3} ) q^{67} + ( 200 \beta_{2} - 16 \beta_{4} - 16 \beta_{6} ) q^{68} + ( -104 - 3 \beta_{1} + 2 \beta_{3} + 12 \beta_{5} ) q^{69} + ( 614 + 26 \beta_{1} + 26 \beta_{3} ) q^{70} + ( -23 \beta_{1} + 23 \beta_{3} + 67 \beta_{5} ) q^{71} + ( -57 \beta_{2} + 15 \beta_{4} - 15 \beta_{6} + 12 \beta_{7} ) q^{72} + ( -20 \beta_{4} + 20 \beta_{6} - 24 \beta_{7} ) q^{73} + ( -198 \beta_{2} + 25 \beta_{4} + 25 \beta_{6} ) q^{74} + ( 237 - 27 \beta_{1} - 21 \beta_{3} + 27 \beta_{5} ) q^{75} + ( -40 \beta_{4} + 40 \beta_{6} + 24 \beta_{7} ) q^{76} + ( -52 \beta_{1} - 30 \beta_{2} + 52 \beta_{3} - 34 \beta_{4} - 21 \beta_{5} - 34 \beta_{6} ) q^{77} + ( -6 + 66 \beta_{1} + 18 \beta_{3} + 15 \beta_{5} ) q^{78} + ( 12 \beta_{4} - 12 \beta_{6} + 25 \beta_{7} ) q^{79} + ( 8 \beta_{1} - 8 \beta_{3} - 29 \beta_{5} ) q^{80} + ( -513 + 90 \beta_{1} - 45 \beta_{3} - 9 \beta_{5} ) q^{81} + ( -802 + 2 \beta_{1} + 2 \beta_{3} ) q^{82} + ( 4 \beta_{2} + 12 \beta_{4} + 12 \beta_{6} ) q^{83} + ( -146 \beta_{2} + 18 \beta_{4} + 2 \beta_{6} - 9 \beta_{7} ) q^{84} + ( 28 \beta_{4} - 28 \beta_{6} ) q^{85} + ( -152 \beta_{1} + 152 \beta_{3} + 66 \beta_{5} ) q^{86} + ( 14 \beta_{2} + 6 \beta_{4} + 22 \beta_{6} - 6 \beta_{7} ) q^{87} + ( -479 - 5 \beta_{1} - 5 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 25 \beta_{7} ) q^{88} + ( 41 \beta_{1} - 41 \beta_{3} - 122 \beta_{5} ) q^{89} + ( -252 \beta_{2} - 6 \beta_{4} + 27 \beta_{6} + 6 \beta_{7} ) q^{90} + ( 626 - 58 \beta_{1} - 58 \beta_{3} ) q^{91} + ( 26 \beta_{1} - 26 \beta_{3} - 17 \beta_{5} ) q^{92} + ( 170 - 9 \beta_{1} - 98 \beta_{3} + 9 \beta_{5} ) q^{93} + ( 34 \beta_{4} - 34 \beta_{6} - 25 \beta_{7} ) q^{94} + ( 304 \beta_{2} - 44 \beta_{4} - 44 \beta_{6} ) q^{95} + ( 111 \beta_{2} + 15 \beta_{4} - 21 \beta_{6} - 15 \beta_{7} ) q^{96} + ( -532 + 105 \beta_{1} + 105 \beta_{3} ) q^{97} + ( -51 \beta_{2} - 58 \beta_{4} - 58 \beta_{6} ) q^{98} + ( 186 - 78 \beta_{1} + 75 \beta_{2} - 129 \beta_{3} - 21 \beta_{4} - 30 \beta_{5} + 15 \beta_{6} - 6 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 6q^{3} + 44q^{4} - 30q^{9} + O(q^{10})$$ $$8q + 6q^{3} + 44q^{4} - 30q^{9} - 144q^{12} + 150q^{15} - 268q^{16} - 300q^{22} + 276q^{25} + 324q^{27} + 820q^{31} + 834q^{33} + 768q^{34} - 696q^{36} - 884q^{37} - 120q^{42} - 1722q^{45} - 732q^{48} - 2032q^{49} - 476q^{55} + 1992q^{58} + 2772q^{60} - 1084q^{64} + 2076q^{66} + 172q^{67} - 834q^{69} + 5016q^{70} + 1800q^{75} + 120q^{78} - 4014q^{81} - 6408q^{82} - 3852q^{88} + 4776q^{91} + 1146q^{93} - 3836q^{97} + 1074q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} - 35 x^{6} + 10 x^{5} + 2614 x^{4} + 16258 x^{3} + 120841 x^{2} + 205270 x + 821047$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$654449 \nu^{7} - 4621500 \nu^{6} - 37910704 \nu^{5} + 258375310 \nu^{4} + 2191936298 \nu^{3} - 3509536790 \nu^{2} - 27353124741 \nu - 251237072504$$$$)/ 43097961858$$ $$\beta_{2}$$ $$=$$ $$($$$$-1117339 \nu^{7} + 7661354 \nu^{6} + 34074596 \nu^{5} - 310366016 \nu^{4} - 2938713578 \nu^{3} + 1344980044 \nu^{2} - 23176332663 \nu + 204737740374$$$$)/ 43097961858$$ $$\beta_{3}$$ $$=$$ $$($$$$1117339 \nu^{7} - 7661354 \nu^{6} - 34074596 \nu^{5} + 310366016 \nu^{4} + 2938713578 \nu^{3} - 1344980044 \nu^{2} + 66274294521 \nu - 204737740374$$$$)/ 43097961858$$ $$\beta_{4}$$ $$=$$ $$($$$$2849712 \nu^{7} + 18044843 \nu^{6} - 255666928 \nu^{5} - 533197318 \nu^{4} + 13402791352 \nu^{3} + 104272839604 \nu^{2} + 378144323188 \nu + 1611381064281$$$$)/ 43097961858$$ $$\beta_{5}$$ $$=$$ $$($$$$-9352895 \nu^{7} - 760108 \nu^{6} + 480817032 \nu^{5} + 759114088 \nu^{4} - 33192323398 \nu^{3} - 193283264130 \nu^{2} - 1043874627293 \nu - 1902320865922$$$$)/ 129293885574$$ $$\beta_{6}$$ $$=$$ $$($$$$-5889566 \nu^{7} + 8072123 \nu^{6} + 295356518 \nu^{5} - 34001274 \nu^{4} - 20257454786 \nu^{3} - 114008586182 \nu^{2} - 441525412450 \nu - 403613661909$$$$)/ 43097961858$$ $$\beta_{7}$$ $$=$$ $$($$$$345396 \nu^{7} - 399831 \nu^{6} - 16270822 \nu^{5} + 12503088 \nu^{4} + 886340512 \nu^{3} + 6517009006 \nu^{2} + 37980909510 \nu + 64671640283$$$$)/ 2268313782$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{3} + \beta_{2}$$ $$\nu^{2}$$ $$=$$ $$-2 \beta_{6} + 3 \beta_{5} + 3 \beta_{3} + 4 \beta_{2} - 2 \beta_{1} + 9$$ $$\nu^{3}$$ $$=$$ $$-9 \beta_{7} - 7 \beta_{6} + 9 \beta_{5} + 8 \beta_{4} + 39 \beta_{3} - \beta_{2} - 33 \beta_{1} + 22$$ $$\nu^{4}$$ $$=$$ $$-12 \beta_{7} - 28 \beta_{6} + 225 \beta_{5} + 116 \beta_{4} + 214 \beta_{3} - 264 \beta_{2} - 381 \beta_{1} - 897$$ $$\nu^{5}$$ $$=$$ $$-285 \beta_{7} - 42 \beta_{6} + 732 \beta_{5} + 653 \beta_{4} - 660 \beta_{3} - 5079 \beta_{2} - 4421 \beta_{1} - 10692$$ $$\nu^{6}$$ $$=$$ $$-288 \beta_{7} + 5656 \beta_{6} + 1206 \beta_{5} + 6910 \beta_{4} - 18636 \beta_{3} - 48980 \beta_{2} - 22362 \beta_{1} - 165641$$ $$\nu^{7}$$ $$=$$ $$16338 \beta_{7} + 61282 \beta_{6} - 51966 \beta_{5} + 14032 \beta_{4} - 327059 \beta_{3} - 469273 \beta_{2} - 97938 \beta_{1} - 1076460$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −0.913072 + 4.51265i −0.913072 − 4.51265i −5.69289 + 3.22272i −5.69289 − 3.22272i −0.459178 + 3.22272i −0.459178 − 3.22272i 8.06514 + 4.51265i 8.06514 − 4.51265i
−4.48911 −2.57603 4.51265i 12.1521 12.2625i 11.5641 + 20.2578i 14.5320i −18.6391 −13.7281 + 23.2495i 55.0476i
32.2 −4.48911 −2.57603 + 4.51265i 12.1521 12.2625i 11.5641 20.2578i 14.5320i −18.6391 −13.7281 23.2495i 55.0476i
32.3 −2.61686 4.07603 3.22272i −1.15207 5.53456i −10.6664 + 8.43340i 31.3500i 23.9496 6.22810 26.2719i 14.4832i
32.4 −2.61686 4.07603 + 3.22272i −1.15207 5.53456i −10.6664 8.43340i 31.3500i 23.9496 6.22810 + 26.2719i 14.4832i
32.5 2.61686 4.07603 3.22272i −1.15207 5.53456i 10.6664 8.43340i 31.3500i −23.9496 6.22810 26.2719i 14.4832i
32.6 2.61686 4.07603 + 3.22272i −1.15207 5.53456i 10.6664 + 8.43340i 31.3500i −23.9496 6.22810 + 26.2719i 14.4832i
32.7 4.48911 −2.57603 4.51265i 12.1521 12.2625i −11.5641 20.2578i 14.5320i 18.6391 −13.7281 + 23.2495i 55.0476i
32.8 4.48911 −2.57603 + 4.51265i 12.1521 12.2625i −11.5641 + 20.2578i 14.5320i 18.6391 −13.7281 23.2495i 55.0476i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.8 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 33.4.d.b 8
3.b odd 2 1 inner 33.4.d.b 8
4.b odd 2 1 528.4.b.e 8
11.b odd 2 1 inner 33.4.d.b 8
12.b even 2 1 528.4.b.e 8
33.d even 2 1 inner 33.4.d.b 8
44.c even 2 1 528.4.b.e 8
132.d odd 2 1 528.4.b.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.d.b 8 1.a even 1 1 trivial
33.4.d.b 8 3.b odd 2 1 inner
33.4.d.b 8 11.b odd 2 1 inner
33.4.d.b 8 33.d even 2 1 inner
528.4.b.e 8 4.b odd 2 1
528.4.b.e 8 12.b even 2 1
528.4.b.e 8 44.c even 2 1
528.4.b.e 8 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 27 T_{2}^{2} + 138$$ acting on $$S_{4}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 5 T^{2} + 90 T^{4} + 320 T^{6} + 4096 T^{8} )^{2}$$
$3$ $$( 1 - 3 T + 12 T^{2} - 81 T^{3} + 729 T^{4} )^{2}$$
$5$ $$( 1 - 319 T^{2} + 53106 T^{4} - 4984375 T^{6} + 244140625 T^{8} )^{2}$$
$7$ $$( 1 - 178 T^{2} + 94362 T^{4} - 20941522 T^{6} + 13841287201 T^{8} )^{2}$$
$11$ $$1 + 872 T^{2} + 3052830 T^{4} + 1544801192 T^{6} + 3138428376721 T^{8}$$
$13$ $$( 1 - 7594 T^{2} + 23921970 T^{4} - 36654787546 T^{6} + 23298085122481 T^{8} )^{2}$$
$17$ $$( 1 + 3812 T^{2} + 51500166 T^{4} + 92012413028 T^{6} + 582622237229761 T^{8} )^{2}$$
$19$ $$( 1 + 3380 T^{2} - 20908650 T^{4} + 159015077780 T^{6} + 2213314919066161 T^{8} )^{2}$$
$23$ $$( 1 - 47119 T^{2} + 851079294 T^{4} - 6975303053791 T^{6} + 21914624432020321 T^{8} )^{2}$$
$29$ $$( 1 + 84752 T^{2} + 2972083566 T^{4} + 50412466101392 T^{6} + 353814783205469041 T^{8} )^{2}$$
$31$ $$( 1 - 205 T + 69690 T^{2} - 6107155 T^{3} + 887503681 T^{4} )^{4}$$
$37$ $$( 1 + 221 T + 85860 T^{2} + 11194313 T^{3} + 2565726409 T^{4} )^{4}$$
$41$ $$( 1 + 148064 T^{2} + 13890568158 T^{4} + 703319434339424 T^{6} + 22563490300366186081 T^{8} )^{2}$$
$43$ $$( 1 - 188068 T^{2} + 17913601926 T^{4} - 1188846105899332 T^{6} + 39959630797262576401 T^{8} )^{2}$$
$47$ $$( 1 - 258454 T^{2} + 33079509450 T^{4} - 2785931318641366 T^{6} +$$$$11\!\cdots\!41$$$$T^{8} )^{2}$$
$53$ $$( 1 - 211354 T^{2} + 21017131314 T^{4} - 4684526382058666 T^{6} +$$$$49\!\cdots\!41$$$$T^{8} )^{2}$$
$59$ $$( 1 - 709753 T^{2} + 209443663428 T^{4} - 29937760293300673 T^{6} +$$$$17\!\cdots\!81$$$$T^{8} )^{2}$$
$61$ $$( 1 - 876394 T^{2} + 295021357554 T^{4} - 45152146967734234 T^{6} +$$$$26\!\cdots\!21$$$$T^{8} )^{2}$$
$67$ $$( 1 - 43 T + 569730 T^{2} - 12932809 T^{3} + 90458382169 T^{4} )^{4}$$
$71$ $$( 1 - 823207 T^{2} + 410231814606 T^{4} - 105453050425754647 T^{6} +$$$$16\!\cdots\!41$$$$T^{8} )^{2}$$
$73$ $$( 1 - 155524 T^{2} + 71302205670 T^{4} - 23536104209370436 T^{6} +$$$$22\!\cdots\!21$$$$T^{8} )^{2}$$
$79$ $$( 1 - 984562 T^{2} + 499365566106 T^{4} - 239334671382666802 T^{6} +$$$$59\!\cdots\!41$$$$T^{8} )^{2}$$
$83$ $$( 1 + 2155676 T^{2} + 1813793947254 T^{4} + 704777516302592444 T^{6} +$$$$10\!\cdots\!61$$$$T^{8} )^{2}$$
$89$ $$( 1 - 801169 T^{2} + 974511119856 T^{4} - 398166003897933409 T^{6} +$$$$24\!\cdots\!21$$$$T^{8} )^{2}$$
$97$ $$( 1 + 959 T + 1567410 T^{2} + 875253407 T^{3} + 832972004929 T^{4} )^{4}$$