# Properties

 Label 320.6.f.c Level 320 Weight 6 Character orbit 320.f Analytic conductor 51.323 Analytic rank 0 Dimension 16 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 320.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.3228223402$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 39122 x^{12} + 391243971 x^{8} + 75462898750 x^{4} + 18538406640625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{42}\cdot 5^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} -\beta_{10} q^{5} + ( -\beta_{8} + \beta_{12} ) q^{7} + ( 119 + \beta_{1} ) q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{3} -\beta_{10} q^{5} + ( -\beta_{8} + \beta_{12} ) q^{7} + ( 119 + \beta_{1} ) q^{9} + ( 16 \beta_{3} - \beta_{4} ) q^{11} + ( 4 \beta_{9} - 4 \beta_{10} + 3 \beta_{13} ) q^{13} + ( -\beta_{5} - 3 \beta_{8} - 5 \beta_{12} ) q^{15} + ( 5 \beta_{11} + \beta_{14} ) q^{17} + ( 17 \beta_{3} - 7 \beta_{4} ) q^{19} + ( \beta_{2} + \beta_{9} + \beta_{10} ) q^{21} + ( 23 \beta_{8} + 36 \beta_{12} ) q^{23} + ( 55 + 10 \beta_{1} - 8 \beta_{11} - \beta_{14} ) q^{25} + ( 53 \beta_{7} - \beta_{15} ) q^{27} + ( 3 \beta_{2} + 8 \beta_{9} + 8 \beta_{10} ) q^{29} + ( -6 \beta_{5} + 5 \beta_{6} ) q^{31} + ( 11 \beta_{11} - \beta_{14} ) q^{33} + ( -440 \beta_{3} - 15 \beta_{4} + 10 \beta_{7} + \beta_{15} ) q^{35} + ( 37 \beta_{9} - 37 \beta_{10} + 50 \beta_{13} ) q^{37} + ( -8 \beta_{5} + 15 \beta_{6} ) q^{39} + ( -6272 - 47 \beta_{1} ) q^{41} + ( -260 \beta_{7} + \beta_{15} ) q^{43} + ( 3 \beta_{2} + 220 \beta_{9} - 131 \beta_{10} + 43 \beta_{13} ) q^{45} + ( -219 \beta_{8} + 207 \beta_{12} ) q^{47} + ( 8017 - 5 \beta_{1} ) q^{49} + ( 1045 \beta_{3} + 226 \beta_{4} ) q^{51} + ( 150 \beta_{9} - 150 \beta_{10} - 101 \beta_{13} ) q^{53} + ( 3 \beta_{5} + 31 \beta_{6} + 8 \beta_{8} - 47 \beta_{12} ) q^{55} + ( -113 \beta_{11} - 7 \beta_{14} ) q^{57} + ( 393 \beta_{3} - 115 \beta_{4} ) q^{59} + ( 335 \beta_{9} + 335 \beta_{10} ) q^{61} + ( -321 \beta_{8} - 68 \beta_{12} ) q^{63} + ( 9360 + 70 \beta_{1} - 161 \beta_{11} + 8 \beta_{14} ) q^{65} + ( 162 \beta_{7} + 9 \beta_{15} ) q^{67} + ( -23 \beta_{2} + 567 \beta_{9} + 567 \beta_{10} ) q^{69} + ( 42 \beta_{5} + 150 \beta_{6} ) q^{71} + ( 561 \beta_{11} + 9 \beta_{14} ) q^{73} + ( -1600 \beta_{3} - 250 \beta_{4} + 1825 \beta_{7} - 10 \beta_{15} ) q^{75} + ( 31 \beta_{9} - 31 \beta_{10} - 105 \beta_{13} ) q^{77} + ( 54 \beta_{5} + 175 \beta_{6} ) q^{79} + ( -9661 - 5 \beta_{1} ) q^{81} + ( -2846 \beta_{7} - 11 \beta_{15} ) q^{83} + ( -24 \beta_{2} + 865 \beta_{9} + 187 \beta_{10} - 469 \beta_{13} ) q^{85} + ( -1662 \beta_{8} + 575 \beta_{12} ) q^{87} + ( -28706 + 144 \beta_{1} ) q^{89} + ( -835 \beta_{3} - 360 \beta_{4} ) q^{91} + ( 1902 \beta_{9} - 1902 \beta_{10} + 388 \beta_{13} ) q^{93} + ( 21 \beta_{5} + 122 \beta_{6} - 324 \beta_{8} - 139 \beta_{12} ) q^{95} + ( -439 \beta_{11} + 7 \beta_{14} ) q^{97} + ( -1973 \beta_{3} + 145 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 1904q^{9} + O(q^{10})$$ $$16q + 1904q^{9} + 880q^{25} - 100352q^{41} + 128272q^{49} + 149760q^{65} - 154576q^{81} - 459296q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 39122 x^{12} + 391243971 x^{8} + 75462898750 x^{4} + 18538406640625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{12} + 29029 \nu^{8} + 5616875 \nu^{4} - 2050595324964$$$$)/ 8101949622$$ $$\beta_{2}$$ $$=$$ $$($$$$14561 \nu^{12} - 630433567 \nu^{8} + 6876956439231 \nu^{4} + 648765163646250$$$$)/ 1687906171250$$ $$\beta_{3}$$ $$=$$ $$($$$$-42872 \nu^{14} + 1660015884 \nu^{10} - 16273459572212 \nu^{6} - 6716350492827500 \nu^{2}$$$$)/ 11180755397828125$$ $$\beta_{4}$$ $$=$$ $$($$$$21443033 \nu^{14} - 838625666026 \nu^{10} + 8292212087562418 \nu^{6} + 3422112718390069375 \nu^{2}$$$$)/ 348839568412237500$$ $$\beta_{5}$$ $$=$$ $$($$$$142076123 \nu^{14} - 5626408460256 \nu^{10} + 58010716758878808 \nu^{6} - 1551252332852933125 \nu^{2}$$$$)/ 1046518705236712500$$ $$\beta_{6}$$ $$=$$ $$($$$$409552846 \nu^{14} - 16108363381212 \nu^{10} + 163621302766477716 \nu^{6} - 4387553578919678750 \nu^{2}$$$$)/ 1308148381545890625$$ $$\beta_{7}$$ $$=$$ $$($$$$368758812 \nu^{15} - 2203274425 \nu^{13} - 14361898838689 \nu^{11} + 81848061919850 \nu^{9} + 142396967402520577 \nu^{7} - 772907074481616050 \nu^{5} + 39045763967463741875 \nu^{3} - 608626187272887734375 \nu$$$$)/$$$$28\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$91205484 \nu^{15} - 2203274425 \nu^{13} - 3614956005673 \nu^{11} + 81848061919850 \nu^{9} + 37042590132020089 \nu^{7} - 772907074481616050 \nu^{5} - 4435889123101493125 \nu^{3} - 29552503708573484375 \nu$$$$)/ 57907368356431425000$$ $$\beta_{9}$$ $$=$$ $$($$$$8401639129 \nu^{15} + 168863684675 \nu^{13} + 3010601501750 \nu^{12} - 330696613108488 \nu^{11} - 6798200017902225 \nu^{9} - 117090837330932250 \nu^{8} + 3382491912594489384 \nu^{7} + 70709994820713844425 \nu^{5} + 1170844192896108074250 \nu^{4} - 405098723407935974375 \nu^{3} + 2730345157724647156250 \nu + 110180743203375915312500$$$$)/$$$$43\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-8401639129 \nu^{15} - 168863684675 \nu^{13} + 3010601501750 \nu^{12} + 330696613108488 \nu^{11} + 6798200017902225 \nu^{9} - 117090837330932250 \nu^{8} - 3382491912594489384 \nu^{7} - 70709994820713844425 \nu^{5} + 1170844192896108074250 \nu^{4} + 405098723407935974375 \nu^{3} - 2730345157724647156250 \nu + 110180743203375915312500$$$$)/$$$$43\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$368758812 \nu^{15} + 2203274425 \nu^{13} - 14361898838689 \nu^{11} - 81848061919850 \nu^{9} + 142396967402520577 \nu^{7} + 772907074481616050 \nu^{5} + 39045763967463741875 \nu^{3} + 608626187272887734375 \nu$$$$)/ 72384210445539281250$$ $$\beta_{12}$$ $$=$$ $$($$$$23643689558 \nu^{15} - 502972951225 \nu^{13} - 932514926642451 \nu^{11} + 19735004679793200 \nu^{9} + 9543178085090485443 \nu^{7} - 199388020227548892600 \nu^{5} - 1142889131048483933125 \nu^{3} - 7677128093592305640625 \nu$$$$)/$$$$21\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-23643689558 \nu^{15} - 502972951225 \nu^{13} + 932514926642451 \nu^{11} + 19735004679793200 \nu^{9} - 9543178085090485443 \nu^{7} - 199388020227548892600 \nu^{5} + 1142889131048483933125 \nu^{3} - 7677128093592305640625 \nu$$$$)/$$$$10\!\cdots\!50$$ $$\beta_{14}$$ $$=$$ $$($$$$-39315884039 \nu^{15} - 221984419225 \nu^{13} + 1535966810987508 \nu^{11} + 8626469159338575 \nu^{9} - 15282593232744638244 \nu^{7} - 82976143759118919975 \nu^{5} - 4190510036689937586875 \nu^{3} - 65323169452491358281250 \nu$$$$)/$$$$14\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$157632294968 \nu^{15} - 890140951325 \nu^{13} - 6158229142788721 \nu^{11} + 34587724699274150 \nu^{9} + 61272769898381073553 \nu^{7} - 332677482110957295950 \nu^{5} + 16801085910727214089375 \nu^{3} - 261901303997238320859375 \nu$$$$)/$$$$28\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{13} + 5 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} + 4 \beta_{8} - 20 \beta_{7}$$$$)/80$$ $$\nu^{2}$$ $$=$$ $$($$$$-5 \beta_{6} + 8 \beta_{5} - 125 \beta_{3}$$$$)/80$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{15} - 10 \beta_{14} + 131 \beta_{13} + 75 \beta_{12} - 485 \beta_{11} - 674 \beta_{10} + 674 \beta_{9} - 674 \beta_{8} - 1945 \beta_{7}$$$$)/80$$ $$\nu^{4}$$ $$=$$ $$($$$$45 \beta_{10} + 45 \beta_{9} - 5 \beta_{2} + 156 \beta_{1} + 39122$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$905 \beta_{15} + 1810 \beta_{14} + 11186 \beta_{13} - 9825 \beta_{12} + 96910 \beta_{11} - 64394 \beta_{10} + 64394 \beta_{9} + 64394 \beta_{8} - 388545 \beta_{7}$$$$)/80$$ $$\nu^{6}$$ $$=$$ $$($$$$-34590 \beta_{6} + 78744 \beta_{5} - 117000 \beta_{4} - 1911875 \beta_{3}$$$$)/40$$ $$\nu^{7}$$ $$=$$ $$($$$$69180 \beta_{15} - 138360 \beta_{14} + 2677141 \beta_{13} + 2207700 \beta_{12} - 7418335 \beta_{11} - 15123964 \beta_{10} + 15123964 \beta_{9} - 15123964 \beta_{8} - 29742520 \beta_{7}$$$$)/80$$ $$\nu^{8}$$ $$=$$ $$($$$$1507365 \beta_{10} + 1507365 \beta_{9} - 199985 \beta_{2} + 2954016 \beta_{1} + 748042942$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$19769705 \beta_{15} + 39539410 \beta_{14} + 160738321 \beta_{13} - 131931825 \beta_{12} + 2121427385 \beta_{11} - 906816934 \beta_{10} + 906816934 \beta_{9} + 906816934 \beta_{8} - 8505479245 \beta_{7}$$$$)/80$$ $$\nu^{10}$$ $$=$$ $$($$$$-1304401855 \beta_{6} + 2943997768 \beta_{5} - 7628790000 \beta_{4} - 124565440625 \beta_{3}$$$$)/80$$ $$\nu^{11}$$ $$=$$ $$($$$$886399855 \beta_{15} - 1772799710 \beta_{14} + 57541845626 \beta_{13} + 47259927825 \beta_{12} - 95129003810 \beta_{11} - 324687238154 \beta_{10} + 324687238154 \beta_{9} - 324687238154 \beta_{8} - 381402415095 \beta_{7}$$$$)/80$$ $$\nu^{12}$$ $$=$$ $$11002514490 \beta_{10} + 11002514490 \beta_{9} - 1458362235 \beta_{2} + 13555141119 \beta_{1} + 3433075161803$$ $$\nu^{13}$$ $$=$$ $$($$$$416939045880 \beta_{15} + 833878091760 \beta_{14} + 1770889401431 \beta_{13} - 1454449858200 \beta_{12} + 44744554623235 \beta_{11} - 9992457322124 \beta_{10} + 9992457322124 \beta_{9} + 9992457322124 \beta_{8} - 179395157538820 \beta_{7}$$$$)/80$$ $$\nu^{14}$$ $$=$$ $$($$$$-23463988448155 \beta_{6} + 52959452609848 \beta_{5} - 206566594416000 \beta_{4} - 3373061336441875 \beta_{3}$$$$)/80$$ $$\nu^{15}$$ $$=$$ $$($$$$7278833579155 \beta_{15} - 14557667158310 \beta_{14} + 1193404460483461 \beta_{13} + 980163927077325 \beta_{12} - 781139728334035 \beta_{11} - 6733945696088494 \beta_{10} + 6733945696088494 \beta_{9} - 6733945696088494 \beta_{8} - 3131837746915295 \beta_{7}$$$$)/80$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$257$$ $$261$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.526271 − 11.8760i 0.526271 + 11.8760i 11.8760 + 0.526271i 11.8760 − 0.526271i 3.34569 + 1.86808i 3.34569 − 1.86808i 1.86808 − 3.34569i 1.86808 + 3.34569i −1.86808 + 3.34569i −1.86808 − 3.34569i −3.34569 − 1.86808i −3.34569 + 1.86808i −11.8760 − 0.526271i −11.8760 + 0.526271i −0.526271 + 11.8760i −0.526271 − 11.8760i
0 −24.8046 0 −53.4447 16.3911i 0 100.281i 0 372.267 0
289.2 0 −24.8046 0 −53.4447 + 16.3911i 0 100.281i 0 372.267 0
289.3 0 −24.8046 0 53.4447 16.3911i 0 100.281i 0 372.267 0
289.4 0 −24.8046 0 53.4447 + 16.3911i 0 100.281i 0 372.267 0
289.5 0 −10.4275 0 −17.9907 52.9276i 0 86.7391i 0 −134.267 0
289.6 0 −10.4275 0 −17.9907 + 52.9276i 0 86.7391i 0 −134.267 0
289.7 0 −10.4275 0 17.9907 52.9276i 0 86.7391i 0 −134.267 0
289.8 0 −10.4275 0 17.9907 + 52.9276i 0 86.7391i 0 −134.267 0
289.9 0 10.4275 0 −17.9907 52.9276i 0 86.7391i 0 −134.267 0
289.10 0 10.4275 0 −17.9907 + 52.9276i 0 86.7391i 0 −134.267 0
289.11 0 10.4275 0 17.9907 52.9276i 0 86.7391i 0 −134.267 0
289.12 0 10.4275 0 17.9907 + 52.9276i 0 86.7391i 0 −134.267 0
289.13 0 24.8046 0 −53.4447 16.3911i 0 100.281i 0 372.267 0
289.14 0 24.8046 0 −53.4447 + 16.3911i 0 100.281i 0 372.267 0
289.15 0 24.8046 0 53.4447 16.3911i 0 100.281i 0 372.267 0
289.16 0 24.8046 0 53.4447 + 16.3911i 0 100.281i 0 372.267 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.f.c 16
4.b odd 2 1 inner 320.6.f.c 16
5.b even 2 1 inner 320.6.f.c 16
8.b even 2 1 inner 320.6.f.c 16
8.d odd 2 1 inner 320.6.f.c 16
20.d odd 2 1 inner 320.6.f.c 16
40.e odd 2 1 inner 320.6.f.c 16
40.f even 2 1 inner 320.6.f.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.f.c 16 1.a even 1 1 trivial
320.6.f.c 16 4.b odd 2 1 inner
320.6.f.c 16 5.b even 2 1 inner
320.6.f.c 16 8.b even 2 1 inner
320.6.f.c 16 8.d odd 2 1 inner
320.6.f.c 16 20.d odd 2 1 inner
320.6.f.c 16 40.e odd 2 1 inner
320.6.f.c 16 40.f even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 724 T_{3}^{2} + 66900$$ acting on $$S_{6}^{\mathrm{new}}(320, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 248 T^{2} + 69330 T^{4} + 14644152 T^{6} + 3486784401 T^{8} )^{4}$$
$5$ $$( 1 - 220 T^{2} - 6114250 T^{4} - 2148437500 T^{6} + 95367431640625 T^{8} )^{2}$$
$7$ $$( 1 - 49648 T^{2} + 1179577874 T^{4} - 14024331162352 T^{6} + 79792266297612001 T^{8} )^{4}$$
$11$ $$( 1 - 577284 T^{2} + 134071408310 T^{4} - 14973260223363684 T^{6} +$$$$67\!\cdots\!01$$$$T^{8} )^{4}$$
$13$ $$( 1 + 1011892 T^{2} + 531287817014 T^{4} + 139497905034068308 T^{6} +$$$$19\!\cdots\!01$$$$T^{8} )^{4}$$
$17$ $$( 1 + 554828 T^{2} + 3105096833510 T^{4} + 1118529863798317772 T^{6} +$$$$40\!\cdots\!01$$$$T^{8} )^{4}$$
$19$ $$( 1 - 8279076 T^{2} + 29313371734870 T^{4} - 50759563509370071876 T^{6} +$$$$37\!\cdots\!01$$$$T^{8} )^{4}$$
$23$ $$( 1 - 265392 T^{2} + 72532989580114 T^{4} - 10994264664012735408 T^{6} +$$$$17\!\cdots\!01$$$$T^{8} )^{4}$$
$29$ $$( 1 - 18262996 T^{2} + 467877918484406 T^{4} -$$$$76\!\cdots\!96$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8} )^{4}$$
$31$ $$( 1 + 39407004 T^{2} + 689994929185606 T^{4} +$$$$32\!\cdots\!04$$$$T^{6} +$$$$67\!\cdots\!01$$$$T^{8} )^{4}$$
$37$ $$( 1 + 171738148 T^{2} + 16366808202619574 T^{4} +$$$$82\!\cdots\!52$$$$T^{6} +$$$$23\!\cdots\!01$$$$T^{8} )^{4}$$
$41$ $$( 1 + 12544 T + 129356290 T^{2} + 1453300185344 T^{3} + 13422659310152401 T^{4} )^{8}$$
$43$ $$( 1 + 515310072 T^{2} + 108506205300479794 T^{4} +$$$$11\!\cdots\!28$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8} )^{4}$$
$47$ $$( 1 - 112031408 T^{2} + 100831510268064114 T^{4} -$$$$58\!\cdots\!92$$$$T^{6} +$$$$27\!\cdots\!01$$$$T^{8} )^{4}$$
$53$ $$( 1 + 823212052 T^{2} + 458582260389137174 T^{4} +$$$$14\!\cdots\!48$$$$T^{6} +$$$$30\!\cdots\!01$$$$T^{8} )^{4}$$
$59$ $$( 1 - 2409567364 T^{2} + 2462718757251169526 T^{4} -$$$$12\!\cdots\!64$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8} )^{4}$$
$61$ $$( 1 - 2000262204 T^{2} + 2103804668157499606 T^{4} -$$$$14\!\cdots\!04$$$$T^{6} +$$$$50\!\cdots\!01$$$$T^{8} )^{4}$$
$67$ $$( 1 + 3461519512 T^{2} + 6000732802449327570 T^{4} +$$$$63\!\cdots\!88$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8} )^{4}$$
$71$ $$( 1 + 823411004 T^{2} + 3069578968927394406 T^{4} +$$$$26\!\cdots\!04$$$$T^{6} +$$$$10\!\cdots\!01$$$$T^{8} )^{4}$$
$73$ $$( 1 - 4148991508 T^{2} + 8650887997979942790 T^{4} -$$$$17\!\cdots\!92$$$$T^{6} +$$$$18\!\cdots\!01$$$$T^{8} )^{4}$$
$79$ $$( 1 + 2847963996 T^{2} + 15373907451301926406 T^{4} +$$$$26\!\cdots\!96$$$$T^{6} +$$$$89\!\cdots\!01$$$$T^{8} )^{4}$$
$83$ $$( 1 + 7032005368 T^{2} + 39846447958779448850 T^{4} +$$$$10\!\cdots\!32$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8} )^{4}$$
$89$ $$( 1 + 57412 T + 10662063350 T^{2} + 320592021085988 T^{3} + 31181719929966183601 T^{4} )^{8}$$
$97$ $$( 1 - 31837059828 T^{2} +$$$$40\!\cdots\!94$$$$T^{4} -$$$$23\!\cdots\!72$$$$T^{6} +$$$$54\!\cdots\!01$$$$T^{8} )^{4}$$