# Properties

 Label 384.8.d.e Level $384$ Weight $8$ Character orbit 384.d Analytic conductor $119.956$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$119.955849786$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 7628 x^{10} + 22070097 x^{8} - 30593373916 x^{6} + 21405948373596 x^{4} + \cdots + 90\!\cdots\!00$$ x^12 - 7628*x^10 + 22070097*x^8 - 30593373916*x^6 + 21405948373596*x^4 - 7160683139999288*x^2 + 907532972347562500 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{57}\cdot 3^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta_1 q^{3} - \beta_{8} q^{5} + ( - \beta_{7} + \beta_{6}) q^{7} - 729 q^{9}+O(q^{10})$$ q + 3*b1 * q^3 - b8 * q^5 + (-b7 + b6) * q^7 - 729 * q^9 $$q + 3 \beta_1 q^{3} - \beta_{8} q^{5} + ( - \beta_{7} + \beta_{6}) q^{7} - 729 q^{9} + (\beta_{5} + \beta_{4} + 10 \beta_1) q^{11} + (\beta_{11} + 12 \beta_{9}) q^{13} + (3 \beta_{10} - 6 \beta_{7} - 9 \beta_{6}) q^{15} + ( - \beta_{3} + 2 \beta_{2} - 6924) q^{17} + ( - 10 \beta_{5} + \beta_{4} + 27 \beta_1) q^{19} + (6 \beta_{11} - 21 \beta_{9} + 15 \beta_{8}) q^{21} + (20 \beta_{10} + 62 \beta_{7} - 4 \beta_{6}) q^{23} + ( - 5 \beta_{3} - 9 \beta_{2} - 10654) q^{25} - 2187 \beta_1 q^{27} + (19 \beta_{11} - 131 \beta_{9} - 183 \beta_{8}) q^{29} + ( - 52 \beta_{10} - 103 \beta_{7} - 318 \beta_{6}) q^{31} + ( - 6 \beta_{3} - 9 \beta_{2} - 2559) q^{33} + ( - 5 \beta_{5} + 25 \beta_{4} - 7000 \beta_1) q^{35} + ( - 65 \beta_{11} + 442 \beta_{9} + 310 \beta_{8}) q^{37} + ( - 39 \beta_{10} - 219 \beta_{7} - 423 \beta_{6}) q^{39} + ( - 47 \beta_{3} - 28 \beta_{2} - 80762) q^{41} + ( - 108 \beta_{5} - 3 \beta_{4} + 23677 \beta_1) q^{43} + 729 \beta_{8} q^{45} + ( - 120 \beta_{10} + 618 \beta_{7} - 1726 \beta_{6}) q^{47} + ( - 31 \beta_{3} - 9 \beta_{2} - 423922) q^{49} + (162 \beta_{5} - 27 \beta_{4} - 20823 \beta_1) q^{51} + ( - 333 \beta_{11} - 855 \beta_{9} - 2285 \beta_{8}) q^{53} + (460 \beta_{10} - 5075 \beta_{6}) q^{55} + ( - 39 \beta_{3} + 90 \beta_{2} - 5766) q^{57} + (458 \beta_{5} + 116 \beta_{4} + 26546 \beta_1) q^{59} + (111 \beta_{11} - 2056 \beta_{9} + 2690 \beta_{8}) q^{61} + (729 \beta_{7} - 729 \beta_{6}) q^{63} + (25 \beta_{3} + 16 \beta_{2} - 20064) q^{65} + ( - 214 \beta_{5} - 584 \beta_{4} - 11338 \beta_1) q^{67} + ( - 138 \beta_{11} + 2616 \beta_{9} + 2382 \beta_{8}) q^{69} + (484 \beta_{10} - 4598 \beta_{7} - 13100 \beta_{6}) q^{71} + (377 \beta_{3} - 225 \beta_{2} + 1506563) q^{73} + ( - 729 \beta_{5} - 648 \beta_{4} - 31590 \beta_1) q^{75} + ( - 313 \beta_{11} + 2123 \beta_{9} - 1530 \beta_{8}) q^{77} + ( - 1532 \beta_{10} + 2545 \beta_{7} - 6561 \beta_{6}) q^{79} + 531441 q^{81} + (229 \beta_{5} + 1819 \beta_{4} + 189624 \beta_1) q^{83} + ( - 1100 \beta_{11} + 380 \beta_{9} - 2086 \beta_{8}) q^{85} + (885 \beta_{10} + 2280 \beta_{7} - 3222 \beta_{6}) q^{87} + (226 \beta_{3} + 804 \beta_{2} + 3630530) q^{89} + (1106 \beta_{5} + 301 \beta_{4} - 331465 \beta_1) q^{91} + ( - 801 \beta_{11} - 6390 \beta_{9} - 5445 \beta_{8}) q^{93} + ( - 1932 \beta_{10} + 5664 \beta_{7} - 5324 \beta_{6}) q^{95} + (700 \beta_{3} + 162 \beta_{2} - 4038140) q^{97} + ( - 729 \beta_{5} - 729 \beta_{4} - 7290 \beta_1) q^{99}+O(q^{100})$$ q + 3*b1 * q^3 - b8 * q^5 + (-b7 + b6) * q^7 - 729 * q^9 + (b5 + b4 + 10*b1) * q^11 + (b11 + 12*b9) * q^13 + (3*b10 - 6*b7 - 9*b6) * q^15 + (-b3 + 2*b2 - 6924) * q^17 + (-10*b5 + b4 + 27*b1) * q^19 + (6*b11 - 21*b9 + 15*b8) * q^21 + (20*b10 + 62*b7 - 4*b6) * q^23 + (-5*b3 - 9*b2 - 10654) * q^25 - 2187*b1 * q^27 + (19*b11 - 131*b9 - 183*b8) * q^29 + (-52*b10 - 103*b7 - 318*b6) * q^31 + (-6*b3 - 9*b2 - 2559) * q^33 + (-5*b5 + 25*b4 - 7000*b1) * q^35 + (-65*b11 + 442*b9 + 310*b8) * q^37 + (-39*b10 - 219*b7 - 423*b6) * q^39 + (-47*b3 - 28*b2 - 80762) * q^41 + (-108*b5 - 3*b4 + 23677*b1) * q^43 + 729*b8 * q^45 + (-120*b10 + 618*b7 - 1726*b6) * q^47 + (-31*b3 - 9*b2 - 423922) * q^49 + (162*b5 - 27*b4 - 20823*b1) * q^51 + (-333*b11 - 855*b9 - 2285*b8) * q^53 + (460*b10 - 5075*b6) * q^55 + (-39*b3 + 90*b2 - 5766) * q^57 + (458*b5 + 116*b4 + 26546*b1) * q^59 + (111*b11 - 2056*b9 + 2690*b8) * q^61 + (729*b7 - 729*b6) * q^63 + (25*b3 + 16*b2 - 20064) * q^65 + (-214*b5 - 584*b4 - 11338*b1) * q^67 + (-138*b11 + 2616*b9 + 2382*b8) * q^69 + (484*b10 - 4598*b7 - 13100*b6) * q^71 + (377*b3 - 225*b2 + 1506563) * q^73 + (-729*b5 - 648*b4 - 31590*b1) * q^75 + (-313*b11 + 2123*b9 - 1530*b8) * q^77 + (-1532*b10 + 2545*b7 - 6561*b6) * q^79 + 531441 * q^81 + (229*b5 + 1819*b4 + 189624*b1) * q^83 + (-1100*b11 + 380*b9 - 2086*b8) * q^85 + (885*b10 + 2280*b7 - 3222*b6) * q^87 + (226*b3 + 804*b2 + 3630530) * q^89 + (1106*b5 + 301*b4 - 331465*b1) * q^91 + (-801*b11 - 6390*b9 - 5445*b8) * q^93 + (-1932*b10 + 5664*b7 - 5324*b6) * q^95 + (700*b3 + 162*b2 - 4038140) * q^97 + (-729*b5 - 729*b4 - 7290*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 8748 q^{9}+O(q^{10})$$ 12 * q - 8748 * q^9 $$12 q - 8748 q^{9} - 83096 q^{17} - 127812 q^{25} - 30672 q^{33} - 969032 q^{41} - 5087028 q^{49} - 69552 q^{57} - 240832 q^{65} + 18079656 q^{73} + 6377292 q^{81} + 43563144 q^{89} - 48458328 q^{97}+O(q^{100})$$ 12 * q - 8748 * q^9 - 83096 * q^17 - 127812 * q^25 - 30672 * q^33 - 969032 * q^41 - 5087028 * q^49 - 69552 * q^57 - 240832 * q^65 + 18079656 * q^73 + 6377292 * q^81 + 43563144 * q^89 - 48458328 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 7628 x^{10} + 22070097 x^{8} - 30593373916 x^{6} + 21405948373596 x^{4} + \cdots + 90\!\cdots\!00$$ :

 $$\beta_{1}$$ $$=$$ $$( - 123191 \nu^{10} + 856517735 \nu^{8} - 2139973212207 \nu^{6} + \cdots + 19\!\cdots\!00 ) / 26\!\cdots\!50$$ (-123191*v^10 + 856517735*v^8 - 2139973212207*v^6 + 2341248558548535*v^4 - 1128037124099579866*v^2 + 192780306068475833800) / 265023946078128650 $$\beta_{2}$$ $$=$$ $$( 196637216072 \nu^{10} + \cdots - 12\!\cdots\!75 ) / 10\!\cdots\!25$$ (196637216072*v^10 - 1314146977902960*v^8 + 3148617488268934904*v^6 - 3301778808132084485360*v^4 + 1433433007237935603758832*v^2 - 125580460509804585424506575) / 10404293132236623194925 $$\beta_{3}$$ $$=$$ $$( 227084275736 \nu^{10} + \cdots - 27\!\cdots\!00 ) / 34\!\cdots\!75$$ (227084275736*v^10 - 1615153601812560*v^8 + 4131836374921854632*v^6 - 4545517499325649390160*v^4 + 2025546791868939542818896*v^2 - 273304976159927711008356800) / 3468097710745541064975 $$\beta_{4}$$ $$=$$ $$( 56\!\cdots\!27 \nu^{10} + \cdots + 48\!\cdots\!00 ) / 86\!\cdots\!50$$ (5672974441096327*v^10 - 33062783492673715015*v^8 + 59851266969044948318559*v^6 - 29004759387320084086979415*v^4 - 8207182554065257328458717318*v^2 + 4822496337857734139863981204600) / 86418058756357392257047050 $$\beta_{5}$$ $$=$$ $$( - 11\!\cdots\!81 \nu^{10} + \cdots + 14\!\cdots\!00 ) / 86\!\cdots\!50$$ (-11371364251156381*v^10 + 76597633961986949965*v^8 - 185142933324686284509957*v^6 + 195406436723521205473042365*v^4 - 90478786966096757622108005726*v^2 + 14929196865981456006738948325400) / 86418058756357392257047050 $$\beta_{6}$$ $$=$$ $$( 1574111264 \nu^{11} - 3149091705952 \nu^{9} + \cdots + 31\!\cdots\!08 \nu ) / 19\!\cdots\!75$$ (1574111264*v^11 - 3149091705952*v^9 - 15969460347850592*v^7 + 50383188262467397856*v^5 - 53645267997583105689856*v^3 + 31678579522542187372463808*v) / 1900914869536904107600875 $$\beta_{7}$$ $$=$$ $$( 15\!\cdots\!89 \nu^{11} + \cdots - 16\!\cdots\!82 \nu ) / 16\!\cdots\!50$$ (154917011633551325689*v^11 - 1076104599890220641909867*v^9 + 2676822927511183918356107333*v^7 - 2855702859755402557234150559849*v^5 + 1232666844376573907311129386265144*v^3 - 169698868922154183267223245231121482*v) / 1646515063769295540721253024180250 $$\beta_{8}$$ $$=$$ $$( 46\!\cdots\!28 \nu^{11} + \cdots - 14\!\cdots\!94 \nu ) / 13\!\cdots\!50$$ (46048231392779628*v^11 - 349768620952085069339*v^9 + 989140056538595817397066*v^7 - 1277556751263475140574114933*v^5 + 736668433860439879409121767938*v^3 - 144565503540614130893388303277094*v) / 132154672427104546169135004750 $$\beta_{9}$$ $$=$$ $$( 12\!\cdots\!18 \nu^{11} + \cdots - 18\!\cdots\!24 \nu ) / 28\!\cdots\!75$$ (12310064902552218*v^11 - 85722368674081532894*v^9 + 213116704559580960923746*v^7 - 229768807499925387252328618*v^5 + 108236009493084474538505554528*v^3 - 18090507330926464813769035324724*v) / 28318858377236688464814643875 $$\beta_{10}$$ $$=$$ $$( - 71\!\cdots\!79 \nu^{11} + \cdots + 42\!\cdots\!82 \nu ) / 16\!\cdots\!50$$ (-717530843903156136079*v^11 + 5006702414859112832278817*v^9 - 12454680772771147126283691563*v^7 + 13091453634663062786658375290699*v^5 - 5301384752612004321185731347013984*v^3 + 428036649770255776049469389288378382*v) / 1646515063769295540721253024180250 $$\beta_{11}$$ $$=$$ $$( - 64\!\cdots\!54 \nu^{11} + \cdots + 85\!\cdots\!82 \nu ) / 19\!\cdots\!25$$ (-646889992428896754*v^11 + 4400925222878197443617*v^9 - 10707544391333607945747088*v^7 + 11316549487480819596007495399*v^5 - 5225577691230718071916754959834*v^3 + 858502849970313973266448572548282*v) / 198232008640656819253702507125
 $$\nu$$ $$=$$ $$( -\beta_{11} - 16\beta_{10} - \beta_{9} + 2\beta_{8} - 112\beta_{7} + 48\beta_{6} ) / 2304$$ (-b11 - 16*b10 - b9 + 2*b8 - 112*b7 + 48*b6) / 2304 $$\nu^{2}$$ $$=$$ $$( -18\beta_{5} - 5\beta_{3} - 111\beta_{2} - 122\beta _1 + 1464539 ) / 1152$$ (-18*b5 - 5*b3 - 111*b2 - 122*b1 + 1464539) / 1152 $$\nu^{3}$$ $$=$$ $$( -6917\beta_{11} - 58144\beta_{10} - 19877\beta_{9} + 1738\beta_{8} - 419104\beta_{7} - 535263\beta_{6} ) / 4608$$ (-6917*b11 - 58144*b10 - 19877*b9 + 1738*b8 - 419104*b7 - 535263*b6) / 4608 $$\nu^{4}$$ $$=$$ $$( -66582\beta_{5} + 756\beta_{4} + 28355\beta_{3} - 325347\beta_{2} + 9720870\beta _1 + 2696760031 ) / 1152$$ (-66582*b5 + 756*b4 + 28355*b3 - 325347*b2 + 9720870*b1 + 2696760031) / 1152 $$\nu^{5}$$ $$=$$ $$( - 18325473 \beta_{11} - 143924608 \beta_{10} - 86840673 \beta_{9} - 1834494 \beta_{8} - 877742656 \beta_{7} - 1870249011 \beta_{6} ) / 4608$$ (-18325473*b11 - 143924608*b10 - 86840673*b9 - 1834494*b8 - 877742656*b7 - 1870249011*b6) / 4608 $$\nu^{6}$$ $$=$$ $$( - 224962497 \beta_{5} - 12158370 \beta_{4} + 126727450 \beta_{3} - 822635634 \beta_{2} + 46369720457 \beta _1 + 5869887545306 ) / 1152$$ (-224962497*b5 - 12158370*b4 + 126727450*b3 - 822635634*b2 + 46369720457*b1 + 5869887545306) / 1152 $$\nu^{7}$$ $$=$$ $$( - 50475122283 \beta_{11} - 360261281216 \beta_{10} - 302582978283 \beta_{9} + 1571254422 \beta_{8} - 1982935947776 \beta_{7} - 5022226211907 \beta_{6} ) / 4608$$ (-50475122283*b11 - 360261281216*b10 - 302582978283*b9 + 1571254422*b8 - 1982935947776*b7 - 5022226211907*b6) / 4608 $$\nu^{8}$$ $$=$$ $$( - 237215237616 \beta_{5} - 22491699552 \beta_{4} + 123733724075 \beta_{3} - 672315701451 \beta_{2} + 54069123344880 \beta _1 + 45\!\cdots\!43 ) / 384$$ (-237215237616*b5 - 22491699552*b4 + 123733724075*b3 - 672315701451*b2 + 54069123344880*b1 + 4541108487911143) / 384 $$\nu^{9}$$ $$=$$ $$( - 143194143041177 \beta_{11} - 886669336602752 \beta_{10} - 952500999343577 \beta_{9} + 25032138200434 \beta_{8} + \cdots - 12\!\cdots\!99 \beta_{6} ) / 4608$$ (-143194143041177*b11 - 886669336602752*b10 - 952500999343577*b9 + 25032138200434*b8 - 4654347443353664*b7 - 12609923233549899*b6) / 4608 $$\nu^{10}$$ $$=$$ $$( - 21\!\cdots\!13 \beta_{5} - 243565265769210 \beta_{4} + 964139649084280 \beta_{3} + \cdots + 32\!\cdots\!24 ) / 1152$$ (-2140609298173713*b5 - 243565265769210*b4 + 964139649084280*b3 - 4900023697115496*b2 + 505676675775548833*b1 + 32397105713661714824) / 1152 $$\nu^{11}$$ $$=$$ $$( - 40\!\cdots\!17 \beta_{11} + \cdots - 30\!\cdots\!73 \beta_{6} ) / 4608$$ (-407470151728479017*b11 - 2159591049293674624*b10 - 2832861836959233257*b9 + 103467526064118418*b8 - 11109033664461310144*b7 - 30924747894485554773*b6) / 4608

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 19.8448 + 0.707107i −31.5402 − 0.707107i −49.2636 + 0.707107i 49.2636 − 0.707107i 31.5402 + 0.707107i −19.8448 − 0.707107i −19.8448 + 0.707107i 31.5402 − 0.707107i 49.2636 + 0.707107i −49.2636 − 0.707107i −31.5402 + 0.707107i 19.8448 − 0.707107i
0 27.0000i 0 467.267i 0 31.2683 0 −729.000 0
193.2 0 27.0000i 0 218.879i 0 −904.541 0 −729.000 0
193.3 0 27.0000i 0 8.99895i 0 −616.197 0 −729.000 0
193.4 0 27.0000i 0 8.99895i 0 616.197 0 −729.000 0
193.5 0 27.0000i 0 218.879i 0 904.541 0 −729.000 0
193.6 0 27.0000i 0 467.267i 0 −31.2683 0 −729.000 0
193.7 0 27.0000i 0 467.267i 0 −31.2683 0 −729.000 0
193.8 0 27.0000i 0 218.879i 0 904.541 0 −729.000 0
193.9 0 27.0000i 0 8.99895i 0 616.197 0 −729.000 0
193.10 0 27.0000i 0 8.99895i 0 −616.197 0 −729.000 0
193.11 0 27.0000i 0 218.879i 0 −904.541 0 −729.000 0
193.12 0 27.0000i 0 467.267i 0 31.2683 0 −729.000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.12 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.8.d.e 12
4.b odd 2 1 inner 384.8.d.e 12
8.b even 2 1 inner 384.8.d.e 12
8.d odd 2 1 inner 384.8.d.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.d.e 12 1.a even 1 1 trivial
384.8.d.e 12 4.b odd 2 1 inner
384.8.d.e 12 8.b even 2 1 inner
384.8.d.e 12 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{8}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{6} + 266328T_{5}^{4} + 10481784000T_{5}^{2} + 847081280000$$ T5^6 + 266328*T5^4 + 10481784000*T5^2 + 847081280000 $$T_{7}^{6} - 1198872T_{7}^{4} + 311839224000T_{7}^{2} - 303742233920000$$ T7^6 - 1198872*T7^4 + 311839224000*T7^2 - 303742233920000

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$(T^{2} + 729)^{6}$$
$5$ $$(T^{6} + 266328 T^{4} + \cdots + 847081280000)^{2}$$
$7$ $$(T^{6} - 1198872 T^{4} + \cdots - 303742233920000)^{2}$$
$11$ $$(T^{6} + 40887216 T^{4} + \cdots + 98\!\cdots\!00)^{2}$$
$13$ $$(T^{6} + 153501024 T^{4} + \cdots + 21\!\cdots\!12)^{2}$$
$17$ $$(T^{3} + 20774 T^{2} + \cdots + 9568373896)^{4}$$
$19$ $$(T^{6} + 3438581424 T^{4} + \cdots + 43\!\cdots\!76)^{2}$$
$23$ $$(T^{6} - 12104042592 T^{4} + \cdots - 16\!\cdots\!52)^{2}$$
$29$ $$(T^{6} + 29283126744 T^{4} + \cdots + 41\!\cdots\!92)^{2}$$
$31$ $$(T^{6} - 72678563928 T^{4} + \cdots - 93\!\cdots\!08)^{2}$$
$37$ $$(T^{6} + 267723127296 T^{4} + \cdots + 52\!\cdots\!00)^{2}$$
$41$ $$(T^{3} + 242258 T^{2} + \cdots + 25\!\cdots\!80)^{4}$$
$43$ $$(T^{6} + 491608617264 T^{4} + \cdots + 62\!\cdots\!76)^{2}$$
$47$ $$(T^{6} - 825788238432 T^{4} + \cdots - 11\!\cdots\!12)^{2}$$
$53$ $$(T^{6} + 2907470155608 T^{4} + \cdots + 89\!\cdots\!68)^{2}$$
$59$ $$(T^{6} + 5758987678128 T^{4} + \cdots + 52\!\cdots\!76)^{2}$$
$61$ $$(T^{6} + 6876216682368 T^{4} + \cdots + 68\!\cdots\!00)^{2}$$
$67$ $$(T^{6} + 10722197211312 T^{4} + \cdots + 72\!\cdots\!64)^{2}$$
$71$ $$(T^{6} - 49555481902176 T^{4} + \cdots - 12\!\cdots\!88)^{2}$$
$73$ $$(T^{3} - 4519914 T^{2} + \cdots + 96\!\cdots\!96)^{4}$$
$79$ $$(T^{6} - 45949368823128 T^{4} + \cdots - 21\!\cdots\!88)^{2}$$
$83$ $$(T^{6} + 122266303705776 T^{4} + \cdots + 30\!\cdots\!64)^{2}$$
$89$ $$(T^{3} - 10890786 T^{2} + \cdots + 21\!\cdots\!96)^{4}$$
$97$ $$(T^{3} + 12114582 T^{2} + \cdots - 19\!\cdots\!08)^{4}$$