# Properties

 Label 162.8.c.q Level $162$ Weight $8$ Character orbit 162.c Analytic conductor $50.606$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,8,Mod(55,162)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("162.55");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 162.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$50.6063741284$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 518x^{5} + 53377x^{4} + 11940x^{3} + 3528x^{2} + 1563408x + 346406544$$ x^8 - 2*x^7 + 2*x^6 + 518*x^5 + 53377*x^4 + 11940*x^3 + 3528*x^2 + 1563408*x + 346406544 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3^{18}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + ( - 8 \beta_1 - 8) q^{2} + 64 \beta_1 q^{4} + ( - \beta_{4} + 132 \beta_1) q^{5} + ( - \beta_{7} + \beta_{3} - 140 \beta_1 - 140) q^{7} + 512 q^{8}+O(q^{10})$$ q + (-8*b1 - 8) * q^2 + 64*b1 * q^4 + (-b4 + 132*b1) * q^5 + (-b7 + b3 - 140*b1 - 140) * q^7 + 512 * q^8 $$q + ( - 8 \beta_1 - 8) q^{2} + 64 \beta_1 q^{4} + ( - \beta_{4} + 132 \beta_1) q^{5} + ( - \beta_{7} + \beta_{3} - 140 \beta_1 - 140) q^{7} + 512 q^{8} + (8 \beta_{5} + 1056) q^{10} + ( - 4 \beta_{7} - 5 \beta_{5} + 5 \beta_{4} + 9 \beta_{3} - 540 \beta_1 - 540) q^{11} + (5 \beta_{6} - 20 \beta_{4} + 7 \beta_{2} + 3365 \beta_1) q^{13} + ( - 8 \beta_{6} + 8 \beta_{2} + 1120 \beta_1) q^{14} + ( - 4096 \beta_1 - 4096) q^{16} + ( - 8 \beta_{7} - 8 \beta_{6} - 78 \beta_{5} + 69 \beta_{3} + 69 \beta_{2} + \cdots + 5640) q^{17}+ \cdots + (448 \beta_{7} + 448 \beta_{6} + 4928 \beta_{5} + 50176 \beta_{3} + \cdots + 2898792) q^{98}+O(q^{100})$$ q + (-8*b1 - 8) * q^2 + 64*b1 * q^4 + (-b4 + 132*b1) * q^5 + (-b7 + b3 - 140*b1 - 140) * q^7 + 512 * q^8 + (8*b5 + 1056) * q^10 + (-4*b7 - 5*b5 + 5*b4 + 9*b3 - 540*b1 - 540) * q^11 + (5*b6 - 20*b4 + 7*b2 + 3365*b1) * q^13 + (-8*b6 + 8*b2 + 1120*b1) * q^14 + (-4096*b1 - 4096) * q^16 + (-8*b7 - 8*b6 - 78*b5 + 69*b3 + 69*b2 + 5640) * q^17 + (-9*b7 - 9*b6 + 16*b5 + 137*b3 + 137*b2 + 9176) * q^19 + (-64*b5 + 64*b4 - 8448*b1 - 8448) * q^20 + (-32*b6 - 40*b4 + 72*b2 + 4320*b1) * q^22 + (28*b6 - 37*b4 + 261*b2 + 15660*b1) * q^23 + (14*b7 - 208*b5 + 208*b4 + 402*b3 - 23674*b1 - 23674) * q^25 + (-40*b7 - 40*b6 + 160*b5 - 56*b3 - 56*b2 + 26920) * q^26 + (64*b7 + 64*b6 - 64*b3 - 64*b2 + 8960) * q^28 + (24*b7 + 245*b5 - 245*b4 - 972*b3 - 17100*b1 - 17100) * q^29 + (58*b6 + 448*b4 - 794*b2 + 56876*b1) * q^31 + 32768*b1 * q^32 + (64*b7 + 624*b5 - 624*b4 - 552*b3 - 45120*b1 - 45120) * q^34 + (188*b7 + 188*b6 + 175*b5 - 1551*b3 - 1551*b2 - 15756) * q^35 + (-109*b7 - 109*b6 - 500*b5 - 1111*b3 - 1111*b2 + 130895) * q^37 + (72*b7 - 128*b5 + 128*b4 - 1096*b3 - 73408*b1 - 73408) * q^38 + (-512*b4 + 67584*b1) * q^40 + (-88*b6 + 880*b4 + 1404*b2 + 16800*b1) * q^41 + (-417*b7 + 480*b5 - 480*b4 + 337*b3 - 140660*b1 - 140660) * q^43 + (256*b7 + 256*b6 + 320*b5 - 576*b3 - 576*b2 + 34560) * q^44 + (-224*b7 - 224*b6 + 296*b5 - 2088*b3 - 2088*b2 + 125280) * q^46 + (-128*b7 + 352*b5 - 352*b4 - 4476*b3 - 128832*b1 - 128832) * q^47 + (-56*b6 - 616*b4 - 6272*b2 + 362349*b1) * q^49 + (112*b6 - 1664*b4 + 3216*b2 + 189392*b1) * q^50 + (320*b7 - 1280*b5 + 1280*b4 + 448*b3 - 215360*b1 - 215360) * q^52 + (-1008*b7 - 1008*b6 - 938*b5 + 654*b3 + 654*b2 + 624504) * q^53 + (817*b7 + 817*b6 + 1600*b5 - 9009*b3 - 9009*b2 + 330696) * q^55 + (-512*b7 + 512*b3 - 71680*b1 - 71680) * q^56 + (192*b6 + 1960*b4 - 7776*b2 + 136800*b1) * q^58 + (-208*b6 - 1260*b4 + 2652*b2 + 1038960*b1) * q^59 + (1139*b7 - 3436*b5 + 3436*b4 + 6753*b3 - 532691*b1 - 532691) * q^61 + (-464*b7 - 464*b6 - 3584*b5 + 6352*b3 + 6352*b2 + 455008) * q^62 + 262144 * q^64 + (896*b7 - 6006*b5 + 6006*b4 + 2193*b3 - 2241624*b1 - 2241624) * q^65 + (-921*b6 - 3120*b4 - 20007*b2 - 301360*b1) * q^67 + (512*b6 + 4992*b4 - 4416*b2 + 360960*b1) * q^68 + (-1504*b7 - 1400*b5 + 1400*b4 + 12408*b3 + 126048*b1 + 126048) * q^70 + (1180*b7 + 1180*b6 + 2475*b5 - 6279*b3 - 6279*b2 + 3121620) * q^71 + (-2080*b7 - 2080*b6 - 1000*b5 - 10488*b3 - 10488*b2 - 1205215) * q^73 + (872*b7 + 4000*b5 - 4000*b4 + 8888*b3 - 1047160*b1 - 1047160) * q^74 + (576*b6 - 1024*b4 - 8768*b2 + 587264*b1) * q^76 + (56*b6 - 4824*b4 - 16908*b2 + 4671648*b1) * q^77 + (105*b7 + 17600*b5 - 17600*b4 + 8215*b3 + 2867920*b1 + 2867920) * q^79 + (4096*b5 + 540672) * q^80 + (704*b7 + 704*b6 - 7040*b5 - 11232*b3 - 11232*b2 + 134400) * q^82 + (-3800*b7 + 14830*b5 - 14830*b4 + 8550*b3 - 4202808*b1 - 4202808) * q^83 + (2059*b6 + 7156*b4 - 53463*b2 - 5251599*b1) * q^85 + (-3336*b6 + 3840*b4 + 2696*b2 + 1125280*b1) * q^86 + (-2048*b7 - 2560*b5 + 2560*b4 + 4608*b3 - 276480*b1 - 276480) * q^88 + (4912*b7 + 4912*b6 + 9960*b5 - 13971*b3 - 13971*b2 + 4214400) * q^89 + (3609*b7 + 3609*b6 + 3984*b5 - 66457*b3 - 66457*b2 - 5800136) * q^91 + (1792*b7 - 2368*b5 + 2368*b4 + 16704*b3 - 1002240*b1 - 1002240) * q^92 + (-1024*b6 + 2816*b4 - 35808*b2 + 1030656*b1) * q^94 + (5372*b6 + 3747*b4 - 27879*b2 + 3522540*b1) * q^95 + (404*b7 - 13400*b5 + 13400*b4 + 30820*b3 + 6769510*b1 + 6769510) * q^97 + (448*b7 + 448*b6 + 4928*b5 + 50176*b3 + 50176*b2 + 2898792) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 32 q^{2} - 256 q^{4} - 528 q^{5} - 560 q^{7} + 4096 q^{8}+O(q^{10})$$ 8 * q - 32 * q^2 - 256 * q^4 - 528 * q^5 - 560 * q^7 + 4096 * q^8 $$8 q - 32 q^{2} - 256 q^{4} - 528 q^{5} - 560 q^{7} + 4096 q^{8} + 8448 q^{10} - 2160 q^{11} - 13460 q^{13} - 4480 q^{14} - 16384 q^{16} + 45120 q^{17} + 73408 q^{19} - 33792 q^{20} - 17280 q^{22} - 62640 q^{23} - 94696 q^{25} + 215360 q^{26} + 71680 q^{28} - 68400 q^{29} - 227504 q^{31} - 131072 q^{32} - 180480 q^{34} - 126048 q^{35} + 1047160 q^{37} - 293632 q^{38} - 270336 q^{40} - 67200 q^{41} - 562640 q^{43} + 276480 q^{44} + 1002240 q^{46} - 515328 q^{47} - 1449396 q^{49} - 757568 q^{50} - 861440 q^{52} + 4996032 q^{53} + 2645568 q^{55} - 286720 q^{56} - 547200 q^{58} - 4155840 q^{59} - 2130764 q^{61} + 3640064 q^{62} + 2097152 q^{64} - 8966496 q^{65} + 1205440 q^{67} - 1443840 q^{68} + 504192 q^{70} + 24972960 q^{71} - 9641720 q^{73} - 4188640 q^{74} - 2349056 q^{76} - 18686592 q^{77} + 11471680 q^{79} + 4325376 q^{80} + 1075200 q^{82} - 16811232 q^{83} + 21006396 q^{85} - 4501120 q^{86} - 1105920 q^{88} + 33715200 q^{89} - 46401088 q^{91} - 4008960 q^{92} - 4122624 q^{94} - 14090160 q^{95} + 27078040 q^{97} + 23190336 q^{98}+O(q^{100})$$ 8 * q - 32 * q^2 - 256 * q^4 - 528 * q^5 - 560 * q^7 + 4096 * q^8 + 8448 * q^10 - 2160 * q^11 - 13460 * q^13 - 4480 * q^14 - 16384 * q^16 + 45120 * q^17 + 73408 * q^19 - 33792 * q^20 - 17280 * q^22 - 62640 * q^23 - 94696 * q^25 + 215360 * q^26 + 71680 * q^28 - 68400 * q^29 - 227504 * q^31 - 131072 * q^32 - 180480 * q^34 - 126048 * q^35 + 1047160 * q^37 - 293632 * q^38 - 270336 * q^40 - 67200 * q^41 - 562640 * q^43 + 276480 * q^44 + 1002240 * q^46 - 515328 * q^47 - 1449396 * q^49 - 757568 * q^50 - 861440 * q^52 + 4996032 * q^53 + 2645568 * q^55 - 286720 * q^56 - 547200 * q^58 - 4155840 * q^59 - 2130764 * q^61 + 3640064 * q^62 + 2097152 * q^64 - 8966496 * q^65 + 1205440 * q^67 - 1443840 * q^68 + 504192 * q^70 + 24972960 * q^71 - 9641720 * q^73 - 4188640 * q^74 - 2349056 * q^76 - 18686592 * q^77 + 11471680 * q^79 + 4325376 * q^80 + 1075200 * q^82 - 16811232 * q^83 + 21006396 * q^85 - 4501120 * q^86 - 1105920 * q^88 + 33715200 * q^89 - 46401088 * q^91 - 4008960 * q^92 - 4122624 * q^94 - 14090160 * q^95 + 27078040 * q^97 + 23190336 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 518x^{5} + 53377x^{4} + 11940x^{3} + 3528x^{2} + 1563408x + 346406544$$ :

 $$\beta_{1}$$ $$=$$ $$( 43855 \nu^{7} - 2805062 \nu^{6} + 9784562 \nu^{5} + 11661362 \nu^{4} + 241042495 \nu^{3} - 91212685860 \nu^{2} + 65476047060 \nu - 3409479049680 ) / 6938096712624$$ (43855*v^7 - 2805062*v^6 + 9784562*v^5 + 11661362*v^4 + 241042495*v^3 - 91212685860*v^2 + 65476047060*v - 3409479049680) / 6938096712624 $$\beta_{2}$$ $$=$$ $$( - 443726439501 \nu^{7} - 39084285383949 \nu^{6} + 196177707589548 \nu^{5} + \cdots - 18\!\cdots\!40 ) / 18\!\cdots\!48$$ (-443726439501*v^7 - 39084285383949*v^6 + 196177707589548*v^5 - 7693934608236408*v^4 - 17160912838823763*v^3 - 2844230406866083035*v^2 + 1305222544459922760*v - 187995427035363234540) / 1822250372373325048 $$\beta_{3}$$ $$=$$ $$( - 1768347693753 \nu^{7} + 81480359331894 \nu^{6} + 96732571818426 \nu^{5} + \cdots - 37\!\cdots\!56 ) / 36\!\cdots\!96$$ (-1768347693753*v^7 + 81480359331894*v^6 + 96732571818426*v^5 - 15536846394272910*v^4 - 9311315888695209*v^3 + 5689229397634797120*v^2 + 3140588638391254092*v - 376930068909068163456) / 3644500744746650096 $$\beta_{4}$$ $$=$$ $$( - 26581205069137 \nu^{7} + 162467854891691 \nu^{6} + \cdots - 12\!\cdots\!20 ) / 54\!\cdots\!44$$ (-26581205069137*v^7 + 162467854891691*v^6 - 6474801265250876*v^5 - 5742322632004688*v^4 - 877327553979120583*v^3 - 946515297109965339*v^2 - 263574408873217816296*v - 127193940482405047020) / 5466751117119975144 $$\beta_{5}$$ $$=$$ $$( 161261082233 \nu^{7} - 262079734900 \nu^{6} - 11218568540246 \nu^{5} + 90127178636698 \nu^{4} + \cdots - 37\!\cdots\!92 ) / 21\!\cdots\!64$$ (161261082233*v^7 - 262079734900*v^6 - 11218568540246*v^5 + 90127178636698*v^4 + 8180010856175885*v^3 + 317422807362978*v^2 - 690865898820764436*v - 374059621790863992) / 21147973373771664 $$\beta_{6}$$ $$=$$ $$( 52230739398496 \nu^{7} - 294121759941161 \nu^{6} + \cdots + 61\!\cdots\!12 ) / 54\!\cdots\!44$$ (52230739398496*v^7 - 294121759941161*v^6 - 2747397883578658*v^5 + 74499173237129618*v^4 - 295507084809269162*v^3 - 3044926280417870337*v^2 - 278324737326601373988*v + 617992564802366238012) / 5466751117119975144 $$\beta_{7}$$ $$=$$ $$( 117817034918123 \nu^{7} + 282711412090070 \nu^{6} + \cdots + 13\!\cdots\!28 ) / 10\!\cdots\!88$$ (117817034918123*v^7 + 282711412090070*v^6 - 26849500279684958*v^5 + 173488660878190858*v^4 + 6813863621624470619*v^3 + 6194777219282785704*v^2 - 83510812414023335028*v + 1317323527421809076928) / 10933502234239950288
 $$\nu$$ $$=$$ $$( \beta_{7} - 2\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 54\beta _1 + 54 ) / 108$$ (b7 - 2*b5 - b4 + b3 + b2 + 54*b1 + 54) / 108 $$\nu^{2}$$ $$=$$ $$( 3\beta_{7} - 3\beta_{6} - 6\beta_{5} + 12\beta_{4} + 203\beta_{3} - 203\beta_{2} + 23652\beta _1 + 11826 ) / 324$$ (3*b7 - 3*b6 - 6*b5 + 12*b4 + 203*b3 - 203*b2 + 23652*b1 + 11826) / 324 $$\nu^{3}$$ $$=$$ $$( - 111 \beta_{7} - 678 \beta_{6} + 924 \beta_{5} + 99 \beta_{4} - 545 \beta_{3} - 1111 \beta_{2} + 90882 \beta _1 - 17658 ) / 324$$ (-111*b7 - 678*b6 + 924*b5 + 99*b4 - 545*b3 - 1111*b2 + 90882*b1 - 17658) / 324 $$\nu^{4}$$ $$=$$ $$( -823\beta_{7} - 823\beta_{6} + 740\beta_{5} - 15217\beta_{3} - 15217\beta_{2} - 2938410 ) / 108$$ (-823*b7 - 823*b6 + 740*b5 - 15217*b3 - 15217*b2 - 2938410) / 108 $$\nu^{5}$$ $$=$$ $$( - 151677 \beta_{7} - 34410 \beta_{6} + 200124 \beta_{5} - 89193 \beta_{4} - 346379 \beta_{3} - 119359 \beta_{2} - 30634362 \beta _1 - 39562830 ) / 324$$ (-151677*b7 - 34410*b6 + 200124*b5 - 89193*b4 - 346379*b3 - 119359*b2 - 30634362*b1 - 39562830) / 324 $$\nu^{6}$$ $$=$$ $$( - 442161 \beta_{7} + 442161 \beta_{6} + 584694 \beta_{5} - 1169388 \beta_{4} - 7517765 \beta_{3} + 7517765 \beta_{2} - 1806933420 \beta _1 - 903466710 ) / 324$$ (-442161*b7 + 442161*b6 + 584694*b5 - 1169388*b4 - 7517765*b3 + 7517765*b2 - 1806933420*b1 - 903466710) / 324 $$\nu^{7}$$ $$=$$ $$( 8586513 \beta_{7} + 34102302 \beta_{6} - 16441776 \beta_{5} - 26003289 \beta_{4} + 29297411 \beta_{3} + 99035917 \beta_{2} - 9030309462 \beta _1 + 3024149094 ) / 324$$ (8586513*b7 + 34102302*b6 - 16441776*b5 - 26003289*b4 + 29297411*b3 + 99035917*b2 - 9030309462*b1 + 3024149094) / 324

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$\beta_{1}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −6.80544 − 6.80544i −9.64382 + 9.64382i 11.0098 − 11.0098i 6.43942 + 6.43942i −6.80544 + 6.80544i −9.64382 − 9.64382i 11.0098 + 11.0098i 6.43942 − 6.43942i
−4.00000 + 6.92820i 0 −32.0000 55.4256i −235.840 408.486i 0 −367.654 + 636.795i 512.000 0 3773.43
55.2 −4.00000 + 6.92820i 0 −32.0000 55.4256i −162.750 281.891i 0 725.227 1256.13i 512.000 0 2604.00
55.3 −4.00000 + 6.92820i 0 −32.0000 55.4256i −5.62316 9.73960i 0 −719.735 + 1246.62i 512.000 0 89.9705
55.4 −4.00000 + 6.92820i 0 −32.0000 55.4256i 140.213 + 242.855i 0 82.1617 142.308i 512.000 0 −2243.40
109.1 −4.00000 6.92820i 0 −32.0000 + 55.4256i −235.840 + 408.486i 0 −367.654 636.795i 512.000 0 3773.43
109.2 −4.00000 6.92820i 0 −32.0000 + 55.4256i −162.750 + 281.891i 0 725.227 + 1256.13i 512.000 0 2604.00
109.3 −4.00000 6.92820i 0 −32.0000 + 55.4256i −5.62316 + 9.73960i 0 −719.735 1246.62i 512.000 0 89.9705
109.4 −4.00000 6.92820i 0 −32.0000 + 55.4256i 140.213 242.855i 0 82.1617 + 142.308i 512.000 0 −2243.40
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.8.c.q 8
3.b odd 2 1 162.8.c.r 8
9.c even 3 1 162.8.a.j yes 4
9.c even 3 1 inner 162.8.c.q 8
9.d odd 6 1 162.8.a.g 4
9.d odd 6 1 162.8.c.r 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.8.a.g 4 9.d odd 6 1
162.8.a.j yes 4 9.c even 3 1
162.8.c.q 8 1.a even 1 1 trivial
162.8.c.q 8 9.c even 3 1 inner
162.8.c.r 8 3.b odd 2 1
162.8.c.r 8 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 528 T_{5}^{7} + 342990 T_{5}^{6} + 53782272 T_{5}^{5} + 27754932771 T_{5}^{4} + 3326203595520 T_{5}^{3} + \cdots + 23\!\cdots\!25$$ acting on $$S_{8}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 8 T + 64)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 528 T^{7} + \cdots + 23\!\cdots\!25$$
$7$ $$T^{8} + 560 T^{7} + \cdots + 63\!\cdots\!96$$
$11$ $$T^{8} + 2160 T^{7} + \cdots + 71\!\cdots\!16$$
$13$ $$T^{8} + 13460 T^{7} + \cdots + 21\!\cdots\!81$$
$17$ $$(T^{4} - 22560 T^{3} + \cdots - 16\!\cdots\!11)^{2}$$
$19$ $$(T^{4} - 36704 T^{3} + \cdots + 66\!\cdots\!04)^{2}$$
$23$ $$T^{8} + 62640 T^{7} + \cdots + 20\!\cdots\!56$$
$29$ $$T^{8} + 68400 T^{7} + \cdots + 11\!\cdots\!61$$
$31$ $$T^{8} + 227504 T^{7} + \cdots + 70\!\cdots\!76$$
$37$ $$(T^{4} - 523580 T^{3} + \cdots - 16\!\cdots\!39)^{2}$$
$41$ $$T^{8} + 67200 T^{7} + \cdots + 37\!\cdots\!56$$
$43$ $$T^{8} + 562640 T^{7} + \cdots + 38\!\cdots\!16$$
$47$ $$T^{8} + 515328 T^{7} + \cdots + 12\!\cdots\!56$$
$53$ $$(T^{4} - 2498016 T^{3} + \cdots + 41\!\cdots\!36)^{2}$$
$59$ $$T^{8} + 4155840 T^{7} + \cdots + 58\!\cdots\!76$$
$61$ $$T^{8} + 2130764 T^{7} + \cdots + 32\!\cdots\!21$$
$67$ $$T^{8} - 1205440 T^{7} + \cdots + 23\!\cdots\!56$$
$71$ $$(T^{4} - 12486480 T^{3} + \cdots + 48\!\cdots\!44)^{2}$$
$73$ $$(T^{4} + 4820860 T^{3} + \cdots + 13\!\cdots\!49)^{2}$$
$79$ $$T^{8} - 11471680 T^{7} + \cdots + 41\!\cdots\!00$$
$83$ $$T^{8} + 16811232 T^{7} + \cdots + 12\!\cdots\!16$$
$89$ $$(T^{4} - 16857600 T^{3} + \cdots + 33\!\cdots\!81)^{2}$$
$97$ $$T^{8} - 27078040 T^{7} + \cdots + 24\!\cdots\!96$$