# Properties

 Label 162.11.d.c Level $162$ Weight $11$ Character orbit 162.d Analytic conductor $102.928$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,11,Mod(53,162)]

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([5]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("162.53");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 162.d (of order $$6$$, 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: $$102.927874933$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} - 862x^{6} + 2600x^{5} + 278207x^{4} - 560752x^{3} - 39833846x^{2} + 40114656x + 2136938124$$ x^8 - 4*x^7 - 862*x^6 + 2600*x^5 + 278207*x^4 - 560752*x^3 - 39833846*x^2 + 40114656*x + 2136938124 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{20}\cdot 3^{12}$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} - 512 \beta_1 q^{4} + ( - 49 \beta_{6} - \beta_{4} + 49 \beta_{3}) q^{5} + (5 \beta_{7} + 5 \beta_{5} - 2879 \beta_1 - 2879) q^{7} - 512 \beta_{6} q^{8}+O(q^{10})$$ q - b3 * q^2 - 512*b1 * q^4 + (-49*b6 - b4 + 49*b3) * q^5 + (5*b7 + 5*b5 - 2879*b1 - 2879) * q^7 - 512*b6 * q^8 $$q - \beta_{3} q^{2} - 512 \beta_1 q^{4} + ( - 49 \beta_{6} - \beta_{4} + 49 \beta_{3}) q^{5} + (5 \beta_{7} + 5 \beta_{5} - 2879 \beta_1 - 2879) q^{7} - 512 \beta_{6} q^{8} + ( - 32 \beta_{5} - 24960) q^{10} + ( - 6049 \beta_{3} - \beta_{2}) q^{11} + ( - 37 \beta_{7} - 337765 \beta_1) q^{13} + ( - 2859 \beta_{6} + 80 \beta_{4} + 2859 \beta_{3}) q^{14} + ( - 262144 \beta_1 - 262144) q^{16} + ( - 139 \beta_{6} + 245 \beta_{4} + 245 \beta_{2}) q^{17} + (1178 \beta_{5} - 212317) q^{19} + (25088 \beta_{3} + 512 \beta_{2}) q^{20} + ( - 32 \beta_{7} - 3096960 \beta_1) q^{22} + (161593 \beta_{6} - 1271 \beta_{4} - 161593 \beta_{3}) q^{23} + (3120 \beta_{7} + 3120 \beta_{5} + 11629895 \beta_1 + 11629895) q^{25} + ( - 337913 \beta_{6} - 592 \beta_{4} - 592 \beta_{2}) q^{26} + (2560 \beta_{5} - 1474048) q^{28} + (93542 \beta_{3} + 2534 \beta_{2}) q^{29} + ( - 8304 \beta_{7} - 10782958 \beta_1) q^{31} + ( - 262144 \beta_{6} + 262144 \beta_{3}) q^{32} + (7840 \beta_{7} + 7840 \beta_{5} - 102528 \beta_1 - 102528) q^{34} + ( - 3012829 \beta_{6} - 1021 \beta_{4} - 1021 \beta_{2}) q^{35} + (31273 \beta_{5} - 7906525) q^{37} + (207605 \beta_{3} - 18848 \beta_{2}) q^{38} + (16384 \beta_{7} + 12779520 \beta_1) q^{40} + (3705914 \beta_{6} + 11066 \beta_{4} - 3705914 \beta_{3}) q^{41} + ( - 5424 \beta_{7} - 5424 \beta_{5} - 135962378 \beta_1 - 135962378) q^{43} + ( - 3097088 \beta_{6} - 512 \beta_{4} - 512 \beta_{2}) q^{44} + ( - 40672 \beta_{5} + 82898304) q^{46} + (13854447 \beta_{3} + 16383 \beta_{2}) q^{47} + ( - 28790 \beta_{7} - 21952608 \beta_1) q^{49} + (11642375 \beta_{6} + 49920 \beta_{4} - 11642375 \beta_{3}) q^{50} + ( - 18944 \beta_{7} - 18944 \beta_{5} - 172935680 \beta_1 - 172935680) q^{52} + ( - 26256578 \beta_{6} + 30814 \beta_{4} + 30814 \beta_{2}) q^{53} + ( - 195120 \beta_{5} - 171155520) q^{55} + (1463808 \beta_{3} - 40960 \beta_{2}) q^{56} + (81088 \beta_{7} + 47569152 \beta_1) q^{58} + (7313221 \beta_{6} - 255995 \beta_{4} - 7313221 \beta_{3}) q^{59} + ( - 232413 \beta_{7} - 232413 \beta_{5} + \cdots - 406320875) q^{61}+ \cdots + ( - 22067768 \beta_{6} - 460640 \beta_{4} + \cdots - 460640 \beta_{2}) q^{98}+O(q^{100})$$ q - b3 * q^2 - 512*b1 * q^4 + (-49*b6 - b4 + 49*b3) * q^5 + (5*b7 + 5*b5 - 2879*b1 - 2879) * q^7 - 512*b6 * q^8 + (-32*b5 - 24960) * q^10 + (-6049*b3 - b2) * q^11 + (-37*b7 - 337765*b1) * q^13 + (-2859*b6 + 80*b4 + 2859*b3) * q^14 + (-262144*b1 - 262144) * q^16 + (-139*b6 + 245*b4 + 245*b2) * q^17 + (1178*b5 - 212317) * q^19 + (25088*b3 + 512*b2) * q^20 + (-32*b7 - 3096960*b1) * q^22 + (161593*b6 - 1271*b4 - 161593*b3) * q^23 + (3120*b7 + 3120*b5 + 11629895*b1 + 11629895) * q^25 + (-337913*b6 - 592*b4 - 592*b2) * q^26 + (2560*b5 - 1474048) * q^28 + (93542*b3 + 2534*b2) * q^29 + (-8304*b7 - 10782958*b1) * q^31 + (-262144*b6 + 262144*b3) * q^32 + (7840*b7 + 7840*b5 - 102528*b1 - 102528) * q^34 + (-3012829*b6 - 1021*b4 - 1021*b2) * q^35 + (31273*b5 - 7906525) * q^37 + (207605*b3 - 18848*b2) * q^38 + (16384*b7 + 12779520*b1) * q^40 + (3705914*b6 + 11066*b4 - 3705914*b3) * q^41 + (-5424*b7 - 5424*b5 - 135962378*b1 - 135962378) * q^43 + (-3097088*b6 - 512*b4 - 512*b2) * q^44 + (-40672*b5 + 82898304) * q^46 + (13854447*b3 + 16383*b2) * q^47 + (-28790*b7 - 21952608*b1) * q^49 + (11642375*b6 + 49920*b4 - 11642375*b3) * q^50 + (-18944*b7 - 18944*b5 - 172935680*b1 - 172935680) * q^52 + (-26256578*b6 + 30814*b4 + 30814*b2) * q^53 + (-195120*b5 - 171155520) * q^55 + (1463808*b3 - 40960*b2) * q^56 + (81088*b7 + 47569152*b1) * q^58 + (7313221*b6 - 255995*b4 - 7313221*b3) * q^59 + (-232413*b7 - 232413*b5 - 406320875*b1 - 406320875) * q^61 + (-10816174*b6 - 132864*b4 - 132864*b2) * q^62 - 134217728 * q^64 + (39889345*b3 + 366625*b2) * q^65 + (247020*b7 - 1666339789*b1) * q^67 + (-71168*b6 + 125440*b4 + 71168*b3) * q^68 + (-32672*b7 - 32672*b5 - 1542437760*b1 - 1542437760) * q^70 + (42457052*b6 + 211580*b4 + 211580*b2) * q^71 + (150318*b5 - 2747014729) * q^73 + (7781433*b3 - 500368*b2) * q^74 + (-603136*b7 + 108706304*b1) * q^76 + (-14141171*b6 + 481021*b4 + 14141171*b3) * q^77 + (1027255*b7 + 1027255*b5 + 408719449*b1 + 408719449) * q^79 + (12845056*b6 + 262144*b4 + 262144*b2) * q^80 + (354112*b5 + 1896011520) * q^82 + (-2318826*b3 - 380682*b2) * q^83 + (-375792*b7 - 4938788160*b1) * q^85 + (-135984074*b6 - 86784*b4 + 135984074*b3) * q^86 + (-16384*b7 - 16384*b5 - 1585643520*b1 - 1585643520) * q^88 + (42112577*b6 - 955711*b4 - 955711*b2) * q^89 + (1582302*b5 + 894106165) * q^91 + (-82735616*b3 + 650752*b2) * q^92 + (524256*b7 + 7091379840*b1) * q^94 + (-732655307*b6 - 706523*b4 + 732655307*b3) * q^95 + (-269270*b7 - 269270*b5 - 5250087935*b1 - 5250087935) * q^97 + (-22067768*b6 - 460640*b4 - 460640*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2048 q^{4} - 11516 q^{7}+O(q^{10})$$ 8 * q + 2048 * q^4 - 11516 * q^7 $$8 q + 2048 q^{4} - 11516 q^{7} - 199680 q^{10} + 1351060 q^{13} - 1048576 q^{16} - 1698536 q^{19} + 12387840 q^{22} + 46519580 q^{25} - 11792384 q^{28} + 43131832 q^{31} - 410112 q^{34} - 63252200 q^{37} - 51118080 q^{40} - 543849512 q^{43} + 663186432 q^{46} + 87810432 q^{49} - 691742720 q^{52} - 1369244160 q^{55} - 190276608 q^{58} - 1625283500 q^{61} - 1073741824 q^{64} + 6665359156 q^{67} - 6169751040 q^{70} - 21976117832 q^{73} - 434825216 q^{76} + 1634877796 q^{79} + 15168092160 q^{82} + 19755152640 q^{85} - 6342574080 q^{88} + 7152849320 q^{91} - 28365519360 q^{94} - 21000351740 q^{97}+O(q^{100})$$ 8 * q + 2048 * q^4 - 11516 * q^7 - 199680 * q^10 + 1351060 * q^13 - 1048576 * q^16 - 1698536 * q^19 + 12387840 * q^22 + 46519580 * q^25 - 11792384 * q^28 + 43131832 * q^31 - 410112 * q^34 - 63252200 * q^37 - 51118080 * q^40 - 543849512 * q^43 + 663186432 * q^46 + 87810432 * q^49 - 691742720 * q^52 - 1369244160 * q^55 - 190276608 * q^58 - 1625283500 * q^61 - 1073741824 * q^64 + 6665359156 * q^67 - 6169751040 * q^70 - 21976117832 * q^73 - 434825216 * q^76 + 1634877796 * q^79 + 15168092160 * q^82 + 19755152640 * q^85 - 6342574080 * q^88 + 7152849320 * q^91 - 28365519360 * q^94 - 21000351740 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 862x^{6} + 2600x^{5} + 278207x^{4} - 560752x^{3} - 39833846x^{2} + 40114656x + 2136938124$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{6} - 6\nu^{5} - 2161\nu^{4} + 4332\nu^{3} + 654065\nu^{2} - 656232\nu - 60780660 ) / 1489506$$ (2*v^6 - 6*v^5 - 2161*v^4 + 4332*v^3 + 654065*v^2 - 656232*v - 60780660) / 1489506 $$\beta_{2}$$ $$=$$ $$( 69136 \nu^{7} + 159893728 \nu^{6} - 543088808 \nu^{5} - 172869318992 \nu^{4} + 369370905424 \nu^{3} + 111966419712928 \nu^{2} + \cdots - 17\!\cdots\!56 ) / 552136786857$$ (69136*v^7 + 159893728*v^6 - 543088808*v^5 - 172869318992*v^4 + 369370905424*v^3 + 111966419712928*v^2 - 114810312310536*v - 17745499966073856) / 552136786857 $$\beta_{3}$$ $$=$$ $$( - 276544 \nu^{7} + 967904 \nu^{6} + 250726784 \nu^{5} - 629236720 \nu^{4} - 90067882240 \nu^{3} + 135731544032 \nu^{2} + \cdots - 5295548274240 ) / 552136786857$$ (-276544*v^7 + 967904*v^6 + 250726784*v^5 - 629236720*v^4 - 90067882240*v^3 + 135731544032*v^2 + 10545810705264*v - 5295548274240) / 552136786857 $$\beta_{4}$$ $$=$$ $$( 2957980 \nu^{7} + 34538925208 \nu^{6} - 105562988192 \nu^{5} - 22417988016200 \nu^{4} + 45428450511820 \nu^{3} + \cdots - 34\!\cdots\!80 ) / 552136786857$$ (2957980*v^7 + 34538925208*v^6 - 105562988192*v^5 - 22417988016200*v^4 + 45428450511820*v^3 + 4813270271254000*v^2 - 4865408279751864*v - 341476150810066080) / 552136786857 $$\beta_{5}$$ $$=$$ $$( 414816 \nu^{7} - 1451856 \nu^{6} - 376090176 \nu^{5} + 943855080 \nu^{4} + 135101823360 \nu^{3} - 203597316048 \nu^{2} + \cdots + 14568963853644 ) / 61348531873$$ (414816*v^7 - 1451856*v^6 - 376090176*v^5 + 943855080*v^4 + 135101823360*v^3 - 203597316048*v^2 - 29069998942464*v + 14568963853644) / 61348531873 $$\beta_{6}$$ $$=$$ $$( - 14096 \nu^{7} + 49336 \nu^{6} + 9209944 \nu^{5} - 23148200 \nu^{4} - 2014517000 \nu^{3} + 3044948368 \nu^{2} + 147278425056 \nu - 74147476704 ) / 642766923$$ (-14096*v^7 + 49336*v^6 + 9209944*v^5 - 23148200*v^4 - 2014517000*v^3 + 3044948368*v^2 + 147278425056*v - 74147476704) / 642766923 $$\beta_{7}$$ $$=$$ $$( - 35955552 \nu^{7} + 125844432 \nu^{6} + 31065504468 \nu^{5} - 77978372250 \nu^{4} - 8395277671272 \nu^{3} + \cdots - 365904406474344 ) / 61348531873$$ (-35955552*v^7 + 125844432*v^6 + 31065504468*v^5 - 77978372250*v^4 - 8395277671272*v^3 + 12670957801374*v^2 + 727579955797488*v - 365904406474344) / 61348531873
 $$\nu$$ $$=$$ $$( -2\beta_{5} - 27\beta_{3} + 216 ) / 432$$ (-2*b5 - 27*b3 + 216) / 432 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} - 13\beta_{3} + 2\beta_{2} - 432\beta _1 + 46764 ) / 216$$ (-b5 - 13*b3 + 2*b2 - 432*b1 + 46764) / 216 $$\nu^{3}$$ $$=$$ $$( 12\beta_{7} - 54\beta_{6} - 434\beta_{5} - 17535\beta_{3} + 6\beta_{2} - 1296\beta _1 + 140184 ) / 432$$ (12*b7 - 54*b6 - 434*b5 - 17535*b3 + 6*b2 - 1296*b1 + 140184) / 432 $$\nu^{4}$$ $$=$$ $$( 12 \beta_{7} - 52 \beta_{6} - 433 \beta_{5} + 8 \beta_{4} - 17306 \beta_{3} + 876 \beta_{2} - 562032 \beta _1 + 10170252 ) / 216$$ (12*b7 - 52*b6 - 433*b5 + 8*b4 - 17306*b3 + 876*b2 - 562032*b1 + 10170252) / 216 $$\nu^{5}$$ $$=$$ $$( 8720 \beta_{7} - 117008 \beta_{6} - 94570 \beta_{5} + 40 \beta_{4} - 6355757 \beta_{3} + 4370 \beta_{2} - 2808000 \beta _1 + 50617656 ) / 432$$ (8720*b7 - 117008*b6 - 94570*b5 + 40*b4 - 6355757*b3 + 4370*b2 - 2808000*b1 + 50617656) / 432 $$\nu^{6}$$ $$=$$ $$( 13050 \beta_{7} - 173216 \beta_{6} - 140773 \beta_{5} + 8704 \beta_{4} - 9420507 \beta_{3} + 292510 \beta_{2} - 307939320 \beta _1 + 2219464476 ) / 216$$ (13050*b7 - 173216*b6 - 140773*b5 + 8704*b4 - 9420507*b3 + 292510*b2 - 307939320*b1 + 2219464476) / 216 $$\nu^{7}$$ $$=$$ $$( 4034380 \beta_{7} - 89473090 \beta_{6} - 20656554 \beta_{5} + 60788 \beta_{4} - 1943503927 \beta_{3} + 2032282 \beta_{2} - 2145748752 \beta _1 + 15359253048 ) / 432$$ (4034380*b7 - 89473090*b6 - 20656554*b5 + 60788*b4 - 1943503927*b3 + 2032282*b2 - 2145748752*b1 + 15359253048) / 432

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 13.9807 + 0.707107i −15.4302 + 0.707107i −12.9807 − 0.707107i 16.4302 − 0.707107i 13.9807 − 0.707107i −15.4302 − 0.707107i −12.9807 + 0.707107i 16.4302 + 0.707107i
−19.5959 + 11.3137i 0 256.000 443.405i −2934.95 1694.49i 0 −9380.44 16247.4i 11585.2i 0 76684.0
53.2 −19.5959 + 11.3137i 0 256.000 443.405i 4845.55 + 2797.58i 0 6501.44 + 11260.8i 11585.2i 0 −126604.
53.3 19.5959 11.3137i 0 256.000 443.405i −4845.55 2797.58i 0 6501.44 + 11260.8i 11585.2i 0 −126604.
53.4 19.5959 11.3137i 0 256.000 443.405i 2934.95 + 1694.49i 0 −9380.44 16247.4i 11585.2i 0 76684.0
107.1 −19.5959 11.3137i 0 256.000 + 443.405i −2934.95 + 1694.49i 0 −9380.44 + 16247.4i 11585.2i 0 76684.0
107.2 −19.5959 11.3137i 0 256.000 + 443.405i 4845.55 2797.58i 0 6501.44 11260.8i 11585.2i 0 −126604.
107.3 19.5959 + 11.3137i 0 256.000 + 443.405i −4845.55 + 2797.58i 0 6501.44 11260.8i 11585.2i 0 −126604.
107.4 19.5959 + 11.3137i 0 256.000 + 443.405i 2934.95 1694.49i 0 −9380.44 + 16247.4i 11585.2i 0 76684.0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.11.d.c 8
3.b odd 2 1 inner 162.11.d.c 8
9.c even 3 1 54.11.b.b 4
9.c even 3 1 inner 162.11.d.c 8
9.d odd 6 1 54.11.b.b 4
9.d odd 6 1 inner 162.11.d.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.11.b.b 4 9.c even 3 1
54.11.b.b 4 9.d odd 6 1
162.11.d.c 8 1.a even 1 1 trivial
162.11.d.c 8 3.b odd 2 1 inner
162.11.d.c 8 9.c even 3 1 inner
162.11.d.c 8 9.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 42791040 T_{5}^{6} + \cdots + 12\!\cdots\!00$$ acting on $$S_{11}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 512 T^{2} + 262144)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 42791040 T^{6} + \cdots + 12\!\cdots\!00$$
$7$ $$(T^{4} + 5758 T^{3} + \cdots + 59\!\cdots\!81)^{2}$$
$11$ $$T^{8} - 37505831040 T^{6} + \cdots + 12\!\cdots\!00$$
$13$ $$(T^{4} - 675530 T^{3} + \cdots + 10\!\cdots\!25)^{2}$$
$17$ $$(T^{4} + 2422496398464 T^{2} + \cdots + 14\!\cdots\!24)^{2}$$
$19$ $$(T^{2} + 424634 T - 13955764933751)^{4}$$
$23$ $$T^{8} - 92039316379776 T^{6} + \cdots + 13\!\cdots\!16$$
$29$ $$T^{8} - 267980575117824 T^{6} + \cdots + 24\!\cdots\!56$$
$31$ $$(T^{4} - 21565916 T^{3} + \cdots + 33\!\cdots\!16)^{2}$$
$37$ $$(T^{2} + 15813050 T - 98\!\cdots\!15)^{4}$$
$41$ $$T^{8} + \cdots + 42\!\cdots\!00$$
$43$ $$(T^{4} + 271924756 T^{3} + \cdots + 33\!\cdots\!76)^{2}$$
$47$ $$T^{8} + \cdots + 74\!\cdots\!00$$
$53$ $$(T^{4} + \cdots + 11\!\cdots\!44)^{2}$$
$59$ $$T^{8} + \cdots + 28\!\cdots\!36$$
$61$ $$(T^{4} + 812641750 T^{3} + \cdots + 14\!\cdots\!25)^{2}$$
$67$ $$(T^{4} - 3332679578 T^{3} + \cdots + 46\!\cdots\!41)^{2}$$
$71$ $$(T^{4} + \cdots + 29\!\cdots\!44)^{2}$$
$73$ $$(T^{2} + 5494029458 T + 73\!\cdots\!01)^{4}$$
$79$ $$(T^{4} - 817438898 T^{3} + \cdots + 10\!\cdots\!01)^{2}$$
$83$ $$T^{8} + \cdots + 72\!\cdots\!96$$
$89$ $$(T^{4} + \cdots + 30\!\cdots\!24)^{2}$$
$97$ $$(T^{4} + 10500175870 T^{3} + \cdots + 71\!\cdots\!25)^{2}$$