# Properties

 Label 378.4.g.c Level $378$ Weight $4$ Character orbit 378.g Analytic conductor $22.303$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 378.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.3027219822$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100$$ x^8 - x^7 + 18*x^6 + 9*x^5 + 283*x^4 - 48*x^3 + 186*x^2 + 40*x + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{4}\cdot 3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (2 \beta_1 - 2) q^{2} - 4 \beta_1 q^{4} + ( - \beta_{7} - \beta_{2} - \beta_1) q^{5} + (\beta_{6} + 3 \beta_1 - 1) q^{7} + 8 q^{8}+O(q^{10})$$ q + (2*b1 - 2) * q^2 - 4*b1 * q^4 + (-b7 - b2 - b1) * q^5 + (b6 + 3*b1 - 1) * q^7 + 8 * q^8 $$q + (2 \beta_1 - 2) q^{2} - 4 \beta_1 q^{4} + ( - \beta_{7} - \beta_{2} - \beta_1) q^{5} + (\beta_{6} + 3 \beta_1 - 1) q^{7} + 8 q^{8} + (2 \beta_{2} + 2 \beta_1) q^{10} + (\beta_{4} + \beta_{3} - 8 \beta_1) q^{11} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1) q^{13} + ( - 2 \beta_{3} - 2 \beta_1 - 4) q^{14} + (16 \beta_1 - 16) q^{16} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{2} + 17 \beta_1 - 2) q^{17} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \cdots + 18) q^{19}+ \cdots + ( - 28 \beta_{7} - 6 \beta_{6} + 14 \beta_{4} + 4 \beta_{3} - 56 \beta_{2} + \cdots - 84) q^{98}+O(q^{100})$$ q + (2*b1 - 2) * q^2 - 4*b1 * q^4 + (-b7 - b2 - b1) * q^5 + (b6 + 3*b1 - 1) * q^7 + 8 * q^8 + (2*b2 + 2*b1) * q^10 + (b4 + b3 - 8*b1) * q^11 + (2*b7 + 2*b6 - b5 - 2*b4 - b3 + b1) * q^13 + (-2*b3 - 2*b1 - 4) * q^14 + (16*b1 - 16) * q^16 + (2*b6 + 2*b5 + 2*b4 + 3*b2 + 17*b1 - 2) * q^17 + (-2*b7 - 2*b6 - b5 - 3*b4 + 3*b3 - 2*b2 - 17*b1 + 18) * q^19 + 4*b7 * q^20 + (2*b6 - 2*b5 - 2*b4 - 2*b3 + 16) * q^22 + (3*b7 - b6 + b5 + 3*b2 + 23*b1 - 20) * q^23 + (-2*b6 - 2*b5 - 2*b4 - 2*b2 - 41*b1 + 2) * q^25 + (-4*b7 - 2*b6 + 4*b5 + 2*b4 - 2*b3 - 4*b2 - 4*b1 - 2) * q^26 + (-4*b6 + 4*b3 - 8*b1 + 12) * q^28 + (5*b7 - 4*b6 + 6*b5 + 4*b4 + 6*b3 + 2*b1 + 28) * q^29 + (-5*b4 - 5*b3 + 6*b2 - 13*b1) * q^31 - 32*b1 * q^32 + (6*b7 - 4*b5 - 4*b3 - 4*b1 - 24) * q^34 + (-7*b7 - 2*b6 + 14*b5 + 7*b4 + b3 + 2*b1 - 10) * q^35 + (-10*b7 - 11*b6 + 6*b5 - 5*b4 + 5*b3 - 10*b2 + 34*b1 - 39) * q^37 + (6*b6 + 6*b5 + 4*b4 - 2*b3 + 4*b2 + 40*b1 - 6) * q^38 + (-8*b7 - 8*b2 - 8*b1) * q^40 + (-b7 + 4*b6 + 6*b5 - 4*b4 + 6*b3 + 10*b1 + 92) * q^41 + (-10*b7 + 2*b6 - 2*b5 - 2*b4 - 2*b3 + 105) * q^43 + (-4*b6 + 4*b5 + 32*b1 - 32) * q^44 + (2*b4 + 2*b3 - 6*b2 - 46*b1) * q^46 + (-11*b7 - 6*b6 + 16*b5 + 10*b4 - 10*b3 - 11*b2 - 87*b1 + 66) * q^47 + (28*b7 + b6 + 7*b5 - 3*b3 + 14*b2 + 70*b1 - 14) * q^49 + (-4*b7 + 4*b5 + 4*b3 + 4*b1 + 74) * q^50 + (-4*b6 - 4*b5 + 4*b4 + 8*b3 + 8*b2 + 4*b1 + 4) * q^52 + (20*b6 + 20*b5 + 9*b4 - 11*b3 - 7*b2 + 29*b1 - 20) * q^53 + (4*b7 + 6*b6 + 6*b5 - 6*b4 + 6*b3 + 12*b1 - 12) * q^55 + (8*b6 + 24*b1 - 8) * q^56 + (-10*b7 + 12*b6 - 8*b5 + 4*b4 - 4*b3 - 10*b2 + 46*b1 - 60) * q^58 + (4*b6 + 4*b5 - 5*b4 - 9*b3 - 6*b2 - 38*b1 - 4) * q^59 + (16*b7 + 11*b6 - 7*b5 + 4*b4 - 4*b3 + 16*b2 + 97*b1 - 85) * q^61 + (12*b7 - 10*b6 + 10*b5 + 10*b4 + 10*b3 + 38) * q^62 + 64 * q^64 + (2*b7 + b6 + 21*b5 + 22*b4 - 22*b3 + 2*b2 + 314*b1 - 334) * q^65 + (7*b6 + 7*b5 + 8*b4 + b3 - 8*b2 - 239*b1 - 7) * q^67 + (-12*b7 - 8*b6 - 8*b4 + 8*b3 - 12*b2 - 60*b1 + 56) * q^68 + (14*b7 + 2*b6 - 14*b5 + 14*b4 + 2*b3 + 14*b2 - 6*b1 + 16) * q^70 + (26*b7 - 19*b6 + 17*b5 + 19*b4 + 17*b3 - 2*b1 + 184) * q^71 + (15*b6 + 15*b5 - 15*b3 + 52*b2 + 20*b1 - 15) * q^73 + (10*b6 + 10*b5 + 22*b4 + 12*b3 + 20*b2 - 58*b1 - 10) * q^74 + (8*b7 - 4*b6 - 8*b5 + 4*b4 - 8*b3 - 12*b1 - 60) * q^76 + (14*b7 - 8*b6 - 7*b4 + 9*b3 + 28*b2 + 20*b1 - 268) * q^77 + (-64*b7 + 7*b6 + 2*b5 + 9*b4 - 9*b3 - 64*b2 + 20*b1 - 93) * q^79 + (16*b2 + 16*b1) * q^80 + (2*b7 + 12*b6 + 8*b5 + 20*b4 - 20*b3 + 2*b2 + 186*b1 - 204) * q^82 + (-42*b7 - 34*b6 + 24*b5 + 34*b4 + 24*b3 - 10*b1 + 376) * q^83 + (-38*b7 - 18*b6 + 36*b5 + 18*b4 + 36*b3 + 18*b1 + 462) * q^85 + (20*b7 - 4*b6 + 4*b5 + 20*b2 + 230*b1 - 210) * q^86 + (8*b4 + 8*b3 - 64*b1) * q^88 + (67*b7 + 24*b6 - 20*b5 + 4*b4 - 4*b3 + 67*b2 - 333*b1 + 396) * q^89 + (42*b7 + 3*b6 - 21*b4 - 8*b3 - 14*b2 - 237*b1 + 618) * q^91 + (-12*b7 + 4*b6 - 4*b5 - 4*b4 - 4*b3 + 80) * q^92 + (-20*b6 - 20*b5 + 12*b4 + 32*b3 + 22*b2 + 154*b1 + 20) * q^94 + (-10*b6 - 10*b5 - 43*b4 - 33*b3 - 45*b2 - 355*b1 + 10) * q^95 + (74*b7 - 34*b6 + 24*b5 + 34*b4 + 24*b3 - 10*b1 - 501) * q^97 + (-28*b7 - 6*b6 + 14*b4 + 4*b3 - 56*b2 - 84*b1 - 84) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8}+O(q^{10})$$ 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8 $$8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8} + 4 q^{10} - 32 q^{11} - 4 q^{13} - 36 q^{14} - 64 q^{16} + 58 q^{17} + 70 q^{19} - 16 q^{20} + 128 q^{22} - 86 q^{23} - 156 q^{25} + 4 q^{26} + 48 q^{28} + 212 q^{29} - 64 q^{31} - 128 q^{32} - 232 q^{34} - 8 q^{35} - 146 q^{37} + 140 q^{38} + 16 q^{40} + 780 q^{41} + 880 q^{43} - 128 q^{44} - 172 q^{46} + 306 q^{47} + 50 q^{49} + 624 q^{50} + 8 q^{52} + 90 q^{53} - 64 q^{55} + 48 q^{56} - 212 q^{58} - 148 q^{59} - 364 q^{61} + 256 q^{62} + 512 q^{64} - 1296 q^{65} - 954 q^{67} + 232 q^{68} + 20 q^{70} + 1360 q^{71} - 54 q^{73} - 292 q^{74} - 560 q^{76} - 2224 q^{77} - 226 q^{79} + 32 q^{80} - 780 q^{82} + 3136 q^{83} + 3920 q^{85} - 880 q^{86} - 256 q^{88} + 1458 q^{89} + 3836 q^{91} + 688 q^{92} + 612 q^{94} - 1310 q^{95} - 4344 q^{97} - 776 q^{98}+O(q^{100})$$ 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8 + 4 * q^10 - 32 * q^11 - 4 * q^13 - 36 * q^14 - 64 * q^16 + 58 * q^17 + 70 * q^19 - 16 * q^20 + 128 * q^22 - 86 * q^23 - 156 * q^25 + 4 * q^26 + 48 * q^28 + 212 * q^29 - 64 * q^31 - 128 * q^32 - 232 * q^34 - 8 * q^35 - 146 * q^37 + 140 * q^38 + 16 * q^40 + 780 * q^41 + 880 * q^43 - 128 * q^44 - 172 * q^46 + 306 * q^47 + 50 * q^49 + 624 * q^50 + 8 * q^52 + 90 * q^53 - 64 * q^55 + 48 * q^56 - 212 * q^58 - 148 * q^59 - 364 * q^61 + 256 * q^62 + 512 * q^64 - 1296 * q^65 - 954 * q^67 + 232 * q^68 + 20 * q^70 + 1360 * q^71 - 54 * q^73 - 292 * q^74 - 560 * q^76 - 2224 * q^77 - 226 * q^79 + 32 * q^80 - 780 * q^82 + 3136 * q^83 + 3920 * q^85 - 880 * q^86 - 256 * q^88 + 1458 * q^89 + 3836 * q^91 + 688 * q^92 + 612 * q^94 - 1310 * q^95 - 4344 * q^97 - 776 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( - 5251 \nu^{7} + 17661 \nu^{6} - 103443 \nu^{5} + 165901 \nu^{4} - 1314483 \nu^{3} + 3812253 \nu^{2} - 811706 \nu + 2144550 ) / 2252390$$ (-5251*v^7 + 17661*v^6 - 103443*v^5 + 165901*v^4 - 1314483*v^3 + 3812253*v^2 - 811706*v + 2144550) / 2252390 $$\beta_{2}$$ $$=$$ $$( - 5533 \nu^{7} + 45633 \nu^{6} - 267279 \nu^{5} + 771473 \nu^{4} - 3396399 \nu^{3} + 9850209 \nu^{2} - 43821428 \nu + 5541150 ) / 2252390$$ (-5533*v^7 + 45633*v^6 - 267279*v^5 + 771473*v^4 - 3396399*v^3 + 9850209*v^2 - 43821428*v + 5541150) / 2252390 $$\beta_{3}$$ $$=$$ $$( 25642 \nu^{7} - 123347 \nu^{6} + 400691 \nu^{5} - 1182462 \nu^{4} + 3388681 \nu^{3} - 26303561 \nu^{2} - 24618118 \nu + 20095080 ) / 2252390$$ (25642*v^7 - 123347*v^6 + 400691*v^5 - 1182462*v^4 + 3388681*v^3 - 26303561*v^2 - 24618118*v + 20095080) / 2252390 $$\beta_{4}$$ $$=$$ $$( - 26206 \nu^{7} + 179291 \nu^{6} - 728363 \nu^{5} + 2393606 \nu^{4} - 7552513 \nu^{3} + 38379473 \nu^{2} - 34372646 \nu - 13301880 ) / 2252390$$ (-26206*v^7 + 179291*v^6 - 728363*v^5 + 2393606*v^4 - 7552513*v^3 + 38379473*v^2 - 34372646*v - 13301880) / 2252390 $$\beta_{5}$$ $$=$$ $$( 16807 \nu^{7} - 3492 \nu^{6} + 296256 \nu^{5} + 417823 \nu^{4} + 4902586 \nu^{3} + 3475204 \nu^{2} + 3795232 \nu + 5873470 ) / 321770$$ (16807*v^7 - 3492*v^6 + 296256*v^5 + 417823*v^4 + 4902586*v^3 + 3475204*v^2 + 3795232*v + 5873470) / 321770 $$\beta_{6}$$ $$=$$ $$( - 42509 \nu^{7} + 56798 \nu^{6} - 783152 \nu^{5} - 84149 \nu^{4} - 11885520 \nu^{3} + 6404040 \nu^{2} - 7027748 \nu + 5545466 ) / 450478$$ (-42509*v^7 + 56798*v^6 - 783152*v^5 - 84149*v^4 - 11885520*v^3 + 6404040*v^2 - 7027748*v + 5545466) / 450478 $$\beta_{7}$$ $$=$$ $$( - 47855 \nu^{7} + 35959 \nu^{6} - 821980 \nu^{5} - 661525 \nu^{4} - 13230433 \nu^{3} - 636190 \nu^{2} - 394100 \nu - 3549092 ) / 450478$$ (-47855*v^7 + 35959*v^6 - 821980*v^5 - 661525*v^4 - 13230433*v^3 - 636190*v^2 - 394100*v - 3549092) / 450478
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta_1 ) / 12$$ (b4 + b3 - 2*b2 + 2*b1) / 12 $$\nu^{2}$$ $$=$$ $$( -2\beta_{7} - \beta_{6} - 3\beta_{5} - 4\beta_{4} + 4\beta_{3} - 2\beta_{2} + 106\beta _1 - 104 ) / 12$$ (-2*b7 - b6 - 3*b5 - 4*b4 + 4*b3 - 2*b2 + 106*b1 - 104) / 12 $$\nu^{3}$$ $$=$$ $$( -14\beta_{7} + 11\beta_{6} - 9\beta_{5} - 11\beta_{4} - 9\beta_{3} + 2\beta _1 - 68 ) / 6$$ (-14*b7 + 11*b6 - 9*b5 - 11*b4 - 9*b3 + 2*b1 - 68) / 6 $$\nu^{4}$$ $$=$$ $$( 24\beta_{6} + 24\beta_{5} + \beta_{4} - 23\beta_{3} + 18\beta_{2} - 570\beta _1 - 24 ) / 4$$ (24*b6 + 24*b5 + b4 - 23*b3 + 18*b2 - 570*b1 - 24) / 4 $$\nu^{5}$$ $$=$$ $$( 502\beta_{7} - 297\beta_{6} + 421\beta_{5} + 124\beta_{4} - 124\beta_{3} + 502\beta_{2} - 3174\beta _1 + 3552 ) / 12$$ (502*b7 - 297*b6 + 421*b5 + 124*b4 - 124*b3 + 502*b2 - 3174*b1 + 3552) / 12 $$\nu^{6}$$ $$=$$ $$( 322\beta_{7} - 369\beta_{6} + 46\beta_{5} + 369\beta_{4} + 46\beta_{3} - 323\beta _1 + 8316 ) / 3$$ (322*b7 - 369*b6 + 46*b5 + 369*b4 + 46*b3 - 323*b1 + 8316) / 3 $$\nu^{7}$$ $$=$$ $$( -3072\beta_{6} - 3072\beta_{5} + 5065\beta_{4} + 8137\beta_{3} - 9326\beta_{2} + 74654\beta _1 + 3072 ) / 12$$ (-3072*b6 - 3072*b5 + 5065*b4 + 8137*b3 - 9326*b2 + 74654*b1 + 3072) / 12

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.445324 − 0.771324i 2.24123 − 3.88192i −1.84763 + 3.20018i −0.338925 + 0.587036i 0.445324 + 0.771324i 2.24123 + 3.88192i −1.84763 − 3.20018i −0.338925 − 0.587036i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −6.50453 11.2662i 0 18.4787 1.24034i 8.00000 0 −13.0091 + 22.5324i
109.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i −5.59791 9.69587i 0 −16.7643 7.87127i 8.00000 0 −11.1958 + 19.3917i
109.3 −1.00000 1.73205i 0 −2.00000 + 3.46410i 5.04834 + 8.74398i 0 −5.48778 + 17.6885i 8.00000 0 10.0967 17.4880i
109.4 −1.00000 1.73205i 0 −2.00000 + 3.46410i 8.05411 + 13.9501i 0 6.77345 17.2372i 8.00000 0 16.1082 27.9003i
163.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −6.50453 + 11.2662i 0 18.4787 + 1.24034i 8.00000 0 −13.0091 22.5324i
163.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i −5.59791 + 9.69587i 0 −16.7643 + 7.87127i 8.00000 0 −11.1958 19.3917i
163.3 −1.00000 + 1.73205i 0 −2.00000 3.46410i 5.04834 8.74398i 0 −5.48778 17.6885i 8.00000 0 10.0967 + 17.4880i
163.4 −1.00000 + 1.73205i 0 −2.00000 3.46410i 8.05411 13.9501i 0 6.77345 + 17.2372i 8.00000 0 16.1082 + 27.9003i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.g.c 8
3.b odd 2 1 378.4.g.f yes 8
7.c even 3 1 inner 378.4.g.c 8
21.h odd 6 1 378.4.g.f yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.g.c 8 1.a even 1 1 trivial
378.4.g.c 8 7.c even 3 1 inner
378.4.g.f yes 8 3.b odd 2 1
378.4.g.f yes 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 2 T_{5}^{7} + 330 T_{5}^{6} + 412 T_{5}^{5} + 82828 T_{5}^{4} + 55632 T_{5}^{3} + 7736688 T_{5}^{2} + 2842560 T_{5} + 561121344$$ acting on $$S_{4}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 2 T^{7} + 330 T^{6} + \cdots + 561121344$$
$7$ $$T^{8} - 6 T^{7} + \cdots + 13841287201$$
$11$ $$T^{8} + 32 T^{7} + \cdots + 53616328704$$
$13$ $$(T^{4} + 2 T^{3} - 4883 T^{2} + \cdots - 513576)^{2}$$
$17$ $$T^{8} - 58 T^{7} + \cdots + 1132436505600$$
$19$ $$T^{8} + \cdots + 543448598016256$$
$23$ $$T^{8} + 86 T^{7} + \cdots + 2210504256$$
$29$ $$(T^{4} - 106 T^{3} - 39170 T^{2} + \cdots - 175957920)^{2}$$
$31$ $$T^{8} + 64 T^{7} + \cdots + 139065597225$$
$37$ $$T^{8} + 146 T^{7} + \cdots + 49\!\cdots\!00$$
$41$ $$(T^{4} - 390 T^{3} - 61074 T^{2} + \cdots + 2132840160)^{2}$$
$43$ $$(T^{4} - 440 T^{3} + 34890 T^{2} + \cdots - 49652003)^{2}$$
$47$ $$T^{8} - 306 T^{7} + \cdots + 73\!\cdots\!36$$
$53$ $$T^{8} - 90 T^{7} + \cdots + 17\!\cdots\!36$$
$59$ $$T^{8} + 148 T^{7} + \cdots + 22\!\cdots\!16$$
$61$ $$T^{8} + 364 T^{7} + \cdots + 80\!\cdots\!25$$
$67$ $$T^{8} + 954 T^{7} + \cdots + 21\!\cdots\!00$$
$71$ $$(T^{4} - 680 T^{3} - 431588 T^{2} + \cdots - 5339649600)^{2}$$
$73$ $$T^{8} + 54 T^{7} + \cdots + 38\!\cdots\!56$$
$79$ $$T^{8} + 226 T^{7} + \cdots + 20\!\cdots\!56$$
$83$ $$(T^{4} - 1568 T^{3} + \cdots - 310761735840)^{2}$$
$89$ $$T^{8} - 1458 T^{7} + \cdots + 29\!\cdots\!00$$
$97$ $$(T^{4} + 2172 T^{3} + \cdots - 1160560203227)^{2}$$