# Properties

 Label 490.4.c.c Level $490$ Weight $4$ Character orbit 490.c Analytic conductor $28.911$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 490.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.9109359028$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.43197465600.1 Defining polynomial: $$x^{6} - 16x^{3} + 1521x^{2} - 624x + 128$$ x^6 - 16*x^3 + 1521*x^2 - 624*x + 128 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{3} q^{2} + ( - \beta_{5} + 2 \beta_{3}) q^{3} - 4 q^{4} + ( - 3 \beta_{3} - \beta_{2} + 3) q^{5} + ( - 2 \beta_1 - 4) q^{6} - 8 \beta_{3} q^{8} + (\beta_{5} + \beta_{4} + 3 \beta_{2} - 4 \beta_1 - 25) q^{9}+O(q^{10})$$ q + 2*b3 * q^2 + (-b5 + 2*b3) * q^3 - 4 * q^4 + (-3*b3 - b2 + 3) * q^5 + (-2*b1 - 4) * q^6 - 8*b3 * q^8 + (b5 + b4 + 3*b2 - 4*b1 - 25) * q^9 $$q + 2 \beta_{3} q^{2} + ( - \beta_{5} + 2 \beta_{3}) q^{3} - 4 q^{4} + ( - 3 \beta_{3} - \beta_{2} + 3) q^{5} + ( - 2 \beta_1 - 4) q^{6} - 8 \beta_{3} q^{8} + (\beta_{5} + \beta_{4} + 3 \beta_{2} - 4 \beta_1 - 25) q^{9} + ( - 2 \beta_{4} + 6 \beta_{3} + 6) q^{10} + (\beta_{5} + \beta_{4} + 3 \beta_{2} - 2 \beta_1 + 6) q^{11} + (4 \beta_{5} - 8 \beta_{3}) q^{12} + ( - 5 \beta_{5} + 12 \beta_{3}) q^{13} + ( - 10 \beta_{5} - 3 \beta_{4} - 9 \beta_{3} - \beta_{2} + 5 \beta_1 + 17) q^{15} + 16 q^{16} + ( - 4 \beta_{5} - 3 \beta_{4} - 52 \beta_{3} + \beta_{2} - \beta_1) q^{17} + (8 \beta_{5} + 6 \beta_{4} - 50 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{18} + (\beta_{5} + \beta_{4} + 3 \beta_{2} - 15 \beta_1 + 56) q^{19} + (12 \beta_{3} + 4 \beta_{2} - 12) q^{20} + (4 \beta_{5} + 6 \beta_{4} + 12 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{22} + (14 \beta_{5} - 6 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{23} + (8 \beta_1 + 16) q^{24} + (5 \beta_{5} + 4 \beta_{4} - 71 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 82) q^{25} + ( - 10 \beta_1 - 24) q^{26} + (28 \beta_{5} + 21 \beta_{4} - 154 \beta_{3} - 7 \beta_{2} + 7 \beta_1) q^{27} + ( - 3 \beta_{5} - 3 \beta_{4} - 9 \beta_{2} + 2 \beta_1 - 28) q^{29} + ( - 10 \beta_{5} - 2 \beta_{4} + 34 \beta_{3} + 6 \beta_{2} - 20 \beta_1 + 18) q^{30} + (4 \beta_{5} + 4 \beta_{4} + 12 \beta_{2} + 10 \beta_1 - 64) q^{31} + 32 \beta_{3} q^{32} + (20 \beta_{5} + 15 \beta_{4} - 50 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{33} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} - 8 \beta_1 + 104) q^{34} + ( - 4 \beta_{5} - 4 \beta_{4} - 12 \beta_{2} + 16 \beta_1 + 100) q^{36} + (16 \beta_{5} + 12 \beta_{4} - 38 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{37} + (30 \beta_{5} + 6 \beta_{4} + 112 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{38} + (5 \beta_{5} + 5 \beta_{4} + 15 \beta_{2} - 22 \beta_1 - 264) q^{39} + (8 \beta_{4} - 24 \beta_{3} - 24) q^{40} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} + 30 \beta_1 - 10) q^{41} + ( - 6 \beta_{5} + 30 \beta_{4} + 60 \beta_{3} - 10 \beta_{2} + 10 \beta_1) q^{43} + ( - 4 \beta_{5} - 4 \beta_{4} - 12 \beta_{2} + 8 \beta_1 - 24) q^{44} + ( - 30 \beta_{5} - 11 \beta_{4} + 211 \beta_{3} + 16 \beta_{2} + \cdots - 325) q^{45}+ \cdots + ( - 8 \beta_{5} - 8 \beta_{4} - 24 \beta_{2} + 146 \beta_1 + 1052) q^{99}+O(q^{100})$$ q + 2*b3 * q^2 + (-b5 + 2*b3) * q^3 - 4 * q^4 + (-3*b3 - b2 + 3) * q^5 + (-2*b1 - 4) * q^6 - 8*b3 * q^8 + (b5 + b4 + 3*b2 - 4*b1 - 25) * q^9 + (-2*b4 + 6*b3 + 6) * q^10 + (b5 + b4 + 3*b2 - 2*b1 + 6) * q^11 + (4*b5 - 8*b3) * q^12 + (-5*b5 + 12*b3) * q^13 + (-10*b5 - 3*b4 - 9*b3 - b2 + 5*b1 + 17) * q^15 + 16 * q^16 + (-4*b5 - 3*b4 - 52*b3 + b2 - b1) * q^17 + (8*b5 + 6*b4 - 50*b3 - 2*b2 + 2*b1) * q^18 + (b5 + b4 + 3*b2 - 15*b1 + 56) * q^19 + (12*b3 + 4*b2 - 12) * q^20 + (4*b5 + 6*b4 + 12*b3 - 2*b2 + 2*b1) * q^22 + (14*b5 - 6*b4 - 10*b3 + 2*b2 - 2*b1) * q^23 + (8*b1 + 16) * q^24 + (5*b5 + 4*b4 - 71*b3 + 2*b2 - 5*b1 + 82) * q^25 + (-10*b1 - 24) * q^26 + (28*b5 + 21*b4 - 154*b3 - 7*b2 + 7*b1) * q^27 + (-3*b5 - 3*b4 - 9*b2 + 2*b1 - 28) * q^29 + (-10*b5 - 2*b4 + 34*b3 + 6*b2 - 20*b1 + 18) * q^30 + (4*b5 + 4*b4 + 12*b2 + 10*b1 - 64) * q^31 + 32*b3 * q^32 + (20*b5 + 15*b4 - 50*b3 - 5*b2 + 5*b1) * q^33 + (2*b5 + 2*b4 + 6*b2 - 8*b1 + 104) * q^34 + (-4*b5 - 4*b4 - 12*b2 + 16*b1 + 100) * q^36 + (16*b5 + 12*b4 - 38*b3 - 4*b2 + 4*b1) * q^37 + (30*b5 + 6*b4 + 112*b3 - 2*b2 + 2*b1) * q^38 + (5*b5 + 5*b4 + 15*b2 - 22*b1 - 264) * q^39 + (8*b4 - 24*b3 - 24) * q^40 + (2*b5 + 2*b4 + 6*b2 + 30*b1 - 10) * q^41 + (-6*b5 + 30*b4 + 60*b3 - 10*b2 + 10*b1) * q^43 + (-4*b5 - 4*b4 - 12*b2 + 8*b1 - 24) * q^44 + (-30*b5 - 11*b4 + 211*b3 + 16*b2 - 35*b1 - 325) * q^45 + (4*b5 + 4*b4 + 12*b2 + 28*b1 + 20) * q^46 + (-38*b5 + 9*b4 - 42*b3 - 3*b2 + 3*b1) * q^47 + (-16*b5 + 32*b3) * q^48 + (10*b5 + 4*b4 + 164*b3 - 8*b2 + 10*b1 + 142) * q^50 + (7*b5 + 7*b4 + 21*b2 + 22*b1 - 54) * q^51 + (20*b5 - 48*b3) * q^52 + (-34*b5 + 18*b4 + 230*b3 - 6*b2 + 6*b1) * q^53 + (-14*b5 - 14*b4 - 42*b2 + 56*b1 + 308) * q^54 + (-20*b5 - 9*b4 + 96*b3 - 17*b2 - 15*b1 - 262) * q^55 + (-4*b5 + 54*b4 - 574*b3 - 18*b2 + 18*b1) * q^57 + (-4*b5 - 18*b4 - 56*b3 + 6*b2 - 6*b1) * q^58 + (9*b5 + 9*b4 + 27*b2 - 49*b1 + 204) * q^59 + (40*b5 + 12*b4 + 36*b3 + 4*b2 - 20*b1 - 68) * q^60 + (-9*b5 - 9*b4 - 27*b2 + 5*b1 - 110) * q^61 + (-20*b5 + 24*b4 - 128*b3 - 8*b2 + 8*b1) * q^62 - 64 * q^64 + (-50*b5 - 17*b4 - 39*b3 - 5*b2 + 25*b1 + 91) * q^65 + (-10*b5 - 10*b4 - 30*b2 + 40*b1 + 100) * q^66 + (-8*b5 - 6*b4 - 420*b3 + 2*b2 - 2*b1) * q^67 + (16*b5 + 12*b4 + 208*b3 - 4*b2 + 4*b1) * q^68 + (-8*b5 - 8*b4 - 24*b2 - 6*b1 + 760) * q^69 + (-74*b1 + 266) * q^71 + (-32*b5 - 24*b4 + 200*b3 + 8*b2 - 8*b1) * q^72 + (30*b5 - 42*b4 - 418*b3 + 14*b2 - 14*b1) * q^73 + (-8*b5 - 8*b4 - 24*b2 + 32*b1 + 76) * q^74 + (-55*b5 + 20*b4 - 90*b3 - 30*b2 + 110*b1 + 300) * q^75 + (-4*b5 - 4*b4 - 12*b2 + 60*b1 - 224) * q^76 + (44*b5 + 30*b4 - 528*b3 - 10*b2 + 10*b1) * q^78 + (19*b5 + 19*b4 + 57*b2 - 26*b1 - 356) * q^79 + (-48*b3 - 16*b2 + 48) * q^80 + (-22*b5 - 22*b4 - 66*b2 + 256*b1 + 739) * q^81 + (-60*b5 + 12*b4 - 20*b3 - 4*b2 + 4*b1) * q^82 + (-53*b5 + 51*b4 - 228*b3 - 17*b2 + 17*b1) * q^83 + (-35*b5 + 61*b4 + 94*b3 - 11*b2 + 30*b1 - 20) * q^85 + (-20*b5 - 20*b4 - 60*b2 - 12*b1 - 120) * q^86 + (-42*b5 - 33*b4 - 62*b3 + 11*b2 - 11*b1) * q^87 + (-16*b5 - 24*b4 - 48*b3 + 8*b2 - 8*b1) * q^88 + (22*b5 + 22*b4 + 66*b2 + 80*b1 - 398) * q^89 + (70*b5 + 32*b4 - 650*b3 + 22*b2 - 60*b1 - 422) * q^90 + (-56*b5 + 24*b4 + 40*b3 - 8*b2 + 8*b1) * q^92 + (132*b5 + 6*b4 + 488*b3 - 2*b2 + 2*b1) * q^93 + (-6*b5 - 6*b4 - 18*b2 - 76*b1 + 84) * q^94 + (-85*b5 - 22*b4 + 89*b3 - 54*b2 - 145*b1 + 83) * q^95 + (-32*b1 - 64) * q^96 + (-16*b5 - 27*b4 + 944*b3 + 9*b2 - 9*b1) * q^97 + (-8*b5 - 8*b4 - 24*b2 + 146*b1 + 1052) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 24 q^{4} + 16 q^{5} - 28 q^{6} - 152 q^{9}+O(q^{10})$$ 6 * q - 24 * q^4 + 16 * q^5 - 28 * q^6 - 152 * q^9 $$6 q - 24 q^{4} + 16 q^{5} - 28 q^{6} - 152 q^{9} + 36 q^{10} + 38 q^{11} + 110 q^{15} + 96 q^{16} + 312 q^{19} - 64 q^{20} + 112 q^{24} + 486 q^{25} - 164 q^{26} - 182 q^{29} + 80 q^{30} - 340 q^{31} + 620 q^{34} + 608 q^{36} - 1598 q^{39} - 144 q^{40} + 12 q^{41} - 152 q^{44} - 1988 q^{45} + 200 q^{46} + 856 q^{50} - 238 q^{51} + 1876 q^{54} - 1636 q^{55} + 1180 q^{59} - 440 q^{60} - 704 q^{61} - 384 q^{64} + 586 q^{65} + 620 q^{66} + 4500 q^{69} + 1448 q^{71} + 472 q^{74} + 1960 q^{75} - 1248 q^{76} - 2074 q^{79} + 256 q^{80} + 4814 q^{81} - 82 q^{85} - 864 q^{86} - 2096 q^{89} - 2608 q^{90} + 316 q^{94} + 100 q^{95} - 448 q^{96} + 6556 q^{99}+O(q^{100})$$ 6 * q - 24 * q^4 + 16 * q^5 - 28 * q^6 - 152 * q^9 + 36 * q^10 + 38 * q^11 + 110 * q^15 + 96 * q^16 + 312 * q^19 - 64 * q^20 + 112 * q^24 + 486 * q^25 - 164 * q^26 - 182 * q^29 + 80 * q^30 - 340 * q^31 + 620 * q^34 + 608 * q^36 - 1598 * q^39 - 144 * q^40 + 12 * q^41 - 152 * q^44 - 1988 * q^45 + 200 * q^46 + 856 * q^50 - 238 * q^51 + 1876 * q^54 - 1636 * q^55 + 1180 * q^59 - 440 * q^60 - 704 * q^61 - 384 * q^64 + 586 * q^65 + 620 * q^66 + 4500 * q^69 + 1448 * q^71 + 472 * q^74 + 1960 * q^75 - 1248 * q^76 - 2074 * q^79 + 256 * q^80 + 4814 * q^81 - 82 * q^85 - 864 * q^86 - 2096 * q^89 - 2608 * q^90 + 316 * q^94 + 100 * q^95 - 448 * q^96 + 6556 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 16x^{3} + 1521x^{2} - 624x + 128$$ :

 $$\beta_{1}$$ $$=$$ $$( 31\nu^{5} - 1513\nu^{4} + 1209\nu^{3} - 248\nu^{2} - 1500719 ) / 177765$$ (31*v^5 - 1513*v^4 + 1209*v^3 - 248*v^2 - 1500719) / 177765 $$\beta_{2}$$ $$=$$ $$( -29\nu^{5} - 223\nu^{4} - 1131\nu^{3} + 232\nu^{2} - 25395\nu - 193529 ) / 25395$$ (-29*v^5 - 223*v^4 - 1131*v^3 + 232*v^2 - 25395*v - 193529) / 25395 $$\beta_{3}$$ $$=$$ $$( 1521\nu^{5} + 312\nu^{4} + 64\nu^{3} - 12168\nu^{2} + 2310945\nu - 475064 ) / 474040$$ (1521*v^5 + 312*v^4 + 64*v^3 - 12168*v^2 + 2310945*v - 475064) / 474040 $$\beta_{4}$$ $$=$$ $$( -5343\nu^{5} - 1096\nu^{4} + 3248\nu^{3} + 110464\nu^{2} - 7779335\nu + 1641032 ) / 203160$$ (-5343*v^5 - 1096*v^4 + 3248*v^3 + 110464*v^2 - 7779335*v + 1641032) / 203160 $$\beta_{5}$$ $$=$$ $$( 38337\nu^{5} + 7864\nu^{4} + 13768\nu^{3} - 780736\nu^{2} + 58721705\nu - 12071288 ) / 1422120$$ (38337*v^5 + 7864*v^4 + 13768*v^3 - 780736*v^2 + 58721705*v - 12071288) / 1422120
 $$\nu$$ $$=$$ $$( 2\beta_{5} + 2\beta_{4} + \beta_{2} - \beta_1 ) / 5$$ (2*b5 + 2*b4 + b2 - b1) / 5 $$\nu^{2}$$ $$=$$ $$( -12\beta_{5} + 3\beta_{4} + 125\beta_{3} - \beta_{2} + \beta_1 ) / 5$$ (-12*b5 + 3*b4 + 125*b3 - b2 + b1) / 5 $$\nu^{3}$$ $$=$$ $$( 39\beta_{5} + 39\beta_{4} - 40\beta_{3} - 78\beta_{2} + 78\beta _1 + 40 ) / 5$$ (39*b5 + 39*b4 - 40*b3 - 78*b2 + 78*b1 + 40) / 5 $$\nu^{4}$$ $$=$$ $$( -31\beta_{5} - 31\beta_{4} - 93\beta_{2} - 492\beta _1 - 4875 ) / 5$$ (-31*b5 - 31*b4 - 93*b2 - 492*b1 - 4875) / 5 $$\nu^{5}$$ $$=$$ $$-626\beta_{5} - 602\beta_{4} + 512\beta_{3} - 301\beta_{2} + 325\beta _1 + 512$$ -626*b5 - 602*b4 + 512*b3 - 301*b2 + 325*b1 + 512

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −4.51508 + 4.51508i 4.30950 − 4.30950i 0.205574 − 0.205574i 0.205574 + 0.205574i 4.30950 + 4.30950i −4.51508 − 4.51508i
2.00000i 10.2673i −4.00000 8.27790 + 7.51508i −20.5347 0 8.00000i −78.4181 15.0302 16.5558i
99.2 2.00000i 3.17488i −4.00000 −11.1034 1.30950i −6.34975 0 8.00000i 16.9202 −2.61901 + 22.2068i
99.3 2.00000i 6.44221i −4.00000 10.8255 + 2.79443i 12.8844 0 8.00000i −14.5021 5.58885 21.6510i
99.4 2.00000i 6.44221i −4.00000 10.8255 2.79443i 12.8844 0 8.00000i −14.5021 5.58885 + 21.6510i
99.5 2.00000i 3.17488i −4.00000 −11.1034 + 1.30950i −6.34975 0 8.00000i 16.9202 −2.61901 22.2068i
99.6 2.00000i 10.2673i −4.00000 8.27790 7.51508i −20.5347 0 8.00000i −78.4181 15.0302 + 16.5558i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 99.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.4.c.c 6
5.b even 2 1 inner 490.4.c.c 6
5.c odd 4 1 2450.4.a.cf 3
5.c odd 4 1 2450.4.a.cg 3
7.b odd 2 1 70.4.c.b 6
21.c even 2 1 630.4.g.j 6
28.d even 2 1 560.4.g.e 6
35.c odd 2 1 70.4.c.b 6
35.f even 4 1 350.4.a.w 3
35.f even 4 1 350.4.a.x 3
105.g even 2 1 630.4.g.j 6
140.c even 2 1 560.4.g.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.c.b 6 7.b odd 2 1
70.4.c.b 6 35.c odd 2 1
350.4.a.w 3 35.f even 4 1
350.4.a.x 3 35.f even 4 1
490.4.c.c 6 1.a even 1 1 trivial
490.4.c.c 6 5.b even 2 1 inner
560.4.g.e 6 28.d even 2 1
560.4.g.e 6 140.c even 2 1
630.4.g.j 6 21.c even 2 1
630.4.g.j 6 105.g even 2 1
2450.4.a.cf 3 5.c odd 4 1
2450.4.a.cg 3 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(490, [\chi])$$:

 $$T_{3}^{6} + 157T_{3}^{4} + 5856T_{3}^{2} + 44100$$ T3^6 + 157*T3^4 + 5856*T3^2 + 44100 $$T_{19}^{3} - 156T_{19}^{2} - 8046T_{19} + 1196840$$ T19^3 - 156*T19^2 - 8046*T19 + 1196840

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 4)^{3}$$
$3$ $$T^{6} + 157 T^{4} + 5856 T^{2} + \cdots + 44100$$
$5$ $$T^{6} - 16 T^{5} - 115 T^{4} + \cdots + 1953125$$
$7$ $$T^{6}$$
$11$ $$(T^{3} - 19 T^{2} - 1560 T - 600)^{2}$$
$13$ $$T^{6} + 4077 T^{4} + \cdots + 829555204$$
$17$ $$T^{6} + 12729 T^{4} + \cdots + 69288976$$
$19$ $$(T^{3} - 156 T^{2} - 8046 T + 1196840)^{2}$$
$23$ $$T^{6} + 45500 T^{4} + \cdots + 2480625000000$$
$29$ $$(T^{3} + 91 T^{2} - 11096 T + 204564)^{2}$$
$31$ $$(T^{3} + 170 T^{2} - 25688 T - 2033232)^{2}$$
$37$ $$T^{6} + 80172 T^{4} + \cdots + 16972878400$$
$41$ $$(T^{3} - 6 T^{2} - 74460 T + 7243272)^{2}$$
$43$ $$T^{6} + \cdots + 125041775755264$$
$47$ $$T^{6} + \cdots + 169390641480256$$
$53$ $$T^{6} + 492124 T^{4} + \cdots + 17\!\cdots\!56$$
$59$ $$(T^{3} - 590 T^{2} - 143278 T + 88782692)^{2}$$
$61$ $$(T^{3} + 352 T^{2} - 83370 T + 3430824)^{2}$$
$67$ $$T^{6} + 546404 T^{4} + \cdots + 48\!\cdots\!96$$
$71$ $$(T^{3} - 724 T^{2} - 210420 T + 55149600)^{2}$$
$73$ $$T^{6} + 1394268 T^{4} + \cdots + 46\!\cdots\!16$$
$79$ $$(T^{3} + 1037 T^{2} - 212824 T - 276622388)^{2}$$
$83$ $$T^{6} + 1558444 T^{4} + \cdots + 83\!\cdots\!36$$
$89$ $$(T^{3} + 1048 T^{2} - 984844 T - 210826480)^{2}$$
$97$ $$T^{6} + 2918281 T^{4} + \cdots + 48\!\cdots\!00$$