# Properties

 Label 507.4.b.g Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.158155776.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18$$ x^6 - 2*x^5 + 2*x^4 + 24*x^3 + 81*x^2 + 54*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 39) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + 3 q^{3} + (\beta_1 - 3) q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + 3 \beta_{4} q^{6} + (6 \beta_{4} - 7 \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{8} + 9 q^{9}+O(q^{10})$$ q + b4 * q^2 + 3 * q^3 + (b1 - 3) * q^4 + (-b5 + b2) * q^5 + 3*b4 * q^6 + (6*b4 - 7*b2) * q^7 + (-b5 + b4 + b2) * q^8 + 9 * q^9 $$q + \beta_{4} q^{2} + 3 q^{3} + (\beta_1 - 3) q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + 3 \beta_{4} q^{6} + (6 \beta_{4} - 7 \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{8} + 9 q^{9} + ( - 3 \beta_{3} - 4 \beta_1 - 1) q^{10} + ( - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_{2}) q^{11} + (3 \beta_1 - 9) q^{12} + ( - 7 \beta_{3} + 6 \beta_1 - 59) q^{14} + ( - 3 \beta_{5} + 3 \beta_{2}) q^{15} + ( - 3 \beta_{3} + 5 \beta_1 - 36) q^{16} + (4 \beta_{3} + 50) q^{17} + 9 \beta_{4} q^{18} + (8 \beta_{5} + 6 \beta_{4} + 11 \beta_{2}) q^{19} + ( - \beta_{5} + 12 \beta_{4} + 37 \beta_{2}) q^{20} + (18 \beta_{4} - 21 \beta_{2}) q^{21} + ( - 9 \beta_{3} - 10 \beta_1 - 25) q^{22} + ( - 16 \beta_{3} - 8 \beta_1 + 8) q^{23} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2}) q^{24} + (12 \beta_{3} - 20 \beta_1 - 51) q^{25} + 27 q^{27} + (\beta_{5} - 42 \beta_{4} + 27 \beta_{2}) q^{28} + (10 \beta_{3} - 8 \beta_1) q^{29} + ( - 9 \beta_{3} - 12 \beta_1 - 3) q^{30} + ( - 2 \beta_{5} + 54 \beta_{4} + 33 \beta_{2}) q^{31} + ( - 10 \beta_{5} - 51 \beta_{4} + 46 \beta_{2}) q^{32} + ( - 9 \beta_{5} + 6 \beta_{4} + 9 \beta_{2}) q^{33} + ( - 4 \beta_{5} + 54 \beta_{4} - 44 \beta_{2}) q^{34} + ( - 18 \beta_{3} + 4 \beta_1 + 22) q^{35} + (9 \beta_1 - 27) q^{36} + (14 \beta_{5} - 48 \beta_{4} - 51 \beta_{2}) q^{37} + (43 \beta_{3} + 38 \beta_1 - 77) q^{38} + (9 \beta_{3} - 24 \beta_1 - 177) q^{40} + (17 \beta_{5} + 4 \beta_{4} + 73 \beta_{2}) q^{41} + ( - 21 \beta_{3} + 18 \beta_1 - 177) q^{42} + (30 \beta_{3} + 4 \beta_1 + 98) q^{43} + ( - 5 \beta_{5} + 22 \beta_{4} + 113 \beta_{2}) q^{44} + ( - 9 \beta_{5} + 9 \beta_{2}) q^{45} + (24 \beta_{5} + 24 \beta_{4} + 168 \beta_{2}) q^{46} + (21 \beta_{5} + 54 \beta_{4} - 21 \beta_{2}) q^{47} + ( - 9 \beta_{3} + 15 \beta_1 - 108) q^{48} + ( - 84 \beta_{3} + 36 \beta_1 - 165) q^{49} + (8 \beta_{5} + 41 \beta_{4} - 152 \beta_{2}) q^{50} + (12 \beta_{3} + 150) q^{51} + ( - 54 \beta_{3} + 12 \beta_1 - 240) q^{53} + 27 \beta_{4} q^{54} + (30 \beta_{3} - 68 \beta_1 - 530) q^{55} + ( - 25 \beta_{3} + 10 \beta_1 - 37) q^{56} + (24 \beta_{5} + 18 \beta_{4} + 33 \beta_{2}) q^{57} + ( - 2 \beta_{5} + 42 \beta_{4} - 118 \beta_{2}) q^{58} + ( - \beta_{5} + 82 \beta_{4} + 271 \beta_{2}) q^{59} + ( - 3 \beta_{5} + 36 \beta_{4} + 111 \beta_{2}) q^{60} + (12 \beta_{3} + 28 \beta_1 + 90) q^{61} + (25 \beta_{3} + 46 \beta_1 - 627) q^{62} + (54 \beta_{4} - 63 \beta_{2}) q^{63} + ( - 18 \beta_{3} - 51 \beta_1 + 227) q^{64} + ( - 27 \beta_{3} - 30 \beta_1 - 75) q^{66} + ( - 38 \beta_{5} - 42 \beta_{4} + 39 \beta_{2}) q^{67} + ( - 28 \beta_{3} + 38 \beta_1 - 150) q^{68} + ( - 48 \beta_{3} - 24 \beta_1 + 24) q^{69} + (14 \beta_{5} - 12 \beta_{4} + 202 \beta_{2}) q^{70} + (7 \beta_{5} + 134 \beta_{4} - 205 \beta_{2}) q^{71} + ( - 9 \beta_{5} + 9 \beta_{4} + 9 \beta_{2}) q^{72} + ( - 6 \beta_{5} - 240 \beta_{4} + 119 \beta_{2}) q^{73} + (5 \beta_{3} + 8 \beta_1 + 579) q^{74} + (36 \beta_{3} - 60 \beta_1 - 153) q^{75} + ( - 17 \beta_{5} - 138 \beta_{4} - 347 \beta_{2}) q^{76} + ( - 68 \beta_{3} + 24 \beta_1 - 52) q^{77} + (24 \beta_{3} + 24 \beta_1 + 8) q^{79} + (7 \beta_{5} + 24 \beta_{4} + 173 \beta_{2}) q^{80} + 81 q^{81} + (141 \beta_{3} + 72 \beta_1 - 117) q^{82} + ( - 5 \beta_{5} - 30 \beta_{4} - 121 \beta_{2}) q^{83} + (3 \beta_{5} - 126 \beta_{4} + 81 \beta_{2}) q^{84} + ( - 38 \beta_{5} + 48 \beta_{4} + 38 \beta_{2}) q^{85} + ( - 34 \beta_{5} + 112 \beta_{4} - 326 \beta_{2}) q^{86} + (30 \beta_{3} - 24 \beta_1) q^{87} + (21 \beta_{3} - 78 \beta_1 - 555) q^{88} + ( - 15 \beta_{5} + 116 \beta_{4} - 273 \beta_{2}) q^{89} + ( - 27 \beta_{3} - 36 \beta_1 - 9) q^{90} + (136 \beta_{3} + 56 \beta_1 - 368) q^{92} + ( - 6 \beta_{5} + 162 \beta_{4} + 99 \beta_{2}) q^{93} + (63 \beta_{3} + 138 \beta_1 - 573) q^{94} + ( - 114 \beta_{3} + 60 \beta_1 + 1326) q^{95} + ( - 30 \beta_{5} - 153 \beta_{4} + 138 \beta_{2}) q^{96} + ( - 2 \beta_{5} + 48 \beta_{4} + 525 \beta_{2}) q^{97} + (48 \beta_{5} - 393 \beta_{4} + 960 \beta_{2}) q^{98} + ( - 27 \beta_{5} + 18 \beta_{4} + 27 \beta_{2}) q^{99}+O(q^{100})$$ q + b4 * q^2 + 3 * q^3 + (b1 - 3) * q^4 + (-b5 + b2) * q^5 + 3*b4 * q^6 + (6*b4 - 7*b2) * q^7 + (-b5 + b4 + b2) * q^8 + 9 * q^9 + (-3*b3 - 4*b1 - 1) * q^10 + (-3*b5 + 2*b4 + 3*b2) * q^11 + (3*b1 - 9) * q^12 + (-7*b3 + 6*b1 - 59) * q^14 + (-3*b5 + 3*b2) * q^15 + (-3*b3 + 5*b1 - 36) * q^16 + (4*b3 + 50) * q^17 + 9*b4 * q^18 + (8*b5 + 6*b4 + 11*b2) * q^19 + (-b5 + 12*b4 + 37*b2) * q^20 + (18*b4 - 21*b2) * q^21 + (-9*b3 - 10*b1 - 25) * q^22 + (-16*b3 - 8*b1 + 8) * q^23 + (-3*b5 + 3*b4 + 3*b2) * q^24 + (12*b3 - 20*b1 - 51) * q^25 + 27 * q^27 + (b5 - 42*b4 + 27*b2) * q^28 + (10*b3 - 8*b1) * q^29 + (-9*b3 - 12*b1 - 3) * q^30 + (-2*b5 + 54*b4 + 33*b2) * q^31 + (-10*b5 - 51*b4 + 46*b2) * q^32 + (-9*b5 + 6*b4 + 9*b2) * q^33 + (-4*b5 + 54*b4 - 44*b2) * q^34 + (-18*b3 + 4*b1 + 22) * q^35 + (9*b1 - 27) * q^36 + (14*b5 - 48*b4 - 51*b2) * q^37 + (43*b3 + 38*b1 - 77) * q^38 + (9*b3 - 24*b1 - 177) * q^40 + (17*b5 + 4*b4 + 73*b2) * q^41 + (-21*b3 + 18*b1 - 177) * q^42 + (30*b3 + 4*b1 + 98) * q^43 + (-5*b5 + 22*b4 + 113*b2) * q^44 + (-9*b5 + 9*b2) * q^45 + (24*b5 + 24*b4 + 168*b2) * q^46 + (21*b5 + 54*b4 - 21*b2) * q^47 + (-9*b3 + 15*b1 - 108) * q^48 + (-84*b3 + 36*b1 - 165) * q^49 + (8*b5 + 41*b4 - 152*b2) * q^50 + (12*b3 + 150) * q^51 + (-54*b3 + 12*b1 - 240) * q^53 + 27*b4 * q^54 + (30*b3 - 68*b1 - 530) * q^55 + (-25*b3 + 10*b1 - 37) * q^56 + (24*b5 + 18*b4 + 33*b2) * q^57 + (-2*b5 + 42*b4 - 118*b2) * q^58 + (-b5 + 82*b4 + 271*b2) * q^59 + (-3*b5 + 36*b4 + 111*b2) * q^60 + (12*b3 + 28*b1 + 90) * q^61 + (25*b3 + 46*b1 - 627) * q^62 + (54*b4 - 63*b2) * q^63 + (-18*b3 - 51*b1 + 227) * q^64 + (-27*b3 - 30*b1 - 75) * q^66 + (-38*b5 - 42*b4 + 39*b2) * q^67 + (-28*b3 + 38*b1 - 150) * q^68 + (-48*b3 - 24*b1 + 24) * q^69 + (14*b5 - 12*b4 + 202*b2) * q^70 + (7*b5 + 134*b4 - 205*b2) * q^71 + (-9*b5 + 9*b4 + 9*b2) * q^72 + (-6*b5 - 240*b4 + 119*b2) * q^73 + (5*b3 + 8*b1 + 579) * q^74 + (36*b3 - 60*b1 - 153) * q^75 + (-17*b5 - 138*b4 - 347*b2) * q^76 + (-68*b3 + 24*b1 - 52) * q^77 + (24*b3 + 24*b1 + 8) * q^79 + (7*b5 + 24*b4 + 173*b2) * q^80 + 81 * q^81 + (141*b3 + 72*b1 - 117) * q^82 + (-5*b5 - 30*b4 - 121*b2) * q^83 + (3*b5 - 126*b4 + 81*b2) * q^84 + (-38*b5 + 48*b4 + 38*b2) * q^85 + (-34*b5 + 112*b4 - 326*b2) * q^86 + (30*b3 - 24*b1) * q^87 + (21*b3 - 78*b1 - 555) * q^88 + (-15*b5 + 116*b4 - 273*b2) * q^89 + (-27*b3 - 36*b1 - 9) * q^90 + (136*b3 + 56*b1 - 368) * q^92 + (-6*b5 + 162*b4 + 99*b2) * q^93 + (63*b3 + 138*b1 - 573) * q^94 + (-114*b3 + 60*b1 + 1326) * q^95 + (-30*b5 - 153*b4 + 138*b2) * q^96 + (-2*b5 + 48*b4 + 525*b2) * q^97 + (48*b5 - 393*b4 + 960*b2) * q^98 + (-27*b5 + 18*b4 + 27*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 18 q^{3} - 20 q^{4} + 54 q^{9}+O(q^{10})$$ 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9 $$6 q + 18 q^{3} - 20 q^{4} + 54 q^{9} + 8 q^{10} - 60 q^{12} - 352 q^{14} - 220 q^{16} + 292 q^{17} - 112 q^{22} + 96 q^{23} - 290 q^{25} + 162 q^{27} - 4 q^{29} + 24 q^{30} + 160 q^{35} - 180 q^{36} - 624 q^{38} - 1032 q^{40} - 1056 q^{42} + 520 q^{43} - 660 q^{48} - 894 q^{49} + 876 q^{51} - 1356 q^{53} - 3104 q^{55} - 192 q^{56} + 460 q^{61} - 3904 q^{62} + 1500 q^{64} - 336 q^{66} - 920 q^{68} + 288 q^{69} + 3448 q^{74} - 870 q^{75} - 224 q^{77} - 48 q^{79} + 486 q^{81} - 1128 q^{82} - 12 q^{87} - 3216 q^{88} + 72 q^{90} - 2592 q^{92} - 3840 q^{94} + 8064 q^{95}+O(q^{100})$$ 6 * q + 18 * q^3 - 20 * q^4 + 54 * q^9 + 8 * q^10 - 60 * q^12 - 352 * q^14 - 220 * q^16 + 292 * q^17 - 112 * q^22 + 96 * q^23 - 290 * q^25 + 162 * q^27 - 4 * q^29 + 24 * q^30 + 160 * q^35 - 180 * q^36 - 624 * q^38 - 1032 * q^40 - 1056 * q^42 + 520 * q^43 - 660 * q^48 - 894 * q^49 + 876 * q^51 - 1356 * q^53 - 3104 * q^55 - 192 * q^56 + 460 * q^61 - 3904 * q^62 + 1500 * q^64 - 336 * q^66 - 920 * q^68 + 288 * q^69 + 3448 * q^74 - 870 * q^75 - 224 * q^77 - 48 * q^79 + 486 * q^81 - 1128 * q^82 - 12 * q^87 - 3216 * q^88 + 72 * q^90 - 2592 * q^92 - 3840 * q^94 + 8064 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$( 4\nu^{5} - 33\nu^{4} + 44\nu^{3} + 48\nu^{2} + 24\nu - 1523 ) / 227$$ (4*v^5 - 33*v^4 + 44*v^3 + 48*v^2 + 24*v - 1523) / 227 $$\beta_{2}$$ $$=$$ $$( 58\nu^{5} - 138\nu^{4} + 184\nu^{3} + 1150\nu^{2} + 4434\nu + 1638 ) / 681$$ (58*v^5 - 138*v^4 + 184*v^3 + 1150*v^2 + 4434*v + 1638) / 681 $$\beta_{3}$$ $$=$$ $$( 68\nu^{5} - 334\nu^{4} + 748\nu^{3} + 816\nu^{2} + 408\nu - 5007 ) / 681$$ (68*v^5 - 334*v^4 + 748*v^3 + 816*v^2 + 408*v - 5007) / 681 $$\beta_{4}$$ $$=$$ $$( 152\nu^{5} - 346\nu^{4} + 310\nu^{3} + 3867\nu^{2} + 10446\nu + 3870 ) / 681$$ (152*v^5 - 346*v^4 + 310*v^3 + 3867*v^2 + 10446*v + 3870) / 681 $$\beta_{5}$$ $$=$$ $$( 406\nu^{5} - 966\nu^{4} + 1288\nu^{3} + 9412\nu^{2} + 31038\nu + 11466 ) / 681$$ (406*v^5 - 966*v^4 + 1288*v^3 + 9412*v^2 + 31038*v + 11466) / 681
 $$\nu$$ $$=$$ $$( \beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + 2\beta _1 + 3 ) / 8$$ (b5 - 2*b4 - b3 - b2 + 2*b1 + 3) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 7\beta_{2} ) / 2$$ (b5 - 7*b2) / 2 $$\nu^{3}$$ $$=$$ $$( 13\beta_{5} - 18\beta_{4} + 9\beta_{3} - 49\beta_{2} - 26\beta _1 - 107 ) / 8$$ (13*b5 - 18*b4 + 9*b3 - 49*b2 - 26*b1 - 107) / 8 $$\nu^{4}$$ $$=$$ $$3\beta_{3} - 17\beta _1 - 92$$ 3*b3 - 17*b1 - 92 $$\nu^{5}$$ $$=$$ $$( -197\beta_{5} + 210\beta_{4} + 105\beta_{3} + 881\beta_{2} - 394\beta _1 - 1867 ) / 8$$ (-197*b5 + 210*b4 + 105*b3 + 881*b2 - 394*b1 - 1867) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 −0.376763 − 0.376763i −1.42234 + 1.42234i 2.79911 + 2.79911i 2.79911 − 2.79911i −1.42234 − 1.42234i −0.376763 + 0.376763i
4.20905i 3.00000 −9.71610 11.4322i 12.6271i 11.2543i 7.22315i 9.00000 48.1187
337.2 3.73549i 3.00000 −5.95388 3.90776i 11.2065i 36.4129i 7.64325i 9.00000 −14.5974
337.3 1.52644i 3.00000 5.66998 19.3400i 4.57932i 4.84136i 20.8664i 9.00000 −29.5213
337.4 1.52644i 3.00000 5.66998 19.3400i 4.57932i 4.84136i 20.8664i 9.00000 −29.5213
337.5 3.73549i 3.00000 −5.95388 3.90776i 11.2065i 36.4129i 7.64325i 9.00000 −14.5974
337.6 4.20905i 3.00000 −9.71610 11.4322i 12.6271i 11.2543i 7.22315i 9.00000 48.1187
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.4.b.g 6
13.b even 2 1 inner 507.4.b.g 6
13.d odd 4 1 39.4.a.c 3
13.d odd 4 1 507.4.a.h 3
39.f even 4 1 117.4.a.f 3
39.f even 4 1 1521.4.a.u 3
52.f even 4 1 624.4.a.t 3
65.g odd 4 1 975.4.a.l 3
91.i even 4 1 1911.4.a.k 3
104.j odd 4 1 2496.4.a.bl 3
104.m even 4 1 2496.4.a.bp 3
156.l odd 4 1 1872.4.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.c 3 13.d odd 4 1
117.4.a.f 3 39.f even 4 1
507.4.a.h 3 13.d odd 4 1
507.4.b.g 6 1.a even 1 1 trivial
507.4.b.g 6 13.b even 2 1 inner
624.4.a.t 3 52.f even 4 1
975.4.a.l 3 65.g odd 4 1
1521.4.a.u 3 39.f even 4 1
1872.4.a.bk 3 156.l odd 4 1
1911.4.a.k 3 91.i even 4 1
2496.4.a.bl 3 104.j odd 4 1
2496.4.a.bp 3 104.m even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{6} + 34T_{2}^{4} + 321T_{2}^{2} + 576$$ T2^6 + 34*T2^4 + 321*T2^2 + 576 $$T_{5}^{6} + 520T_{5}^{4} + 56592T_{5}^{2} + 746496$$ T5^6 + 520*T5^4 + 56592*T5^2 + 746496

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 34 T^{4} + 321 T^{2} + \cdots + 576$$
$3$ $$(T - 3)^{6}$$
$5$ $$T^{6} + 520 T^{4} + 56592 T^{2} + \cdots + 746496$$
$7$ $$T^{6} + 1476 T^{4} + \cdots + 3936256$$
$11$ $$T^{6} + 4768 T^{4} + \cdots + 920272896$$
$13$ $$T^{6}$$
$17$ $$(T^{3} - 146 T^{2} + 6060 T - 71256)^{2}$$
$19$ $$T^{6} + 37700 T^{4} + \cdots + 607801107456$$
$23$ $$(T^{3} - 48 T^{2} - 20928 T - 534528)^{2}$$
$29$ $$(T^{3} + 2 T^{2} - 10116 T - 199176)^{2}$$
$31$ $$T^{6} + 126276 T^{4} + \cdots + 51800378773504$$
$37$ $$T^{6} + 213804 T^{4} + \cdots + 60188177674816$$
$41$ $$T^{6} + \cdots + 166921852190976$$
$43$ $$(T^{3} - 260 T^{2} - 38096 T + 3663168)^{2}$$
$47$ $$T^{6} + \cdots + 327701519990784$$
$53$ $$(T^{3} + 678 T^{2} - 42228 T - 1471608)^{2}$$
$59$ $$T^{6} + 1284720 T^{4} + \cdots + 18\!\cdots\!64$$
$61$ $$(T^{3} - 230 T^{2} - 44452 T + 6279512)^{2}$$
$67$ $$T^{6} + 823908 T^{4} + \cdots + 18155234722816$$
$71$ $$T^{6} + 923856 T^{4} + \cdots + 49\!\cdots\!56$$
$73$ $$T^{6} + \cdots + 518953549424704$$
$79$ $$(T^{3} + 24 T^{2} - 78336 T + 7757824)^{2}$$
$83$ $$T^{6} + \cdots + 194992402305024$$
$89$ $$T^{6} + \cdots + 907810150892544$$
$97$ $$T^{6} + 3579564 T^{4} + \cdots + 14\!\cdots\!04$$