# Properties

 Label 72.4.d.d Level $72$ Weight $4$ Character orbit 72.d Analytic conductor $4.248$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 72.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.24813752041$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.8248384.1 Defining polynomial: $$x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64$$ x^6 + x^4 - 12*x^3 + 4*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{5} + 3) q^{4} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 6) q^{7} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + 4 \beta_1 + 14) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b5 + 3) * q^4 + (b4 + b3 + b2 - b1) * q^5 + (-b5 - b3 + 2*b2 + 3*b1 + 6) * q^7 + (-b5 - 2*b4 + b2 + 4*b1 + 14) * q^8 $$q + \beta_1 q^{2} + (\beta_{5} + 3) q^{4} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 6) q^{7} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + 4 \beta_1 + 14) q^{8} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 10) q^{10} + (4 \beta_{5} - 4 \beta_{3} - 4 \beta_1) q^{11} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2}) q^{13} + (2 \beta_{5} + 4 \beta_{4} - 6 \beta_{2} + 2 \beta_1 + 16) q^{14} + (4 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 12 \beta_1 + 14) q^{16} + ( - 8 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} - 12 \beta_1 - 14) q^{17} + (4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} - 12 \beta_1) q^{19} + (12 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 8 \beta_1 - 6) q^{20} + ( - 8 \beta_{5} - 16 \beta_{4} - 8 \beta_{2} + 32) q^{22} + ( - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 6 \beta_1 - 52) q^{23} + ( - 8 \beta_{5} - 8 \beta_{2} - 48 \beta_1 - 39) q^{25} + ( - 16 \beta_{4} + 4 \beta_{3} - 8 \beta_{2} - 12) q^{26} + (6 \beta_{5} - 16 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} + 24 \beta_1 - 46) q^{28} + (4 \beta_{5} - 11 \beta_{4} - 7 \beta_{3} - 3 \beta_{2} - \beta_1) q^{29} + (9 \beta_{5} + \beta_{3} + 6 \beta_{2} + 45 \beta_1 - 86) q^{31} + (10 \beta_{5} - 20 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} + 16 \beta_1 + 48) q^{32} + ( - 8 \beta_{5} + 16 \beta_{4} - 24 \beta_{2} - 30 \beta_1 - 112) q^{34} + (4 \beta_{5} + 2 \beta_{4} + 12 \beta_{3} + 16 \beta_{2} - 20 \beta_1) q^{35} + ( - 22 \beta_{5} - 20 \beta_{4} + 14 \beta_{3} - 8 \beta_{2} + 30 \beta_1) q^{37} + ( - 24 \beta_{5} + 16 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} + 124) q^{38} + (6 \beta_{5} + 20 \beta_{4} + 24 \beta_{3} + 22 \beta_{2} + 48) q^{40} + (8 \beta_{5} + 12 \beta_{3} - 28 \beta_{2} - 60 \beta_1 - 66) q^{41} + (20 \beta_{5} - 18 \beta_{4} - 28 \beta_{3} - 8 \beta_{2} - 12 \beta_1) q^{43} + (16 \beta_{5} - 32 \beta_{3} + 32 \beta_1 - 176) q^{44} + (4 \beta_{5} + 8 \beta_{4} - 12 \beta_{2} - 60 \beta_1 + 32) q^{46} + (6 \beta_{5} - 2 \beta_{3} + 12 \beta_{2} + 54 \beta_1 + 92) q^{47} + ( - 24 \beta_{5} - 8 \beta_{3} - 72 \beta_1 + 77) q^{49} + ( - 32 \beta_{5} - 39 \beta_1 - 352) q^{50} + (8 \beta_{5} - 8 \beta_{4} - 32 \beta_{3} + 20 \beta_{2} - 52) q^{52} + ( - 20 \beta_{5} + 51 \beta_{4} - \beta_{3} - 21 \beta_{2} + 41 \beta_1) q^{53} + (32 \beta_{5} + 8 \beta_{3} + 8 \beta_{2} + 120 \beta_1 + 224) q^{55} + (10 \beta_{5} + 20 \beta_{4} - 32 \beta_{3} + 22 \beta_{2} - 40 \beta_1 + 308) q^{56} + ( - 2 \beta_{5} - 28 \beta_{4} - 22 \beta_{3} - 14 \beta_{2} + 18) q^{58} + ( - 16 \beta_{5} - 18 \beta_{4} - 16 \beta_{2} + 32 \beta_1) q^{59} + ( - 18 \beta_{5} + 80 \beta_{4} + 6 \beta_{3} - 12 \beta_{2} + 30 \beta_1) q^{61} + (30 \beta_{5} - 4 \beta_{4} + 6 \beta_{2} - 82 \beta_1 + 336) q^{62} + (12 \beta_{5} - 16 \beta_{4} - 40 \beta_{3} + 40 \beta_{2} + 72 \beta_1 + 180) q^{64} + (8 \beta_{5} - 4 \beta_{3} + 20 \beta_{2} + 84 \beta_1 + 328) q^{65} + (58 \beta_{4} + 48 \beta_{3} + 48 \beta_{2} - 48 \beta_1) q^{67} + (2 \beta_{5} - 32 \beta_{4} + 32 \beta_{3} + 16 \beta_{2} - 96 \beta_1 - 522) q^{68} + ( - 40 \beta_{5} + 48 \beta_{4} + 4 \beta_{3} + 24 \beta_{2} + 220) q^{70} + (42 \beta_{5} + 18 \beta_{3} - 12 \beta_{2} + 90 \beta_1 + 324) q^{71} + (40 \beta_{5} + 24 \beta_{3} - 32 \beta_{2} + 24 \beta_1 + 170) q^{73} + (60 \beta_{5} + 56 \beta_{4} - 40 \beta_{3} + 28 \beta_{2} - 232) q^{74} + (16 \beta_{5} + 72 \beta_{4} + 32 \beta_{3} - 20 \beta_{2} + 96 \beta_1 - 260) q^{76} + ( - 40 \beta_{5} - 128 \beta_{4} + 40 \beta_{3} + 40 \beta_1) q^{77} + ( - 19 \beta_{5} - 11 \beta_{3} + 14 \beta_{2} - 15 \beta_1 - 14) q^{79} + ( - 28 \beta_{5} + 80 \beta_{4} + 40 \beta_{3} + 56 \beta_{2} + 56 \beta_1 + 380) q^{80} + ( - 40 \beta_{5} - 48 \beta_{4} + 72 \beta_{2} - 18 \beta_1 - 368) q^{82} + ( - 4 \beta_{5} + 56 \beta_{4} - 12 \beta_{3} - 16 \beta_{2} + 20 \beta_1) q^{83} + (56 \beta_{5} - 158 \beta_{4} - 70 \beta_{3} - 14 \beta_{2} - 42 \beta_1) q^{85} + ( - 24 \beta_{5} - 112 \beta_{4} - 36 \beta_{3} - 56 \beta_{2} + \cdots + 100) q^{86}+ \cdots + ( - 48 \beta_{5} + 32 \beta_{4} - 48 \beta_{2} + 45 \beta_1 - 576) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b5 + 3) * q^4 + (b4 + b3 + b2 - b1) * q^5 + (-b5 - b3 + 2*b2 + 3*b1 + 6) * q^7 + (-b5 - 2*b4 + b2 + 4*b1 + 14) * q^8 + (-2*b5 + 4*b4 + 2*b3 + 2*b2 + 10) * q^10 + (4*b5 - 4*b3 - 4*b1) * q^11 + (2*b5 + 2*b4 - 4*b3 - 2*b2) * q^13 + (2*b5 + 4*b4 - 6*b2 + 2*b1 + 16) * q^14 + (4*b5 + 4*b4 - 4*b3 - 2*b2 + 12*b1 + 14) * q^16 + (-8*b5 - 4*b3 + 4*b2 - 12*b1 - 14) * q^17 + (4*b5 + 2*b4 + 4*b3 + 8*b2 - 12*b1) * q^19 + (12*b4 + 8*b3 + 2*b2 + 8*b1 - 6) * q^20 + (-8*b5 - 16*b4 - 8*b2 + 32) * q^22 + (-2*b5 - 2*b3 + 4*b2 + 6*b1 - 52) * q^23 + (-8*b5 - 8*b2 - 48*b1 - 39) * q^25 + (-16*b4 + 4*b3 - 8*b2 - 12) * q^26 + (6*b5 - 16*b4 + 8*b3 + 8*b2 + 24*b1 - 46) * q^28 + (4*b5 - 11*b4 - 7*b3 - 3*b2 - b1) * q^29 + (9*b5 + b3 + 6*b2 + 45*b1 - 86) * q^31 + (10*b5 - 20*b4 + 8*b3 - 6*b2 + 16*b1 + 48) * q^32 + (-8*b5 + 16*b4 - 24*b2 - 30*b1 - 112) * q^34 + (4*b5 + 2*b4 + 12*b3 + 16*b2 - 20*b1) * q^35 + (-22*b5 - 20*b4 + 14*b3 - 8*b2 + 30*b1) * q^37 + (-24*b5 + 16*b4 + 4*b3 + 8*b2 + 124) * q^38 + (6*b5 + 20*b4 + 24*b3 + 22*b2 + 48) * q^40 + (8*b5 + 12*b3 - 28*b2 - 60*b1 - 66) * q^41 + (20*b5 - 18*b4 - 28*b3 - 8*b2 - 12*b1) * q^43 + (16*b5 - 32*b3 + 32*b1 - 176) * q^44 + (4*b5 + 8*b4 - 12*b2 - 60*b1 + 32) * q^46 + (6*b5 - 2*b3 + 12*b2 + 54*b1 + 92) * q^47 + (-24*b5 - 8*b3 - 72*b1 + 77) * q^49 + (-32*b5 - 39*b1 - 352) * q^50 + (8*b5 - 8*b4 - 32*b3 + 20*b2 - 52) * q^52 + (-20*b5 + 51*b4 - b3 - 21*b2 + 41*b1) * q^53 + (32*b5 + 8*b3 + 8*b2 + 120*b1 + 224) * q^55 + (10*b5 + 20*b4 - 32*b3 + 22*b2 - 40*b1 + 308) * q^56 + (-2*b5 - 28*b4 - 22*b3 - 14*b2 + 18) * q^58 + (-16*b5 - 18*b4 - 16*b2 + 32*b1) * q^59 + (-18*b5 + 80*b4 + 6*b3 - 12*b2 + 30*b1) * q^61 + (30*b5 - 4*b4 + 6*b2 - 82*b1 + 336) * q^62 + (12*b5 - 16*b4 - 40*b3 + 40*b2 + 72*b1 + 180) * q^64 + (8*b5 - 4*b3 + 20*b2 + 84*b1 + 328) * q^65 + (58*b4 + 48*b3 + 48*b2 - 48*b1) * q^67 + (2*b5 - 32*b4 + 32*b3 + 16*b2 - 96*b1 - 522) * q^68 + (-40*b5 + 48*b4 + 4*b3 + 24*b2 + 220) * q^70 + (42*b5 + 18*b3 - 12*b2 + 90*b1 + 324) * q^71 + (40*b5 + 24*b3 - 32*b2 + 24*b1 + 170) * q^73 + (60*b5 + 56*b4 - 40*b3 + 28*b2 - 232) * q^74 + (16*b5 + 72*b4 + 32*b3 - 20*b2 + 96*b1 - 260) * q^76 + (-40*b5 - 128*b4 + 40*b3 + 40*b1) * q^77 + (-19*b5 - 11*b3 + 14*b2 - 15*b1 - 14) * q^79 + (-28*b5 + 80*b4 + 40*b3 + 56*b2 + 56*b1 + 380) * q^80 + (-40*b5 - 48*b4 + 72*b2 - 18*b1 - 368) * q^82 + (-4*b5 + 56*b4 - 12*b3 - 16*b2 + 20*b1) * q^83 + (56*b5 - 158*b4 - 70*b3 - 14*b2 - 42*b1) * q^85 + (-24*b5 - 112*b4 - 36*b3 - 56*b2 + 100) * q^86 + (16*b5 - 96*b4 - 80*b2 - 192*b1 + 224) * q^88 + (-16*b5 - 8*b3 + 8*b2 - 24*b1 + 26) * q^89 + (-76*b5 - 60*b4 + 20*b3 - 56*b2 + 132*b1) * q^91 + (-52*b5 - 32*b4 + 16*b3 + 16*b2 + 48*b1 - 284) * q^92 + (36*b5 + 8*b4 - 12*b2 + 84*b1 + 384) * q^94 + (4*b5 + 20*b3 - 56*b2 - 156*b1 - 920) * q^95 + (32*b5 + 8*b3 + 8*b2 + 120*b1 - 354) * q^97 + (-48*b5 + 32*b4 - 48*b2 + 45*b1 - 576) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 76 q^{8}+O(q^{10})$$ 6 * q - 2 * q^2 + 16 * q^4 + 28 * q^7 + 76 * q^8 $$6 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 76 q^{8} + 60 q^{10} + 100 q^{14} + 56 q^{16} - 52 q^{17} - 56 q^{20} + 224 q^{22} - 328 q^{23} - 106 q^{25} - 56 q^{26} - 352 q^{28} - 636 q^{31} + 248 q^{32} - 548 q^{34} + 776 q^{38} + 232 q^{40} - 236 q^{41} - 1152 q^{44} + 328 q^{46} + 408 q^{47} + 654 q^{49} - 1970 q^{50} - 368 q^{52} + 1024 q^{55} + 1864 q^{56} + 140 q^{58} + 2108 q^{62} + 832 q^{64} + 1744 q^{65} - 2976 q^{68} + 1352 q^{70} + 1704 q^{71} + 956 q^{73} - 1568 q^{74} - 1744 q^{76} - 44 q^{79} + 2112 q^{80} - 2236 q^{82} + 760 q^{86} + 1856 q^{88} + 220 q^{89} - 1728 q^{92} + 2088 q^{94} - 5104 q^{95} - 2444 q^{97} - 3354 q^{98}+O(q^{100})$$ 6 * q - 2 * q^2 + 16 * q^4 + 28 * q^7 + 76 * q^8 + 60 * q^10 + 100 * q^14 + 56 * q^16 - 52 * q^17 - 56 * q^20 + 224 * q^22 - 328 * q^23 - 106 * q^25 - 56 * q^26 - 352 * q^28 - 636 * q^31 + 248 * q^32 - 548 * q^34 + 776 * q^38 + 232 * q^40 - 236 * q^41 - 1152 * q^44 + 328 * q^46 + 408 * q^47 + 654 * q^49 - 1970 * q^50 - 368 * q^52 + 1024 * q^55 + 1864 * q^56 + 140 * q^58 + 2108 * q^62 + 832 * q^64 + 1744 * q^65 - 2976 * q^68 + 1352 * q^70 + 1704 * q^71 + 956 * q^73 - 1568 * q^74 - 1744 * q^76 - 44 * q^79 + 2112 * q^80 - 2236 * q^82 + 760 * q^86 + 1856 * q^88 + 220 * q^89 - 1728 * q^92 + 2088 * q^94 - 5104 * q^95 - 2444 * q^97 - 3354 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} - 32 ) / 16$$ (v^5 - 2*v^4 + 5*v^3 - 6*v^2 - 32) / 16 $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 2\nu^{4} - 3\nu^{3} - 18\nu^{2} - 8\nu + 24 ) / 8$$ (v^5 + 2*v^4 - 3*v^3 - 18*v^2 - 8*v + 24) / 8 $$\beta_{3}$$ $$=$$ $$( 3\nu^{5} - 6\nu^{4} + 15\nu^{3} - 18\nu^{2} + 96\nu - 80 ) / 16$$ (3*v^5 - 6*v^4 + 15*v^3 - 18*v^2 + 96*v - 80) / 16 $$\beta_{4}$$ $$=$$ $$( -3\nu^{5} - 6\nu^{4} + 9\nu^{3} + 6\nu^{2} + 24\nu - 96 ) / 16$$ (-3*v^5 - 6*v^4 + 9*v^3 + 6*v^2 + 24*v - 96) / 16 $$\beta_{5}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} - 3\nu^{3} - 6\nu^{2} + 16\nu + 20 ) / 4$$ (v^5 - 2*v^4 - 3*v^3 - 6*v^2 + 16*v + 20) / 4
 $$\nu$$ $$=$$ $$( \beta_{3} - 3\beta _1 - 1 ) / 6$$ (b3 - 3*b1 - 1) / 6 $$\nu^{2}$$ $$=$$ $$( -2\beta_{4} - 3\beta_{2} - 3 ) / 6$$ (-2*b4 - 3*b2 - 3) / 6 $$\nu^{3}$$ $$=$$ $$( -3\beta_{5} + 2\beta_{3} + 6\beta _1 + 37 ) / 6$$ (-3*b5 + 2*b3 + 6*b1 + 37) / 6 $$\nu^{4}$$ $$=$$ $$( -2\beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} - 6\beta _1 - 7 ) / 2$$ (-2*b5 - 2*b4 + 2*b3 + b2 - 6*b1 - 7) / 2 $$\nu^{5}$$ $$=$$ $$( 3\beta_{5} - 24\beta_{4} + 2\beta_{3} - 12\beta_{2} + 30\beta _1 - 53 ) / 6$$ (3*b5 - 24*b4 + 2*b3 - 12*b2 + 30*b1 - 53) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 1.88322 − 0.673417i 1.88322 + 0.673417i −0.641412 + 1.89436i −0.641412 − 1.89436i −1.24181 + 1.56777i −1.24181 − 1.56777i
−2.55664 1.20980i 0 5.07277 + 6.18604i 0.612661i 0 −22.7441 −5.48534 21.9525i 0 0.741198 1.56635i
37.2 −2.55664 + 1.20980i 0 5.07277 6.18604i 0.612661i 0 −22.7441 −5.48534 + 21.9525i 0 0.741198 + 1.56635i
37.3 −1.25295 2.53577i 0 −4.86025 + 6.35436i 9.15486i 0 27.4175 22.2028 + 4.36281i 0 23.2146 11.4705i
37.4 −1.25295 + 2.53577i 0 −4.86025 6.35436i 9.15486i 0 27.4175 22.2028 4.36281i 0 23.2146 + 11.4705i
37.5 2.80958 0.325969i 0 7.78749 1.83167i 18.5422i 0 9.32669 21.2825 7.68472i 0 6.04419 + 52.0958i
37.6 2.80958 + 0.325969i 0 7.78749 + 1.83167i 18.5422i 0 9.32669 21.2825 + 7.68472i 0 6.04419 52.0958i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.4.d.d 6
3.b odd 2 1 24.4.d.a 6
4.b odd 2 1 288.4.d.d 6
8.b even 2 1 inner 72.4.d.d 6
8.d odd 2 1 288.4.d.d 6
12.b even 2 1 96.4.d.a 6
16.e even 4 1 2304.4.a.bt 3
16.e even 4 1 2304.4.a.bv 3
16.f odd 4 1 2304.4.a.bu 3
16.f odd 4 1 2304.4.a.bw 3
24.f even 2 1 96.4.d.a 6
24.h odd 2 1 24.4.d.a 6
48.i odd 4 1 768.4.a.r 3
48.i odd 4 1 768.4.a.s 3
48.k even 4 1 768.4.a.q 3
48.k even 4 1 768.4.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 3.b odd 2 1
24.4.d.a 6 24.h odd 2 1
72.4.d.d 6 1.a even 1 1 trivial
72.4.d.d 6 8.b even 2 1 inner
96.4.d.a 6 12.b even 2 1
96.4.d.a 6 24.f even 2 1
288.4.d.d 6 4.b odd 2 1
288.4.d.d 6 8.d odd 2 1
768.4.a.q 3 48.k even 4 1
768.4.a.r 3 48.i odd 4 1
768.4.a.s 3 48.i odd 4 1
768.4.a.t 3 48.k even 4 1
2304.4.a.bt 3 16.e even 4 1
2304.4.a.bu 3 16.f odd 4 1
2304.4.a.bv 3 16.e even 4 1
2304.4.a.bw 3 16.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 428T_{5}^{4} + 28976T_{5}^{2} + 10816$$ acting on $$S_{4}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 2 T^{5} - 6 T^{4} - 40 T^{3} + \cdots + 512$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 428 T^{4} + 28976 T^{2} + \cdots + 10816$$
$7$ $$(T^{3} - 14 T^{2} - 580 T + 5816)^{2}$$
$11$ $$T^{6} + 5632 T^{4} + \cdots + 2415919104$$
$13$ $$T^{6} + 4912 T^{4} + \cdots + 3121680384$$
$17$ $$(T^{3} + 26 T^{2} - 11124 T - 477576)^{2}$$
$19$ $$T^{6} + 22960 T^{4} + \cdots + 75488661504$$
$23$ $$(T^{3} + 164 T^{2} + 6384 T + 45504)^{2}$$
$29$ $$T^{6} + 22348 T^{4} + \cdots + 3766031424$$
$31$ $$(T^{3} + 318 T^{2} + 4476 T - 3749624)^{2}$$
$37$ $$T^{6} + 179776 T^{4} + \cdots + 6879707136$$
$41$ $$(T^{3} + 118 T^{2} - 117300 T - 19985976)^{2}$$
$43$ $$T^{6} + 229552 T^{4} + \cdots + 73984219582464$$
$47$ $$(T^{3} - 204 T^{2} - 27792 T + 1964736)^{2}$$
$53$ $$T^{6} + \cdots + 427051482970176$$
$59$ $$T^{6} + 138416 T^{4} + \cdots + 72651484205056$$
$61$ $$T^{6} + 902016 T^{4} + \cdots + 10\!\cdots\!56$$
$67$ $$T^{6} + 1054512 T^{4} + \cdots + 10\!\cdots\!84$$
$71$ $$(T^{3} - 852 T^{2} - 66960 T + 85084992)^{2}$$
$73$ $$(T^{3} - 478 T^{2} - 255956 T + 120833304)^{2}$$
$79$ $$(T^{3} + 22 T^{2} - 71524 T - 7902616)^{2}$$
$83$ $$T^{6} + 520448 T^{4} + \cdots + 14\!\cdots\!96$$
$89$ $$(T^{3} - 110 T^{2} - 41364 T - 1423656)^{2}$$
$97$ $$(T^{3} + 1222 T^{2} + 251660 T - 74802424)^{2}$$