# Properties

 Label 507.2.j.g Level $507$ Weight $2$ Character orbit 507.j Analytic conductor $4.048$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.1731891456.1 Defining polynomial: $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ x^8 - 9*x^6 + 65*x^4 - 144*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + 2) q^{4} + ( - 2 \beta_{7} - \beta_{6}) q^{5} + (\beta_{6} - \beta_1) q^{6} + ( - \beta_{7} + \beta_{6} - \beta_{3} - \beta_1) q^{7} + ( - 4 \beta_{7} + \beta_{6}) q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + b2 * q^3 + (b5 - b4 + 3*b2 + 2) * q^4 + (-2*b7 - b6) * q^5 + (b6 - b1) * q^6 + (-b7 + b6 - b3 - b1) * q^7 + (-4*b7 + b6) * q^8 + (-b2 - 1) * q^9 $$q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + 2) q^{4} + ( - 2 \beta_{7} - \beta_{6}) q^{5} + (\beta_{6} - \beta_1) q^{6} + ( - \beta_{7} + \beta_{6} - \beta_{3} - \beta_1) q^{7} + ( - 4 \beta_{7} + \beta_{6}) q^{8} + ( - \beta_{2} - 1) q^{9} + (\beta_{5} - 3 \beta_{2}) q^{10} + 2 \beta_{3} q^{11} + (\beta_{4} - 2) q^{12} + (2 \beta_{4} - 4) q^{14} + ( - 2 \beta_{3} + \beta_1) q^{15} + (3 \beta_{5} + 3 \beta_{2}) q^{16} + (\beta_{5} - \beta_{4} + \beta_{2}) q^{17} - \beta_{6} q^{18} + ( - 4 \beta_{7} - 2 \beta_{6} - 4 \beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{6} + \beta_1) q^{20} + (\beta_{7} - \beta_{6}) q^{21} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{22} - 2 \beta_{2} q^{23} + ( - 4 \beta_{3} - \beta_1) q^{24} + ( - 3 \beta_{4} - 3) q^{25} + q^{27} + ( - 6 \beta_{3} - 4 \beta_1) q^{28} + ( - 3 \beta_{5} - \beta_{2}) q^{29} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} + 4) q^{30} + (\beta_{7} + \beta_{6}) q^{31} + ( - 4 \beta_{7} + \beta_{6} - 4 \beta_{3} - \beta_1) q^{32} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{33} + ( - 4 \beta_{7} + \beta_{6}) q^{34} + (2 \beta_{2} + 2) q^{35} + ( - \beta_{5} - 3 \beta_{2}) q^{36} + ( - 6 \beta_{3} + \beta_1) q^{37} + (2 \beta_{4} + 8) q^{38} + ( - 3 \beta_{4} - 4) q^{40} + \beta_1 q^{41} + ( - 2 \beta_{5} - 6 \beta_{2}) q^{42} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} + 3) q^{43} + ( - 4 \beta_{7} + 2 \beta_{6}) q^{44} + (2 \beta_{7} + \beta_{6} + 2 \beta_{3} - \beta_1) q^{45} + ( - 2 \beta_{6} + 2 \beta_1) q^{46} + ( - 2 \beta_{7} - 4 \beta_{6}) q^{47} + ( - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{2}) q^{48} + ( - 3 \beta_{5} - \beta_{2}) q^{49} + 12 \beta_{3} q^{50} + \beta_{4} q^{51} + ( - 3 \beta_{4} + 4) q^{53} + \beta_1 q^{54} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{55} + ( - 6 \beta_{5} + 6 \beta_{4} - 14 \beta_{2} - 8) q^{56} + (4 \beta_{7} + 2 \beta_{6}) q^{57} + (12 \beta_{7} - \beta_{6} + 12 \beta_{3} + \beta_1) q^{58} + ( - 6 \beta_{7} + 2 \beta_{6} - 6 \beta_{3} - 2 \beta_1) q^{59} + \beta_{6} q^{60} + ( - 2 \beta_{5} + 2 \beta_{4} - 9 \beta_{2} - 7) q^{61} + 4 \beta_{2} q^{62} + (\beta_{3} + \beta_1) q^{63} + ( - \beta_{4} - 4) q^{64} + 2 \beta_{4} q^{66} + (3 \beta_{3} - \beta_1) q^{67} + (3 \beta_{5} + 7 \beta_{2}) q^{68} + (2 \beta_{2} + 2) q^{69} + 2 \beta_{6} q^{70} + ( - 14 \beta_{7} - 14 \beta_{3}) q^{71} + (4 \beta_{7} - \beta_{6} + 4 \beta_{3} + \beta_1) q^{72} + (7 \beta_{7} + 2 \beta_{6}) q^{73} + ( - 5 \beta_{5} + 5 \beta_{4} - \beta_{2} + 4) q^{74} + 3 \beta_{5} q^{75} + 2 \beta_1 q^{76} + (2 \beta_{4} - 2) q^{77} + ( - \beta_{4} + 7) q^{79} + (12 \beta_{3} - 3 \beta_1) q^{80} + \beta_{2} q^{81} + (\beta_{5} - \beta_{4} + 5 \beta_{2} + 4) q^{82} + ( - 4 \beta_{7} + 2 \beta_{6}) q^{83} + (6 \beta_{7} - 4 \beta_{6} + 6 \beta_{3} + 4 \beta_1) q^{84} + (4 \beta_{7} + \beta_{6} + 4 \beta_{3} - \beta_1) q^{85} + (4 \beta_{7} + 2 \beta_{6}) q^{86} + (3 \beta_{5} - 3 \beta_{4} + \beta_{2} - 2) q^{87} + (2 \beta_{5} + 10 \beta_{2}) q^{88} + ( - 8 \beta_{3} - 2 \beta_1) q^{89} + ( - \beta_{4} - 4) q^{90} + ( - 2 \beta_{4} + 4) q^{92} + (\beta_{3} - \beta_1) q^{93} + ( - 2 \beta_{5} - 18 \beta_{2}) q^{94} + (6 \beta_{5} - 6 \beta_{4} - 10 \beta_{2} - 16) q^{95} + (4 \beta_{7} - \beta_{6}) q^{96} + (7 \beta_{7} + \beta_{6} + 7 \beta_{3} - \beta_1) q^{97} + (12 \beta_{7} - \beta_{6} + 12 \beta_{3} + \beta_1) q^{98} + 2 \beta_{7} q^{99}+O(q^{100})$$ q + b1 * q^2 + b2 * q^3 + (b5 - b4 + 3*b2 + 2) * q^4 + (-2*b7 - b6) * q^5 + (b6 - b1) * q^6 + (-b7 + b6 - b3 - b1) * q^7 + (-4*b7 + b6) * q^8 + (-b2 - 1) * q^9 + (b5 - 3*b2) * q^10 + 2*b3 * q^11 + (b4 - 2) * q^12 + (2*b4 - 4) * q^14 + (-2*b3 + b1) * q^15 + (3*b5 + 3*b2) * q^16 + (b5 - b4 + b2) * q^17 - b6 * q^18 + (-4*b7 - 2*b6 - 4*b3 + 2*b1) * q^19 + (-b6 + b1) * q^20 + (b7 - b6) * q^21 + (2*b5 - 2*b4 + 2*b2) * q^22 - 2*b2 * q^23 + (-4*b3 - b1) * q^24 + (-3*b4 - 3) * q^25 + q^27 + (-6*b3 - 4*b1) * q^28 + (-3*b5 - b2) * q^29 + (-b5 + b4 + 3*b2 + 4) * q^30 + (b7 + b6) * q^31 + (-4*b7 + b6 - 4*b3 - b1) * q^32 + (-2*b7 - 2*b3) * q^33 + (-4*b7 + b6) * q^34 + (2*b2 + 2) * q^35 + (-b5 - 3*b2) * q^36 + (-6*b3 + b1) * q^37 + (2*b4 + 8) * q^38 + (-3*b4 - 4) * q^40 + b1 * q^41 + (-2*b5 - 6*b2) * q^42 + (-b5 + b4 + 2*b2 + 3) * q^43 + (-4*b7 + 2*b6) * q^44 + (2*b7 + b6 + 2*b3 - b1) * q^45 + (-2*b6 + 2*b1) * q^46 + (-2*b7 - 4*b6) * q^47 + (-3*b5 + 3*b4 - 3*b2) * q^48 + (-3*b5 - b2) * q^49 + 12*b3 * q^50 + b4 * q^51 + (-3*b4 + 4) * q^53 + b1 * q^54 + (-2*b5 + 2*b2) * q^55 + (-6*b5 + 6*b4 - 14*b2 - 8) * q^56 + (4*b7 + 2*b6) * q^57 + (12*b7 - b6 + 12*b3 + b1) * q^58 + (-6*b7 + 2*b6 - 6*b3 - 2*b1) * q^59 + b6 * q^60 + (-2*b5 + 2*b4 - 9*b2 - 7) * q^61 + 4*b2 * q^62 + (b3 + b1) * q^63 + (-b4 - 4) * q^64 + 2*b4 * q^66 + (3*b3 - b1) * q^67 + (3*b5 + 7*b2) * q^68 + (2*b2 + 2) * q^69 + 2*b6 * q^70 + (-14*b7 - 14*b3) * q^71 + (4*b7 - b6 + 4*b3 + b1) * q^72 + (7*b7 + 2*b6) * q^73 + (-5*b5 + 5*b4 - b2 + 4) * q^74 + 3*b5 * q^75 + 2*b1 * q^76 + (2*b4 - 2) * q^77 + (-b4 + 7) * q^79 + (12*b3 - 3*b1) * q^80 + b2 * q^81 + (b5 - b4 + 5*b2 + 4) * q^82 + (-4*b7 + 2*b6) * q^83 + (6*b7 - 4*b6 + 6*b3 + 4*b1) * q^84 + (4*b7 + b6 + 4*b3 - b1) * q^85 + (4*b7 + 2*b6) * q^86 + (3*b5 - 3*b4 + b2 - 2) * q^87 + (2*b5 + 10*b2) * q^88 + (-8*b3 - 2*b1) * q^89 + (-b4 - 4) * q^90 + (-2*b4 + 4) * q^92 + (b3 - b1) * q^93 + (-2*b5 - 18*b2) * q^94 + (6*b5 - 6*b4 - 10*b2 - 16) * q^95 + (4*b7 - b6) * q^96 + (7*b7 + b6 + 7*b3 - b1) * q^97 + (12*b7 - b6 + 12*b3 + b1) * q^98 + 2*b7 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{3} + 10 q^{4} - 4 q^{9}+O(q^{10})$$ 8 * q - 4 * q^3 + 10 * q^4 - 4 * q^9 $$8 q - 4 q^{3} + 10 q^{4} - 4 q^{9} + 14 q^{10} - 20 q^{12} - 40 q^{14} - 6 q^{16} + 2 q^{17} + 4 q^{22} + 8 q^{23} - 12 q^{25} + 8 q^{27} - 2 q^{29} + 14 q^{30} + 8 q^{35} + 10 q^{36} + 56 q^{38} - 20 q^{40} + 20 q^{42} + 10 q^{43} - 6 q^{48} - 2 q^{49} - 4 q^{51} + 44 q^{53} - 12 q^{55} - 44 q^{56} - 32 q^{61} - 16 q^{62} - 28 q^{64} - 8 q^{66} - 22 q^{68} + 8 q^{69} + 6 q^{74} + 6 q^{75} - 24 q^{77} + 60 q^{79} - 4 q^{81} + 18 q^{82} - 2 q^{87} - 36 q^{88} - 28 q^{90} + 40 q^{92} + 68 q^{94} - 52 q^{95}+O(q^{100})$$ 8 * q - 4 * q^3 + 10 * q^4 - 4 * q^9 + 14 * q^10 - 20 * q^12 - 40 * q^14 - 6 * q^16 + 2 * q^17 + 4 * q^22 + 8 * q^23 - 12 * q^25 + 8 * q^27 - 2 * q^29 + 14 * q^30 + 8 * q^35 + 10 * q^36 + 56 * q^38 - 20 * q^40 + 20 * q^42 + 10 * q^43 - 6 * q^48 - 2 * q^49 - 4 * q^51 + 44 * q^53 - 12 * q^55 - 44 * q^56 - 32 * q^61 - 16 * q^62 - 28 * q^64 - 8 * q^66 - 22 * q^68 + 8 * q^69 + 6 * q^74 + 6 * q^75 - 24 * q^77 + 60 * q^79 - 4 * q^81 + 18 * q^82 - 2 * q^87 - 36 * q^88 - 28 * q^90 + 40 * q^92 + 68 * q^94 - 52 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040$$ (9*v^6 - 65*v^4 + 585*v^2 - 1296) / 1040 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} - 181\nu ) / 260$$ (-v^7 - 181*v) / 260 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 116 ) / 65$$ (v^6 + 116) / 65 $$\beta_{5}$$ $$=$$ $$( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040$$ (-29*v^6 + 325*v^4 - 1885*v^2 + 4176) / 1040 $$\beta_{6}$$ $$=$$ $$( 9\nu^{7} - 65\nu^{5} + 585\nu^{3} - 256\nu ) / 1040$$ (9*v^7 - 65*v^5 + 585*v^3 - 256*v) / 1040 $$\beta_{7}$$ $$=$$ $$( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 832$$ (9*v^7 - 65*v^5 + 377*v^3 - 256*v) / 832
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + 5\beta_{2} + 4$$ b5 - b4 + 5*b2 + 4 $$\nu^{3}$$ $$=$$ $$-4\beta_{7} + 5\beta_{6}$$ -4*b7 + 5*b6 $$\nu^{4}$$ $$=$$ $$9\beta_{5} + 29\beta_{2}$$ 9*b5 + 29*b2 $$\nu^{5}$$ $$=$$ $$-36\beta_{7} + 29\beta_{6} - 36\beta_{3} - 29\beta_1$$ -36*b7 + 29*b6 - 36*b3 - 29*b1 $$\nu^{6}$$ $$=$$ $$65\beta_{4} - 116$$ 65*b4 - 116 $$\nu^{7}$$ $$=$$ $$-260\beta_{3} - 181\beta_1$$ -260*b3 - 181*b1

## Character values

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

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
316.1
 −2.21837 − 1.28078i −1.35234 − 0.780776i 1.35234 + 0.780776i 2.21837 + 1.28078i −2.21837 + 1.28078i −1.35234 + 0.780776i 1.35234 − 0.780776i 2.21837 − 1.28078i
−2.21837 1.28078i −0.500000 + 0.866025i 2.28078 + 3.95042i 0.561553i 2.21837 1.28078i 3.08440 1.78078i 6.56155i −0.500000 0.866025i 0.719224 1.24573i
316.2 −1.35234 0.780776i −0.500000 + 0.866025i 0.219224 + 0.379706i 3.56155i 1.35234 0.780776i 0.486319 0.280776i 2.43845i −0.500000 0.866025i 2.78078 4.81645i
316.3 1.35234 + 0.780776i −0.500000 + 0.866025i 0.219224 + 0.379706i 3.56155i −1.35234 + 0.780776i −0.486319 + 0.280776i 2.43845i −0.500000 0.866025i 2.78078 4.81645i
316.4 2.21837 + 1.28078i −0.500000 + 0.866025i 2.28078 + 3.95042i 0.561553i −2.21837 + 1.28078i −3.08440 + 1.78078i 6.56155i −0.500000 0.866025i 0.719224 1.24573i
361.1 −2.21837 + 1.28078i −0.500000 0.866025i 2.28078 3.95042i 0.561553i 2.21837 + 1.28078i 3.08440 + 1.78078i 6.56155i −0.500000 + 0.866025i 0.719224 + 1.24573i
361.2 −1.35234 + 0.780776i −0.500000 0.866025i 0.219224 0.379706i 3.56155i 1.35234 + 0.780776i 0.486319 + 0.280776i 2.43845i −0.500000 + 0.866025i 2.78078 + 4.81645i
361.3 1.35234 0.780776i −0.500000 0.866025i 0.219224 0.379706i 3.56155i −1.35234 0.780776i −0.486319 0.280776i 2.43845i −0.500000 + 0.866025i 2.78078 + 4.81645i
361.4 2.21837 1.28078i −0.500000 0.866025i 2.28078 3.95042i 0.561553i −2.21837 1.28078i −3.08440 1.78078i 6.56155i −0.500000 + 0.866025i 0.719224 + 1.24573i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.g 8
13.b even 2 1 inner 507.2.j.g 8
13.c even 3 1 507.2.b.d 4
13.c even 3 1 inner 507.2.j.g 8
13.d odd 4 1 39.2.e.b 4
13.d odd 4 1 507.2.e.g 4
13.e even 6 1 507.2.b.d 4
13.e even 6 1 inner 507.2.j.g 8
13.f odd 12 1 39.2.e.b 4
13.f odd 12 1 507.2.a.d 2
13.f odd 12 1 507.2.a.g 2
13.f odd 12 1 507.2.e.g 4
39.f even 4 1 117.2.g.c 4
39.h odd 6 1 1521.2.b.h 4
39.i odd 6 1 1521.2.b.h 4
39.k even 12 1 117.2.g.c 4
39.k even 12 1 1521.2.a.g 2
39.k even 12 1 1521.2.a.m 2
52.f even 4 1 624.2.q.h 4
52.l even 12 1 624.2.q.h 4
52.l even 12 1 8112.2.a.bk 2
52.l even 12 1 8112.2.a.bo 2
65.f even 4 1 975.2.bb.i 8
65.g odd 4 1 975.2.i.k 4
65.k even 4 1 975.2.bb.i 8
65.o even 12 1 975.2.bb.i 8
65.s odd 12 1 975.2.i.k 4
65.t even 12 1 975.2.bb.i 8
156.l odd 4 1 1872.2.t.r 4
156.v odd 12 1 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 13.d odd 4 1
39.2.e.b 4 13.f odd 12 1
117.2.g.c 4 39.f even 4 1
117.2.g.c 4 39.k even 12 1
507.2.a.d 2 13.f odd 12 1
507.2.a.g 2 13.f odd 12 1
507.2.b.d 4 13.c even 3 1
507.2.b.d 4 13.e even 6 1
507.2.e.g 4 13.d odd 4 1
507.2.e.g 4 13.f odd 12 1
507.2.j.g 8 1.a even 1 1 trivial
507.2.j.g 8 13.b even 2 1 inner
507.2.j.g 8 13.c even 3 1 inner
507.2.j.g 8 13.e even 6 1 inner
624.2.q.h 4 52.f even 4 1
624.2.q.h 4 52.l even 12 1
975.2.i.k 4 65.g odd 4 1
975.2.i.k 4 65.s odd 12 1
975.2.bb.i 8 65.f even 4 1
975.2.bb.i 8 65.k even 4 1
975.2.bb.i 8 65.o even 12 1
975.2.bb.i 8 65.t even 12 1
1521.2.a.g 2 39.k even 12 1
1521.2.a.m 2 39.k even 12 1
1521.2.b.h 4 39.h odd 6 1
1521.2.b.h 4 39.i odd 6 1
1872.2.t.r 4 156.l odd 4 1
1872.2.t.r 4 156.v odd 12 1
8112.2.a.bk 2 52.l even 12 1
8112.2.a.bo 2 52.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{8} - 9T_{2}^{6} + 65T_{2}^{4} - 144T_{2}^{2} + 256$$ T2^8 - 9*T2^6 + 65*T2^4 - 144*T2^2 + 256 $$T_{5}^{4} + 13T_{5}^{2} + 4$$ T5^4 + 13*T5^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 9 T^{6} + 65 T^{4} - 144 T^{2} + \cdots + 256$$
$3$ $$(T^{2} + T + 1)^{4}$$
$5$ $$(T^{4} + 13 T^{2} + 4)^{2}$$
$7$ $$T^{8} - 13 T^{6} + 165 T^{4} + \cdots + 16$$
$11$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$13$ $$T^{8}$$
$17$ $$(T^{4} - T^{3} + 5 T^{2} + 4 T + 16)^{2}$$
$19$ $$T^{8} - 52 T^{6} + 2640 T^{4} + \cdots + 4096$$
$23$ $$(T^{2} - 2 T + 4)^{4}$$
$29$ $$(T^{4} + T^{3} + 39 T^{2} - 38 T + 1444)^{2}$$
$31$ $$(T^{4} + 9 T^{2} + 16)^{2}$$
$37$ $$T^{8} - 69 T^{6} + 4085 T^{4} + \cdots + 456976$$
$41$ $$T^{8} - 9 T^{6} + 65 T^{4} - 144 T^{2} + \cdots + 256$$
$43$ $$(T^{4} - 5 T^{3} + 23 T^{2} - 10 T + 4)^{2}$$
$47$ $$(T^{2} + 68)^{4}$$
$53$ $$(T^{2} - 11 T - 8)^{4}$$
$59$ $$T^{8} - 132 T^{6} + 16400 T^{4} + \cdots + 1048576$$
$61$ $$(T^{4} + 16 T^{3} + 209 T^{2} + 752 T + 2209)^{2}$$
$67$ $$T^{8} - 21 T^{6} + 437 T^{4} + \cdots + 16$$
$71$ $$(T^{4} - 196 T^{2} + 38416)^{2}$$
$73$ $$(T^{4} + 106 T^{2} + 361)^{2}$$
$79$ $$(T^{2} - 15 T + 52)^{4}$$
$83$ $$(T^{4} + 84 T^{2} + 64)^{2}$$
$89$ $$T^{8} - 196 T^{6} + \cdots + 16777216$$
$97$ $$T^{8} - 93 T^{6} + 7205 T^{4} + \cdots + 2085136$$