# Properties

 Label 507.2.j.f 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: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}$$ 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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{2}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}^{2}$$ -v^7 + v^5 + v^3 - v^2 $$\beta_{3}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{2} + \zeta_{24}$$ -v^7 + v^6 - v^2 + v $$\beta_{4}$$ $$=$$ $$-2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24}$$ -2*v^5 + 2*v^3 + 2*v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2}$$ -v^7 + v^5 + v^3 + v^2 $$\beta_{6}$$ $$=$$ $$2\zeta_{24}^{7} + 2\zeta_{24}$$ 2*v^7 + 2*v $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + \zeta_{24}$$ -v^7 - v^6 + v^2 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{3} ) / 4$$ (b7 + b6 + b3) / 4 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{5} - \beta_{2} ) / 2$$ (b5 - b2) / 2 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4$$ (-b7 + b5 + b4 - b3 + b2) / 4 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{5}$$ $$=$$ $$( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} ) / 4$$ (b6 + b5 - b4 + b2) / 4 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} ) / 2$$ (-b7 + b5 + b3 - b2) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} + \beta_{6} - \beta_{3} ) / 4$$ (-b7 + b6 - b3) / 4

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
316.1
 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i
−2.09077 1.20711i −0.500000 + 0.866025i 1.91421 + 3.31552i 2.82843i 2.09077 1.20711i −2.44949 + 1.41421i 4.41421i −0.500000 0.866025i −3.41421 + 5.91359i
316.2 −0.358719 0.207107i −0.500000 + 0.866025i −0.914214 1.58346i 2.82843i 0.358719 0.207107i −2.44949 + 1.41421i 1.58579i −0.500000 0.866025i −0.585786 + 1.01461i
316.3 0.358719 + 0.207107i −0.500000 + 0.866025i −0.914214 1.58346i 2.82843i −0.358719 + 0.207107i 2.44949 1.41421i 1.58579i −0.500000 0.866025i −0.585786 + 1.01461i
316.4 2.09077 + 1.20711i −0.500000 + 0.866025i 1.91421 + 3.31552i 2.82843i −2.09077 + 1.20711i 2.44949 1.41421i 4.41421i −0.500000 0.866025i −3.41421 + 5.91359i
361.1 −2.09077 + 1.20711i −0.500000 0.866025i 1.91421 3.31552i 2.82843i 2.09077 + 1.20711i −2.44949 1.41421i 4.41421i −0.500000 + 0.866025i −3.41421 5.91359i
361.2 −0.358719 + 0.207107i −0.500000 0.866025i −0.914214 + 1.58346i 2.82843i 0.358719 + 0.207107i −2.44949 1.41421i 1.58579i −0.500000 + 0.866025i −0.585786 1.01461i
361.3 0.358719 0.207107i −0.500000 0.866025i −0.914214 + 1.58346i 2.82843i −0.358719 0.207107i 2.44949 + 1.41421i 1.58579i −0.500000 + 0.866025i −0.585786 1.01461i
361.4 2.09077 1.20711i −0.500000 0.866025i 1.91421 3.31552i 2.82843i −2.09077 1.20711i 2.44949 + 1.41421i 4.41421i −0.500000 + 0.866025i −3.41421 5.91359i
 $$n$$: 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.f 8
13.b even 2 1 inner 507.2.j.f 8
13.c even 3 1 507.2.b.e 4
13.c even 3 1 inner 507.2.j.f 8
13.d odd 4 1 507.2.e.d 4
13.d odd 4 1 507.2.e.h 4
13.e even 6 1 507.2.b.e 4
13.e even 6 1 inner 507.2.j.f 8
13.f odd 12 1 39.2.a.b 2
13.f odd 12 1 507.2.a.h 2
13.f odd 12 1 507.2.e.d 4
13.f odd 12 1 507.2.e.h 4
39.h odd 6 1 1521.2.b.j 4
39.i odd 6 1 1521.2.b.j 4
39.k even 12 1 117.2.a.c 2
39.k even 12 1 1521.2.a.f 2
52.l even 12 1 624.2.a.k 2
52.l even 12 1 8112.2.a.bm 2
65.o even 12 1 975.2.c.h 4
65.s odd 12 1 975.2.a.l 2
65.t even 12 1 975.2.c.h 4
91.bc even 12 1 1911.2.a.h 2
104.u even 12 1 2496.2.a.bi 2
104.x odd 12 1 2496.2.a.bf 2
117.w odd 12 1 1053.2.e.m 4
117.x even 12 1 1053.2.e.e 4
117.bb odd 12 1 1053.2.e.m 4
117.bc even 12 1 1053.2.e.e 4
143.o even 12 1 4719.2.a.p 2
156.v odd 12 1 1872.2.a.w 2
195.bc odd 12 1 2925.2.c.u 4
195.bh even 12 1 2925.2.a.v 2
195.bn odd 12 1 2925.2.c.u 4
273.ca odd 12 1 5733.2.a.u 2
312.bo even 12 1 7488.2.a.cl 2
312.bq odd 12 1 7488.2.a.co 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 13.f odd 12 1
117.2.a.c 2 39.k even 12 1
507.2.a.h 2 13.f odd 12 1
507.2.b.e 4 13.c even 3 1
507.2.b.e 4 13.e even 6 1
507.2.e.d 4 13.d odd 4 1
507.2.e.d 4 13.f odd 12 1
507.2.e.h 4 13.d odd 4 1
507.2.e.h 4 13.f odd 12 1
507.2.j.f 8 1.a even 1 1 trivial
507.2.j.f 8 13.b even 2 1 inner
507.2.j.f 8 13.c even 3 1 inner
507.2.j.f 8 13.e even 6 1 inner
624.2.a.k 2 52.l even 12 1
975.2.a.l 2 65.s odd 12 1
975.2.c.h 4 65.o even 12 1
975.2.c.h 4 65.t even 12 1
1053.2.e.e 4 117.x even 12 1
1053.2.e.e 4 117.bc even 12 1
1053.2.e.m 4 117.w odd 12 1
1053.2.e.m 4 117.bb odd 12 1
1521.2.a.f 2 39.k even 12 1
1521.2.b.j 4 39.h odd 6 1
1521.2.b.j 4 39.i odd 6 1
1872.2.a.w 2 156.v odd 12 1
1911.2.a.h 2 91.bc even 12 1
2496.2.a.bf 2 104.x odd 12 1
2496.2.a.bi 2 104.u even 12 1
2925.2.a.v 2 195.bh even 12 1
2925.2.c.u 4 195.bc odd 12 1
2925.2.c.u 4 195.bn odd 12 1
4719.2.a.p 2 143.o even 12 1
5733.2.a.u 2 273.ca odd 12 1
7488.2.a.cl 2 312.bo even 12 1
7488.2.a.co 2 312.bq odd 12 1
8112.2.a.bm 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1$$ T2^8 - 6*T2^6 + 35*T2^4 - 6*T2^2 + 1 $$T_{5}^{2} + 8$$ T5^2 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 6 T^{6} + 35 T^{4} - 6 T^{2} + \cdots + 1$$
$3$ $$(T^{2} + T + 1)^{4}$$
$5$ $$(T^{2} + 8)^{4}$$
$7$ $$(T^{4} - 8 T^{2} + 64)^{2}$$
$11$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$13$ $$T^{8}$$
$17$ $$(T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784)^{2}$$
$19$ $$(T^{4} - 8 T^{2} + 64)^{2}$$
$23$ $$(T^{2} + 4 T + 16)^{4}$$
$29$ $$(T^{2} + 2 T + 4)^{4}$$
$31$ $$(T^{4} + 48 T^{2} + 64)^{2}$$
$37$ $$T^{8} - 72 T^{6} + 4400 T^{4} + \cdots + 614656$$
$41$ $$T^{8} - 144 T^{6} + 17600 T^{4} + \cdots + 9834496$$
$43$ $$(T^{4} - 8 T^{3} + 80 T^{2} + 128 T + 256)^{2}$$
$47$ $$(T^{4} + 136 T^{2} + 16)^{2}$$
$53$ $$(T + 2)^{8}$$
$59$ $$T^{8} - 72 T^{6} + 4400 T^{4} + \cdots + 614656$$
$61$ $$(T^{4} + 4 T^{3} + 140 T^{2} - 496 T + 15376)^{2}$$
$67$ $$T^{8} - 48 T^{6} + 2240 T^{4} + \cdots + 4096$$
$71$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$73$ $$(T^{4} + 136 T^{2} + 16)^{2}$$
$79$ $$(T^{2} - 128)^{4}$$
$83$ $$(T^{4} + 72 T^{2} + 784)^{2}$$
$89$ $$T^{8} - 304 T^{6} + \cdots + 342102016$$
$97$ $$T^{8} - 72 T^{6} + 4400 T^{4} + \cdots + 614656$$