# Properties

 Label 570.2.c.f Level $570$ Weight $2$ Character orbit 570.c Analytic conductor $4.551$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1499238400.2 Defining polynomial: $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5$$ x^8 - 4*x^7 + 16*x^6 - 34*x^5 + 59*x^4 - 66*x^3 + 54*x^2 - 26*x + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{3} - q^{4} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{6} - \beta_{3}) q^{6} + (\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} - 3 \beta_1 + 1) q^{7} - \beta_{6} q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 3) q^{9}+O(q^{10})$$ q + b6 * q^2 + (b7 + b6 + b4 - b3) * q^3 - q^4 + (-b7 - 2*b6 - b4 + b3 - b2 + 1) * q^5 + (b6 - b3) * q^6 + (b7 + 2*b6 - b4 + b2 - 3*b1 + 1) * q^7 - b6 * q^8 + (b7 + b6 + b4 - b3 - 3) * q^9 $$q + \beta_{6} q^{2} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{3} - q^{4} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{6} - \beta_{3}) q^{6} + (\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} - 3 \beta_1 + 1) q^{7} - \beta_{6} q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - 3) q^{9} + ( - \beta_{6} + \beta_{3} + \beta_1) q^{10} + (2 \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 1) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3}) q^{12} + ( - \beta_{6} + 2 \beta_{3}) q^{13} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{2} + \beta_1) q^{14} + ( - \beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 + 4) q^{15} + q^{16} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{6} - \beta_{3}) q^{18} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{20} + (3 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} + 2 \beta_{4} + \beta_1 - 3) q^{21} + (\beta_{6} - 2 \beta_{3} + 1) q^{22} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 5 \beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{6} + \beta_{3}) q^{24} + (\beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{25} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 1) q^{26} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 3) q^{27} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{2} + 3 \beta_1 - 1) q^{28} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 3 \beta_{2} + \beta_1 - 3) q^{29} + (2 \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{30} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - 4 \beta_1 + 1) q^{31} + \beta_{6} q^{32} + (\beta_{7} + \beta_{4} - 6) q^{33} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{2} + 3 \beta_1 - 1) q^{34} + ( - 2 \beta_{7} - 4 \beta_{6} + 6 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{35}+ \cdots + ( - 5 \beta_{7} - 3 \beta_{6} - 5 \beta_{4} + 6 \beta_{3} - 3) q^{99}+O(q^{100})$$ q + b6 * q^2 + (b7 + b6 + b4 - b3) * q^3 - q^4 + (-b7 - 2*b6 - b4 + b3 - b2 + 1) * q^5 + (b6 - b3) * q^6 + (b7 + 2*b6 - b4 + b2 - 3*b1 + 1) * q^7 - b6 * q^8 + (b7 + b6 + b4 - b3 - 3) * q^9 + (-b6 + b3 + b1) * q^10 + (2*b7 + b6 + 2*b4 - 2*b3 - 1) * q^11 + (-b7 - b6 - b4 + b3) * q^12 + (-b6 + 2*b3) * q^13 + (b7 - b6 - 2*b5 + b4 - 3*b2 + b1) * q^14 + (-b7 - b6 + b3 - b2 - b1 + 4) * q^15 + q^16 + (b7 - b6 - 2*b5 + b4 - 3*b2 + b1) * q^17 + (-2*b6 - b3) * q^18 + (-b7 - 2*b6 + b4 + 2*b3 - b2 + b1 - 1) * q^19 + (b7 + 2*b6 + b4 - b3 + b2 - 1) * q^20 + (3*b7 + 2*b6 - 5*b5 + 2*b4 + b1 - 3) * q^21 + (b6 - 2*b3 + 1) * q^22 + (b7 - 2*b6 - 2*b5 + b4 - 5*b2 + b1 + 1) * q^23 + (-b6 + b3) * q^24 + (b7 + b6 - b4 - 2*b3 + b2 + b1 - 2) * q^25 + (2*b7 + 2*b6 + 2*b4 - 2*b3 - 1) * q^26 + (-2*b7 - 2*b6 - 2*b4 + 2*b3 - 3) * q^27 + (-b7 - 2*b6 + b4 - b2 + 3*b1 - 1) * q^28 + (b7 + 2*b6 - 2*b5 + b4 + 3*b2 + b1 - 3) * q^29 + (2*b6 + b5 - b2 + 1) * q^30 + (2*b7 + 3*b6 - 2*b4 + 2*b2 - 4*b1 + 1) * q^31 + b6 * q^32 + (b7 + b4 - 6) * q^33 + (-b7 - 2*b6 + b4 - b2 + 3*b1 - 1) * q^34 + (-2*b7 - 4*b6 + 6*b5 - 2*b4 - b3 + 2*b2 - 3*b1 + 3) * q^35 + (-b7 - b6 - b4 + b3 + 3) * q^36 + 2 * q^37 + (b7 + b6 + 2*b5 + b4 - 2*b3 + b2 - b1) * q^38 + (5*b6 + b3) * q^39 + (b6 - b3 - b1) * q^40 + (-3*b6 - 6*b2 + 3) * q^41 + (-2*b7 - 2*b6 - b5 + 3*b4 - 4*b2 + b1 + 1) * q^42 + (b6 - 2*b1 + 1) * q^43 + (-2*b7 - b6 - 2*b4 + 2*b3 + 1) * q^44 + (2*b7 + 5*b6 + 3*b4 - 2*b3 + 2*b2 - b1 + 1) * q^45 + (-b7 - 3*b6 + b4 - b2 + 5*b1 - 2) * q^46 + (-2*b7 - 2*b6 + 4*b5 - 2*b4 - 2*b2 - 2*b1 + 4) * q^47 + (b7 + b6 + b4 - b3) * q^48 + (2*b6 - 4*b3 - 10) * q^49 + (-b7 - 2*b6 - 2*b5 - b4 + 2*b3 + b2 + b1 - 1) * q^50 + (-2*b7 - 2*b6 - b5 + 3*b4 - 4*b2 + b1 + 1) * q^51 + (b6 - 2*b3) * q^52 + b6 * q^53 + (-5*b6 + 2*b3) * q^54 + (-b7 + b6 + b4 - b2 - 3*b1 + 7) * q^55 + (-b7 + b6 + 2*b5 - b4 + 3*b2 - b1) * q^56 + (-3*b7 + 4*b6 + 3*b5 - 2*b4 + 2*b3 + 2*b2 - b1 + 1) * q^57 + (-b7 + b6 + b4 - b2 - 3*b1 + 2) * q^58 + (-3*b7 - b6 + 6*b5 - 3*b4 + b2 - 3*b1 + 4) * q^59 + (b7 + b6 - b3 + b2 + b1 - 4) * q^60 + (-b6 + 2*b3 + 1) * q^61 + (2*b7 - b6 - 4*b5 + 2*b4 - 4*b2 + 2*b1 - 1) * q^62 + (-4*b6 - 5*b5 + 5*b4 - 3*b2 + 10*b1 - 6) * q^63 - q^64 + (-5*b6 - 2*b5 + b3 + 2*b2 + b1 - 2) * q^65 + (b7 - 4*b6 + b4 - 2*b3) * q^66 + 7 * q^67 + (-b7 + b6 + 2*b5 - b4 + 3*b2 - b1) * q^68 + (-3*b7 - 2*b6 - b5 + 4*b4 - 6*b2 - b1 + 3) * q^69 + (3*b7 + 3*b6 - 3*b4 - b3 + 3*b2 - 2*b1 + 4) * q^70 + (-2*b7 + 3*b6 + 4*b5 - 2*b4 + 8*b2 - 2*b1 - 1) * q^71 + (2*b6 + b3) * q^72 + (3*b7 + 3*b6 - 3*b4 + 3*b2 - 3*b1) * q^73 + 2*b6 * q^74 + (-8*b6 - b5 - b4 - 4*b2 + b1 + 1) * q^75 + (b7 + 2*b6 - b4 - 2*b3 + b2 - b1 + 1) * q^76 + (4*b7 + 3*b6 - 8*b5 + 4*b4 + 2*b2 + 4*b1 - 7) * q^77 + (b7 + b6 + b4 - b3 - 6) * q^78 + (-4*b6 + 8*b1 - 4) * q^79 + (-b7 - 2*b6 - b4 + b3 - b2 + 1) * q^80 + (-5*b7 - 5*b6 - 5*b4 + 5*b3 + 6) * q^81 + (-3*b6 + 6*b1 - 3) * q^82 + (-2*b7 + 2*b6 + 4*b5 - 2*b4 + 6*b2 - 2*b1) * q^83 + (-3*b7 - 2*b6 + 5*b5 - 2*b4 - b1 + 3) * q^84 + (3*b7 + 3*b6 - 3*b4 - b3 + 3*b2 - 2*b1 + 4) * q^85 + (-b6 - 2*b2 + 1) * q^86 + (b7 - 2*b6 - b5 + 2*b2 + 7*b1 - 5) * q^87 + (-b6 + 2*b3 - 1) * q^88 + (-2*b7 + b6 + 4*b5 - 2*b4 + 4*b2 - 2*b1 + 1) * q^89 + (5*b6 + b5 - 3*b3 - b2 - 3*b1 + 1) * q^90 + (5*b7 + 3*b6 - 5*b4 + 5*b2 - b1 - 2) * q^91 + (-b7 + 2*b6 + 2*b5 - b4 + 5*b2 - b1 - 1) * q^92 + (5*b7 + 2*b6 - 8*b5 + 3*b4 - 2*b2 + 2*b1 - 4) * q^93 + (2*b7 - 2*b4 + 2*b2 + 2*b1 - 2) * q^94 + (4*b7 - 2*b6 - 6*b5 + 4*b4 - b3 + 3*b1 - 5) * q^95 + (b6 - b3) * q^96 + (-2*b6 + 4*b3 + 10) * q^97 + (-4*b7 - 14*b6 - 4*b4 + 4*b3 + 2) * q^98 + (-5*b7 - 3*b6 - 5*b4 + 6*b3 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{3} - 8 q^{4} - 20 q^{9}+O(q^{10})$$ 8 * q + 4 * q^3 - 8 * q^4 - 20 * q^9 $$8 q + 4 q^{3} - 8 q^{4} - 20 q^{9} + 4 q^{10} - 4 q^{12} + 22 q^{15} + 8 q^{16} - 8 q^{19} + 8 q^{22} - 8 q^{25} - 32 q^{27} + 2 q^{30} - 44 q^{33} + 20 q^{36} + 16 q^{37} - 4 q^{40} + 22 q^{45} + 4 q^{48} - 80 q^{49} + 40 q^{55} - 4 q^{57} - 22 q^{60} + 8 q^{61} - 8 q^{64} + 4 q^{66} + 56 q^{67} + 36 q^{70} - 4 q^{75} + 8 q^{76} - 44 q^{78} + 28 q^{81} + 36 q^{85} - 8 q^{88} - 10 q^{90} + 80 q^{97} - 44 q^{99}+O(q^{100})$$ 8 * q + 4 * q^3 - 8 * q^4 - 20 * q^9 + 4 * q^10 - 4 * q^12 + 22 * q^15 + 8 * q^16 - 8 * q^19 + 8 * q^22 - 8 * q^25 - 32 * q^27 + 2 * q^30 - 44 * q^33 + 20 * q^36 + 16 * q^37 - 4 * q^40 + 22 * q^45 + 4 * q^48 - 80 * q^49 + 40 * q^55 - 4 * q^57 - 22 * q^60 + 8 * q^61 - 8 * q^64 + 4 * q^66 + 56 * q^67 + 36 * q^70 - 4 * q^75 + 8 * q^76 - 44 * q^78 + 28 * q^81 + 36 * q^85 - 8 * q^88 - 10 * q^90 + 80 * q^97 - 44 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 3\nu^{7} - 8\nu^{6} + 34\nu^{5} - 50\nu^{4} + 82\nu^{3} - 62\nu^{2} + 46\nu - 10 ) / 5$$ (3*v^7 - 8*v^6 + 34*v^5 - 50*v^4 + 82*v^3 - 62*v^2 + 46*v - 10) / 5 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} + 8\nu^{6} - 34\nu^{5} + 50\nu^{4} - 82\nu^{3} + 67\nu^{2} - 51\nu + 25 ) / 5$$ (-3*v^7 + 8*v^6 - 34*v^5 + 50*v^4 - 82*v^3 + 67*v^2 - 51*v + 25) / 5 $$\beta_{4}$$ $$=$$ $$( 2\nu^{7} - 7\nu^{6} + 31\nu^{5} - 60\nu^{4} + 118\nu^{3} - 118\nu^{2} + 109\nu - 30 ) / 5$$ (2*v^7 - 7*v^6 + 31*v^5 - 60*v^4 + 118*v^3 - 118*v^2 + 109*v - 30) / 5 $$\beta_{5}$$ $$=$$ $$( -4\nu^{7} + 14\nu^{6} - 57\nu^{5} + 110\nu^{4} - 186\nu^{3} + 191\nu^{2} - 138\nu + 50 ) / 5$$ (-4*v^7 + 14*v^6 - 57*v^5 + 110*v^4 - 186*v^3 + 191*v^2 - 138*v + 50) / 5 $$\beta_{6}$$ $$=$$ $$( -6\nu^{7} + 21\nu^{6} - 83\nu^{5} + 155\nu^{4} - 249\nu^{3} + 229\nu^{2} - 157\nu + 45 ) / 5$$ (-6*v^7 + 21*v^6 - 83*v^5 + 155*v^4 - 249*v^3 + 229*v^2 - 157*v + 45) / 5 $$\beta_{7}$$ $$=$$ $$( -7\nu^{7} + 22\nu^{6} - 96\nu^{5} + 170\nu^{4} - 313\nu^{3} + 298\nu^{2} - 249\nu + 85 ) / 5$$ (-7*v^7 + 22*v^6 - 96*v^5 + 170*v^4 - 313*v^3 + 298*v^2 - 249*v + 85) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 - 3$$ b3 + b2 + b1 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2\beta_{4} + \beta_{2} - 3\beta _1 - 3$$ b7 + 2*b4 + b2 - 3*b1 - 3 $$\nu^{4}$$ $$=$$ $$\beta_{7} - \beta_{6} + 2\beta_{5} + 3\beta_{4} - 7\beta_{3} - 6\beta_{2} - 8\beta _1 + 13$$ b7 - b6 + 2*b5 + 3*b4 - 7*b3 - 6*b2 - 8*b1 + 13 $$\nu^{5}$$ $$=$$ $$-8\beta_{7} - 2\beta_{6} + 5\beta_{5} - 12\beta_{4} - 5\beta_{3} - 13\beta_{2} + 7\beta _1 + 31$$ -8*b7 - 2*b6 + 5*b5 - 12*b4 - 5*b3 - 13*b2 + 7*b1 + 31 $$\nu^{6}$$ $$=$$ $$-18\beta_{7} + 6\beta_{6} - 7\beta_{5} - 35\beta_{4} + 41\beta_{3} + 25\beta_{2} + 50\beta _1 - 43$$ -18*b7 + 6*b6 - 7*b5 - 35*b4 + 41*b3 + 25*b2 + 50*b1 - 43 $$\nu^{7}$$ $$=$$ $$32\beta_{7} + 22\beta_{6} - 42\beta_{5} + 38\beta_{4} + 70\beta_{3} + 109\beta_{2} + 8\beta _1 - 226$$ 32*b7 + 22*b6 - 42*b5 + 38*b4 + 70*b3 + 109*b2 + 8*b1 - 226

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/570\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
569.1
 0.5 + 0.0845405i 0.5 − 1.08454i 0.5 + 1.41267i 0.5 − 2.41267i 0.5 − 1.41267i 0.5 + 2.41267i 0.5 − 0.0845405i 0.5 + 1.08454i
1.00000i 0.500000 1.65831i −1.00000 −0.584541 + 2.15831i −1.65831 0.500000i 4.86140i 1.00000i −2.50000 1.65831i 2.15831 + 0.584541i
569.2 1.00000i 0.500000 1.65831i −1.00000 0.584541 + 2.15831i −1.65831 0.500000i 4.86140i 1.00000i −2.50000 1.65831i 2.15831 0.584541i
569.3 1.00000i 0.500000 + 1.65831i −1.00000 −1.91267 1.15831i 1.65831 0.500000i 3.21974i 1.00000i −2.50000 + 1.65831i −1.15831 + 1.91267i
569.4 1.00000i 0.500000 + 1.65831i −1.00000 1.91267 1.15831i 1.65831 0.500000i 3.21974i 1.00000i −2.50000 + 1.65831i −1.15831 1.91267i
569.5 1.00000i 0.500000 1.65831i −1.00000 −1.91267 + 1.15831i 1.65831 + 0.500000i 3.21974i 1.00000i −2.50000 1.65831i −1.15831 1.91267i
569.6 1.00000i 0.500000 1.65831i −1.00000 1.91267 + 1.15831i 1.65831 + 0.500000i 3.21974i 1.00000i −2.50000 1.65831i −1.15831 + 1.91267i
569.7 1.00000i 0.500000 + 1.65831i −1.00000 −0.584541 2.15831i −1.65831 + 0.500000i 4.86140i 1.00000i −2.50000 + 1.65831i 2.15831 0.584541i
569.8 1.00000i 0.500000 + 1.65831i −1.00000 0.584541 2.15831i −1.65831 + 0.500000i 4.86140i 1.00000i −2.50000 + 1.65831i 2.15831 + 0.584541i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 569.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
95.d odd 2 1 inner
285.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.c.f yes 8
3.b odd 2 1 inner 570.2.c.f yes 8
5.b even 2 1 570.2.c.c 8
15.d odd 2 1 570.2.c.c 8
19.b odd 2 1 570.2.c.c 8
57.d even 2 1 570.2.c.c 8
95.d odd 2 1 inner 570.2.c.f yes 8
285.b even 2 1 inner 570.2.c.f yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.c.c 8 5.b even 2 1
570.2.c.c 8 15.d odd 2 1
570.2.c.c 8 19.b odd 2 1
570.2.c.c 8 57.d even 2 1
570.2.c.f yes 8 1.a even 1 1 trivial
570.2.c.f yes 8 3.b odd 2 1 inner
570.2.c.f yes 8 95.d odd 2 1 inner
570.2.c.f yes 8 285.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$:

 $$T_{7}^{4} + 34T_{7}^{2} + 245$$ T7^4 + 34*T7^2 + 245 $$T_{11}^{4} + 24T_{11}^{2} + 100$$ T11^4 + 24*T11^2 + 100 $$T_{29}^{4} - 70T_{29}^{2} + 125$$ T29^4 - 70*T29^2 + 125 $$T_{37} - 2$$ T37 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{4}$$
$3$ $$(T^{2} - T + 3)^{4}$$
$5$ $$T^{8} + 4 T^{6} + 10 T^{4} + 100 T^{2} + \cdots + 625$$
$7$ $$(T^{4} + 34 T^{2} + 245)^{2}$$
$11$ $$(T^{4} + 24 T^{2} + 100)^{2}$$
$13$ $$(T^{2} - 11)^{4}$$
$17$ $$(T^{4} - 34 T^{2} + 245)^{2}$$
$19$ $$(T^{4} + 4 T^{3} - 2 T^{2} + 76 T + 361)^{2}$$
$23$ $$(T^{4} - 86 T^{2} + 1805)^{2}$$
$29$ $$(T^{4} - 70 T^{2} + 125)^{2}$$
$31$ $$(T^{4} + 80 T^{2} + 500)^{2}$$
$37$ $$(T - 2)^{8}$$
$41$ $$(T^{4} - 144 T^{2} + 1620)^{2}$$
$43$ $$(T^{4} + 16 T^{2} + 20)^{2}$$
$47$ $$(T^{4} - 104 T^{2} + 2000)^{2}$$
$53$ $$(T^{2} + 1)^{4}$$
$59$ $$(T^{4} - 130 T^{2} + 3125)^{2}$$
$61$ $$(T^{2} - 2 T - 10)^{4}$$
$67$ $$(T - 7)^{8}$$
$71$ $$(T^{4} - 224 T^{2} + 12500)^{2}$$
$73$ $$(T^{4} + 126 T^{2} + 405)^{2}$$
$79$ $$(T^{4} + 256 T^{2} + 5120)^{2}$$
$83$ $$(T^{4} - 136 T^{2} + 3920)^{2}$$
$89$ $$(T^{4} - 80 T^{2} + 500)^{2}$$
$97$ $$(T^{2} - 20 T + 56)^{4}$$