# Properties

 Label 775.2.b.e Level $775$ Weight $2$ Character orbit 775.b Analytic conductor $6.188$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 775.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.18840615665$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2x^{5} + 28x^{4} - 12x^{3} + 2x^{2} + 8x + 16$$ x^8 - 2*x^5 + 28*x^4 - 12*x^3 + 2*x^2 + 8*x + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 155) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{7} q^{2} - \beta_{4} q^{3} + (\beta_{6} + \beta_{3} - 2) q^{4} + (\beta_{3} + 2) q^{6} - \beta_{2} q^{7} + (3 \beta_{7} + 2 \beta_{4} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{6} - 2 \beta_{3} - 2) q^{9}+O(q^{10})$$ q - b7 * q^2 - b4 * q^3 + (b6 + b3 - 2) * q^4 + (b3 + 2) * q^6 - b2 * q^7 + (3*b7 + 2*b4 - b2 - b1) * q^8 + (-b6 - 2*b3 - 2) * q^9 $$q - \beta_{7} q^{2} - \beta_{4} q^{3} + (\beta_{6} + \beta_{3} - 2) q^{4} + (\beta_{3} + 2) q^{6} - \beta_{2} q^{7} + (3 \beta_{7} + 2 \beta_{4} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{6} - 2 \beta_{3} - 2) q^{9} + ( - 2 \beta_{6} - \beta_{3} - 2) q^{11} + \beta_1 q^{12} + ( - \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{6} - \beta_{5} - 2) q^{14} + ( - 4 \beta_{6} - \beta_{5} - 3 \beta_{3} + 2) q^{16} + (2 \beta_{7} + \beta_{4} - \beta_{2}) q^{17} + ( - 3 \beta_{7} - 4 \beta_{4} + \beta_{2}) q^{18} + ( - \beta_{5} - 1) q^{19} + ( - \beta_{6} + \beta_{5} + \beta_{3}) q^{21} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2} + 3 \beta_1) q^{22} + ( - 2 \beta_{7} - 2 \beta_{4} - \beta_{2}) q^{23} + (2 \beta_{6} + 2 \beta_{3} + 4) q^{24} + (3 \beta_{6} - \beta_{5} - 2) q^{26} + (3 \beta_{4} - \beta_{2} - \beta_1) q^{27} + (2 \beta_{2} + 2 \beta_1) q^{28} + ( - \beta_{6} + \beta_{5} + \beta_{3} - 2) q^{29} + q^{31} + ( - 7 \beta_{7} - 2 \beta_{4} + 5 \beta_{2} + 3 \beta_1) q^{32} + (4 \beta_{4} + \beta_{2} - 2 \beta_1) q^{33} + ( - 3 \beta_{6} - \beta_{5} - 3 \beta_{3} + 4) q^{34} + (2 \beta_{6} + \beta_{5} + 3 \beta_{3} - 6) q^{36} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{37} + 3 \beta_{2} q^{38} + ( - 3 \beta_{6} + 3 \beta_{5} + \beta_{3}) q^{39} + (\beta_{5} - \beta_{3} + 3) q^{41} + (2 \beta_{7} + 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{42} + (2 \beta_{7} + \beta_{4} - \beta_{2} + \beta_1) q^{43} + (6 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 4) q^{44} + (\beta_{6} - \beta_{5} + 4 \beta_{3} - 6) q^{46} + ( - 2 \beta_{4} + 2 \beta_1) q^{47} + (2 \beta_{7} + 4 \beta_{4} - 2 \beta_{2}) q^{48} + ( - \beta_{6} - \beta_{5} + 1) q^{49} + (\beta_{5} + \beta_{3} + 1) q^{51} + (4 \beta_{7} - 2 \beta_{2} - 2 \beta_1) q^{52} + (3 \beta_{4} + 2 \beta_{2} + \beta_1) q^{53} + ( - 3 \beta_{6} - \beta_{5} - 3 \beta_{3} - 8) q^{54} + 4 \beta_{6} q^{56} + (2 \beta_{7} + \beta_{4} - 3 \beta_{2} + 4 \beta_1) q^{57} + (4 \beta_{7} + 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{58} + ( - \beta_{6} - 2 \beta_{5} - \beta_{3} - 3) q^{59} + (\beta_{6} - \beta_{5} + 6) q^{61} - \beta_{7} q^{62} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_{2} - 5 \beta_1) q^{63} + (10 \beta_{6} + 3 \beta_{5} + 3 \beta_{3} - 10) q^{64} + ( - 3 \beta_{6} + \beta_{5} - 4 \beta_{3} - 6) q^{66} + ( - 2 \beta_{7} + \beta_{2} + \beta_1) q^{67} + ( - 10 \beta_{7} - 4 \beta_{4} + 4 \beta_{2} + 3 \beta_1) q^{68} + ( - 3 \beta_{6} + \beta_{5} - \beta_{3} - 6) q^{69} + (\beta_{6} + 2 \beta_{3} + 1) q^{71} + (9 \beta_{7} - 2 \beta_{4} - 3 \beta_{2} - \beta_1) q^{72} + ( - 4 \beta_{7} - 3 \beta_{4} + \beta_1) q^{73} + ( - \beta_{6} + \beta_{5} + \beta_{3} + 4) q^{74} + (3 \beta_{6} + \beta_{5} + 4) q^{76} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{2} - \beta_1) q^{77} + (2 \beta_{7} + 2 \beta_{4} - 6 \beta_{2} + 7 \beta_1) q^{78} + ( - 3 \beta_{3} + 4) q^{79} + (\beta_{3} + 9) q^{81} + ( - 4 \beta_{7} - 2 \beta_{4} - 3 \beta_{2} - \beta_1) q^{82} + ( - \beta_{4} - 3 \beta_{2} - 2 \beta_1) q^{83} - 2 \beta_{3} q^{84} + ( - \beta_{6} - \beta_{5} - 3 \beta_{3} + 4) q^{86} + ( - 2 \beta_{7} + 5 \beta_{2} - 5 \beta_1) q^{87} + (12 \beta_{7} - 8 \beta_{2} - 4 \beta_1) q^{88} + ( - 4 \beta_{6} + \beta_{3} + 2) q^{89} + ( - \beta_{6} - \beta_{5} - 4 \beta_{3} - 6) q^{91} + (10 \beta_{7} + 4 \beta_{4} + 2 \beta_1) q^{92} - \beta_{4} q^{93} + (4 \beta_{6} + 2 \beta_{3} + 4) q^{94} + ( - 2 \beta_{5} - 2 \beta_{3} + 4) q^{96} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{97} + ( - 3 \beta_{7} + 4 \beta_{2} + 2 \beta_1) q^{98} + (\beta_{6} - 3 \beta_{5} + 4 \beta_{3} + 14) q^{99}+O(q^{100})$$ q - b7 * q^2 - b4 * q^3 + (b6 + b3 - 2) * q^4 + (b3 + 2) * q^6 - b2 * q^7 + (3*b7 + 2*b4 - b2 - b1) * q^8 + (-b6 - 2*b3 - 2) * q^9 + (-2*b6 - b3 - 2) * q^11 + b1 * q^12 + (-b2 + 2*b1) * q^13 + (-b6 - b5 - 2) * q^14 + (-4*b6 - b5 - 3*b3 + 2) * q^16 + (2*b7 + b4 - b2) * q^17 + (-3*b7 - 4*b4 + b2) * q^18 + (-b5 - 1) * q^19 + (-b6 + b5 + b3) * q^21 + (-2*b7 - 2*b4 + 2*b2 + 3*b1) * q^22 + (-2*b7 - 2*b4 - b2) * q^23 + (2*b6 + 2*b3 + 4) * q^24 + (3*b6 - b5 - 2) * q^26 + (3*b4 - b2 - b1) * q^27 + (2*b2 + 2*b1) * q^28 + (-b6 + b5 + b3 - 2) * q^29 + q^31 + (-7*b7 - 2*b4 + 5*b2 + 3*b1) * q^32 + (4*b4 + b2 - 2*b1) * q^33 + (-3*b6 - b5 - 3*b3 + 4) * q^34 + (2*b6 + b5 + 3*b3 - 6) * q^36 + (-b4 + b2 - b1) * q^37 + 3*b2 * q^38 + (-3*b6 + 3*b5 + b3) * q^39 + (b5 - b3 + 3) * q^41 + (2*b7 + 2*b4 - 2*b2 + 3*b1) * q^42 + (2*b7 + b4 - b2 + b1) * q^43 + (6*b6 + 2*b5 + 2*b3 - 4) * q^44 + (b6 - b5 + 4*b3 - 6) * q^46 + (-2*b4 + 2*b1) * q^47 + (2*b7 + 4*b4 - 2*b2) * q^48 + (-b6 - b5 + 1) * q^49 + (b5 + b3 + 1) * q^51 + (4*b7 - 2*b2 - 2*b1) * q^52 + (3*b4 + 2*b2 + b1) * q^53 + (-3*b6 - b5 - 3*b3 - 8) * q^54 + 4*b6 * q^56 + (2*b7 + b4 - 3*b2 + 4*b1) * q^57 + (4*b7 + 2*b4 - 2*b2 + 3*b1) * q^58 + (-b6 - 2*b5 - b3 - 3) * q^59 + (b6 - b5 + 6) * q^61 - b7 * q^62 + (-2*b7 - 2*b4 + 2*b2 - 5*b1) * q^63 + (10*b6 + 3*b5 + 3*b3 - 10) * q^64 + (-3*b6 + b5 - 4*b3 - 6) * q^66 + (-2*b7 + b2 + b1) * q^67 + (-10*b7 - 4*b4 + 4*b2 + 3*b1) * q^68 + (-3*b6 + b5 - b3 - 6) * q^69 + (b6 + 2*b3 + 1) * q^71 + (9*b7 - 2*b4 - 3*b2 - b1) * q^72 + (-4*b7 - 3*b4 + b1) * q^73 + (-b6 + b5 + b3 + 4) * q^74 + (3*b6 + b5 + 4) * q^76 + (2*b7 + 2*b4 + 2*b2 - b1) * q^77 + (2*b7 + 2*b4 - 6*b2 + 7*b1) * q^78 + (-3*b3 + 4) * q^79 + (b3 + 9) * q^81 + (-4*b7 - 2*b4 - 3*b2 - b1) * q^82 + (-b4 - 3*b2 - 2*b1) * q^83 - 2*b3 * q^84 + (-b6 - b5 - 3*b3 + 4) * q^86 + (-2*b7 + 5*b2 - 5*b1) * q^87 + (12*b7 - 8*b2 - 4*b1) * q^88 + (-4*b6 + b3 + 2) * q^89 + (-b6 - b5 - 4*b3 - 6) * q^91 + (10*b7 + 4*b4 + 2*b1) * q^92 - b4 * q^93 + (4*b6 + 2*b3 + 4) * q^94 + (-2*b5 - 2*b3 + 4) * q^96 + (-2*b7 - 2*b4 - 2*b1) * q^97 + (-3*b7 + 4*b2 + 2*b1) * q^98 + (b6 - 3*b5 + 4*b3 + 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 18 q^{4} + 16 q^{6} - 14 q^{9}+O(q^{10})$$ 8 * q - 18 * q^4 + 16 * q^6 - 14 * q^9 $$8 q - 18 q^{4} + 16 q^{6} - 14 q^{9} - 12 q^{11} - 16 q^{14} + 22 q^{16} - 10 q^{19} + 4 q^{21} + 28 q^{24} - 24 q^{26} - 12 q^{29} + 8 q^{31} + 36 q^{34} - 50 q^{36} + 12 q^{39} + 26 q^{41} - 40 q^{44} - 52 q^{46} + 8 q^{49} + 10 q^{51} - 60 q^{54} - 8 q^{56} - 26 q^{59} + 44 q^{61} - 94 q^{64} - 40 q^{66} - 40 q^{69} + 6 q^{71} + 36 q^{74} + 28 q^{76} + 32 q^{79} + 72 q^{81} + 32 q^{86} + 24 q^{89} - 48 q^{91} + 24 q^{94} + 28 q^{96} + 104 q^{99}+O(q^{100})$$ 8 * q - 18 * q^4 + 16 * q^6 - 14 * q^9 - 12 * q^11 - 16 * q^14 + 22 * q^16 - 10 * q^19 + 4 * q^21 + 28 * q^24 - 24 * q^26 - 12 * q^29 + 8 * q^31 + 36 * q^34 - 50 * q^36 + 12 * q^39 + 26 * q^41 - 40 * q^44 - 52 * q^46 + 8 * q^49 + 10 * q^51 - 60 * q^54 - 8 * q^56 - 26 * q^59 + 44 * q^61 - 94 * q^64 - 40 * q^66 - 40 * q^69 + 6 * q^71 + 36 * q^74 + 28 * q^76 + 32 * q^79 + 72 * q^81 + 32 * q^86 + 24 * q^89 - 48 * q^91 + 24 * q^94 + 28 * q^96 + 104 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{5} + 28x^{4} - 12x^{3} + 2x^{2} + 8x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -215\nu^{7} + 864\nu^{6} + 144\nu^{5} + 454\nu^{4} - 7744\nu^{3} + 30084\nu^{2} - 6214\nu - 956 ) / 10798$$ (-215*v^7 + 864*v^6 + 144*v^5 + 454*v^4 - 7744*v^3 + 30084*v^2 - 6214*v - 956) / 10798 $$\beta_{2}$$ $$=$$ $$( -216\nu^{7} - 36\nu^{6} - 6\nu^{5} + 431\nu^{4} - 6876\nu^{3} + 1446\nu^{2} - 5590\nu - 860 ) / 5399$$ (-216*v^7 - 36*v^6 - 6*v^5 + 431*v^4 - 6876*v^3 + 1446*v^2 - 5590*v - 860) / 5399 $$\beta_{3}$$ $$=$$ $$( -216\nu^{7} - 36\nu^{6} - 6\nu^{5} + 431\nu^{4} - 6876\nu^{3} + 1446\nu^{2} + 5208\nu - 860 ) / 5399$$ (-216*v^7 - 36*v^6 - 6*v^5 + 431*v^4 - 6876*v^3 + 1446*v^2 + 5208*v - 860) / 5399 $$\beta_{4}$$ $$=$$ $$( -655\nu^{7} - 1009\nu^{6} - 1068\nu^{5} + 1132\nu^{4} - 14552\nu^{3} - 12562\nu^{2} - 12402\nu - 1908 ) / 10798$$ (-655*v^7 - 1009*v^6 - 1068*v^5 + 1132*v^4 - 14552*v^3 - 12562*v^2 - 12402*v - 1908) / 10798 $$\beta_{5}$$ $$=$$ $$( 714\nu^{7} + 119\nu^{6} - 880\nu^{5} - 5174\nu^{4} + 17330\nu^{3} - 3880\nu^{2} - 13616\nu - 36150 ) / 10798$$ (714*v^7 + 119*v^6 - 880*v^5 - 5174*v^4 + 17330*v^3 - 3880*v^2 - 13616*v - 36150) / 10798 $$\beta_{6}$$ $$=$$ $$( -858\nu^{7} - 143\nu^{6} + 876\nu^{5} + 1862\nu^{4} - 21914\nu^{3} + 4844\nu^{2} + 17088\nu - 14814 ) / 10798$$ (-858*v^7 - 143*v^6 + 876*v^5 + 1862*v^4 - 21914*v^3 + 4844*v^2 + 17088*v - 14814) / 10798 $$\beta_{7}$$ $$=$$ $$( 1955\nu^{7} - 574\nu^{6} + 1704\nu^{5} - 3626\nu^{4} + 52336\nu^{3} - 43532\nu^{2} + 43446\nu + 6684 ) / 21596$$ (1955*v^7 - 574*v^6 + 1704*v^5 - 3626*v^4 + 52336*v^3 - 43532*v^2 + 43446*v + 6684) / 21596
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} ) / 2$$ (b3 - b2) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} + 2\beta_{4} + 3\beta_1 ) / 2$$ (2*b7 + 2*b4 + 3*b1) / 2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{7} + 2\beta_{6} - 4\beta_{3} - 4\beta_{2} - \beta _1 + 2 ) / 2$$ (-2*b7 + 2*b6 - 4*b3 - 4*b2 - b1 + 2) / 2 $$\nu^{4}$$ $$=$$ $$-3\beta_{6} - 3\beta_{5} + \beta_{3} - 14$$ -3*b6 - 3*b5 + b3 - 14 $$\nu^{5}$$ $$=$$ $$( 14\beta_{7} + 13\beta_{6} + \beta_{5} + 2\beta_{4} - 20\beta_{3} + 20\beta_{2} + 9\beta _1 + 18 ) / 2$$ (14*b7 + 13*b6 + b5 + 2*b4 - 20*b3 + 20*b2 + 9*b1 + 18) / 2 $$\nu^{6}$$ $$=$$ $$-34\beta_{7} - 32\beta_{4} - 10\beta_{2} - 37\beta_1$$ -34*b7 - 32*b4 - 10*b2 - 37*b1 $$\nu^{7}$$ $$=$$ $$44\beta_{7} - 38\beta_{6} - 6\beta_{5} + 12\beta_{4} + 53\beta_{3} + 53\beta_{2} + 32\beta _1 - 64$$ 44*b7 - 38*b6 - 6*b5 + 12*b4 + 53*b3 + 53*b2 + 32*b1 - 64

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$$.

 $$n$$ $$251$$ $$652$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1
 −0.520627 + 0.520627i −1.71822 − 1.71822i 1.48716 − 1.48716i 0.751690 + 0.751690i 0.751690 − 0.751690i 1.48716 + 1.48716i −1.71822 + 1.71822i −0.520627 − 0.520627i
2.80027i 0.342376i −5.84153 0 0.958747 1.04125i 10.7573i 2.88278 0
249.2 2.27244i 0.632112i −3.16400 0 −1.43644 3.43644i 2.64511i 2.60043 0
249.3 1.62946i 3.05273i −0.655151 0 4.97431 2.97431i 2.19138i −6.31916 0
249.4 1.15729i 3.02722i 0.660672 0 3.50338 1.50338i 3.07918i −6.16405 0
249.5 1.15729i 3.02722i 0.660672 0 3.50338 1.50338i 3.07918i −6.16405 0
249.6 1.62946i 3.05273i −0.655151 0 4.97431 2.97431i 2.19138i −6.31916 0
249.7 2.27244i 0.632112i −3.16400 0 −1.43644 3.43644i 2.64511i 2.60043 0
249.8 2.80027i 0.342376i −5.84153 0 0.958747 1.04125i 10.7573i 2.88278 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 249.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.b.e 8
5.b even 2 1 inner 775.2.b.e 8
5.c odd 4 1 155.2.a.d 4
5.c odd 4 1 775.2.a.g 4
15.e even 4 1 1395.2.a.m 4
15.e even 4 1 6975.2.a.bj 4
20.e even 4 1 2480.2.a.z 4
35.f even 4 1 7595.2.a.q 4
40.i odd 4 1 9920.2.a.ch 4
40.k even 4 1 9920.2.a.cd 4
155.f even 4 1 4805.2.a.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.d 4 5.c odd 4 1
775.2.a.g 4 5.c odd 4 1
775.2.b.e 8 1.a even 1 1 trivial
775.2.b.e 8 5.b even 2 1 inner
1395.2.a.m 4 15.e even 4 1
2480.2.a.z 4 20.e even 4 1
4805.2.a.j 4 155.f even 4 1
6975.2.a.bj 4 15.e even 4 1
7595.2.a.q 4 35.f even 4 1
9920.2.a.cd 4 40.k even 4 1
9920.2.a.ch 4 40.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 17T_{2}^{6} + 96T_{2}^{4} + 208T_{2}^{2} + 144$$ acting on $$S_{2}^{\mathrm{new}}(775, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 17 T^{6} + 96 T^{4} + \cdots + 144$$
$3$ $$T^{8} + 19 T^{6} + 95 T^{4} + 45 T^{2} + \cdots + 4$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 24 T^{6} + 176 T^{4} + \cdots + 256$$
$11$ $$(T^{4} + 6 T^{3} - 16 T^{2} - 124 T - 144)^{2}$$
$13$ $$T^{8} + 88 T^{6} + 2192 T^{4} + \cdots + 4096$$
$17$ $$T^{8} + 51 T^{6} + 479 T^{4} + \cdots + 576$$
$19$ $$(T^{4} + 5 T^{3} - 21 T^{2} - 81 T + 108)^{2}$$
$23$ $$T^{8} + 128 T^{6} + 4048 T^{4} + \cdots + 576$$
$29$ $$(T^{4} + 6 T^{3} - 40 T^{2} - 308 T - 456)^{2}$$
$31$ $$(T - 1)^{8}$$
$37$ $$T^{8} + 67 T^{6} + 167 T^{4} + \cdots + 16$$
$41$ $$(T^{4} - 13 T^{3} + 17 T^{2} + 161 T - 294)^{2}$$
$43$ $$T^{8} + 63 T^{6} + 1427 T^{4} + \cdots + 45796$$
$47$ $$T^{8} + 124 T^{6} + 2480 T^{4} + \cdots + 36864$$
$53$ $$T^{8} + 271 T^{6} + 24107 T^{4} + \cdots + 8363664$$
$59$ $$(T^{4} + 13 T^{3} - 65 T^{2} - 625 T + 2484)^{2}$$
$61$ $$(T^{4} - 22 T^{3} + 144 T^{2} - 288 T - 32)^{2}$$
$67$ $$T^{8} + 84 T^{6} + 720 T^{4} + \cdots + 1024$$
$71$ $$(T^{4} - 3 T^{3} - 37 T^{2} + 59 T + 384)^{2}$$
$73$ $$T^{8} + 271 T^{6} + 21611 T^{4} + \cdots + 204304$$
$79$ $$(T^{4} - 16 T^{3} - 12 T^{2} + 500 T + 256)^{2}$$
$83$ $$T^{8} + 295 T^{6} + 16955 T^{4} + \cdots + 544644$$
$89$ $$(T^{4} - 12 T^{3} - 124 T^{2} + 1348 T + 1656)^{2}$$
$97$ $$T^{8} + 144 T^{6} + 4640 T^{4} + \cdots + 256$$