# Properties

 Label 144.3.q.e Level $144$ Weight $3$ Character orbit 144.q Analytic conductor $3.924$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.19269881856.9 Defining polynomial: $$x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144$$ x^8 - 2*x^7 + 15*x^6 - 2*x^5 + 133*x^4 - 84*x^3 + 276*x^2 + 144*x + 144 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 72) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_{2} - 1) q^{3} + ( - \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 3) q^{9}+O(q^{10})$$ q + (-b3 + b2 - 1) * q^3 + (-b7 + b6 + b3 - b2 + b1 - 1) * q^5 + (-b7 - b5 + b3 + b2 + b1 - 1) * q^7 + (-b7 - 2*b6 + b5 - b4 + b3 - 3*b2 - 3) * q^9 $$q + ( - \beta_{3} + \beta_{2} - 1) q^{3} + ( - \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 3) q^{9} + ( - 2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 5) q^{11} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{13} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 7 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{15} + ( - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{3} + 4 \beta_{2} + 5 \beta_1 + 1) q^{17} + ( - 6 \beta_{7} - 3 \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{19} + (\beta_{7} - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 5) q^{21} + (2 \beta_{7} - 4 \beta_{6} + 6 \beta_{5} - 5 \beta_{3} - 7 \beta_{2} + \beta_1 + 14) q^{23} + (\beta_{7} - 6 \beta_{6} + \beta_{5} - 3 \beta_{4} - 7 \beta_{3} + 9 \beta_{2} - 6 \beta_1 + 2) q^{25} + (\beta_{7} + 10 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_1 + 2) q^{27} + (\beta_{7} + 3 \beta_{5} - 3 \beta_{4} - 10 \beta_{3} + \beta_{2} + 2 \beta_1 - 11) q^{29} + ( - 6 \beta_{6} - 12 \beta_{4} + 13 \beta_{3} - \beta_{2} - \beta_1) q^{31} + (4 \beta_{7} + \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + 20 \beta_{3} - 6 \beta_{2} + \cdots - 20) q^{33}+ \cdots + ( - 9 \beta_{7} + 14 \beta_{6} - 16 \beta_{5} + 22 \beta_{4} + 10 \beta_{3} + \cdots + 80) q^{99}+O(q^{100})$$ q + (-b3 + b2 - 1) * q^3 + (-b7 + b6 + b3 - b2 + b1 - 1) * q^5 + (-b7 - b5 + b3 + b2 + b1 - 1) * q^7 + (-b7 - 2*b6 + b5 - b4 + b3 - 3*b2 - 3) * q^9 + (-2*b7 + 4*b5 + 2*b4 - 2*b3 - 2*b2 + b1 - 5) * q^11 + (-b7 + 3*b6 + 2*b5 + 6*b4 + 3*b3 + b2 + b1 - 1) * q^13 + (2*b7 + 2*b6 - 2*b5 + 4*b4 + 7*b3 + b2 + b1 - 4) * q^15 + (-2*b7 + b6 - 3*b5 + b4 - b3 + 4*b2 + 5*b1 + 1) * q^17 + (-6*b7 - 3*b5 + b3 - 2*b2 + b1 + 1) * q^19 + (b7 - b5 + 3*b4 - 2*b3 - b2 - 2*b1 - 5) * q^21 + (2*b7 - 4*b6 + 6*b5 - 5*b3 - 7*b2 + b1 + 14) * q^23 + (b7 - 6*b6 + b5 - 3*b4 - 7*b3 + 9*b2 - 6*b1 + 2) * q^25 + (b7 + 10*b6 - 2*b5 + 2*b4 + 3*b3 - b1 + 2) * q^27 + (b7 + 3*b5 - 3*b4 - 10*b3 + b2 + 2*b1 - 11) * q^29 + (-6*b6 - 12*b4 + 13*b3 - b2 - b1) * q^31 + (4*b7 + b6 - 3*b5 - 7*b4 + 20*b3 - 6*b2 - b1 - 20) * q^33 + (-b7 + b5 - 12*b3 + 2*b2 + 5) * q^35 + (-2*b7 + 6*b6 - 2*b5 - 6*b4 + 16) * q^37 + (10*b7 - 12*b6 - 6*b5 - 12*b4 - 3*b3 + b2 - 5*b1 - 6) * q^39 + (4*b5 - 19*b3 - 6*b2 + 2*b1 + 42) * q^41 + (12*b6 + 6*b4 - 14*b3 + 14*b2 - 7*b1 + 7) * q^43 + (7*b7 - 11*b6 - 6*b5 - 4*b4 - 3*b3 + 5*b2 - 7*b1 + 29) * q^45 + (5*b7 + b5 - 6*b4 - 15*b3 + 5*b2 + 3*b1 - 13) * q^47 + (b7 + 3*b6 - 2*b5 + 6*b4 + 26*b3 - 5*b2 - 5*b1 + 1) * q^49 + (3*b7 - 8*b6 - b5 + 8*b4 + 19*b3 - b2 - 8*b1 - 30) * q^51 + (6*b7 - 4*b6 + 2*b5 - 4*b4 + 16*b3 - 12*b2 - 8*b1 - 6) * q^53 + (13*b7 - 12*b6 - 5*b5 + 12*b4 - 6*b3 + 12*b2 - 6*b1 + 33) * q^55 + (11*b7 + 18*b6 + b5 + 15*b4 - 29*b3 - b2 - 4*b1 - 8) * q^57 + (-9*b7 + 14*b6 - 6*b5 - 9*b3 + 6*b1 + 21) * q^59 + (-7*b7 - 18*b6 - 7*b5 - 9*b4 - 2*b3 - 3*b2 + 12*b1 + 7) * q^61 + (7*b7 + 5*b5 - 6*b4 - 11*b3 - 7*b2 - 5*b1 + 31) * q^63 + (-9*b7 + 21*b5 + 7*b4 - 12*b3 - 9*b2 + 6*b1 - 27) * q^65 + (-5*b7 + 10*b5 + 5*b3 - 5) * q^67 + (-b7 + 7*b6 - 2*b5 - 10*b4 + 39*b3 + 19*b2 - 3*b1 - 25) * q^69 + (-4*b6 - 12*b5 - 4*b4 + 72*b3 + 12*b1 - 30) * q^71 + (-4*b7 + 9*b6 - 25*b5 - 9*b4 - 7*b3 + 14*b2 - 7*b1 + 3) * q^73 + (-17*b7 - 6*b6 + 12*b5 - 18*b4 - 58*b3 - 19*b2 + b1 - 13) * q^75 + (7*b7 - 9*b6 + 8*b5 - 3*b3 - 5*b2 - 3*b1 + 7) * q^77 + (b7 + 24*b6 + b5 + 12*b4 + 13*b3 - 13*b2 + 5*b1 - 7) * q^79 + (5*b7 - 11*b6 + 8*b4 - 2*b3 + 5*b2 + 15*b1 + 24) * q^81 + (-3*b7 + b5 - 2*b4 - 19*b3 - 3*b2 - b1 - 21) * q^83 + (-6*b7 + 12*b5 - 58*b3 + 4*b2 + 4*b1 - 6) * q^85 + (-11*b7 - 6*b6 + 5*b5 + 67*b3 - 19*b2 + 7*b1 - 29) * q^87 + (-10*b7 + 14*b5 + 20*b3 + 20*b2 - 4*b1 - 22) * q^89 + (-b7 - 6*b6 + 11*b5 + 6*b4 + 4*b3 - 8*b2 + 4*b1 - 19) * q^91 + (-23*b7 + 21*b6 - 12*b5 + 18*b4 - 21*b3 + 13*b2 + 13*b1 - 3) * q^93 + (-2*b7 - 2*b6 - 12*b5 - 46*b3 + 16*b2 - 4*b1 + 82) * q^95 + (-4*b7 - 4*b5 + 63*b3 - 8*b2 + 10*b1 - 57) * q^97 + (-9*b7 + 14*b6 - 16*b5 + 22*b4 + 10*b3 + 9*b2 + 14*b1 + 80) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 10 q^{3} - 6 q^{5} - 6 q^{7} - 22 q^{9}+O(q^{10})$$ 8 * q - 10 * q^3 - 6 * q^5 - 6 * q^7 - 22 * q^9 $$8 q - 10 q^{3} - 6 q^{5} - 6 q^{7} - 22 q^{9} - 36 q^{11} + 14 q^{13} - 10 q^{15} - 4 q^{19} - 54 q^{21} + 102 q^{23} + 10 q^{25} + 20 q^{27} - 114 q^{29} + 50 q^{31} - 104 q^{33} + 120 q^{37} - 82 q^{39} + 264 q^{41} + 28 q^{43} + 206 q^{45} - 150 q^{47} + 94 q^{49} - 170 q^{51} + 244 q^{55} - 178 q^{57} + 108 q^{59} + 14 q^{61} + 210 q^{63} - 198 q^{65} + 20 q^{67} - 14 q^{69} - 76 q^{73} - 326 q^{75} + 66 q^{77} - 26 q^{79} + 194 q^{81} - 246 q^{83} - 224 q^{85} + 18 q^{87} - 108 q^{91} - 130 q^{93} + 456 q^{95} - 236 q^{97} + 634 q^{99}+O(q^{100})$$ 8 * q - 10 * q^3 - 6 * q^5 - 6 * q^7 - 22 * q^9 - 36 * q^11 + 14 * q^13 - 10 * q^15 - 4 * q^19 - 54 * q^21 + 102 * q^23 + 10 * q^25 + 20 * q^27 - 114 * q^29 + 50 * q^31 - 104 * q^33 + 120 * q^37 - 82 * q^39 + 264 * q^41 + 28 * q^43 + 206 * q^45 - 150 * q^47 + 94 * q^49 - 170 * q^51 + 244 * q^55 - 178 * q^57 + 108 * q^59 + 14 * q^61 + 210 * q^63 - 198 * q^65 + 20 * q^67 - 14 * q^69 - 76 * q^73 - 326 * q^75 + 66 * q^77 - 26 * q^79 + 194 * q^81 - 246 * q^83 - 224 * q^85 + 18 * q^87 - 108 * q^91 - 130 * q^93 + 456 * q^95 - 236 * q^97 + 634 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144$$ :

 $$\beta_{1}$$ $$=$$ $$( -6\nu^{7} - 169\nu^{6} + 466\nu^{5} - 2647\nu^{4} + 3180\nu^{3} - 20091\nu^{2} + 64128\nu - 19236 ) / 17700$$ (-6*v^7 - 169*v^6 + 466*v^5 - 2647*v^4 + 3180*v^3 - 20091*v^2 + 64128*v - 19236) / 17700 $$\beta_{2}$$ $$=$$ $$( 331\nu^{7} - 756\nu^{6} + 8709\nu^{5} - 4178\nu^{4} + 73845\nu^{3} + 16116\nu^{2} + 432972\nu + 72936 ) / 159300$$ (331*v^7 - 756*v^6 + 8709*v^5 - 4178*v^4 + 73845*v^3 + 16116*v^2 + 432972*v + 72936) / 159300 $$\beta_{3}$$ $$=$$ $$( - 677 \nu^{7} + 1827 \nu^{6} - 10353 \nu^{5} + 6901 \nu^{4} - 82215 \nu^{3} + 132153 \nu^{2} - 156924 \nu + 80388 ) / 159300$$ (-677*v^7 + 1827*v^6 - 10353*v^5 + 6901*v^4 - 82215*v^3 + 132153*v^2 - 156924*v + 80388) / 159300 $$\beta_{4}$$ $$=$$ $$( 1013 \nu^{7} - 2688 \nu^{6} + 16707 \nu^{5} - 19444 \nu^{4} + 143085 \nu^{3} - 232782 \nu^{2} + 247356 \nu - 401472 ) / 159300$$ (1013*v^7 - 2688*v^6 + 16707*v^5 - 19444*v^4 + 143085*v^3 - 232782*v^2 + 247356*v - 401472) / 159300 $$\beta_{5}$$ $$=$$ $$( -602\nu^{7} + 252\nu^{6} - 5853\nu^{5} - 12374\nu^{4} - 64440\nu^{3} - 39297\nu^{2} + 41526\nu - 95112 ) / 79650$$ (-602*v^7 + 252*v^6 - 5853*v^5 - 12374*v^4 - 64440*v^3 - 39297*v^2 + 41526*v - 95112) / 79650 $$\beta_{6}$$ $$=$$ $$( 644\nu^{7} - 2019\nu^{6} + 9966\nu^{5} - 7447\nu^{4} + 68730\nu^{3} - 107691\nu^{2} + 129078\nu + 194364 ) / 79650$$ (644*v^7 - 2019*v^6 + 9966*v^5 - 7447*v^4 + 68730*v^3 - 107691*v^2 + 129078*v + 194364) / 79650 $$\beta_{7}$$ $$=$$ $$( - 1004 \nu^{7} + 729 \nu^{6} - 12981 \nu^{5} - 14198 \nu^{4} - 139005 \nu^{3} - 62319 \nu^{2} - 104598 \nu - 259974 ) / 79650$$ (-1004*v^7 + 729*v^6 - 12981*v^5 - 14198*v^4 - 139005*v^3 - 62319*v^2 - 104598*v - 259974) / 79650
 $$\nu$$ $$=$$ $$( \beta_{7} - 2\beta_{5} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 3$$ (b7 - 2*b5 + b3 + b2 + b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} + 6\beta_{6} - \beta_{5} + 3\beta_{4} + 20\beta_{3} - \beta_{2} + 2\beta _1 - 19 ) / 3$$ (-b7 + 6*b6 - b5 + 3*b4 + 20*b3 - b2 + 2*b1 - 19) / 3 $$\nu^{3}$$ $$=$$ $$( -20\beta_{7} + 3\beta_{6} + 19\beta_{5} - 3\beta_{4} + 13\beta_{3} - 26\beta_{2} + 13\beta _1 - 38 ) / 3$$ (-20*b7 + 3*b6 + 19*b5 - 3*b4 + 13*b3 - 26*b2 + 13*b1 - 38) / 3 $$\nu^{4}$$ $$=$$ $$( -25\beta_{7} - 39\beta_{6} + 50\beta_{5} - 78\beta_{4} - 211\beta_{3} - 13\beta_{2} - 13\beta _1 - 25 ) / 3$$ (-25*b7 - 39*b6 + 50*b5 - 78*b4 - 211*b3 - 13*b2 - 13*b1 - 25) / 3 $$\nu^{5}$$ $$=$$ $$( 79\beta_{7} - 150\beta_{6} + 79\beta_{5} - 75\beta_{4} - 626\beta_{3} + 259\beta_{2} - 248\beta _1 + 457 ) / 3$$ (79*b7 - 150*b6 + 79*b5 - 75*b4 - 626*b3 + 259*b2 - 248*b1 + 457) / 3 $$\nu^{6}$$ $$=$$ $$( 650\beta_{7} - 507\beta_{6} - 685\beta_{5} + 507\beta_{4} - 445\beta_{3} + 890\beta_{2} - 445\beta _1 + 3152 ) / 3$$ (650*b7 - 507*b6 - 685*b5 + 507*b4 - 445*b3 + 890*b2 - 445*b1 + 3152) / 3 $$\nu^{7}$$ $$=$$ $$( 2293 \beta_{7} + 1335 \beta_{6} - 4586 \beta_{5} + 2670 \beta_{4} + 7609 \beta_{3} + 1039 \beta_{2} + 1039 \beta _1 + 2293 ) / 3$$ (2293*b7 + 1335*b6 - 4586*b5 + 2670*b4 + 7609*b3 + 1039*b2 + 1039*b1 + 2293) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −1.41950 − 2.45865i −0.331167 − 0.573598i 0.831167 + 1.43962i 1.91950 + 3.32468i −1.41950 + 2.45865i −0.331167 + 0.573598i 0.831167 − 1.43962i 1.91950 − 3.32468i
0 −2.91950 + 0.690286i 0 1.80902 1.04444i 0 0.781452 1.35351i 0 8.04701 4.03058i 0
65.2 0 −1.83117 2.37631i 0 3.44299 1.98781i 0 1.80469 3.12582i 0 −2.29365 + 8.70282i 0
65.3 0 −0.668833 + 2.92449i 0 −0.0440114 + 0.0254100i 0 −4.52944 + 7.84521i 0 −8.10532 3.91200i 0
65.4 0 0.419504 2.97052i 0 −8.20800 + 4.73889i 0 −1.05671 + 1.83027i 0 −8.64803 2.49230i 0
113.1 0 −2.91950 0.690286i 0 1.80902 + 1.04444i 0 0.781452 + 1.35351i 0 8.04701 + 4.03058i 0
113.2 0 −1.83117 + 2.37631i 0 3.44299 + 1.98781i 0 1.80469 + 3.12582i 0 −2.29365 8.70282i 0
113.3 0 −0.668833 2.92449i 0 −0.0440114 0.0254100i 0 −4.52944 7.84521i 0 −8.10532 + 3.91200i 0
113.4 0 0.419504 + 2.97052i 0 −8.20800 4.73889i 0 −1.05671 1.83027i 0 −8.64803 + 2.49230i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 113.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.q.e 8
3.b odd 2 1 432.3.q.e 8
4.b odd 2 1 72.3.m.b 8
8.b even 2 1 576.3.q.j 8
8.d odd 2 1 576.3.q.i 8
9.c even 3 1 432.3.q.e 8
9.c even 3 1 1296.3.e.i 8
9.d odd 6 1 inner 144.3.q.e 8
9.d odd 6 1 1296.3.e.i 8
12.b even 2 1 216.3.m.b 8
24.f even 2 1 1728.3.q.j 8
24.h odd 2 1 1728.3.q.i 8
36.f odd 6 1 216.3.m.b 8
36.f odd 6 1 648.3.e.c 8
36.h even 6 1 72.3.m.b 8
36.h even 6 1 648.3.e.c 8
72.j odd 6 1 576.3.q.j 8
72.l even 6 1 576.3.q.i 8
72.n even 6 1 1728.3.q.i 8
72.p odd 6 1 1728.3.q.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.b 8 4.b odd 2 1
72.3.m.b 8 36.h even 6 1
144.3.q.e 8 1.a even 1 1 trivial
144.3.q.e 8 9.d odd 6 1 inner
216.3.m.b 8 12.b even 2 1
216.3.m.b 8 36.f odd 6 1
432.3.q.e 8 3.b odd 2 1
432.3.q.e 8 9.c even 3 1
576.3.q.i 8 8.d odd 2 1
576.3.q.i 8 72.l even 6 1
576.3.q.j 8 8.b even 2 1
576.3.q.j 8 72.j odd 6 1
648.3.e.c 8 36.f odd 6 1
648.3.e.c 8 36.h even 6 1
1296.3.e.i 8 9.c even 3 1
1296.3.e.i 8 9.d odd 6 1
1728.3.q.i 8 24.h odd 2 1
1728.3.q.i 8 72.n even 6 1
1728.3.q.j 8 24.f even 2 1
1728.3.q.j 8 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 6T_{5}^{7} - 37T_{5}^{6} - 294T_{5}^{5} + 2661T_{5}^{4} - 6468T_{5}^{3} + 5612T_{5}^{2} + 528T_{5} + 16$$ acting on $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 10 T^{7} + 61 T^{6} + \cdots + 6561$$
$5$ $$T^{8} + 6 T^{7} - 37 T^{6} - 294 T^{5} + \cdots + 16$$
$7$ $$T^{8} + 6 T^{7} + 69 T^{6} + \cdots + 11664$$
$11$ $$T^{8} + 36 T^{7} + \cdots + 105616729$$
$13$ $$T^{8} - 14 T^{7} + 547 T^{6} + \cdots + 2611456$$
$17$ $$T^{8} + 1454 T^{6} + \cdots + 7020428944$$
$19$ $$(T^{4} + 2 T^{3} - 1179 T^{2} + \cdots + 226348)^{2}$$
$23$ $$T^{8} - 102 T^{7} + \cdots + 11198718976$$
$29$ $$T^{8} + 114 T^{7} + \cdots + 106450807824$$
$31$ $$T^{8} - 50 T^{7} + \cdots + 152712134656$$
$37$ $$(T^{4} - 60 T^{3} - 276 T^{2} + \cdots + 206496)^{2}$$
$41$ $$T^{8} - 264 T^{7} + \cdots + 1919025613521$$
$43$ $$T^{8} - 28 T^{7} + \cdots + 1352729498761$$
$47$ $$T^{8} + 150 T^{7} + \cdots + 4615347568896$$
$53$ $$T^{8} + 7016 T^{6} + \cdots + 78435844096$$
$59$ $$T^{8} - 108 T^{7} + \cdots + 127589696809$$
$61$ $$T^{8} - 14 T^{7} + \cdots + 133593174016$$
$67$ $$T^{8} - 20 T^{7} + \cdots + 17391015625$$
$71$ $$T^{8} + \cdots + 114698616545536$$
$73$ $$(T^{4} + 38 T^{3} - 9831 T^{2} + \cdots + 2961976)^{2}$$
$79$ $$T^{8} + \cdots + 103529078405776$$
$83$ $$T^{8} + 246 T^{7} + \cdots + 1085363908864$$
$89$ $$T^{8} + \cdots + 309931236458496$$
$97$ $$T^{8} + 236 T^{7} + \cdots + 3435006304129$$