# Properties

 Label 912.2.f.i Level $912$ Weight $2$ Character orbit 912.f Analytic conductor $7.282$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: 10.0.20322144469993472.1 Defining polynomial: $$x^{10} - x^{9} - x^{8} - 2x^{7} - 2x^{6} + 22x^{5} - 6x^{4} - 18x^{3} - 27x^{2} - 81x + 243$$ x^10 - x^9 - x^8 - 2*x^7 - 2*x^6 + 22*x^5 - 6*x^4 - 18*x^3 - 27*x^2 - 81*x + 243 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 456) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{6} q^{5} + \beta_{4} q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 - b6 * q^5 + b4 * q^7 + b2 * q^9 $$q + \beta_1 q^{3} - \beta_{6} q^{5} + \beta_{4} q^{7} + \beta_{2} q^{9} - \beta_{9} q^{11} + (\beta_{9} + \beta_{6} + \beta_{5} + \beta_1) q^{13} + ( - \beta_{8} - \beta_{6} - 1) q^{15} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_1) q^{17} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{19} + ( - \beta_{7} + \beta_{4} - \beta_{3}) q^{21} + (\beta_{9} + \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} - \beta_1 - 1) q^{25} + ( - \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} + 1) q^{27} + (\beta_{3} - \beta_{2}) q^{29} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6}) q^{31} + (\beta_{9} - \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{33} + ( - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{35} + ( - \beta_{9} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{37} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 - 1) q^{39} + ( - \beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{41} + ( - \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_1 - 2) q^{43} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_1) q^{45} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{47} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_1 + 1) q^{49} + (2 \beta_{9} - \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{51} + ( - \beta_{8} + \beta_{7} + \beta_{3} - \beta_{2} - 2) q^{53} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{55} + ( - \beta_{9} - \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{57} + (\beta_{8} - \beta_{7} + 2 \beta_{4} + \beta_{3} - \beta_{2}) q^{59} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{61} + (\beta_{8} - \beta_{7} - 2 \beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{63} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{65} + ( - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{67} + ( - 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 3) q^{69} + (\beta_{8} - \beta_{7} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{71} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{73} + ( - \beta_{9} - \beta_{8} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 3) q^{75} + (\beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{77} + (\beta_{9} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{79} + (\beta_{9} - 2 \beta_{7} + 4 \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{81} + (\beta_{8} + \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{83} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{85} + (\beta_{9} + 4 \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{87} + ( - \beta_{8} + \beta_{7} + 2 \beta_{4}) q^{89} + (\beta_{8} + \beta_{7} + 4 \beta_{6} - \beta_{3} - \beta_{2}) q^{91} + (2 \beta_{9} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{93} + (\beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{95} + (\beta_{8} + \beta_{7} - 2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2}) q^{97} + ( - 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 - b6 * q^5 + b4 * q^7 + b2 * q^9 - b9 * q^11 + (b9 + b6 + b5 + b1) * q^13 + (-b8 - b6 - 1) * q^15 + (-b9 + b8 + b7 - b5 - b1) * q^17 + (-b8 + b3 + b1) * q^19 + (-b7 + b4 - b3) * q^21 + (b9 + b6 + 2*b5 + b3 + b2 + 2*b1) * q^23 + (-b8 + b7 + b5 - b4 - b1 - 1) * q^25 + (-b9 - b5 + b4 + b3 + 1) * q^27 + (b3 - b2) * q^29 + (-b9 + b8 + b7 + b6) * q^31 + (b9 - b7 + b5 + b3 + b2 + b1 - 1) * q^33 + (-b9 - b8 - b7 - 2*b6 - b5 + b3 + b2 - b1) * q^35 + (-b9 + b6 - b5 - b3 - b2 - b1) * q^37 + (-b9 + b8 + b7 + b6 - b5 - b3 - b1 - 1) * q^39 + (-b5 + b3 - b2 + b1) * q^41 + (-b8 + b7 - b5 - b4 + b1 - 2) * q^43 + (-b9 - b8 - b7 - b6 - b5 - b4 - b1) * q^45 + (-b8 - b7 - b6 + b5 - b3 - b2 + b1) * q^47 + (-2*b5 + b4 + 2*b1 + 1) * q^49 + (2*b9 - b8 + 2*b6 + 2*b5 + b4 + b3 + 2*b1) * q^51 + (-b8 + b7 + b3 - b2 - 2) * q^53 + (-b4 + b3 - b2 + 2) * q^55 + (-b9 - b7 + 2*b5 - b4 + b2 + 1) * q^57 + (b8 - b7 + 2*b4 + b3 - b2) * q^59 + (-2*b5 - b4 + b3 - b2 + 2*b1) * q^61 + (b8 - b7 - 2*b6 - 3*b5 + b4 - b3 - b1 + 1) * q^63 + (-b5 + 2*b4 - b3 + b2 + b1 + 2) * q^65 + (-b8 - b7 - 2*b6 + b5 - 2*b3 - 2*b2 + b1) * q^67 + (-2*b9 + b8 + b7 + b6 + b5 + b4 + b2 - b1 - 3) * q^69 + (b8 - b7 - 2*b3 + 2*b2 - 2) * q^71 + (-b8 + b7 + b5 + b4 + b3 - b2 - b1 - 2) * q^73 + (-b9 - b8 + 2*b6 - b5 - 2*b4 + b3 - b2 - 3) * q^75 + (b9 - b8 - b7 - 2*b6 + b5 - b3 - b2 + b1) * q^77 + (b9 + b6 - 2*b5 - 2*b3 - 2*b2 - 2*b1) * q^79 + (b9 - 2*b7 + 4*b5 + b4 + b2 + 2*b1 + 2) * q^81 + (b8 + b7 + b5 + b3 + b2 + b1) * q^83 + (-2*b5 + b4 - b3 + b2 + 2*b1 - 2) * q^85 + (b9 + 4*b5 - b4 - b3 - 1) * q^87 + (-b8 + b7 + 2*b4) * q^89 + (b8 + b7 + 4*b6 - b3 - b2) * q^91 + (2*b9 + 3*b6 + 2*b5 + b4 + b3 + b2 + 2*b1 - 2) * q^93 + (b6 - b5 - 2*b4 + b3 - b2 - 3*b1 + 2) * q^95 + (b8 + b7 - 2*b6 - 2*b3 - 2*b2) * q^97 + (-2*b9 + b8 + b7 - 2*b6 + b5 + b4 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 10 * q + q^3 + 2 * q^7 + 3 * q^9 $$10 q + q^{3} + 2 q^{7} + 3 q^{9} - 10 q^{15} - 2 q^{19} + 5 q^{21} - 14 q^{25} + 10 q^{27} - 6 q^{29} - 10 q^{33} - 7 q^{39} - 4 q^{41} - 20 q^{43} - 2 q^{45} + 16 q^{49} - q^{51} - 26 q^{53} + 12 q^{55} + 9 q^{57} - 2 q^{59} - 4 q^{61} + 17 q^{63} + 32 q^{65} - 27 q^{69} - 8 q^{71} - 26 q^{73} - 39 q^{75} + 23 q^{81} - 8 q^{85} - 13 q^{87} + 4 q^{89} - 18 q^{93} + 8 q^{95} - 2 q^{99}+O(q^{100})$$ 10 * q + q^3 + 2 * q^7 + 3 * q^9 - 10 * q^15 - 2 * q^19 + 5 * q^21 - 14 * q^25 + 10 * q^27 - 6 * q^29 - 10 * q^33 - 7 * q^39 - 4 * q^41 - 20 * q^43 - 2 * q^45 + 16 * q^49 - q^51 - 26 * q^53 + 12 * q^55 + 9 * q^57 - 2 * q^59 - 4 * q^61 + 17 * q^63 + 32 * q^65 - 27 * q^69 - 8 * q^71 - 26 * q^73 - 39 * q^75 + 23 * q^81 - 8 * q^85 - 13 * q^87 + 4 * q^89 - 18 * q^93 + 8 * q^95 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} - x^{8} - 2x^{7} - 2x^{6} + 22x^{5} - 6x^{4} - 18x^{3} - 27x^{2} - 81x + 243$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$( \nu^{9} + 2\nu^{8} - 4\nu^{7} - 5\nu^{6} - 8\nu^{5} + 16\nu^{4} + 60\nu^{3} - 36\nu^{2} - 81\nu - 162 ) / 81$$ (v^9 + 2*v^8 - 4*v^7 - 5*v^6 - 8*v^5 + 16*v^4 + 60*v^3 - 36*v^2 - 81*v - 162) / 81 $$\beta_{4}$$ $$=$$ $$( -2\nu^{9} - 4\nu^{8} - \nu^{7} + 19\nu^{6} + 25\nu^{5} - 14\nu^{4} - 21\nu^{3} - 45\nu^{2} + 135\nu + 324 ) / 162$$ (-2*v^9 - 4*v^8 - v^7 + 19*v^6 + 25*v^5 - 14*v^4 - 21*v^3 - 45*v^2 + 135*v + 324) / 162 $$\beta_{5}$$ $$=$$ $$( \nu^{9} - \nu^{8} - \nu^{7} - 2\nu^{6} - 2\nu^{5} + 22\nu^{4} - 6\nu^{3} - 18\nu^{2} - 27\nu - 81 ) / 81$$ (v^9 - v^8 - v^7 - 2*v^6 - 2*v^5 + 22*v^4 - 6*v^3 - 18*v^2 - 27*v - 81) / 81 $$\beta_{6}$$ $$=$$ $$( -5\nu^{9} - 4\nu^{8} + 5\nu^{7} + \nu^{6} - 17\nu^{5} - 47\nu^{4} - 15\nu^{3} + 81\nu^{2} + 81 ) / 324$$ (-5*v^9 - 4*v^8 + 5*v^7 + v^6 - 17*v^5 - 47*v^4 - 15*v^3 + 81*v^2 + 81) / 324 $$\beta_{7}$$ $$=$$ $$( 2\nu^{9} - 5\nu^{8} - 8\nu^{7} + 8\nu^{6} + 11\nu^{5} - 13\nu^{4} - 60\nu^{3} - 54\nu^{2} + 135\nu + 162 ) / 162$$ (2*v^9 - 5*v^8 - 8*v^7 + 8*v^6 + 11*v^5 - 13*v^4 - 60*v^3 - 54*v^2 + 135*v + 162) / 162 $$\beta_{8}$$ $$=$$ $$( -2\nu^{9} + 2\nu^{8} - 7\nu^{7} - 14\nu^{6} + 40\nu^{5} + \nu^{4} + 3\nu^{3} - 108\nu^{2} - 162\nu + 405 ) / 162$$ (-2*v^9 + 2*v^8 - 7*v^7 - 14*v^6 + 40*v^5 + v^4 + 3*v^3 - 108*v^2 - 162*v + 405) / 162 $$\beta_{9}$$ $$=$$ $$( -2\nu^{9} + 2\nu^{8} - 7\nu^{7} + 13\nu^{6} + 13\nu^{5} - 26\nu^{4} - 51\nu^{3} - 81\nu^{2} + 27\nu + 324 ) / 162$$ (-2*v^9 + 2*v^8 - 7*v^7 + 13*v^6 + 13*v^5 - 26*v^4 - 51*v^3 - 81*v^2 + 27*v + 324) / 162
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} + 1$$ -b9 - b5 + b4 + b3 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{9} - 2\beta_{7} + 4\beta_{5} + \beta_{4} + \beta_{2} + 2\beta _1 + 2$$ b9 - 2*b7 + 4*b5 + b4 + b2 + 2*b1 + 2 $$\nu^{5}$$ $$=$$ $$-2\beta_{9} + 2\beta_{8} - 4\beta_{6} - 2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 8$$ -2*b9 + 2*b8 - 4*b6 - 2*b5 + 2*b4 - b3 + b2 - b1 - 8 $$\nu^{6}$$ $$=$$ $$3\beta_{9} - 4\beta_{8} - 2\beta_{7} - 4\beta_{6} + 5\beta_{4} + \beta_{3} + \beta_{2} - 6\beta _1 - 1$$ 3*b9 - 4*b8 - 2*b7 - 4*b6 + 5*b4 + b3 + b2 - 6*b1 - 1 $$\nu^{7}$$ $$=$$ $$-8\beta_{9} - 2\beta_{8} - 6\beta_{7} - 8\beta_{6} - 5\beta_{5} + 2\beta_{4} - 7\beta_{3} - 9\beta_{2} - 6\beta _1 + 6$$ -8*b9 - 2*b8 - 6*b7 - 8*b6 - 5*b5 + 2*b4 - 7*b3 - 9*b2 - 6*b1 + 6 $$\nu^{8}$$ $$=$$ $$15 \beta_{9} - 2 \beta_{8} - 12 \beta_{7} - 20 \beta_{6} - 6 \beta_{5} - 9 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 8 \beta _1 - 2$$ 15*b9 - 2*b8 - 12*b7 - 20*b6 - 6*b5 - 9*b4 - 3*b3 + 2*b2 + 8*b1 - 2 $$\nu^{9}$$ $$=$$ $$- 19 \beta_{9} - 8 \beta_{8} + 22 \beta_{7} - 44 \beta_{6} - 28 \beta_{5} - 9 \beta_{4} - 4 \beta_{3} - 7 \beta_{2} - 29 \beta _1 + 29$$ -19*b9 - 8*b8 + 22*b7 - 44*b6 - 28*b5 - 9*b4 - 4*b3 - 7*b2 - 29*b1 + 29

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 −1.69134 − 0.373340i −1.69134 + 0.373340i −0.729858 − 1.57077i −0.729858 + 1.57077i −0.171092 − 1.72358i −0.171092 + 1.72358i 1.39399 − 1.02801i 1.39399 + 1.02801i 1.69830 − 0.340259i 1.69830 + 0.340259i
0 −1.69134 0.373340i 0 0.568907i 0 −0.718465 0 2.72123 + 1.26289i 0
113.2 0 −1.69134 + 0.373340i 0 0.568907i 0 −0.718465 0 2.72123 1.26289i 0
113.3 0 −0.729858 1.57077i 0 1.22502i 0 2.80880 0 −1.93462 + 2.29287i 0
113.4 0 −0.729858 + 1.57077i 0 1.22502i 0 2.80880 0 −1.93462 2.29287i 0
113.5 0 −0.171092 1.72358i 0 3.81594i 0 −2.25057 0 −2.94146 + 0.589781i 0
113.6 0 −0.171092 + 1.72358i 0 3.81594i 0 −2.25057 0 −2.94146 0.589781i 0
113.7 0 1.39399 1.02801i 0 1.12291i 0 −3.21832 0 0.886389 2.86606i 0
113.8 0 1.39399 + 1.02801i 0 1.12291i 0 −3.21832 0 0.886389 + 2.86606i 0
113.9 0 1.69830 0.340259i 0 3.78858i 0 4.37856 0 2.76845 1.15572i 0
113.10 0 1.69830 + 0.340259i 0 3.78858i 0 4.37856 0 2.76845 + 1.15572i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 113.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.f.i 10
3.b odd 2 1 912.2.f.h 10
4.b odd 2 1 456.2.f.a 10
12.b even 2 1 456.2.f.b yes 10
19.b odd 2 1 912.2.f.h 10
57.d even 2 1 inner 912.2.f.i 10
76.d even 2 1 456.2.f.b yes 10
228.b odd 2 1 456.2.f.a 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.f.a 10 4.b odd 2 1
456.2.f.a 10 228.b odd 2 1
456.2.f.b yes 10 12.b even 2 1
456.2.f.b yes 10 76.d even 2 1
912.2.f.h 10 3.b odd 2 1
912.2.f.h 10 19.b odd 2 1
912.2.f.i 10 1.a even 1 1 trivial
912.2.f.i 10 57.d 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 $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{10} + 32T_{5}^{8} + 301T_{5}^{6} + 726T_{5}^{4} + 600T_{5}^{2} + 128$$ T5^10 + 32*T5^8 + 301*T5^6 + 726*T5^4 + 600*T5^2 + 128 $$T_{7}^{5} - T_{7}^{4} - 21T_{7}^{3} + T_{7}^{2} + 100T_{7} + 64$$ T7^5 - T7^4 - 21*T7^3 + T7^2 + 100*T7 + 64 $$T_{29}^{5} + 3T_{29}^{4} - 52T_{29}^{3} - 140T_{29}^{2} + 608T_{29} + 1216$$ T29^5 + 3*T29^4 - 52*T29^3 - 140*T29^2 + 608*T29 + 1216

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} - T^{9} - T^{8} - 2 T^{7} - 2 T^{6} + \cdots + 243$$
$5$ $$T^{10} + 32 T^{8} + 301 T^{6} + \cdots + 128$$
$7$ $$(T^{5} - T^{4} - 21 T^{3} + T^{2} + 100 T + 64)^{2}$$
$11$ $$T^{10} + 50 T^{8} + 889 T^{6} + \cdots + 41472$$
$13$ $$T^{10} + 57 T^{8} + 1014 T^{6} + \cdots + 128$$
$17$ $$T^{10} + 101 T^{8} + 3391 T^{6} + \cdots + 524288$$
$19$ $$T^{10} + 2 T^{9} + 3 T^{8} + \cdots + 2476099$$
$23$ $$T^{10} + 129 T^{8} + 5314 T^{6} + \cdots + 46208$$
$29$ $$(T^{5} + 3 T^{4} - 52 T^{3} - 140 T^{2} + \cdots + 1216)^{2}$$
$31$ $$T^{10} + 134 T^{8} + 6028 T^{6} + \cdots + 107648$$
$37$ $$T^{10} + 174 T^{8} + \cdots + 20891648$$
$41$ $$(T^{5} + 2 T^{4} - 58 T^{3} + 52 T^{2} + \cdots - 256)^{2}$$
$43$ $$(T^{5} + 10 T^{4} - 59 T^{3} - 616 T^{2} + \cdots - 432)^{2}$$
$47$ $$T^{10} + 204 T^{8} + 12093 T^{6} + \cdots + 30752$$
$53$ $$(T^{5} + 13 T^{4} - 22 T^{3} - 798 T^{2} + \cdots + 1432)^{2}$$
$59$ $$(T^{5} + T^{4} - 206 T^{3} - 42 T^{2} + \cdots - 18336)^{2}$$
$61$ $$(T^{5} + 2 T^{4} - 123 T^{3} + 16 T^{2} + \cdots + 3488)^{2}$$
$67$ $$T^{10} + 471 T^{8} + \cdots + 757071872$$
$71$ $$(T^{5} + 4 T^{4} - 194 T^{3} + 48 T^{2} + \cdots + 7936)^{2}$$
$73$ $$(T^{5} + 13 T^{4} - 75 T^{3} - 1837 T^{2} + \cdots - 13752)^{2}$$
$79$ $$T^{10} + 486 T^{8} + \cdots + 11829248$$
$83$ $$T^{10} + 240 T^{8} + 17552 T^{6} + \cdots + 2654208$$
$89$ $$(T^{5} - 2 T^{4} - 222 T^{3} + 1044 T^{2} + \cdots - 432)^{2}$$
$97$ $$T^{10} + 620 T^{8} + \cdots + 169869312$$