# Properties

 Label 1024.2.g.g Level $1024$ Weight $2$ Character orbit 1024.g Analytic conductor $8.177$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.g (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.17668116698$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ x^16 - x^8 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{14} + \beta_{11}) q^{3} + (\beta_{13} + \beta_{10} + \beta_{8} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{5} + ( - \beta_{15} + \beta_{12} + \beta_{7} + \beta_{5}) q^{7} + (2 \beta_{10} + \beta_{8} + \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + (b14 + b11) * q^3 + (b13 + b10 + b8 + b4 + b3 + b2 + b1) * q^5 + (-b15 + b12 + b7 + b5) * q^7 + (2*b10 + b8 + b2 + b1) * q^9 $$q + (\beta_{14} + \beta_{11}) q^{3} + (\beta_{13} + \beta_{10} + \beta_{8} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{5} + ( - \beta_{15} + \beta_{12} + \beta_{7} + \beta_{5}) q^{7} + (2 \beta_{10} + \beta_{8} + \beta_{2} + \beta_1) q^{9} + ( - \beta_{12} + \beta_{11} - \beta_{7} + 2 \beta_{6}) q^{11} + ( - \beta_{13} - \beta_{10} - 2 \beta_{8} - \beta_{4} - \beta_1 - 1) q^{13} + (\beta_{14} + \beta_{11} - \beta_{9} - \beta_{7} + \beta_{6}) q^{15} + (\beta_{13} - \beta_{8} - \beta_{4} + 3 \beta_1 + 1) q^{17} + (\beta_{15} + \beta_{12} - \beta_{11} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5}) q^{19} + ( - 2 \beta_{13} - \beta_{8} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{21} + (3 \beta_{14} + 2 \beta_{12} + \beta_{11} - \beta_{9} - 3 \beta_{7} - 2 \beta_{6} - 3 \beta_{5}) q^{23} + (\beta_{13} + \beta_{10} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{25} + (3 \beta_{15} + 3 \beta_{14} + \beta_{11} - \beta_{9} - \beta_{7} - 2 \beta_{5}) q^{27} + ( - 3 \beta_{10} - \beta_{8} - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{29} + ( - 4 \beta_{15} + 2 \beta_{12} + \beta_{11} - \beta_{9} - 4 \beta_{7} - 3 \beta_{6} - \beta_{5}) q^{31} + ( - \beta_{13} - 2 \beta_{10} - 2 \beta_{8} + \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 5) q^{33} + (2 \beta_{14} + \beta_{11} - \beta_{7}) q^{35} + (2 \beta_{13} + 3 \beta_{10} + 4 \beta_{8} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{37}+ \cdots + ( - \beta_{14} + \beta_{12} - 7 \beta_{11} - \beta_{9} - 6 \beta_{7} - 2 \beta_{6} - 2 \beta_{5}) q^{99}+O(q^{100})$$ q + (b14 + b11) * q^3 + (b13 + b10 + b8 + b4 + b3 + b2 + b1) * q^5 + (-b15 + b12 + b7 + b5) * q^7 + (2*b10 + b8 + b2 + b1) * q^9 + (-b12 + b11 - b7 + 2*b6) * q^11 + (-b13 - b10 - 2*b8 - b4 - b1 - 1) * q^13 + (b14 + b11 - b9 - b7 + b6) * q^15 + (b13 - b8 - b4 + 3*b1 + 1) * q^17 + (b15 + b12 - b11 + b9 + b7 + b6 - b5) * q^19 + (-2*b13 - b8 - 2*b4 + b3 + 2*b2 - 3*b1 + 3) * q^21 + (3*b14 + 2*b12 + b11 - b9 - 3*b7 - 2*b6 - 3*b5) * q^23 + (b13 + b10 + 2*b4 + 2*b3 + 2*b2 - b1 + 1) * q^25 + (3*b15 + 3*b14 + b11 - b9 - b7 - 2*b5) * q^27 + (-3*b10 - b8 - 2*b4 + b3 - 3*b2 + b1 + 3) * q^29 + (-4*b15 + 2*b12 + b11 - b9 - 4*b7 - 3*b6 - b5) * q^31 + (-b13 - 2*b10 - 2*b8 + b4 - 2*b3 - 4*b2 - 2*b1 - 5) * q^33 + (2*b14 + b11 - b7) * q^35 + (2*b13 + 3*b10 + 4*b8 + 2*b4 + 4*b3 + 2*b2 + b1 - 2) * q^37 + (-2*b15 - b14 + b7 - b6) * q^39 + (2*b13 + 2*b10 + 6*b8 + b3 + 2*b2 + 2*b1 + 1) * q^41 + (b12 - b11 + 5*b9 + 2*b7 - 2*b6) * q^43 + (b13 - b3 + b2 + 2*b1) * q^45 + (b15 - b14 - 4*b11 - 2*b9 + b7 + 2*b6 - 4*b5) * q^47 + (-b13 - 5*b10 - 4*b4 - 2*b1 + 4) * q^49 + (2*b15 + 2*b12 + b11 + 2*b9 - b7 - b6 + b5) * q^51 + (-b13 + 2*b8 + 2*b4 - 5*b3 - 2*b2 + b1 - 4) * q^53 + (-3*b15 + 2*b14 - 2*b12 + 3*b11 - 5*b9 - 6*b7 + 2*b6 - 3*b5) * q^55 + (b13 + b10 + 2*b4 + b3 + b2 - b1) * q^57 + (b15 + b14 + 5*b12 + b11 + b9 + b7 + 2*b5) * q^59 + (-4*b10 - 2*b8 + b4 - 3*b3 - 4*b2 - 3*b1 + 1) * q^61 + (-3*b15 + 4*b14 + 3*b12 + 5*b11 + b9 + 3*b7 - 2*b6 + b5) * q^63 + (-4*b13 - 2*b10 - 5*b8 - 2*b4 - 2*b3 - 4*b2 - 5*b1 - 2) * q^65 + (3*b14 - 2*b12 + 4*b11 + 2*b9 + b7 + b6 + b5) * q^67 + (2*b13 + 6*b10 - b8 + 2*b4 - b3 + 2*b2 + 3*b1 - 3) * q^69 + (b15 - 2*b14 + 2*b12 + 2*b7 + 3*b6 + 2*b5) * q^71 + (-5*b13 - b10 - 2*b8 - b3 - 3*b2 - 3*b1 - 4) * q^73 + (-3*b15 + 3*b14 - b12 + 4*b11 - 3*b9 + 2*b6) * q^75 + (-2*b13 + 2*b10 - 3*b8 + 2*b4 - 5*b3 - 4*b2 + 3*b1 - 5) * q^77 + (-b15 + b14 + 3*b11 + b9 - b7 - b6 + 3*b5) * q^79 + (-b13 - 4*b8 + b4 - 3*b3 + 2*b1 - 1) * q^81 + (3*b15 - 4*b12 + b11 - 4*b9 - b7 + 6*b6 - 6*b5) * q^83 + (b13 + 3*b8 + b4 - 3*b3 - b2 - 2*b1 + 2) * q^85 + (-2*b15 - 3*b14 + 2*b12 - 5*b11 + 5*b9 + b7 - 2*b6 + 3*b5) * q^87 + (-2*b3 + 3*b2 - b1 - 2) * q^89 + (-b15 - b14 - 3*b12 - 2*b11 - b9 - b7 - 2*b5) * q^91 + (4*b8 - 2*b4 - 4*b3 - 4*b1 - 2) * q^93 + (2*b15 + 3*b14 + 3*b12 - 3*b11 - b9 - 2*b7 - 4*b6 - b5) * q^95 + (3*b13 + 5*b8 + 3*b4 + 5*b1 - 1) * q^97 + (-b14 + b12 - 7*b11 - b9 - 6*b7 - 2*b6 - 2*b5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{5} + 16 q^{9}+O(q^{10})$$ 16 * q + 8 * q^5 + 16 * q^9 $$16 q + 8 q^{5} + 16 q^{9} - 24 q^{13} + 48 q^{21} + 32 q^{25} + 8 q^{29} - 80 q^{33} - 8 q^{37} + 16 q^{41} - 8 q^{45} - 40 q^{53} + 16 q^{57} - 8 q^{61} - 32 q^{65} - 32 q^{73} - 32 q^{77} + 32 q^{85} - 32 q^{89} - 48 q^{93} - 16 q^{97}+O(q^{100})$$ 16 * q + 8 * q^5 + 16 * q^9 - 24 * q^13 + 48 * q^21 + 32 * q^25 + 8 * q^29 - 80 * q^33 - 8 * q^37 + 16 * q^41 - 8 * q^45 - 40 * q^53 + 16 * q^57 - 8 * q^61 - 32 * q^65 - 32 * q^73 - 32 * q^77 + 32 * q^85 - 32 * q^89 - 48 * q^93 - 16 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{48}^{6}$$ v^6 $$\beta_{2}$$ $$=$$ $$\zeta_{48}^{10} + \zeta_{48}^{2}$$ v^10 + v^2 $$\beta_{3}$$ $$=$$ $$\zeta_{48}^{12}$$ v^12 $$\beta_{4}$$ $$=$$ $$\zeta_{48}^{14} + \zeta_{48}^{8} + \zeta_{48}^{4} + \zeta_{48}^{2}$$ v^14 + v^8 + v^4 + v^2 $$\beta_{5}$$ $$=$$ $$\zeta_{48}^{15} + \zeta_{48}^{5} + \zeta_{48}$$ v^15 + v^5 + v $$\beta_{6}$$ $$=$$ $$\zeta_{48}^{13} + \zeta_{48}^{9} + \zeta_{48}^{3} - \zeta_{48}$$ v^13 + v^9 + v^3 - v $$\beta_{7}$$ $$=$$ $$-\zeta_{48}^{15} - \zeta_{48}^{5} + \zeta_{48}$$ -v^15 - v^5 + v $$\beta_{8}$$ $$=$$ $$-\zeta_{48}^{10} + \zeta_{48}^{2}$$ -v^10 + v^2 $$\beta_{9}$$ $$=$$ $$-\zeta_{48}^{15} + \zeta_{48}^{5} - \zeta_{48}$$ -v^15 + v^5 - v $$\beta_{10}$$ $$=$$ $$-\zeta_{48}^{14} + \zeta_{48}^{8} - \zeta_{48}^{4} - \zeta_{48}^{2}$$ -v^14 + v^8 - v^4 - v^2 $$\beta_{11}$$ $$=$$ $$-\zeta_{48}^{13} + \zeta_{48}^{9} - \zeta_{48}^{3} - \zeta_{48}$$ -v^13 + v^9 - v^3 - v $$\beta_{12}$$ $$=$$ $$-\zeta_{48}^{13} + \zeta_{48}^{9} + \zeta_{48}^{3} - \zeta_{48}$$ -v^13 + v^9 + v^3 - v $$\beta_{13}$$ $$=$$ $$-\zeta_{48}^{14} - \zeta_{48}^{12} - \zeta_{48}^{8} + \zeta_{48}^{4} - \zeta_{48}^{2}$$ -v^14 - v^12 - v^8 + v^4 - v^2 $$\beta_{14}$$ $$=$$ $$-\zeta_{48}^{15} + \zeta_{48}^{13} + \zeta_{48}^{11} + \zeta_{48}^{7} + \zeta_{48}$$ -v^15 + v^13 + v^11 + v^7 + v $$\beta_{15}$$ $$=$$ $$-\zeta_{48}^{13} - \zeta_{48}^{11} + \zeta_{48}^{7} + \zeta_{48}^{5}$$ -v^13 - v^11 + v^7 + v^5
 $$\zeta_{48}$$ $$=$$ $$( \beta_{7} + \beta_{5} ) / 2$$ (b7 + b5) / 2 $$\zeta_{48}^{2}$$ $$=$$ $$( \beta_{8} + \beta_{2} ) / 2$$ (b8 + b2) / 2 $$\zeta_{48}^{3}$$ $$=$$ $$( \beta_{12} - \beta_{11} ) / 2$$ (b12 - b11) / 2 $$\zeta_{48}^{4}$$ $$=$$ $$( \beta_{13} + \beta_{4} + \beta_{3} ) / 2$$ (b13 + b4 + b3) / 2 $$\zeta_{48}^{5}$$ $$=$$ $$( \beta_{9} + \beta_{5} ) / 2$$ (b9 + b5) / 2 $$\zeta_{48}^{6}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{48}^{7}$$ $$=$$ $$( \beta_{15} + \beta_{14} - \beta_{9} - \beta_{7} - \beta_{5} ) / 2$$ (b15 + b14 - b9 - b7 - b5) / 2 $$\zeta_{48}^{8}$$ $$=$$ $$( \beta_{10} + \beta_{4} ) / 2$$ (b10 + b4) / 2 $$\zeta_{48}^{9}$$ $$=$$ $$( \beta_{11} + \beta_{7} + \beta_{6} + \beta_{5} ) / 2$$ (b11 + b7 + b6 + b5) / 2 $$\zeta_{48}^{10}$$ $$=$$ $$( -\beta_{8} + \beta_{2} ) / 2$$ (-b8 + b2) / 2 $$\zeta_{48}^{11}$$ $$=$$ $$( -\beta_{15} + \beta_{14} + \beta_{12} - \beta_{7} - \beta_{6} ) / 2$$ (-b15 + b14 + b12 - b7 - b6) / 2 $$\zeta_{48}^{12}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{48}^{13}$$ $$=$$ $$( -\beta_{12} + \beta_{6} ) / 2$$ (-b12 + b6) / 2 $$\zeta_{48}^{14}$$ $$=$$ $$( -\beta_{13} - \beta_{10} - \beta_{8} - \beta_{3} - \beta_{2} ) / 2$$ (-b13 - b10 - b8 - b3 - b2) / 2 $$\zeta_{48}^{15}$$ $$=$$ $$( -\beta_{9} - \beta_{7} ) / 2$$ (-b9 - b7) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 −0.991445 + 0.130526i −0.608761 − 0.793353i 0.608761 + 0.793353i 0.991445 − 0.130526i −0.793353 − 0.608761i −0.130526 + 0.991445i 0.130526 − 0.991445i 0.793353 + 0.608761i −0.793353 + 0.608761i −0.130526 − 0.991445i 0.130526 + 0.991445i 0.793353 − 0.608761i −0.991445 − 0.130526i −0.608761 + 0.793353i 0.608761 − 0.793353i 0.991445 + 0.130526i
0 −0.580775 + 1.40211i 0 3.29788 1.36603i 0 −1.02642 + 1.02642i 0 0.492694 + 0.492694i 0
129.2 0 −0.356604 + 0.860919i 0 −0.883663 + 0.366025i 0 2.35207 2.35207i 0 1.50731 + 1.50731i 0
129.3 0 0.356604 0.860919i 0 −0.883663 + 0.366025i 0 −2.35207 + 2.35207i 0 1.50731 + 1.50731i 0
129.4 0 0.580775 1.40211i 0 3.29788 1.36603i 0 1.02642 1.02642i 0 0.492694 + 0.492694i 0
385.1 0 −2.70868 + 1.12197i 0 0.151613 0.366025i 0 −3.06528 3.06528i 0 3.95680 3.95680i 0
385.2 0 −0.445644 + 0.184592i 0 −0.565826 + 1.36603i 0 0.135131 + 0.135131i 0 −1.95680 + 1.95680i 0
385.3 0 0.445644 0.184592i 0 −0.565826 + 1.36603i 0 −0.135131 0.135131i 0 −1.95680 + 1.95680i 0
385.4 0 2.70868 1.12197i 0 0.151613 0.366025i 0 3.06528 + 3.06528i 0 3.95680 3.95680i 0
641.1 0 −2.70868 1.12197i 0 0.151613 + 0.366025i 0 −3.06528 + 3.06528i 0 3.95680 + 3.95680i 0
641.2 0 −0.445644 0.184592i 0 −0.565826 1.36603i 0 0.135131 0.135131i 0 −1.95680 1.95680i 0
641.3 0 0.445644 + 0.184592i 0 −0.565826 1.36603i 0 −0.135131 + 0.135131i 0 −1.95680 1.95680i 0
641.4 0 2.70868 + 1.12197i 0 0.151613 + 0.366025i 0 3.06528 3.06528i 0 3.95680 + 3.95680i 0
897.1 0 −0.580775 1.40211i 0 3.29788 + 1.36603i 0 −1.02642 1.02642i 0 0.492694 0.492694i 0
897.2 0 −0.356604 0.860919i 0 −0.883663 0.366025i 0 2.35207 + 2.35207i 0 1.50731 1.50731i 0
897.3 0 0.356604 + 0.860919i 0 −0.883663 0.366025i 0 −2.35207 2.35207i 0 1.50731 1.50731i 0
897.4 0 0.580775 + 1.40211i 0 3.29788 + 1.36603i 0 1.02642 + 1.02642i 0 0.492694 0.492694i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 897.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
32.g even 8 1 inner
32.h odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.g.g yes 16
4.b odd 2 1 inner 1024.2.g.g yes 16
8.b even 2 1 1024.2.g.d yes 16
8.d odd 2 1 1024.2.g.d yes 16
16.e even 4 1 1024.2.g.a 16
16.e even 4 1 1024.2.g.f yes 16
16.f odd 4 1 1024.2.g.a 16
16.f odd 4 1 1024.2.g.f yes 16
32.g even 8 1 1024.2.g.a 16
32.g even 8 1 1024.2.g.d yes 16
32.g even 8 1 1024.2.g.f yes 16
32.g even 8 1 inner 1024.2.g.g yes 16
32.h odd 8 1 1024.2.g.a 16
32.h odd 8 1 1024.2.g.d yes 16
32.h odd 8 1 1024.2.g.f yes 16
32.h odd 8 1 inner 1024.2.g.g yes 16
64.i even 16 1 4096.2.a.i 8
64.i even 16 1 4096.2.a.s 8
64.j odd 16 1 4096.2.a.i 8
64.j odd 16 1 4096.2.a.s 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1024.2.g.a 16 16.e even 4 1
1024.2.g.a 16 16.f odd 4 1
1024.2.g.a 16 32.g even 8 1
1024.2.g.a 16 32.h odd 8 1
1024.2.g.d yes 16 8.b even 2 1
1024.2.g.d yes 16 8.d odd 2 1
1024.2.g.d yes 16 32.g even 8 1
1024.2.g.d yes 16 32.h odd 8 1
1024.2.g.f yes 16 16.e even 4 1
1024.2.g.f yes 16 16.f odd 4 1
1024.2.g.f yes 16 32.g even 8 1
1024.2.g.f yes 16 32.h odd 8 1
1024.2.g.g yes 16 1.a even 1 1 trivial
1024.2.g.g yes 16 4.b odd 2 1 inner
1024.2.g.g yes 16 32.g even 8 1 inner
1024.2.g.g yes 16 32.h odd 8 1 inner
4096.2.a.i 8 64.i even 16 1
4096.2.a.i 8 64.j odd 16 1
4096.2.a.s 8 64.i even 16 1
4096.2.a.s 8 64.j odd 16 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1024, [\chi])$$:

 $$T_{3}^{16} - 8T_{3}^{14} + 32T_{3}^{12} + 208T_{3}^{10} + 568T_{3}^{8} + 416T_{3}^{6} + 128T_{3}^{4} - 64T_{3}^{2} + 16$$ T3^16 - 8*T3^14 + 32*T3^12 + 208*T3^10 + 568*T3^8 + 416*T3^6 + 128*T3^4 - 64*T3^2 + 16 $$T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{5} + 32T_{5}^{4} + 40T_{5}^{3} + 16T_{5}^{2} + 4$$ T5^8 - 4*T5^7 + 8*T5^5 + 32*T5^4 + 40*T5^3 + 16*T5^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} - 8 T^{14} + 32 T^{12} + 208 T^{10} + \cdots + 16$$
$5$ $$(T^{8} - 4 T^{7} + 8 T^{5} + 32 T^{4} + 40 T^{3} + \cdots + 4)^{2}$$
$7$ $$T^{16} + 480 T^{12} + 45344 T^{8} + \cdots + 256$$
$11$ $$T^{16} + 8 T^{14} + 32 T^{12} + \cdots + 406586896$$
$13$ $$(T^{8} + 12 T^{7} + 64 T^{6} + 208 T^{5} + \cdots + 2116)^{2}$$
$17$ $$(T^{8} + 72 T^{6} + 1664 T^{4} + \cdots + 40000)^{2}$$
$19$ $$T^{16} + 24 T^{14} + 288 T^{12} + \cdots + 4477456$$
$23$ $$T^{16} + 4704 T^{12} + \cdots + 6505390336$$
$29$ $$(T^{8} - 4 T^{7} + 64 T^{6} + \cdots + 1119364)^{2}$$
$31$ $$(T^{8} - 224 T^{6} + 15680 T^{4} + \cdots + 1024)^{2}$$
$37$ $$(T^{8} + 4 T^{7} + 48 T^{6} - 320 T^{5} + \cdots + 8836)^{2}$$
$41$ $$(T^{8} - 8 T^{7} + 32 T^{6} + 144 T^{5} + \cdots + 8464)^{2}$$
$43$ $$T^{16} + 280 T^{14} + \cdots + 1749006250000$$
$47$ $$(T^{8} + 256 T^{6} + 17984 T^{4} + \cdots + 1364224)^{2}$$
$53$ $$(T^{8} + 20 T^{7} + 160 T^{6} + \cdots + 21316)^{2}$$
$59$ $$T^{16} - 216 T^{14} + \cdots + 36030006250000$$
$61$ $$(T^{8} + 4 T^{7} - 128 T^{6} - 464 T^{5} + \cdots + 2500)^{2}$$
$67$ $$T^{16} - 24 T^{14} + \cdots + 78650008458256$$
$71$ $$T^{16} + 16608 T^{12} + \cdots + 355196928256$$
$73$ $$(T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 2062096)^{2}$$
$79$ $$(T^{8} + 128 T^{6} + 4928 T^{4} + \cdots + 160000)^{2}$$
$83$ $$T^{16} + 168 T^{14} + \cdots + 21848556971536$$
$89$ $$(T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 85264)^{2}$$
$97$ $$(T^{4} + 4 T^{3} - 148 T^{2} - 304 T + 376)^{4}$$