Properties

 Label 363.2.f.h Level $363$ Weight $2$ Character orbit 363.f Analytic conductor $2.899$ Analytic rank $0$ Dimension $16$ CM no Inner twists $16$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.f (of order $$10$$, degree $$4$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.89856959337$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: 16.0.26873856000000000000.1 Defining polynomial: $$x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256$$ x^16 + 2*x^14 - 8*x^10 - 16*x^8 - 32*x^6 + 128*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{13} + \beta_{12} - \beta_{8} - 2 \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{15} + 2 \beta_{5}) q^{3} + (4 \beta_{15} + 4 \beta_{9} + 4 \beta_{7} - 4 \beta_{3} - 4 \beta_1 - 4) q^{4} + (\beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{5} - 3 \beta_{14} q^{6} - \beta_{8} q^{7} + (2 \beta_{13} + 4 \beta_{10}) q^{8} + (3 \beta_{11} - 3 \beta_1) q^{9}+O(q^{10})$$ q + (b13 + b12 - b8 - 2*b6 + b2) * q^2 + (-b15 + 2*b5) * q^3 + (4*b15 + 4*b9 + 4*b7 - 4*b3 - 4*b1 - 4) * q^4 + (b15 + b11 + b9 - b7 - b5 - b3 - b1) * q^5 - 3*b14 * q^6 - b8 * q^7 + (2*b13 + 4*b10) * q^8 + (3*b11 - 3*b1) * q^9 $$q + (\beta_{13} + \beta_{12} - \beta_{8} - 2 \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{15} + 2 \beta_{5}) q^{3} + (4 \beta_{15} + 4 \beta_{9} + 4 \beta_{7} - 4 \beta_{3} - 4 \beta_1 - 4) q^{4} + (\beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{5} - 3 \beta_{14} q^{6} - \beta_{8} q^{7} + (2 \beta_{13} + 4 \beta_{10}) q^{8} + (3 \beta_{11} - 3 \beta_1) q^{9} + 3 \beta_{12} q^{10} + ( - 4 \beta_{9} - 4) q^{12} + ( - 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{8} - 2 \beta_{2}) q^{13} + ( - 4 \beta_{15} + 2 \beta_{5}) q^{14} + ( - 3 \beta_{15} - 3 \beta_{9} - 3 \beta_{7} + 3 \beta_1 + 3) q^{15} + ( - 4 \beta_{15} - 4 \beta_{11} - 4 \beta_{9} - 4 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} + \cdots + 4 \beta_1) q^{16}+ \cdots + (5 \beta_{12} - 10 \beta_{4}) q^{98}+O(q^{100})$$ q + (b13 + b12 - b8 - 2*b6 + b2) * q^2 + (-b15 + 2*b5) * q^3 + (4*b15 + 4*b9 + 4*b7 - 4*b3 - 4*b1 - 4) * q^4 + (b15 + b11 + b9 - b7 - b5 - b3 - b1) * q^5 - 3*b14 * q^6 - b8 * q^7 + (2*b13 + 4*b10) * q^8 + (3*b11 - 3*b1) * q^9 + 3*b12 * q^10 + (-4*b9 - 4) * q^12 + (-2*b13 - 2*b12 + 2*b8 - 2*b2) * q^13 + (-4*b15 + 2*b5) * q^14 + (-3*b15 - 3*b9 - 3*b7 + 3*b1 + 3) * q^15 + (-4*b15 - 4*b11 - 4*b9 - 4*b7 + 4*b5 + 4*b3 + 4*b1) * q^16 + (2*b14 + b2) * q^17 + (-3*b14 - 3*b13 - 3*b12 - 3*b10 + 6*b8 + 3*b6 + 3*b4 - 3*b2) * q^18 - 2*b13 * q^19 + (-4*b11 + 8*b1) * q^20 + (-2*b12 + b4) * q^21 + (10*b9 - 5) * q^23 + (-6*b13 - 6*b12 + 6*b8 + 6*b6 - 6*b2) * q^24 - 2*b5 * q^25 + (4*b15 + 4*b9 + 4*b7 + 4*b3 - 4*b1 - 4) * q^26 + (3*b15 + 3*b11 + 3*b9 - 3*b7 - 3*b5 - 3*b3 - 3*b1) * q^27 + 4*b2 * q^28 + (4*b14 + 4*b13 + 4*b12 + 4*b10 - 2*b8 - 4*b6 - 4*b4 + 4*b2) * q^29 + (-6*b13 - 3*b10) * q^30 + 3*b11 * q^31 + 6 * q^34 + (-b13 - b12 + b8 + 2*b6 - b2) * q^35 + (12*b15 - 12*b5) * q^36 + (3*b15 + 3*b9 + 3*b7 - 3*b3 - 3*b1 - 3) * q^37 + (-4*b15 - 4*b11 - 4*b9 + 4*b7 + 4*b5 + 4*b3 + 4*b1) * q^38 + (-2*b14 - 4*b2) * q^39 - 6*b8 * q^40 + (-4*b13 - 8*b10) * q^41 - 6*b1 * q^42 - 7*b12 * q^43 + (-3*b9 + 6) * q^45 + (15*b13 + 15*b12 - 15*b8 + 15*b2) * q^46 + (8*b15 - 4*b5) * q^47 + (-4*b15 - 4*b9 - 4*b7 + 8*b3 + 4*b1 + 4) * q^48 + (-5*b15 - 5*b11 - 5*b9 - 5*b7 + 5*b5 + 5*b3 + 5*b1) * q^49 + (4*b14 + 2*b2) * q^50 + (3*b14 + 3*b13 + 3*b12 + 3*b10 - 3*b8 - 3*b6 - 3*b4 + 3*b2) * q^51 + 8*b13 * q^52 + (-2*b11 + 4*b1) * q^53 + 9*b12 * q^54 + (-8*b9 + 4) * q^56 + (-2*b13 - 2*b12 + 2*b8 - 2*b6 - 2*b2) * q^57 + 12*b5 * q^58 + (b15 + b9 + b7 + b3 - b1 - 1) * q^59 + 12*b7 * q^60 + 2*b2 * q^61 + (-6*b14 - 6*b13 - 6*b12 - 6*b10 + 3*b8 + 6*b6 + 6*b4 - 6*b2) * q^62 + (3*b13 + 3*b10) * q^63 - 8*b11 * q^64 + (-2*b12 + 4*b4) * q^65 - 3 * q^67 + (4*b13 + 4*b12 - 4*b8 - 8*b6 + 4*b2) * q^68 - 15*b15 * q^69 + (-6*b15 - 6*b9 - 6*b7 + 6*b3 + 6*b1 + 6) * q^70 + (b15 + b11 + b9 - b7 - b5 - b3 - b1) * q^71 + (6*b14 - 6*b2) * q^72 + 4*b8 * q^73 + (3*b13 + 6*b10) * q^74 + (-4*b11 + 2*b1) * q^75 - 8*b12 * q^76 + (12*b9 - 12) * q^78 + (-7*b13 - 7*b12 + 7*b8 - 7*b2) * q^79 + (-8*b15 + 4*b5) * q^80 + (-9*b15 - 9*b9 - 9*b7 + 9*b1 + 9) * q^81 + (24*b15 + 24*b11 + 24*b9 + 24*b7 - 24*b5 - 24*b3 - 24*b1) * q^82 + (-2*b14 - b2) * q^83 + (4*b14 + 4*b13 + 4*b12 + 4*b10 + 4*b8 - 4*b6 - 4*b4 + 4*b2) * q^84 + 3*b13 * q^85 + (14*b11 - 28*b1) * q^86 - 6*b4 * q^87 + (-10*b9 + 5) * q^89 - 9*b6 * q^90 + 4*b5 * q^91 + (-20*b15 - 20*b9 - 20*b7 - 20*b3 + 20*b1 + 20) * q^92 + (6*b15 + 6*b11 + 6*b9 + 3*b7 - 6*b5 - 6*b3 - 6*b1) * q^93 - 12*b2 * q^94 + (-4*b14 - 4*b13 - 4*b12 - 4*b10 + 2*b8 + 4*b6 + 4*b4 - 4*b2) * q^95 + 13*b11 * q^97 + (5*b12 - 10*b4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 6 q^{3} - 16 q^{4} - 6 q^{9}+O(q^{10})$$ 16 * q + 6 * q^3 - 16 * q^4 - 6 * q^9 $$16 q + 6 q^{3} - 16 q^{4} - 6 q^{9} - 96 q^{12} + 6 q^{15} - 16 q^{16} - 8 q^{25} - 12 q^{31} + 96 q^{34} - 24 q^{36} - 12 q^{37} + 12 q^{42} + 72 q^{45} + 24 q^{48} - 20 q^{49} + 48 q^{58} + 24 q^{60} + 32 q^{64} - 48 q^{67} - 30 q^{69} + 24 q^{70} + 12 q^{75} - 96 q^{78} + 18 q^{81} + 96 q^{82} + 16 q^{91} + 18 q^{93} - 52 q^{97}+O(q^{100})$$ 16 * q + 6 * q^3 - 16 * q^4 - 6 * q^9 - 96 * q^12 + 6 * q^15 - 16 * q^16 - 8 * q^25 - 12 * q^31 + 96 * q^34 - 24 * q^36 - 12 * q^37 + 12 * q^42 + 72 * q^45 + 24 * q^48 - 20 * q^49 + 48 * q^58 + 24 * q^60 + 32 * q^64 - 48 * q^67 - 30 * q^69 + 24 * q^70 + 12 * q^75 - 96 * q^78 + 18 * q^81 + 96 * q^82 + 16 * q^91 + 18 * q^93 - 52 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{4} ) / 4$$ (v^4) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{5} ) / 4$$ (v^5) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{6} ) / 8$$ (v^6) / 8 $$\beta_{6}$$ $$=$$ $$( \nu^{7} ) / 8$$ (v^7) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{8} ) / 16$$ (v^8) / 16 $$\beta_{8}$$ $$=$$ $$( \nu^{9} ) / 16$$ (v^9) / 16 $$\beta_{9}$$ $$=$$ $$( \nu^{10} ) / 32$$ (v^10) / 32 $$\beta_{10}$$ $$=$$ $$( \nu^{11} ) / 32$$ (v^11) / 32 $$\beta_{11}$$ $$=$$ $$( \nu^{12} ) / 64$$ (v^12) / 64 $$\beta_{12}$$ $$=$$ $$( \nu^{15} ) / 128$$ (v^15) / 128 $$\beta_{13}$$ $$=$$ $$( -\nu^{11} + 32\nu ) / 32$$ (-v^11 + 32*v) / 32 $$\beta_{14}$$ $$=$$ $$( \nu^{13} - 32\nu^{3} ) / 64$$ (v^13 - 32*v^3) / 64 $$\beta_{15}$$ $$=$$ $$( \nu^{14} - 4\nu^{10} - 8\nu^{8} + 64\nu^{2} + 128 ) / 128$$ (v^14 - 4*v^10 - 8*v^8 + 64*v^2 + 128) / 128
 $$\nu$$ $$=$$ $$\beta_{13} + \beta_{10}$$ b13 + b10 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{4}$$ $$=$$ $$4\beta_{3}$$ 4*b3 $$\nu^{5}$$ $$=$$ $$4\beta_{4}$$ 4*b4 $$\nu^{6}$$ $$=$$ $$8\beta_{5}$$ 8*b5 $$\nu^{7}$$ $$=$$ $$8\beta_{6}$$ 8*b6 $$\nu^{8}$$ $$=$$ $$16\beta_{7}$$ 16*b7 $$\nu^{9}$$ $$=$$ $$16\beta_{8}$$ 16*b8 $$\nu^{10}$$ $$=$$ $$32\beta_{9}$$ 32*b9 $$\nu^{11}$$ $$=$$ $$32\beta_{10}$$ 32*b10 $$\nu^{12}$$ $$=$$ $$64\beta_{11}$$ 64*b11 $$\nu^{13}$$ $$=$$ $$64\beta_{14} + 64\beta_{2}$$ 64*b14 + 64*b2 $$\nu^{14}$$ $$=$$ $$128\beta_{15} + 128\beta_{9} + 128\beta_{7} - 128\beta _1 - 128$$ 128*b15 + 128*b9 + 128*b7 - 128*b1 - 128 $$\nu^{15}$$ $$=$$ $$128\beta_{12}$$ 128*b12

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$1 + \beta_{1} + \beta_{3} - \beta_{7} - \beta_{9} - \beta_{15}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 −0.294032 − 1.38331i 1.05097 − 0.946294i 0.294032 + 1.38331i −1.05097 + 0.946294i 1.40647 + 0.147826i 0.575212 + 1.29195i −1.40647 − 0.147826i −0.575212 − 1.29195i 1.40647 − 0.147826i 0.575212 − 1.29195i −1.40647 + 0.147826i −0.575212 + 1.29195i −0.294032 + 1.38331i 1.05097 + 0.946294i 0.294032 − 1.38331i −1.05097 − 0.946294i
−1.98168 1.43977i −1.28716 + 1.15897i 1.23607 + 3.80423i 1.01807 + 1.40126i 4.21940 0.443477i 1.34500 0.437016i 1.51387 4.65921i 0.313585 2.98357i 4.24264i
161.2 −1.98168 1.43977i 0.360114 + 1.69420i 1.23607 + 3.80423i −1.01807 1.40126i 1.72564 3.87585i −1.34500 + 0.437016i 1.51387 4.65921i −2.74064 + 1.22021i 4.24264i
161.3 1.98168 + 1.43977i −1.28716 + 1.15897i 1.23607 + 3.80423i 1.01807 + 1.40126i −4.21940 + 0.443477i −1.34500 + 0.437016i −1.51387 + 4.65921i 0.313585 2.98357i 4.24264i
161.4 1.98168 + 1.43977i 0.360114 + 1.69420i 1.23607 + 3.80423i −1.01807 1.40126i −1.72564 + 3.87585i 1.34500 0.437016i −1.51387 + 4.65921i −2.74064 + 1.22021i 4.24264i
215.1 −0.756934 2.32960i 0.704489 + 1.58231i −3.23607 + 2.35114i −1.64728 0.535233i 3.15290 2.83888i −0.831254 1.14412i 3.96336 + 2.87955i −2.00739 + 2.22943i 4.24264i
215.2 −0.756934 2.32960i 1.72256 + 0.181049i −3.23607 + 2.35114i 1.64728 + 0.535233i −0.882095 4.14993i 0.831254 + 1.14412i 3.96336 + 2.87955i 2.93444 + 0.623735i 4.24264i
215.3 0.756934 + 2.32960i 0.704489 + 1.58231i −3.23607 + 2.35114i −1.64728 0.535233i −3.15290 + 2.83888i 0.831254 + 1.14412i −3.96336 2.87955i −2.00739 + 2.22943i 4.24264i
215.4 0.756934 + 2.32960i 1.72256 + 0.181049i −3.23607 + 2.35114i 1.64728 + 0.535233i 0.882095 + 4.14993i −0.831254 1.14412i −3.96336 2.87955i 2.93444 + 0.623735i 4.24264i
233.1 −0.756934 + 2.32960i 0.704489 1.58231i −3.23607 2.35114i −1.64728 + 0.535233i 3.15290 + 2.83888i −0.831254 + 1.14412i 3.96336 2.87955i −2.00739 2.22943i 4.24264i
233.2 −0.756934 + 2.32960i 1.72256 0.181049i −3.23607 2.35114i 1.64728 0.535233i −0.882095 + 4.14993i 0.831254 1.14412i 3.96336 2.87955i 2.93444 0.623735i 4.24264i
233.3 0.756934 2.32960i 0.704489 1.58231i −3.23607 2.35114i −1.64728 + 0.535233i −3.15290 2.83888i 0.831254 1.14412i −3.96336 + 2.87955i −2.00739 2.22943i 4.24264i
233.4 0.756934 2.32960i 1.72256 0.181049i −3.23607 2.35114i 1.64728 0.535233i 0.882095 4.14993i −0.831254 + 1.14412i −3.96336 + 2.87955i 2.93444 0.623735i 4.24264i
239.1 −1.98168 + 1.43977i −1.28716 1.15897i 1.23607 3.80423i 1.01807 1.40126i 4.21940 + 0.443477i 1.34500 + 0.437016i 1.51387 + 4.65921i 0.313585 + 2.98357i 4.24264i
239.2 −1.98168 + 1.43977i 0.360114 1.69420i 1.23607 3.80423i −1.01807 + 1.40126i 1.72564 + 3.87585i −1.34500 0.437016i 1.51387 + 4.65921i −2.74064 1.22021i 4.24264i
239.3 1.98168 1.43977i −1.28716 1.15897i 1.23607 3.80423i 1.01807 1.40126i −4.21940 0.443477i −1.34500 0.437016i −1.51387 4.65921i 0.313585 + 2.98357i 4.24264i
239.4 1.98168 1.43977i 0.360114 1.69420i 1.23607 3.80423i −1.01807 + 1.40126i −1.72564 3.87585i 1.34500 + 0.437016i −1.51387 4.65921i −2.74064 1.22021i 4.24264i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.f.h 16
3.b odd 2 1 inner 363.2.f.h 16
11.b odd 2 1 inner 363.2.f.h 16
11.c even 5 1 363.2.d.c 4
11.c even 5 3 inner 363.2.f.h 16
11.d odd 10 1 363.2.d.c 4
11.d odd 10 3 inner 363.2.f.h 16
33.d even 2 1 inner 363.2.f.h 16
33.f even 10 1 363.2.d.c 4
33.f even 10 3 inner 363.2.f.h 16
33.h odd 10 1 363.2.d.c 4
33.h odd 10 3 inner 363.2.f.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.d.c 4 11.c even 5 1
363.2.d.c 4 11.d odd 10 1
363.2.d.c 4 33.f even 10 1
363.2.d.c 4 33.h odd 10 1
363.2.f.h 16 1.a even 1 1 trivial
363.2.f.h 16 3.b odd 2 1 inner
363.2.f.h 16 11.b odd 2 1 inner
363.2.f.h 16 11.c even 5 3 inner
363.2.f.h 16 11.d odd 10 3 inner
363.2.f.h 16 33.d even 2 1 inner
363.2.f.h 16 33.f even 10 3 inner
363.2.f.h 16 33.h odd 10 3 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{8} + 6T_{2}^{6} + 36T_{2}^{4} + 216T_{2}^{2} + 1296$$ T2^8 + 6*T2^6 + 36*T2^4 + 216*T2^2 + 1296 $$T_{5}^{8} - 3T_{5}^{6} + 9T_{5}^{4} - 27T_{5}^{2} + 81$$ T5^8 - 3*T5^6 + 9*T5^4 - 27*T5^2 + 81 $$T_{7}^{8} - 2T_{7}^{6} + 4T_{7}^{4} - 8T_{7}^{2} + 16$$ T7^8 - 2*T7^6 + 4*T7^4 - 8*T7^2 + 16

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} + 6 T^{6} + 36 T^{4} + 216 T^{2} + \cdots + 1296)^{2}$$
$3$ $$(T^{8} - 3 T^{7} + 6 T^{6} - 9 T^{5} + 9 T^{4} + \cdots + 81)^{2}$$
$5$ $$(T^{8} - 3 T^{6} + 9 T^{4} - 27 T^{2} + \cdots + 81)^{2}$$
$7$ $$(T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2}$$
$11$ $$T^{16}$$
$13$ $$(T^{8} - 8 T^{6} + 64 T^{4} - 512 T^{2} + \cdots + 4096)^{2}$$
$17$ $$(T^{8} + 6 T^{6} + 36 T^{4} + 216 T^{2} + \cdots + 1296)^{2}$$
$19$ $$(T^{8} - 8 T^{6} + 64 T^{4} - 512 T^{2} + \cdots + 4096)^{2}$$
$23$ $$(T^{2} + 75)^{8}$$
$29$ $$(T^{8} + 24 T^{6} + 576 T^{4} + \cdots + 331776)^{2}$$
$31$ $$(T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{4}$$
$37$ $$(T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{4}$$
$41$ $$(T^{8} + 96 T^{6} + 9216 T^{4} + \cdots + 84934656)^{2}$$
$43$ $$(T^{2} + 98)^{8}$$
$47$ $$(T^{8} - 48 T^{6} + 2304 T^{4} + \cdots + 5308416)^{2}$$
$53$ $$(T^{8} - 12 T^{6} + 144 T^{4} + \cdots + 20736)^{2}$$
$59$ $$(T^{8} - 3 T^{6} + 9 T^{4} - 27 T^{2} + \cdots + 81)^{2}$$
$61$ $$(T^{8} - 8 T^{6} + 64 T^{4} - 512 T^{2} + \cdots + 4096)^{2}$$
$67$ $$(T + 3)^{16}$$
$71$ $$(T^{8} - 3 T^{6} + 9 T^{4} - 27 T^{2} + \cdots + 81)^{2}$$
$73$ $$(T^{8} - 32 T^{6} + 1024 T^{4} + \cdots + 1048576)^{2}$$
$79$ $$(T^{8} - 98 T^{6} + 9604 T^{4} + \cdots + 92236816)^{2}$$
$83$ $$(T^{8} + 6 T^{6} + 36 T^{4} + 216 T^{2} + \cdots + 1296)^{2}$$
$89$ $$(T^{2} + 75)^{8}$$
$97$ $$(T^{4} + 13 T^{3} + 169 T^{2} + \cdots + 28561)^{4}$$