# Properties

 Label 192.2.k.a Level $192$ Weight $2$ Character orbit 192.k Analytic conductor $1.533$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 192.k (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.53312771881$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.163368480538624.2 Defining polynomial: $$x^{12} - 2x^{10} - 2x^{8} + 16x^{6} - 8x^{4} - 32x^{2} + 64$$ x^12 - 2*x^10 - 2*x^8 + 16*x^6 - 8*x^4 - 32*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{10} q^{3} + \beta_{7} q^{5} + ( - \beta_{11} + 1) q^{7} + ( - \beta_{9} - \beta_{8} - \beta_{5} + \beta_{3}) q^{9}+O(q^{10})$$ q - b10 * q^3 + b7 * q^5 + (-b11 + 1) * q^7 + (-b9 - b8 - b5 + b3) * q^9 $$q - \beta_{10} q^{3} + \beta_{7} q^{5} + ( - \beta_{11} + 1) q^{7} + ( - \beta_{9} - \beta_{8} - \beta_{5} + \beta_{3}) q^{9} + ( - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{11} + \beta_{5}) q^{13} + ( - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_1) q^{15} + ( - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3}) q^{17} + (\beta_{11} + \beta_{10} + \beta_{5} - \beta_{3} + \beta_1 + 1) q^{19} + ( - 2 \beta_{10} + \beta_{8} - \beta_{7} - \beta_{2} - \beta_1 - 1) q^{21} + (\beta_{10} + \beta_{9} + \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{23} + (\beta_{10} + \beta_{9} - \beta_{4} - \beta_{3} + \beta_1) q^{25} + (\beta_{11} + \beta_{9} + \beta_{8} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{2} + \beta_1 - 1) q^{27} + (2 \beta_{9} - \beta_{6} + 2 \beta_{4}) q^{29} + (\beta_{10} + \beta_{9} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{31} + (\beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{33} + (\beta_{8} - \beta_{2}) q^{35} + (\beta_{11} + 2 \beta_{10} + \beta_{5} - 2 \beta_{3}) q^{37} + ( - \beta_{10} - \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{2} - 2) q^{39} + (2 \beta_{8} - \beta_{7} - \beta_{6}) q^{41} + ( - \beta_{11} + \beta_{9} + \beta_{5} - \beta_{4} + \beta_1 - 1) q^{43} + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} + \beta_1 - 1) q^{45} + ( - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3}) q^{47} + (\beta_{10} - \beta_{9} + \beta_{4} - \beta_{3} - 1) q^{49} + ( - \beta_{8} - 2 \beta_{7} + \beta_{2} - 2 \beta_1 - 2) q^{51} + (4 \beta_{10} - 2 \beta_{8} + \beta_{7} + 4 \beta_{3} + 2 \beta_{2}) q^{53} + (2 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{4} - \beta_{3} - 2) q^{55} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{57} + ( - \beta_{9} + 4 \beta_{6} - \beta_{4}) q^{59} + (\beta_{11} + 2 \beta_{9} - \beta_{5} - 2 \beta_{4}) q^{61} + (2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{63} + ( - \beta_{7} + \beta_{6} - 2 \beta_{2}) q^{65} + ( - 2 \beta_{11} - \beta_{10} - 2 \beta_{5} + \beta_{3} - 2 \beta_1 - 2) q^{67} + ( - \beta_{11} - 2 \beta_{10} + \beta_{8} - 2 \beta_{7} - \beta_{5} - 4 \beta_{3} - \beta_{2} + \cdots + 1) q^{69}+ \cdots + ( - \beta_{11} + \beta_{10} + 2 \beta_{7} - \beta_{5} - \beta_{3} + 5 \beta_1 + 5) q^{99}+O(q^{100})$$ q - b10 * q^3 + b7 * q^5 + (-b11 + 1) * q^7 + (-b9 - b8 - b5 + b3) * q^9 + (-b9 - b8 - b4 - b2) * q^11 + (-b11 + b5) * q^13 + (-b9 + b7 + b6 - b5 + b3 - b1) * q^15 + (-b10 - b9 - b7 + b6 - b4 - b3) * q^17 + (b11 + b10 + b5 - b3 + b1 + 1) * q^19 + (-2*b10 + b8 - b7 - b2 - b1 - 1) * q^21 + (b10 + b9 + b4 + b3 + 2*b2) * q^23 + (b10 + b9 - b4 - b3 + b1) * q^25 + (b11 + b9 + b8 - 2*b6 - b5 + 2*b4 + b2 + b1 - 1) * q^27 + (2*b9 - b6 + 2*b4) * q^29 + (b10 + b9 + b5 - b4 - b3 - b1) * q^31 + (b11 + b10 + 2*b9 + b7 - b6 + b4 + 2*b3 + b2 - 1) * q^33 + (b8 - b2) * q^35 + (b11 + 2*b10 + b5 - 2*b3) * q^37 + (-b10 - b7 + b6 - b4 - 2*b2 - 2) * q^39 + (2*b8 - b7 - b6) * q^41 + (-b11 + b9 + b5 - b4 + b1 - 1) * q^43 + (b11 - 2*b9 + b8 + b6 - b5 + b2 + b1 - 1) * q^45 + (-2*b10 + 2*b9 + 2*b8 - 2*b7 - 2*b6 + 2*b4 - 2*b3) * q^47 + (b10 - b9 + b4 - b3 - 1) * q^49 + (-b8 - 2*b7 + b2 - 2*b1 - 2) * q^51 + (4*b10 - 2*b8 + b7 + 4*b3 + 2*b2) * q^53 + (2*b11 + b10 - b9 + b4 - b3 - 2) * q^55 + (b9 - b8 + b7 + b6 + b5 - b3 - b1) * q^57 + (-b9 + 4*b6 - b4) * q^59 + (b11 + 2*b9 - b5 - 2*b4) * q^61 + (2*b10 - 2*b9 - 2*b8 + b5 - 2*b4 + 2*b3 + 3*b1) * q^63 + (-b7 + b6 - 2*b2) * q^65 + (-2*b11 - b10 - 2*b5 + b3 - 2*b1 - 2) * q^67 + (-b11 - 2*b10 + b8 - 2*b7 - b5 - 4*b3 - b2 + b1 + 1) * q^69 + (-b10 - b9 + 2*b7 - 2*b6 - b4 - b3) * q^71 + (-2*b10 - 2*b9 + 2*b5 + 2*b4 + 2*b3 - 2*b1) * q^73 + (-b11 + 2*b9 + b8 + b5 + b4 + b2 - 3*b1 + 3) * q^75 + (-4*b9 - 2*b8 - 4*b4 - 2*b2) * q^77 + (-b10 - b9 - b5 + b4 + b3 + 5*b1) * q^79 + (-2*b11 + b10 - b9 + 2*b7 - 2*b6 + b4 - b3 + 1) * q^81 + (3*b10 - b8 + 4*b7 + 3*b3 + b2) * q^83 + (-2*b11 - 4*b10 - 2*b5 + 4*b3 + 2*b1 + 2) * q^85 + (b11 - b9 + b7 - b6 - b3 - 2*b2 + 5) * q^87 + (-3*b10 + 3*b9 + 2*b8 + 3*b4 - 3*b3) * q^89 + (b11 - 2*b9 - b5 + 2*b4 - 5*b1 + 5) * q^91 + (-b11 + 2*b9 + b6 + b5 - 2*b4 - 2*b1 + 2) * q^93 + (-b10 + b9 - 2*b8 + b4 - b3) * q^95 + (-2*b11 - 3*b10 + 3*b9 - 3*b4 + 3*b3 - 2) * q^97 + (-b11 + b10 + 2*b7 - b5 - b3 + 5*b1 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 2 q^{3} + 8 q^{7}+O(q^{10})$$ 12 * q + 2 * q^3 + 8 * q^7 $$12 q + 2 q^{3} + 8 q^{7} - 4 q^{13} + 12 q^{19} - 8 q^{21} - 10 q^{27} - 4 q^{33} - 4 q^{37} - 20 q^{39} - 12 q^{43} - 12 q^{45} - 20 q^{49} - 24 q^{51} - 24 q^{55} + 12 q^{61} - 28 q^{67} + 4 q^{69} + 34 q^{75} - 4 q^{81} + 32 q^{85} + 60 q^{87} + 56 q^{91} + 28 q^{93} - 8 q^{97} + 52 q^{99}+O(q^{100})$$ 12 * q + 2 * q^3 + 8 * q^7 - 4 * q^13 + 12 * q^19 - 8 * q^21 - 10 * q^27 - 4 * q^33 - 4 * q^37 - 20 * q^39 - 12 * q^43 - 12 * q^45 - 20 * q^49 - 24 * q^51 - 24 * q^55 + 12 * q^61 - 28 * q^67 + 4 * q^69 + 34 * q^75 - 4 * q^81 + 32 * q^85 + 60 * q^87 + 56 * q^91 + 28 * q^93 - 8 * q^97 + 52 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2x^{10} - 2x^{8} + 16x^{6} - 8x^{4} - 32x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{8} - 2\nu^{4} + 4\nu^{2} + 8 ) / 8$$ (v^8 - 2*v^4 + 4*v^2 + 8) / 8 $$\beta_{2}$$ $$=$$ $$( -\nu^{9} - 2\nu^{7} + 2\nu^{5} - 24\nu ) / 8$$ (-v^9 - 2*v^7 + 2*v^5 - 24*v) / 8 $$\beta_{3}$$ $$=$$ $$( 2\nu^{11} - \nu^{10} - 8\nu^{7} + 2\nu^{6} + 16\nu^{5} + 4\nu^{4} + 24\nu^{3} - 16\nu^{2} - 32\nu ) / 32$$ (2*v^11 - v^10 - 8*v^7 + 2*v^6 + 16*v^5 + 4*v^4 + 24*v^3 - 16*v^2 - 32*v) / 32 $$\beta_{4}$$ $$=$$ $$( -\nu^{11} + \nu^{10} + 6\nu^{7} - 10\nu^{6} - 4\nu^{5} + 12\nu^{4} - 8\nu^{3} + 32\nu^{2} + 32\nu - 64 ) / 32$$ (-v^11 + v^10 + 6*v^7 - 10*v^6 - 4*v^5 + 12*v^4 - 8*v^3 + 32*v^2 + 32*v - 64) / 32 $$\beta_{5}$$ $$=$$ $$( \nu^{10} - 2\nu^{6} + 12\nu^{4} + 16\nu^{2} - 16 ) / 16$$ (v^10 - 2*v^6 + 12*v^4 + 16*v^2 - 16) / 16 $$\beta_{6}$$ $$=$$ $$( -\nu^{11} + 2\nu^{9} + 2\nu^{7} - 8\nu^{5} + 8\nu^{3} + 16\nu ) / 16$$ (-v^11 + 2*v^9 + 2*v^7 - 8*v^5 + 8*v^3 + 16*v) / 16 $$\beta_{7}$$ $$=$$ $$( -\nu^{11} + 2\nu^{7} - 4\nu^{5} - 16\nu^{3} + 16\nu ) / 16$$ (-v^11 + 2*v^7 - 4*v^5 - 16*v^3 + 16*v) / 16 $$\beta_{8}$$ $$=$$ $$( \nu^{11} - 6\nu^{7} + 12\nu^{5} + 8\nu^{3} - 48\nu ) / 16$$ (v^11 - 6*v^7 + 12*v^5 + 8*v^3 - 48*v) / 16 $$\beta_{9}$$ $$=$$ $$( -\nu^{11} - \nu^{10} + 6\nu^{7} + 10\nu^{6} - 4\nu^{5} - 12\nu^{4} - 8\nu^{3} - 32\nu^{2} + 32\nu + 64 ) / 32$$ (-v^11 - v^10 + 6*v^7 + 10*v^6 - 4*v^5 - 12*v^4 - 8*v^3 - 32*v^2 + 32*v + 64) / 32 $$\beta_{10}$$ $$=$$ $$( 2\nu^{11} + \nu^{10} - 8\nu^{7} - 2\nu^{6} + 16\nu^{5} - 4\nu^{4} + 24\nu^{3} + 16\nu^{2} - 32\nu ) / 32$$ (2*v^11 + v^10 - 8*v^7 - 2*v^6 + 16*v^5 - 4*v^4 + 24*v^3 + 16*v^2 - 32*v) / 32 $$\beta_{11}$$ $$=$$ $$( -3\nu^{10} + 2\nu^{8} + 14\nu^{6} - 24\nu^{4} - 24\nu^{2} + 80 ) / 16$$ (-3*v^10 + 2*v^8 + 14*v^6 - 24*v^4 - 24*v^2 + 80) / 16
 $$\nu$$ $$=$$ $$( \beta_{10} - \beta_{8} + \beta_{7} + \beta_{3} ) / 2$$ (b10 - b8 + b7 + b3) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{3} - \beta _1 + 1 ) / 2$$ (b11 + b10 - b9 + b5 + b4 - b3 - b1 + 1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} ) / 2$$ (b10 + b9 - b8 - b7 + b6 + b4 + b3 + b2) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{10} + \beta_{5} + \beta_{3} + 1$$ -b10 + b5 + b3 + 1 $$\nu^{5}$$ $$=$$ $$\beta_{10} + 2\beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{4} + \beta_{3}$$ b10 + 2*b9 + b8 + b7 + 2*b4 + b3 $$\nu^{6}$$ $$=$$ $$\beta_{11} + \beta_{10} + \beta_{9} + 3\beta_{5} - \beta_{4} - \beta_{3} - \beta _1 - 5$$ b11 + b10 + b9 + 3*b5 - b4 - b3 - b1 - 5 $$\nu^{7}$$ $$=$$ $$-3\beta_{10} + 3\beta_{9} + 3\beta_{8} - 5\beta_{7} - \beta_{6} + 3\beta_{4} - 3\beta_{3} - \beta_{2}$$ -3*b10 + 3*b9 + 3*b8 - 5*b7 - b6 + 3*b4 - 3*b3 - b2 $$\nu^{8}$$ $$=$$ $$-2\beta_{11} - 4\beta_{10} + 2\beta_{9} - 2\beta_{4} + 4\beta_{3} + 10\beta _1 - 8$$ -2*b11 - 4*b10 + 2*b9 - 2*b4 + 4*b3 + 10*b1 - 8 $$\nu^{9}$$ $$=$$ $$-4\beta_{10} - 2\beta_{9} + 8\beta_{8} + 2\beta_{6} - 2\beta_{4} - 4\beta_{3} - 6\beta_{2}$$ -4*b10 - 2*b9 + 8*b8 + 2*b6 - 2*b4 - 4*b3 - 6*b2 $$\nu^{10}$$ $$=$$ $$-6\beta_{11} + 6\beta_{10} + 10\beta_{9} + 2\beta_{5} - 10\beta_{4} - 6\beta_{3} + 6\beta _1 - 14$$ -6*b11 + 6*b10 + 10*b9 + 2*b5 - 10*b4 - 6*b3 + 6*b1 - 14 $$\nu^{11}$$ $$=$$ $$-10\beta_{10} - 10\beta_{9} + 2\beta_{8} - 14\beta_{7} - 10\beta_{6} - 10\beta_{4} - 10\beta_{3} - 10\beta_{2}$$ -10*b10 - 10*b9 + 2*b8 - 14*b7 - 10*b6 - 10*b4 - 10*b3 - 10*b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 1.27715 + 0.607364i −1.35164 + 0.416001i −1.27715 − 0.607364i 0.204810 + 1.39930i 1.35164 − 0.416001i −0.204810 − 1.39930i 1.27715 − 0.607364i −1.35164 − 0.416001i −1.27715 + 0.607364i 0.204810 − 1.39930i 1.35164 + 0.416001i −0.204810 + 1.39930i
0 −1.73003 + 0.0835731i 0 −0.431733 0.431733i 0 3.10278 0 2.98603 0.289169i 0
47.2 0 −0.966579 1.43726i 0 −1.57184 1.57184i 0 −2.24914 0 −1.13145 + 2.77846i 0
47.3 0 −0.0835731 + 1.73003i 0 0.431733 + 0.431733i 0 3.10278 0 −2.98603 0.289169i 0
47.4 0 0.814141 1.52878i 0 2.08397 + 2.08397i 0 1.14637 0 −1.67435 2.48929i 0
47.5 0 1.43726 + 0.966579i 0 1.57184 + 1.57184i 0 −2.24914 0 1.13145 + 2.77846i 0
47.6 0 1.52878 0.814141i 0 −2.08397 2.08397i 0 1.14637 0 1.67435 2.48929i 0
143.1 0 −1.73003 0.0835731i 0 −0.431733 + 0.431733i 0 3.10278 0 2.98603 + 0.289169i 0
143.2 0 −0.966579 + 1.43726i 0 −1.57184 + 1.57184i 0 −2.24914 0 −1.13145 2.77846i 0
143.3 0 −0.0835731 1.73003i 0 0.431733 0.431733i 0 3.10278 0 −2.98603 + 0.289169i 0
143.4 0 0.814141 + 1.52878i 0 2.08397 2.08397i 0 1.14637 0 −1.67435 + 2.48929i 0
143.5 0 1.43726 0.966579i 0 1.57184 1.57184i 0 −2.24914 0 1.13145 2.77846i 0
143.6 0 1.52878 + 0.814141i 0 −2.08397 + 2.08397i 0 1.14637 0 1.67435 + 2.48929i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 143.6 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
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.2.k.a 12
3.b odd 2 1 inner 192.2.k.a 12
4.b odd 2 1 48.2.k.a 12
8.b even 2 1 384.2.k.a 12
8.d odd 2 1 384.2.k.b 12
12.b even 2 1 48.2.k.a 12
16.e even 4 1 48.2.k.a 12
16.e even 4 1 384.2.k.b 12
16.f odd 4 1 inner 192.2.k.a 12
16.f odd 4 1 384.2.k.a 12
24.f even 2 1 384.2.k.b 12
24.h odd 2 1 384.2.k.a 12
48.i odd 4 1 48.2.k.a 12
48.i odd 4 1 384.2.k.b 12
48.k even 4 1 inner 192.2.k.a 12
48.k even 4 1 384.2.k.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.k.a 12 4.b odd 2 1
48.2.k.a 12 12.b even 2 1
48.2.k.a 12 16.e even 4 1
48.2.k.a 12 48.i odd 4 1
192.2.k.a 12 1.a even 1 1 trivial
192.2.k.a 12 3.b odd 2 1 inner
192.2.k.a 12 16.f odd 4 1 inner
192.2.k.a 12 48.k even 4 1 inner
384.2.k.a 12 8.b even 2 1
384.2.k.a 12 16.f odd 4 1
384.2.k.a 12 24.h odd 2 1
384.2.k.a 12 48.k even 4 1
384.2.k.b 12 8.d odd 2 1
384.2.k.b 12 16.e even 4 1
384.2.k.b 12 24.f even 2 1
384.2.k.b 12 48.i odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - 2 T^{11} + 2 T^{10} + 2 T^{9} + \cdots + 729$$
$5$ $$T^{12} + 100 T^{8} + 1856 T^{4} + \cdots + 256$$
$7$ $$(T^{3} - 2 T^{2} - 6 T + 8)^{4}$$
$11$ $$T^{12} + 356 T^{8} + 12288 T^{4} + \cdots + 65536$$
$13$ $$(T^{6} + 2 T^{5} + 2 T^{4} - 32 T^{3} + 196 T^{2} + \cdots + 8)^{2}$$
$17$ $$(T^{6} + 40 T^{4} + 288 T^{2} + 512)^{2}$$
$19$ $$(T^{6} - 6 T^{5} + 18 T^{4} + 32 T^{3} + \cdots + 128)^{2}$$
$23$ $$(T^{6} + 52 T^{4} + 640 T^{2} + 2048)^{2}$$
$29$ $$T^{12} + 1316 T^{8} + \cdots + 71639296$$
$31$ $$(T^{6} + 36 T^{4} + 68 T^{2} + 16)^{2}$$
$37$ $$(T^{6} + 2 T^{5} + 2 T^{4} - 128 T^{3} + \cdots + 2312)^{2}$$
$41$ $$(T^{6} - 108 T^{4} + 384 T^{2} - 128)^{2}$$
$43$ $$(T^{6} + 6 T^{5} + 18 T^{4} - 32 T^{3} + \cdots + 128)^{2}$$
$47$ $$(T^{6} - 112 T^{4} + 3712 T^{2} + \cdots - 32768)^{2}$$
$53$ $$T^{12} + 24356 T^{8} + 87360 T^{4} + \cdots + 256$$
$59$ $$T^{12} + 27748 T^{8} + 24418304 T^{4} + \cdots + 4096$$
$61$ $$(T^{6} - 6 T^{5} + 18 T^{4} - 64 T^{3} + \cdots + 95048)^{2}$$
$67$ $$(T^{6} + 14 T^{5} + 98 T^{4} - 232 T^{3} + \cdots + 32)^{2}$$
$71$ $$(T^{6} + 196 T^{4} + 10176 T^{2} + \cdots + 147968)^{2}$$
$73$ $$(T^{6} + 272 T^{4} + 18496 T^{2} + \cdots + 369664)^{2}$$
$79$ $$(T^{6} + 116 T^{4} + 1956 T^{2} + \cdots + 8464)^{2}$$
$83$ $$T^{12} + 19044 T^{8} + \cdots + 5473632256$$
$89$ $$(T^{6} - 212 T^{4} + 8576 T^{2} + \cdots - 2048)^{2}$$
$97$ $$(T^{3} + 2 T^{2} - 128 T - 608)^{4}$$