# Properties

 Label 1800.2.k.u Level $1800$ Weight $2$ Character orbit 1800.k Analytic conductor $14.373$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1800.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3730723638$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.180227832610816.1 Defining polynomial: $$x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64$$ x^12 + x^10 - 8*x^6 + 16*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{4} q^{4} + ( - \beta_{8} - \beta_1) q^{7} + (\beta_{8} - \beta_{3}) q^{8}+O(q^{10})$$ q + b1 * q^2 + b4 * q^4 + (-b8 - b1) * q^7 + (b8 - b3) * q^8 $$q + \beta_1 q^{2} + \beta_{4} q^{4} + ( - \beta_{8} - \beta_1) q^{7} + (\beta_{8} - \beta_{3}) q^{8} + (\beta_{6} - \beta_{4}) q^{11} + ( - \beta_{9} + \beta_{7} - \beta_{2} - \beta_1) q^{13} + ( - \beta_{11} + \beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} - 2) q^{14} + ( - \beta_{10} + \beta_{6} - \beta_{5}) q^{16} + ( - \beta_{7} - \beta_{2} - \beta_1) q^{17} + (\beta_{11} - \beta_{10} + \beta_{6} - 2 \beta_{5}) q^{19} + ( - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{3} - \beta_{2}) q^{22} + ( - \beta_{7} + \beta_{3} - 2 \beta_{2}) q^{23} + ( - \beta_{11} + \beta_{10} + \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{26} + ( - 2 \beta_{7} - 2 \beta_1) q^{28} + (\beta_{11} - \beta_{10} + \beta_{5} - 2 \beta_{4}) q^{29} + (\beta_{11} + \beta_{10} + \beta_{6} - 2) q^{31} + ( - 2 \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{3}) q^{32} + (\beta_{11} - \beta_{5} - \beta_{4} - 2) q^{34} + (\beta_{9} + \beta_{7} - \beta_{2} - \beta_1) q^{37} + ( - 2 \beta_{8} + 2 \beta_{7}) q^{38} + (2 \beta_{11} + 2 \beta_{10} + 2 \beta_{6} + 2) q^{41} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{43} + (2 \beta_{10} + 4) q^{44} + (2 \beta_{11} - 2 \beta_{5}) q^{46} + ( - 2 \beta_{8} + \beta_{7} + \beta_{3}) q^{47} + ( - 2 \beta_{10} + 2 \beta_{4} + 1) q^{49} + ( - 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{2} + 2 \beta_1) q^{52} + ( - 2 \beta_{9} + \beta_{3} + \beta_{2} - \beta_1) q^{53} + (2 \beta_{11} - 2 \beta_{4} - 4) q^{56} + (\beta_{9} - \beta_{8} + \beta_{7} + 4 \beta_{3} + \beta_{2}) q^{58} + (3 \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{59} + ( - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{6}) q^{61} + (2 \beta_{9} - 2 \beta_{2} - 2 \beta_1) q^{62} + ( - 2 \beta_{11} + 3 \beta_{10} + \beta_{6} + \beta_{5} + 4) q^{64} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{67} + (2 \beta_{9} - 2 \beta_{8} - 2 \beta_1) q^{68} + ( - 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{6} - 4) q^{71} + ( - 2 \beta_{8} + 4 \beta_{7} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{73} + ( - \beta_{11} - \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{74} + ( - 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{6} + 2 \beta_{5}) q^{76} + 2 \beta_{9} q^{77} + (\beta_{11} + \beta_{10} + \beta_{6} - 2) q^{79} + (4 \beta_{9} - 4 \beta_{2} + 2 \beta_1) q^{82} + (2 \beta_{9} - 2 \beta_{7} + 2 \beta_{2} + 2 \beta_1) q^{83} + ( - 2 \beta_{11} - 2 \beta_{5} + 2 \beta_{4} - 4) q^{86} + (2 \beta_{9} - 2 \beta_{7} - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{88} + (2 \beta_{11} - 2 \beta_{10} + 2 \beta_{6} + 4 \beta_{4} - 2) q^{89} + (4 \beta_{11} - 4 \beta_{10} - 2 \beta_{6} - 2 \beta_{4}) q^{91} + (4 \beta_{9} - 2 \beta_{8} - 2 \beta_{3}) q^{92} + ( - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{6} + 2 \beta_{5}) q^{94} + ( - 4 \beta_{8} + 2 \beta_{7} + 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{97} + ( - 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{2} + \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + b4 * q^4 + (-b8 - b1) * q^7 + (b8 - b3) * q^8 + (b6 - b4) * q^11 + (-b9 + b7 - b2 - b1) * q^13 + (-b11 + b10 - b6 + b5 - b4 - 2) * q^14 + (-b10 + b6 - b5) * q^16 + (-b7 - b2 - b1) * q^17 + (b11 - b10 + b6 - 2*b5) * q^19 + (-b9 - b8 + b7 + 2*b3 - b2) * q^22 + (-b7 + b3 - 2*b2) * q^23 + (-b11 + b10 + b6 - b5 - b4 + 2) * q^26 + (-2*b7 - 2*b1) * q^28 + (b11 - b10 + b5 - 2*b4) * q^29 + (b11 + b10 + b6 - 2) * q^31 + (-2*b9 - b8 + 2*b7 + b3) * q^32 + (b11 - b5 - b4 - 2) * q^34 + (b9 + b7 - b2 - b1) * q^37 + (-2*b8 + 2*b7) * q^38 + (2*b11 + 2*b10 + 2*b6 + 2) * q^41 + (-2*b3 - 2*b2 + 2*b1) * q^43 + (2*b10 + 4) * q^44 + (2*b11 - 2*b5) * q^46 + (-2*b8 + b7 + b3) * q^47 + (-2*b10 + 2*b4 + 1) * q^49 + (-2*b9 - 2*b8 - 2*b2 + 2*b1) * q^52 + (-2*b9 + b3 + b2 - b1) * q^53 + (2*b11 - 2*b4 - 4) * q^56 + (b9 - b8 + b7 + 4*b3 + b2) * q^58 + (3*b6 - 2*b5 - b4) * q^59 + (-2*b11 + 2*b10 + 2*b6) * q^61 + (2*b9 - 2*b2 - 2*b1) * q^62 + (-2*b11 + 3*b10 + b6 + b5 + 4) * q^64 + (-2*b3 - 2*b2 + 2*b1) * q^67 + (2*b9 - 2*b8 - 2*b1) * q^68 + (-2*b11 - 2*b10 - 2*b6 - 4) * q^71 + (-2*b8 + 4*b7 + 2*b3 + 2*b2 + 4*b1) * q^73 + (-b11 - b10 - b6 - b5 - b4 + 2) * q^74 + (-4*b11 + 2*b10 - 2*b6 + 2*b5) * q^76 + 2*b9 * q^77 + (b11 + b10 + b6 - 2) * q^79 + (4*b9 - 4*b2 + 2*b1) * q^82 + (2*b9 - 2*b7 + 2*b2 + 2*b1) * q^83 + (-2*b11 - 2*b5 + 2*b4 - 4) * q^86 + (2*b9 - 2*b7 - 2*b3 - 2*b2 + 4*b1) * q^88 + (2*b11 - 2*b10 + 2*b6 + 4*b4 - 2) * q^89 + (4*b11 - 4*b10 - 2*b6 - 2*b4) * q^91 + (4*b9 - 2*b8 - 2*b3) * q^92 + (-2*b11 + 2*b10 - 2*b6 + 2*b5) * q^94 + (-4*b8 + 2*b7 + 4*b3 - 2*b2 + 2*b1) * q^97 + (-2*b9 + 2*b8 + 2*b7 + 2*b2 + b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{4}+O(q^{10})$$ 12 * q - 2 * q^4 $$12 q - 2 q^{4} - 20 q^{14} + 2 q^{16} + 28 q^{26} - 32 q^{31} - 24 q^{34} + 8 q^{41} + 44 q^{44} - 4 q^{46} + 12 q^{49} - 52 q^{56} + 46 q^{64} - 32 q^{71} + 36 q^{74} + 12 q^{76} - 32 q^{79} - 40 q^{86} - 40 q^{89} + 4 q^{94}+O(q^{100})$$ 12 * q - 2 * q^4 - 20 * q^14 + 2 * q^16 + 28 * q^26 - 32 * q^31 - 24 * q^34 + 8 * q^41 + 44 * q^44 - 4 * q^46 + 12 * q^49 - 52 * q^56 + 46 * q^64 - 32 * q^71 + 36 * q^74 + 12 * q^76 - 32 * q^79 - 40 * q^86 - 40 * q^89 + 4 * q^94

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

 $$\beta_{1}$$ $$=$$ $$( -\nu^{11} - \nu^{9} + 8\nu^{5} - 16\nu ) / 32$$ (-v^11 - v^9 + 8*v^5 - 16*v) / 32 $$\beta_{2}$$ $$=$$ $$( \nu^{9} - \nu^{7} + 2\nu^{5} + 4\nu^{3} + 8\nu ) / 16$$ (v^9 - v^7 + 2*v^5 + 4*v^3 + 8*v) / 16 $$\beta_{3}$$ $$=$$ $$( -\nu^{11} + \nu^{9} + 6\nu^{7} + 4\nu^{5} - 8\nu^{3} ) / 32$$ (-v^11 + v^9 + 6*v^7 + 4*v^5 - 8*v^3) / 32 $$\beta_{4}$$ $$=$$ $$( -\nu^{10} - \nu^{8} + 8\nu^{4} - 16 ) / 16$$ (-v^10 - v^8 + 8*v^4 - 16) / 16 $$\beta_{5}$$ $$=$$ $$( \nu^{8} - \nu^{6} + 2\nu^{4} + 4\nu^{2} + 8 ) / 8$$ (v^8 - v^6 + 2*v^4 + 4*v^2 + 8) / 8 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - \nu^{4} + 4\nu^{2} + 4 ) / 4$$ (-v^6 - v^4 + 4*v^2 + 4) / 4 $$\beta_{7}$$ $$=$$ $$( \nu^{11} - \nu^{9} - 6\nu^{7} - 4\nu^{5} + 8\nu^{3} + 64\nu ) / 32$$ (v^11 - v^9 - 6*v^7 - 4*v^5 + 8*v^3 + 64*v) / 32 $$\beta_{8}$$ $$=$$ $$( -\nu^{9} + \nu^{7} - 2\nu^{5} + 12\nu^{3} + 8\nu ) / 16$$ (-v^9 + v^7 - 2*v^5 + 12*v^3 + 8*v) / 16 $$\beta_{9}$$ $$=$$ $$( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 32$$ (v^11 - v^9 + 2*v^7 + 4*v^5 + 8*v^3) / 32 $$\beta_{10}$$ $$=$$ $$( -\nu^{10} + \nu^{8} + 2\nu^{6} - 24\nu^{2} - 16 ) / 16$$ (-v^10 + v^8 + 2*v^6 - 24*v^2 - 16) / 16 $$\beta_{11}$$ $$=$$ $$( -\nu^{10} + \nu^{8} + 6\nu^{6} + 4\nu^{4} - 8\nu^{2} - 32 ) / 16$$ (-v^10 + v^8 + 6*v^6 + 4*v^4 - 8*v^2 - 32) / 16
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{3} ) / 2$$ (b7 + b3) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{11} - \beta_{10} + \beta_{6} ) / 2$$ (b11 - b10 + b6) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{8} - \beta_{7} - \beta_{3} + 2\beta_{2} ) / 2$$ (2*b8 - b7 - b3 + 2*b2) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{11} - \beta_{10} - 3\beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 2$$ (-b11 - b10 - 3*b6 + 2*b5 + 2*b4) / 2 $$\nu^{5}$$ $$=$$ $$( 4\beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{3} + 2\beta_{2} + 4\beta_1 ) / 2$$ (4*b9 - 2*b8 + b7 + b3 + 2*b2 + 4*b1) / 2 $$\nu^{6}$$ $$=$$ $$( 5\beta_{11} - 3\beta_{10} - \beta_{6} - 2\beta_{5} - 2\beta_{4} + 8 ) / 2$$ (5*b11 - 3*b10 - b6 - 2*b5 - 2*b4 + 8) / 2 $$\nu^{7}$$ $$=$$ $$( 4\beta_{9} + 2\beta_{8} - \beta_{7} + 7\beta_{3} - 2\beta_{2} - 4\beta_1 ) / 2$$ (4*b9 + 2*b8 - b7 + 7*b3 - 2*b2 - 4*b1) / 2 $$\nu^{8}$$ $$=$$ $$( 3\beta_{11} + 3\beta_{10} + \beta_{6} + 10\beta_{5} - 6\beta_{4} - 8 ) / 2$$ (3*b11 + 3*b10 + b6 + 10*b5 - 6*b4 - 8) / 2 $$\nu^{9}$$ $$=$$ $$( -4\beta_{9} - 2\beta_{8} - 7\beta_{7} + \beta_{3} + 18\beta_{2} - 12\beta_1 ) / 2$$ (-4*b9 - 2*b8 - 7*b7 + b3 + 18*b2 - 12*b1) / 2 $$\nu^{10}$$ $$=$$ $$( -11\beta_{11} - 11\beta_{10} - 25\beta_{6} + 6\beta_{5} - 10\beta_{4} - 24 ) / 2$$ (-11*b11 - 11*b10 - 25*b6 + 6*b5 - 10*b4 - 24) / 2 $$\nu^{11}$$ $$=$$ $$( 36\beta_{9} - 14\beta_{8} - \beta_{7} - 9\beta_{3} - 2\beta_{2} - 20\beta_1 ) / 2$$ (36*b9 - 14*b8 - b7 - 9*b3 - 2*b2 - 20*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1
 −1.37729 + 0.321037i −1.37729 − 0.321037i −0.806504 + 1.16170i −0.806504 − 1.16170i −0.450129 + 1.34067i −0.450129 − 1.34067i 0.450129 + 1.34067i 0.450129 − 1.34067i 0.806504 + 1.16170i 0.806504 − 1.16170i 1.37729 + 0.321037i 1.37729 − 0.321037i
−1.37729 0.321037i 0 1.79387 + 0.884323i 0 0 4.05705 −2.18678 1.79387i 0 0
901.2 −1.37729 + 0.321037i 0 1.79387 0.884323i 0 0 4.05705 −2.18678 + 1.79387i 0 0
901.3 −0.806504 1.16170i 0 −0.699104 + 1.87383i 0 0 0.746175 2.74067 0.699104i 0 0
901.4 −0.806504 + 1.16170i 0 −0.699104 1.87383i 0 0 0.746175 2.74067 + 0.699104i 0 0
901.5 −0.450129 1.34067i 0 −1.59477 + 1.20695i 0 0 −2.64265 2.33596 + 1.59477i 0 0
901.6 −0.450129 + 1.34067i 0 −1.59477 1.20695i 0 0 −2.64265 2.33596 1.59477i 0 0
901.7 0.450129 1.34067i 0 −1.59477 1.20695i 0 0 2.64265 −2.33596 + 1.59477i 0 0
901.8 0.450129 + 1.34067i 0 −1.59477 + 1.20695i 0 0 2.64265 −2.33596 1.59477i 0 0
901.9 0.806504 1.16170i 0 −0.699104 1.87383i 0 0 −0.746175 −2.74067 0.699104i 0 0
901.10 0.806504 + 1.16170i 0 −0.699104 + 1.87383i 0 0 −0.746175 −2.74067 + 0.699104i 0 0
901.11 1.37729 0.321037i 0 1.79387 0.884323i 0 0 −4.05705 2.18678 1.79387i 0 0
901.12 1.37729 + 0.321037i 0 1.79387 + 0.884323i 0 0 −4.05705 2.18678 + 1.79387i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.k.u 12
3.b odd 2 1 600.2.k.f 12
4.b odd 2 1 7200.2.k.u 12
5.b even 2 1 inner 1800.2.k.u 12
5.c odd 4 1 360.2.d.e 6
5.c odd 4 1 360.2.d.f 6
8.b even 2 1 inner 1800.2.k.u 12
8.d odd 2 1 7200.2.k.u 12
12.b even 2 1 2400.2.k.f 12
15.d odd 2 1 600.2.k.f 12
15.e even 4 1 120.2.d.a 6
15.e even 4 1 120.2.d.b yes 6
20.d odd 2 1 7200.2.k.u 12
20.e even 4 1 1440.2.d.e 6
20.e even 4 1 1440.2.d.f 6
24.f even 2 1 2400.2.k.f 12
24.h odd 2 1 600.2.k.f 12
40.e odd 2 1 7200.2.k.u 12
40.f even 2 1 inner 1800.2.k.u 12
40.i odd 4 1 360.2.d.e 6
40.i odd 4 1 360.2.d.f 6
40.k even 4 1 1440.2.d.e 6
40.k even 4 1 1440.2.d.f 6
60.h even 2 1 2400.2.k.f 12
60.l odd 4 1 480.2.d.a 6
60.l odd 4 1 480.2.d.b 6
120.i odd 2 1 600.2.k.f 12
120.m even 2 1 2400.2.k.f 12
120.q odd 4 1 480.2.d.a 6
120.q odd 4 1 480.2.d.b 6
120.w even 4 1 120.2.d.a 6
120.w even 4 1 120.2.d.b yes 6
240.z odd 4 2 3840.2.f.m 12
240.bb even 4 2 3840.2.f.l 12
240.bd odd 4 2 3840.2.f.m 12
240.bf even 4 2 3840.2.f.l 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 15.e even 4 1
120.2.d.a 6 120.w even 4 1
120.2.d.b yes 6 15.e even 4 1
120.2.d.b yes 6 120.w even 4 1
360.2.d.e 6 5.c odd 4 1
360.2.d.e 6 40.i odd 4 1
360.2.d.f 6 5.c odd 4 1
360.2.d.f 6 40.i odd 4 1
480.2.d.a 6 60.l odd 4 1
480.2.d.a 6 120.q odd 4 1
480.2.d.b 6 60.l odd 4 1
480.2.d.b 6 120.q odd 4 1
600.2.k.f 12 3.b odd 2 1
600.2.k.f 12 15.d odd 2 1
600.2.k.f 12 24.h odd 2 1
600.2.k.f 12 120.i odd 2 1
1440.2.d.e 6 20.e even 4 1
1440.2.d.e 6 40.k even 4 1
1440.2.d.f 6 20.e even 4 1
1440.2.d.f 6 40.k even 4 1
1800.2.k.u 12 1.a even 1 1 trivial
1800.2.k.u 12 5.b even 2 1 inner
1800.2.k.u 12 8.b even 2 1 inner
1800.2.k.u 12 40.f even 2 1 inner
2400.2.k.f 12 12.b even 2 1
2400.2.k.f 12 24.f even 2 1
2400.2.k.f 12 60.h even 2 1
2400.2.k.f 12 120.m even 2 1
3840.2.f.l 12 240.bb even 4 2
3840.2.f.l 12 240.bf even 4 2
3840.2.f.m 12 240.z odd 4 2
3840.2.f.m 12 240.bd odd 4 2
7200.2.k.u 12 4.b odd 2 1
7200.2.k.u 12 8.d odd 2 1
7200.2.k.u 12 20.d odd 2 1
7200.2.k.u 12 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{6} - 24T_{7}^{4} + 128T_{7}^{2} - 64$$ T7^6 - 24*T7^4 + 128*T7^2 - 64 $$T_{11}^{6} + 32T_{11}^{4} + 96T_{11}^{2} + 64$$ T11^6 + 32*T11^4 + 96*T11^2 + 64 $$T_{17}^{6} - 36T_{17}^{4} + 368T_{17}^{2} - 1024$$ T17^6 - 36*T17^4 + 368*T17^2 - 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + T^{10} - 8 T^{6} + 16 T^{2} + \cdots + 64$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$(T^{6} - 24 T^{4} + 128 T^{2} - 64)^{2}$$
$11$ $$(T^{6} + 32 T^{4} + 96 T^{2} + 64)^{2}$$
$13$ $$(T^{6} + 48 T^{4} + 704 T^{2} + 3136)^{2}$$
$17$ $$(T^{6} - 36 T^{4} + 368 T^{2} - 1024)^{2}$$
$19$ $$(T^{6} + 60 T^{4} + 512 T^{2} + 1024)^{2}$$
$23$ $$(T^{6} - 92 T^{4} + 2304 T^{2} + \cdots - 16384)^{2}$$
$29$ $$(T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544)^{2}$$
$31$ $$(T^{3} + 8 T^{2} - 4 T - 64)^{4}$$
$37$ $$(T^{6} + 64 T^{4} + 128 T^{2} + 64)^{2}$$
$41$ $$(T^{3} - 2 T^{2} - 100 T - 56)^{4}$$
$43$ $$(T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2}$$
$47$ $$(T^{6} - 60 T^{4} + 512 T^{2} - 1024)^{2}$$
$53$ $$(T^{6} + 80 T^{4} + 1216 T^{2} + 64)^{2}$$
$59$ $$(T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776)^{2}$$
$61$ $$(T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536)^{2}$$
$67$ $$(T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2}$$
$71$ $$(T^{3} + 8 T^{2} - 80 T - 128)^{4}$$
$73$ $$(T^{6} - 384 T^{4} + 34560 T^{2} + \cdots - 16384)^{2}$$
$79$ $$(T^{3} + 8 T^{2} - 4 T - 64)^{4}$$
$83$ $$(T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704)^{2}$$
$89$ $$(T^{3} + 10 T^{2} - 164 T - 1384)^{4}$$
$97$ $$(T^{6} - 336 T^{4} + 28416 T^{2} + \cdots - 262144)^{2}$$