Properties

 Label 2320.2.d.h Level $2320$ Weight $2$ Character orbit 2320.d Analytic conductor $18.525$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(929,2320)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2320.929");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2320 = 2^{4} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2320.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$18.5252932689$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} + 24x^{8} + 152x^{6} + 377x^{4} + 352x^{2} + 64$$ x^10 + 24*x^8 + 152*x^6 + 377*x^4 + 352*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 290) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} + \beta_{5} q^{5} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{7} + (\beta_{6} - \beta_{4} - \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10})$$ q + b7 * q^3 + b5 * q^5 + (b9 - b8 - b7 + b5 - b3 - b2 - b1) * q^7 + (b6 - b4 - b2 + b1 - 2) * q^9 $$q + \beta_{7} q^{3} + \beta_{5} q^{5} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{7} + (\beta_{6} - \beta_{4} - \beta_{2} + \beta_1 - 2) q^{9} + (\beta_{2} - \beta_1) q^{11} + (\beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{13} + (\beta_{8} - \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{15} + ( - \beta_{9} - \beta_{8} - 3 \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{17} - 2 \beta_{4} q^{19} + ( - 4 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{21} + ( - \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{2} - \beta_1) q^{23} + ( - \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{25} + (\beta_{8} - 2 \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{27} - q^{29} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 1) q^{31} + (2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{5} + \beta_{3}) q^{33} + (\beta_{9} - 3 \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{35} + (\beta_{9} - 2 \beta_{8} - 4 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{37} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{39} + ( - 2 \beta_{6} + 2 \beta_{2} - 2 \beta_1) q^{41} + ( - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 3) q^{45} + (2 \beta_{9} - 2 \beta_{8} - \beta_{2} - \beta_1) q^{47} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{49} + ( - 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 6) q^{51} + (2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{2} - \beta_1) q^{53} + (2 \beta_{9} - \beta_{8} + 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{55} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{57} + (\beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} - 3) q^{59} + (5 \beta_{6} - \beta_{5} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{61} + ( - 3 \beta_{9} + 3 \beta_{8} - \beta_{5} + \beta_{3}) q^{63} + ( - \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + \cdots + 2) q^{65}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{99}+O(q^{100})$$ q + b7 * q^3 + b5 * q^5 + (b9 - b8 - b7 + b5 - b3 - b2 - b1) * q^7 + (b6 - b4 - b2 + b1 - 2) * q^9 + (b2 - b1) * q^11 + (b7 - b5 + b3 + b2 + b1) * q^13 + (b8 - b6 - b4 + b2 - 1) * q^15 + (-b9 - b8 - 3*b7 + b5 - b3 - b2 - b1) * q^17 - 2*b4 * q^19 + (-4*b6 + 2*b5 + b4 + 2*b3 + b2 - b1) * q^21 + (-b9 - 2*b8 - b7 - b2 - b1) * q^23 + (-b9 + b8 + 2*b7 - b5 - b4 + 2*b3 + b2 + b1) * q^25 + (b8 - 2*b7 + b5 - b3 - b2 - b1) * q^27 - q^29 + (-b6 + b5 + b4 + b3 - 1) * q^31 + (2*b9 - 2*b8 + 2*b7 - b5 + b3) * q^33 + (b9 - 3*b6 + b5 + b4 + 3*b3 + b2 - b1 - 1) * q^35 + (b9 - 2*b8 - 4*b7 + 2*b5 - 2*b3 - 2*b2 - 2*b1) * q^37 + (b6 - b5 - b4 - b3 - 2*b2 + 2*b1 + 1) * q^39 + (-2*b6 + 2*b2 - 2*b1) * q^41 + (-2*b9 - b8 + b7 - b5 + b3 + 2*b2 + 2*b1) * q^43 + (-b9 - b7 - b5 - b4 + b3 - 2*b1 + 3) * q^45 + (2*b9 - 2*b8 - b2 - b1) * q^47 + (-b6 + b5 - 2*b4 + b3 + 2*b2 - 2*b1 - 2) * q^49 + (-2*b6 + 2*b5 + 3*b4 + 2*b3 + 3*b2 - 3*b1 + 6) * q^51 + (2*b9 + b8 + b7 - b2 - b1) * q^53 + (2*b9 - b8 + 2*b6 - b5 - b4 - b3 - b2 + 2*b1 - 2) * q^55 + (-2*b7 - 2*b5 + 2*b3 - 2*b2 - 2*b1) * q^57 + (b6 + b5 + 2*b4 + b3 - 3) * q^59 + (5*b6 - b5 - b3 - 2*b2 + 2*b1 - 1) * q^61 + (-3*b9 + 3*b8 - b5 + b3) * q^63 + (-b9 - b8 - 2*b7 + b6 - b4 - 3*b3 - 3*b2 + b1 + 2) * q^65 + (-b9 - 2*b8 + 4*b7 - 2*b5 + 2*b3 + 2*b2 + 2*b1) * q^67 + (b6 + 3*b5 + 3*b4 + 3*b3 - b2 + b1 - 3) * q^69 + (2*b6 - 2*b5 - 4*b4 - 2*b3 - 2*b2 + 2*b1 + 2) * q^71 + (3*b9 + b8 + 3*b7 - 3*b5 + 3*b3 - b2 - b1) * q^73 + (b8 - b7 + 6*b6 - 3*b5 - b4 - b3 - 3*b2 + 2*b1 - 4) * q^75 + (2*b8 + 6*b7 - 2*b5 + 2*b3 + 4*b2 + 4*b1) * q^77 + (-3*b6 + b5 + 2*b4 + b3 + b2 - b1 - 3) * q^79 + (2*b6 - b4 + b2 - b1 - 1) * q^81 + (-b9 + 4*b8 + 4*b7 - 2*b5 + 2*b3) * q^83 + (-b9 - 2*b8 - b6 + b5 + 3*b4 + 3*b3 + 3*b2 - b1 + 1) * q^85 - b7 * q^87 + (-2*b5 - 2*b3 - 2*b2 + 2*b1 - 2) * q^89 + (4*b6 - 4*b5 - 3*b4 - 4*b3 - 5*b2 + 5*b1 + 4) * q^91 + (-2*b9 + 3*b8 + 2*b2 + 2*b1) * q^93 + (-4*b7 + 4*b6 - 2*b4 - 2*b2) * q^95 + (3*b9 - 2*b8 + 3*b7 - 2*b5 + 2*b3 - b2 - b1) * q^97 + (-2*b6 + 2*b5 + 2*b3 - 2*b2 + 2*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 20 q^{9}+O(q^{10})$$ 10 * q - 20 * q^9 $$10 q - 20 q^{9} - 18 q^{15} - 8 q^{19} - 12 q^{21} - 4 q^{25} - 10 q^{29} - 10 q^{31} - 18 q^{35} + 10 q^{39} - 8 q^{41} + 26 q^{45} - 32 q^{49} + 64 q^{51} - 16 q^{55} - 18 q^{59} + 10 q^{61} + 20 q^{65} - 14 q^{69} + 12 q^{71} - 20 q^{75} - 34 q^{79} - 6 q^{81} + 18 q^{85} - 20 q^{89} + 44 q^{91} + 8 q^{95} - 68 q^{99}+O(q^{100})$$ 10 * q - 20 * q^9 - 18 * q^15 - 8 * q^19 - 12 * q^21 - 4 * q^25 - 10 * q^29 - 10 * q^31 - 18 * q^35 + 10 * q^39 - 8 * q^41 + 26 * q^45 - 32 * q^49 + 64 * q^51 - 16 * q^55 - 18 * q^59 + 10 * q^61 + 20 * q^65 - 14 * q^69 + 12 * q^71 - 20 * q^75 - 34 * q^79 - 6 * q^81 + 18 * q^85 - 20 * q^89 + 44 * q^91 + 8 * q^95 - 68 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 24x^{8} + 152x^{6} + 377x^{4} + 352x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( - 27 \nu^{9} + 20 \nu^{8} - 688 \nu^{7} + 352 \nu^{6} - 4808 \nu^{5} + 544 \nu^{4} - 11267 \nu^{3} - 76 \nu^{2} - 8136 \nu + 2176 ) / 1216$$ (-27*v^9 + 20*v^8 - 688*v^7 + 352*v^6 - 4808*v^5 + 544*v^4 - 11267*v^3 - 76*v^2 - 8136*v + 2176) / 1216 $$\beta_{2}$$ $$=$$ $$( - 27 \nu^{9} - 20 \nu^{8} - 688 \nu^{7} - 352 \nu^{6} - 4808 \nu^{5} - 544 \nu^{4} - 11267 \nu^{3} + 76 \nu^{2} - 8136 \nu - 2176 ) / 1216$$ (-27*v^9 - 20*v^8 - 688*v^7 - 352*v^6 - 4808*v^5 - 544*v^4 - 11267*v^3 + 76*v^2 - 8136*v - 2176) / 1216 $$\beta_{3}$$ $$=$$ $$( - 11 \nu^{9} + 200 \nu^{8} - 224 \nu^{7} + 4128 \nu^{6} - 968 \nu^{5} + 16384 \nu^{4} - 3059 \nu^{3} + 17480 \nu^{2} - 5240 \nu + 1696 ) / 1216$$ (-11*v^9 + 200*v^8 - 224*v^7 + 4128*v^6 - 968*v^5 + 16384*v^4 - 3059*v^3 + 17480*v^2 - 5240*v + 1696) / 1216 $$\beta_{4}$$ $$=$$ $$( -29\nu^{8} - 632\nu^{6} - 3008\nu^{4} - 4237\nu^{2} - 784 ) / 152$$ (-29*v^8 - 632*v^6 - 3008*v^4 - 4237*v^2 - 784) / 152 $$\beta_{5}$$ $$=$$ $$( 11 \nu^{9} + 200 \nu^{8} + 224 \nu^{7} + 4128 \nu^{6} + 968 \nu^{5} + 16384 \nu^{4} + 3059 \nu^{3} + 17480 \nu^{2} + 5240 \nu + 1696 ) / 1216$$ (11*v^9 + 200*v^8 + 224*v^7 + 4128*v^6 + 968*v^5 + 16384*v^4 + 3059*v^3 + 17480*v^2 + 5240*v + 1696) / 1216 $$\beta_{6}$$ $$=$$ $$( 17\nu^{8} + 360\nu^{6} + 1572\nu^{4} + 2033\nu^{2} + 360 ) / 76$$ (17*v^8 + 360*v^6 + 1572*v^4 + 2033*v^2 + 360) / 76 $$\beta_{7}$$ $$=$$ $$( -45\nu^{9} - 944\nu^{7} - 3960\nu^{5} - 4389\nu^{3} + 1032\nu ) / 608$$ (-45*v^9 - 944*v^7 - 3960*v^5 - 4389*v^3 + 1032*v) / 608 $$\beta_{8}$$ $$=$$ $$( 55\nu^{9} + 1120\nu^{7} + 4232\nu^{5} + 4351\nu^{3} + 1272\nu ) / 608$$ (55*v^9 + 1120*v^7 + 4232*v^5 + 4351*v^3 + 1272*v) / 608 $$\beta_{9}$$ $$=$$ $$( 45\nu^{9} + 944\nu^{7} + 3960\nu^{5} + 4389\nu^{3} - 424\nu ) / 304$$ (45*v^9 + 944*v^7 + 3960*v^5 + 4389*v^3 - 424*v) / 304
 $$\nu$$ $$=$$ $$( \beta_{9} + 2\beta_{7} ) / 2$$ (b9 + 2*b7) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{6} - \beta_{4} - \beta_{2} + \beta _1 - 4$$ -b6 - b4 - b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$( -13\beta_{9} + 8\beta_{8} - 16\beta_{7} - 4\beta_{5} + 4\beta_{3} - 2\beta_{2} - 2\beta_1 ) / 2$$ (-13*b9 + 8*b8 - 16*b7 - 4*b5 + 4*b3 - 2*b2 - 2*b1) / 2 $$\nu^{4}$$ $$=$$ $$23\beta_{6} - 4\beta_{5} + 18\beta_{4} - 4\beta_{3} + 12\beta_{2} - 12\beta _1 + 38$$ 23*b6 - 4*b5 + 18*b4 - 4*b3 + 12*b2 - 12*b1 + 38 $$\nu^{5}$$ $$=$$ $$( 193\beta_{9} - 146\beta_{8} + 206\beta_{7} + 82\beta_{5} - 82\beta_{3} + 36\beta_{2} + 36\beta_1 ) / 2$$ (193*b9 - 146*b8 + 206*b7 + 82*b5 - 82*b3 + 36*b2 + 36*b1) / 2 $$\nu^{6}$$ $$=$$ $$-384\beta_{6} + 73\beta_{5} - 294\beta_{4} + 73\beta_{3} - 176\beta_{2} + 176\beta _1 - 531$$ -384*b6 + 73*b5 - 294*b4 + 73*b3 - 176*b2 + 176*b1 - 531 $$\nu^{7}$$ $$=$$ $$( -3009\beta_{9} + 2352\beta_{8} - 3122\beta_{7} - 1356\beta_{5} + 1356\beta_{3} - 588\beta_{2} - 588\beta_1 ) / 2$$ (-3009*b9 + 2352*b8 - 3122*b7 - 1356*b5 + 1356*b3 - 588*b2 - 588*b1) / 2 $$\nu^{8}$$ $$=$$ $$6129\beta_{6} - 1176\beta_{5} + 4681\beta_{4} - 1176\beta_{3} + 2737\beta_{2} - 2737\beta _1 + 8188$$ 6129*b6 - 1176*b5 + 4681*b4 - 1176*b3 + 2737*b2 - 2737*b1 + 8188 $$\nu^{9}$$ $$=$$ $$( 47429 \beta_{9} - 37272 \beta_{8} + 48944 \beta_{7} + 21620 \beta_{5} - 21620 \beta_{3} + 9362 \beta_{2} + 9362 \beta_1 ) / 2$$ (47429*b9 - 37272*b8 + 48944*b7 + 21620*b5 - 21620*b3 + 9362*b2 + 9362*b1) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$$.

 $$n$$ $$321$$ $$581$$ $$1857$$ $$2031$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
929.1
 − 3.97530i − 1.81278i − 1.35864i − 0.485804i − 1.68193i 1.68193i 0.485804i 1.35864i 1.81278i 3.97530i
0 2.97530i 0 −0.639550 2.14266i 0 3.57331i 0 −5.85241 0
929.2 0 2.81278i 0 −1.63193 + 1.52866i 0 0.647892i 0 −4.91176 0
929.3 0 2.35864i 0 1.32739 1.79945i 0 4.21797i 0 −2.56319 0
929.4 0 1.48580i 0 −1.29150 1.82539i 0 4.37538i 0 0.792385 0
929.5 0 0.681929i 0 2.23558 + 0.0464742i 0 0.936197i 0 2.53497 0
929.6 0 0.681929i 0 2.23558 0.0464742i 0 0.936197i 0 2.53497 0
929.7 0 1.48580i 0 −1.29150 + 1.82539i 0 4.37538i 0 0.792385 0
929.8 0 2.35864i 0 1.32739 + 1.79945i 0 4.21797i 0 −2.56319 0
929.9 0 2.81278i 0 −1.63193 1.52866i 0 0.647892i 0 −4.91176 0
929.10 0 2.97530i 0 −0.639550 + 2.14266i 0 3.57331i 0 −5.85241 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 929.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2320.2.d.h 10
4.b odd 2 1 290.2.b.b 10
5.b even 2 1 inner 2320.2.d.h 10
12.b even 2 1 2610.2.e.i 10
20.d odd 2 1 290.2.b.b 10
20.e even 4 1 1450.2.a.t 5
20.e even 4 1 1450.2.a.u 5
60.h even 2 1 2610.2.e.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.b.b 10 4.b odd 2 1
290.2.b.b 10 20.d odd 2 1
1450.2.a.t 5 20.e even 4 1
1450.2.a.u 5 20.e even 4 1
2320.2.d.h 10 1.a even 1 1 trivial
2320.2.d.h 10 5.b even 2 1 inner
2610.2.e.i 10 12.b even 2 1
2610.2.e.i 10 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2320, [\chi])$$:

 $$T_{3}^{10} + 25T_{3}^{8} + 224T_{3}^{6} + 849T_{3}^{4} + 1209T_{3}^{2} + 400$$ T3^10 + 25*T3^8 + 224*T3^6 + 849*T3^4 + 1209*T3^2 + 400 $$T_{7}^{10} + 51T_{7}^{8} + 877T_{7}^{6} + 5420T_{7}^{4} + 5936T_{7}^{2} + 1600$$ T7^10 + 51*T7^8 + 877*T7^6 + 5420*T7^4 + 5936*T7^2 + 1600

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} + 25 T^{8} + 224 T^{6} + \cdots + 400$$
$5$ $$T^{10} + 2 T^{8} - 18 T^{7} + \cdots + 3125$$
$7$ $$T^{10} + 51 T^{8} + 877 T^{6} + \cdots + 1600$$
$11$ $$(T^{5} - 27 T^{3} - 4 T^{2} + 172 T + 16)^{2}$$
$13$ $$T^{10} + 53 T^{8} + 356 T^{6} + \cdots + 64$$
$17$ $$T^{10} + 119 T^{8} + 5325 T^{6} + \cdots + 2930944$$
$19$ $$(T^{5} + 4 T^{4} - 56 T^{3} - 144 T^{2} + \cdots + 640)^{2}$$
$23$ $$T^{10} + 167 T^{8} + 9813 T^{6} + \cdots + 2611456$$
$29$ $$(T + 1)^{10}$$
$31$ $$(T^{5} + 5 T^{4} - 22 T^{3} - 175 T^{2} + \cdots - 184)^{2}$$
$37$ $$T^{10} + 212 T^{8} + 12608 T^{6} + \cdots + 262144$$
$41$ $$(T^{5} + 4 T^{4} - 76 T^{3} - 504 T^{2} + \cdots + 608)^{2}$$
$43$ $$T^{10} + 205 T^{8} + 11856 T^{6} + \cdots + 440896$$
$47$ $$T^{10} + 194 T^{8} + 11057 T^{6} + \cdots + 4096$$
$53$ $$T^{10} + 109 T^{8} + 1084 T^{6} + \cdots + 16$$
$59$ $$(T^{5} + 9 T^{4} - 79 T^{3} - 448 T^{2} + \cdots + 4000)^{2}$$
$61$ $$(T^{5} - 5 T^{4} - 169 T^{3} + 998 T^{2} + \cdots - 9808)^{2}$$
$67$ $$T^{10} + 564 T^{8} + 112832 T^{6} + \cdots + 1048576$$
$71$ $$(T^{5} - 6 T^{4} - 208 T^{3} + 1192 T^{2} + \cdots - 62464)^{2}$$
$73$ $$T^{10} + 543 T^{8} + \cdots + 3008303104$$
$79$ $$(T^{5} + 17 T^{4} + 48 T^{3} - 471 T^{2} + \cdots - 3020)^{2}$$
$83$ $$T^{10} + 684 T^{8} + \cdots + 3399356416$$
$89$ $$(T^{5} + 10 T^{4} - 96 T^{3} - 1088 T^{2} + \cdots - 160)^{2}$$
$97$ $$T^{10} + 623 T^{8} + \cdots + 3351946816$$