# Properties

 Label 3330.2.d.o Level $3330$ Weight $2$ Character orbit 3330.d Analytic conductor $26.590$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1999,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3330.1999");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.d (of order $$2$$, degree $$1$$, 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: $$26.5901838731$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.57815240704.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16$$ x^8 - 2*x^7 + 2*x^6 + 89*x^4 - 170*x^3 + 162*x^2 - 72*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1110) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - q^{4} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{7} - \beta_{6} + \beta_{3} - \beta_{2}) q^{7} - \beta_{3} q^{8}+O(q^{10})$$ q + b3 * q^2 - q^4 + (-b5 - b1) * q^5 + (b7 - b6 + b3 - b2) * q^7 - b3 * q^8 $$q + \beta_{3} q^{2} - q^{4} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{7} - \beta_{6} + \beta_{3} - \beta_{2}) q^{7} - \beta_{3} q^{8} + (\beta_{6} + \beta_{2} - \beta_1) q^{10} + ( - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{6} + \beta_{3} - \beta_{2}) q^{13} + ( - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{14} + q^{16} + ( - \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{17} + (\beta_{5} + 2 \beta_{4} + 3) q^{19} + (\beta_{5} + \beta_1) q^{20} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2}) q^{22} + ( - 3 \beta_{6} + 3 \beta_{3} + 3 \beta_1) q^{23} + (\beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1) q^{25} + ( - \beta_{5} + \beta_{2} - \beta_1 - 1) q^{26} + ( - \beta_{7} + \beta_{6} - \beta_{3} + \beta_{2}) q^{28} + ( - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{29} + (4 \beta_{5} + \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 4) q^{31} + \beta_{3} q^{32} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{34} + (2 \beta_{7} + \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{35} + \beta_{3} q^{37} + ( - 2 \beta_{7} - \beta_{6} + 3 \beta_{3} + \beta_1) q^{38} + ( - \beta_{6} - \beta_{2} + \beta_1) q^{40} + (2 \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 2) q^{41} + \beta_{7} q^{43} + (\beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{44} + ( - 3 \beta_{5} - 3) q^{46} + ( - 2 \beta_{6} - 6 \beta_{3} - \beta_{2} + \beta_1) q^{47} + (3 \beta_{5} - \beta_{4} - 2) q^{49} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_1 + 1) q^{50} + (\beta_{6} - \beta_{3} + \beta_{2}) q^{52} + (\beta_{7} + 3 \beta_{6} + 7 \beta_{3} + 2 \beta_{2} - \beta_1) q^{53} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{55} + (\beta_{5} - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{56} + (\beta_{7} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{58} + ( - 2 \beta_{5} + \beta_{2} - \beta_1 + 2) q^{59} - 3 \beta_{4} q^{61} + ( - \beta_{7} - 4 \beta_{6} - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{62} - q^{64} + (\beta_{7} + 2 \beta_{6} + \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{65} + ( - 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2}) q^{67} + (\beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{68} + ( - \beta_{7} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{70} + ( - 6 \beta_{5} - 4 \beta_{4} + \beta_{2} - \beta_1) q^{71} + (2 \beta_{7} + \beta_{6} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{73} - q^{74} + ( - \beta_{5} - 2 \beta_{4} - 3) q^{76} + (3 \beta_{7} - 5 \beta_{6} - 7 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{77} + ( - 6 \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1) q^{79} + ( - \beta_{5} - \beta_1) q^{80} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{82} + (2 \beta_{7} - 5 \beta_{6} + 7 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{83} + (2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2} + 5) q^{85} + \beta_{4} q^{86} + (\beta_{7} - \beta_{6} - \beta_{3} - \beta_{2}) q^{88} + (\beta_{5} + \beta_{2} - \beta_1 + 9) q^{89} + ( - \beta_{5} + 2 \beta_{2} - 2 \beta_1 - 5) q^{91} + (3 \beta_{6} - 3 \beta_{3} - 3 \beta_1) q^{92} + ( - 2 \beta_{5} + \beta_{2} - \beta_1 + 6) q^{94} + ( - \beta_{7} - \beta_{6} - 7 \beta_{5} + \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{95} + (3 \beta_{7} + \beta_{2} + \beta_1) q^{97} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{3} + 3 \beta_1) q^{98}+O(q^{100})$$ q + b3 * q^2 - q^4 + (-b5 - b1) * q^5 + (b7 - b6 + b3 - b2) * q^7 - b3 * q^8 + (b6 + b2 - b1) * q^10 + (-b5 + b4 + b2 - b1 + 1) * q^11 + (-b6 + b3 - b2) * q^13 + (-b5 + b4 + b2 - b1 - 1) * q^14 + q^16 + (-b7 - b6 + b3 + b2 + 2*b1) * q^17 + (b5 + 2*b4 + 3) * q^19 + (b5 + b1) * q^20 + (-b7 + b6 + b3 + b2) * q^22 + (-3*b6 + 3*b3 + 3*b1) * q^23 + (b6 + b5 + 2*b4 - b3 - b1) * q^25 + (-b5 + b2 - b1 - 1) * q^26 + (-b7 + b6 - b3 + b2) * q^28 + (-b4 + b2 - b1 + 2) * q^29 + (4*b5 + b4 - 2*b2 + 2*b1 - 4) * q^31 + b3 * q^32 + (-b5 - b4 - b2 + b1 - 1) * q^34 + (2*b7 + b4 - 3*b3 + b2 - 2*b1 - 1) * q^35 + b3 * q^37 + (-2*b7 - b6 + 3*b3 + b1) * q^38 + (-b6 - b2 + b1) * q^40 + (2*b5 + b4 + b2 - b1 - 2) * q^41 + b7 * q^43 + (b5 - b4 - b2 + b1 - 1) * q^44 + (-3*b5 - 3) * q^46 + (-2*b6 - 6*b3 - b2 + b1) * q^47 + (3*b5 - b4 - 2) * q^49 + (-2*b7 - b6 + b5 + b1 + 1) * q^50 + (b6 - b3 + b2) * q^52 + (b7 + 3*b6 + 7*b3 + 2*b2 - b1) * q^53 + (-b7 - 2*b5 + 2*b4 - b3 + 2*b2 - b1 + 3) * q^55 + (b5 - b4 - b2 + b1 + 1) * q^56 + (b7 + 2*b3 + b2 + b1) * q^58 + (-2*b5 + b2 - b1 + 2) * q^59 - 3*b4 * q^61 + (-b7 - 4*b6 - 4*b3 - 2*b2 + 2*b1) * q^62 - q^64 + (b7 + 2*b6 + b4 - 3*b3 + b2 - 2*b1 - 3) * q^65 + (-4*b7 - 2*b6 + 2*b3 - 2*b2) * q^67 + (b7 + b6 - b3 - b2 - 2*b1) * q^68 + (-b7 + 2*b4 - b3 + 2*b2 + b1 + 3) * q^70 + (-6*b5 - 4*b4 + b2 - b1) * q^71 + (2*b7 + b6 - b3 + 2*b2 + b1) * q^73 - q^74 + (-b5 - 2*b4 - 3) * q^76 + (3*b7 - 5*b6 - 7*b3 - 2*b2 + 3*b1) * q^77 + (-6*b5 - 2*b4 + b2 - b1) * q^79 + (-b5 - b1) * q^80 + (-b7 - 2*b6 - 2*b3 + b2 + 3*b1) * q^82 + (2*b7 - 5*b6 + 7*b3 - 3*b2 + 2*b1) * q^83 + (2*b6 - 2*b5 - 3*b4 - b3 + b2 + 5) * q^85 + b4 * q^86 + (b7 - b6 - b3 - b2) * q^88 + (b5 + b2 - b1 + 9) * q^89 + (-b5 + 2*b2 - 2*b1 - 5) * q^91 + (3*b6 - 3*b3 - 3*b1) * q^92 + (-2*b5 + b2 - b1 + 6) * q^94 + (-b7 - b6 - 7*b5 + b4 + 2*b3 + 4*b2 - 2*b1 - 2) * q^95 + (3*b7 + b2 + b1) * q^97 + (b7 - 3*b6 - 2*b3 + 3*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4} + 2 q^{5}+O(q^{10})$$ 8 * q - 8 * q^4 + 2 * q^5 $$8 q - 8 q^{4} + 2 q^{5} - 2 q^{10} + 10 q^{11} - 6 q^{14} + 8 q^{16} + 24 q^{19} - 2 q^{20} - 8 q^{26} + 10 q^{29} - 38 q^{31} - 2 q^{34} - 12 q^{35} + 2 q^{40} - 26 q^{41} - 10 q^{44} - 12 q^{46} - 30 q^{49} + 4 q^{50} + 30 q^{55} + 6 q^{56} + 20 q^{59} - 6 q^{61} - 8 q^{64} - 24 q^{65} + 26 q^{70} + 12 q^{71} - 8 q^{74} - 24 q^{76} + 16 q^{79} + 2 q^{80} + 44 q^{85} + 2 q^{86} + 64 q^{89} - 44 q^{91} + 52 q^{94}+O(q^{100})$$ 8 * q - 8 * q^4 + 2 * q^5 - 2 * q^10 + 10 * q^11 - 6 * q^14 + 8 * q^16 + 24 * q^19 - 2 * q^20 - 8 * q^26 + 10 * q^29 - 38 * q^31 - 2 * q^34 - 12 * q^35 + 2 * q^40 - 26 * q^41 - 10 * q^44 - 12 * q^46 - 30 * q^49 + 4 * q^50 + 30 * q^55 + 6 * q^56 + 20 * q^59 - 6 * q^61 - 8 * q^64 - 24 * q^65 + 26 * q^70 + 12 * q^71 - 8 * q^74 - 24 * q^76 + 16 * q^79 + 2 * q^80 + 44 * q^85 + 2 * q^86 + 64 * q^89 - 44 * q^91 + 52 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 198\nu^{7} - 217\nu^{6} + 209\nu^{5} + 88\nu^{4} + 18711\nu^{3} - 17653\nu^{2} + 16897\nu - 7524 ) / 9085$$ (198*v^7 - 217*v^6 + 209*v^5 + 88*v^4 + 18711*v^3 - 17653*v^2 + 16897*v - 7524) / 9085 $$\beta_{3}$$ $$=$$ $$( -1881\nu^{7} + 2970\nu^{6} - 2894\nu^{5} - 836\nu^{4} - 167761\nu^{3} + 244926\nu^{2} - 234110\nu + 67844 ) / 36340$$ (-1881*v^7 + 2970*v^6 - 2894*v^5 - 836*v^4 - 167761*v^3 + 244926*v^2 - 234110*v + 67844) / 36340 $$\beta_{4}$$ $$=$$ $$( 1969\nu^{7} - 2057\nu^{6} + 968\nu^{5} + 2894\nu^{4} + 176077\nu^{3} - 166969\nu^{2} + 74052\nu + 56002 ) / 18170$$ (1969*v^7 - 2057*v^6 + 968*v^5 + 2894*v^4 + 176077*v^3 - 166969*v^2 + 74052*v + 56002) / 18170 $$\beta_{5}$$ $$=$$ $$( -358\nu^{7} + 374\nu^{6} - 176\nu^{5} - 361\nu^{4} - 32014\nu^{3} + 30358\nu^{2} - 13464\nu - 2749 ) / 1817$$ (-358*v^7 + 374*v^6 - 176*v^5 - 361*v^4 - 32014*v^3 + 30358*v^2 - 13464*v - 2749) / 1817 $$\beta_{6}$$ $$=$$ $$( - 9117 \nu^{7} + 17838 \nu^{6} - 14166 \nu^{5} - 4052 \nu^{4} - 811589 \nu^{3} + 1516102 \nu^{2} - 1096418 \nu + 328276 ) / 36340$$ (-9117*v^7 + 17838*v^6 - 14166*v^5 - 4052*v^4 - 811589*v^3 + 1516102*v^2 - 1096418*v + 328276) / 36340 $$\beta_{7}$$ $$=$$ $$( 9261 \nu^{7} - 16344 \nu^{6} + 14318 \nu^{5} + 4116 \nu^{4} + 825197 \nu^{3} - 1388536 \nu^{2} + 1151654 \nu - 333748 ) / 36340$$ (9261*v^7 - 16344*v^6 + 14318*v^5 + 4116*v^4 + 825197*v^3 - 1388536*v^2 + 1151654*v - 333748) / 36340
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-2\beta_{7} - \beta_{6} - 5\beta_{3} + \beta_1$$ -2*b7 - b6 - 5*b3 + b1 $$\nu^{3}$$ $$=$$ $$-\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 9\beta_{2} + \beta _1 + 2$$ -b7 - b6 + b5 + b4 + 2*b3 + 9*b2 + b1 + 2 $$\nu^{4}$$ $$=$$ $$11\beta_{5} + 20\beta_{4} - 45$$ 11*b5 + 20*b4 - 45 $$\nu^{5}$$ $$=$$ $$11\beta_{7} + 11\beta_{6} + 11\beta_{5} + 11\beta_{4} - 18\beta_{3} - 96\beta _1 + 18$$ 11*b7 + 11*b6 + 11*b5 + 11*b4 - 18*b3 - 96*b1 + 18 $$\nu^{6}$$ $$=$$ $$192\beta_{7} + 107\beta_{6} + 425\beta_{3} - 4\beta_{2} - 111\beta_1$$ 192*b7 + 107*b6 + 425*b3 - 4*b2 - 111*b1 $$\nu^{7}$$ $$=$$ $$115\beta_{7} + 111\beta_{6} - 111\beta_{5} - 115\beta_{4} - 150\beta_{3} - 809\beta_{2} - 111\beta _1 - 150$$ 115*b7 + 111*b6 - 111*b5 - 115*b4 - 150*b3 - 809*b2 - 111*b1 - 150

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1999.1
 0.594137 + 0.594137i 2.20793 + 2.20793i −2.15569 − 2.15569i 0.353624 + 0.353624i 0.594137 − 0.594137i 2.20793 − 2.20793i −2.15569 + 2.15569i 0.353624 − 0.353624i
1.00000i 0 −1.00000 −2.15569 0.594137i 0 2.38360i 1.00000i 0 −0.594137 + 2.15569i
1999.2 1.00000i 0 −1.00000 0.353624 2.20793i 0 3.94848i 1.00000i 0 −2.20793 0.353624i
1999.3 1.00000i 0 −1.00000 0.594137 + 2.15569i 0 1.17795i 1.00000i 0 2.15569 0.594137i
1999.4 1.00000i 0 −1.00000 2.20793 0.353624i 0 4.51003i 1.00000i 0 −0.353624 2.20793i
1999.5 1.00000i 0 −1.00000 −2.15569 + 0.594137i 0 2.38360i 1.00000i 0 −0.594137 2.15569i
1999.6 1.00000i 0 −1.00000 0.353624 + 2.20793i 0 3.94848i 1.00000i 0 −2.20793 + 0.353624i
1999.7 1.00000i 0 −1.00000 0.594137 2.15569i 0 1.17795i 1.00000i 0 2.15569 + 0.594137i
1999.8 1.00000i 0 −1.00000 2.20793 + 0.353624i 0 4.51003i 1.00000i 0 −0.353624 + 2.20793i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1999.8 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 3330.2.d.o 8
3.b odd 2 1 1110.2.d.j 8
5.b even 2 1 inner 3330.2.d.o 8
15.d odd 2 1 1110.2.d.j 8
15.e even 4 1 5550.2.a.cl 4
15.e even 4 1 5550.2.a.cm 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.j 8 3.b odd 2 1
1110.2.d.j 8 15.d odd 2 1
3330.2.d.o 8 1.a even 1 1 trivial
3330.2.d.o 8 5.b even 2 1 inner
5550.2.a.cl 4 15.e even 4 1
5550.2.a.cm 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3330, [\chi])$$:

 $$T_{7}^{8} + 43T_{7}^{6} + 579T_{7}^{4} + 2525T_{7}^{2} + 2500$$ T7^8 + 43*T7^6 + 579*T7^4 + 2525*T7^2 + 2500 $$T_{11}^{4} - 5T_{11}^{3} - 11T_{11}^{2} + 7T_{11} + 4$$ T11^4 - 5*T11^3 - 11*T11^2 + 7*T11 + 4 $$T_{17}^{8} + 75T_{17}^{6} + 1411T_{17}^{4} + 8285T_{17}^{2} + 11236$$ T17^8 + 75*T17^6 + 1411*T17^4 + 8285*T17^2 + 11236 $$T_{29}^{4} - 5T_{29}^{3} - 34T_{29}^{2} - 46T_{29} - 16$$ T29^4 - 5*T29^3 - 34*T29^2 - 46*T29 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 2 T^{7} + 2 T^{6} + 6 T^{5} + \cdots + 625$$
$7$ $$T^{8} + 43 T^{6} + 579 T^{4} + \cdots + 2500$$
$11$ $$(T^{4} - 5 T^{3} - 11 T^{2} + 7 T + 4)^{2}$$
$13$ $$T^{8} + 26 T^{6} + 201 T^{4} + \cdots + 256$$
$17$ $$T^{8} + 75 T^{6} + 1411 T^{4} + \cdots + 11236$$
$19$ $$(T^{4} - 12 T^{3} + 13 T^{2} + 138 T + 128)^{2}$$
$23$ $$(T^{4} + 81 T^{2} + 1296)^{2}$$
$29$ $$(T^{4} - 5 T^{3} - 34 T^{2} - 46 T - 16)^{2}$$
$31$ $$(T^{4} + 19 T^{3} + 56 T^{2} - 676 T - 3328)^{2}$$
$37$ $$(T^{2} + 1)^{4}$$
$41$ $$(T^{4} + 13 T^{3} - 8 T^{2} - 146 T + 172)^{2}$$
$43$ $$T^{8} + 25 T^{6} + 184 T^{4} + \cdots + 256$$
$47$ $$T^{8} + 208 T^{6} + 13396 T^{4} + \cdots + 719104$$
$53$ $$T^{8} + 335 T^{6} + 32191 T^{4} + \cdots + 1747684$$
$59$ $$(T^{4} - 10 T^{3} + 18 T^{2} + 52 T - 128)^{2}$$
$61$ $$(T^{4} + 3 T^{3} - 108 T^{2} - 108 T + 1296)^{2}$$
$67$ $$T^{8} + 568 T^{6} + \cdots + 11505664$$
$71$ $$(T^{4} - 6 T^{3} - 250 T^{2} + 1644 T - 64)^{2}$$
$73$ $$T^{8} + 278 T^{6} + \cdots + 11075584$$
$79$ $$(T^{4} - 8 T^{3} - 170 T^{2} + 1424 T - 1024)^{2}$$
$83$ $$T^{8} + 518 T^{6} + \cdots + 163840000$$
$89$ $$(T^{4} - 32 T^{3} + 339 T^{2} - 1362 T + 1616)^{2}$$
$97$ $$T^{8} + 337 T^{6} + 32404 T^{4} + \cdots + 839056$$