# Properties

 Label 637.2.f.h Level $637$ Weight $2$ Character orbit 637.f Analytic conductor $5.086$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(295,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("637.295");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.f (of order $$3$$, degree $$2$$, 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: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.100088711424.6 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 13x^{6} + 130x^{4} - 507x^{2} + 1521$$ x^8 - 13*x^6 + 130*x^4 - 507*x^2 + 1521 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} - \beta_{2} - 1) q^{2} + (\beta_{7} - \beta_{3}) q^{3} + (\beta_{5} + 2 \beta_{2}) q^{4} + \beta_{3} q^{5} + (2 \beta_{4} + \beta_1) q^{6} + 3 q^{8} + ( - \beta_{5} + 3 \beta_{2}) q^{9}+O(q^{10})$$ q + (-b6 - b2 - 1) * q^2 + (b7 - b3) * q^3 + (b5 + 2*b2) * q^4 + b3 * q^5 + (2*b4 + b1) * q^6 + 3 * q^8 + (-b5 + 3*b2) * q^9 $$q + ( - \beta_{6} - \beta_{2} - 1) q^{2} + (\beta_{7} - \beta_{3}) q^{3} + (\beta_{5} + 2 \beta_{2}) q^{4} + \beta_{3} q^{5} + (2 \beta_{4} + \beta_1) q^{6} + 3 q^{8} + ( - \beta_{5} + 3 \beta_{2}) q^{9} + ( - \beta_{4} + \beta_1) q^{10} + ( - 3 \beta_{6} - \beta_{2} - 1) q^{11} + ( - \beta_{4} + \beta_{3} - 2 \beta_1) q^{12} + (\beta_{7} + \beta_1) q^{13} + (\beta_{6} - 6 \beta_{2} - 6) q^{15} + ( - \beta_{6} + \beta_{2} + 1) q^{16} + ( - \beta_{7} + 2 \beta_{4} + \beta_1) q^{17} + (3 \beta_{6} - 3 \beta_{5}) q^{18} - \beta_{7} q^{19} + ( - \beta_{7} + 2 \beta_{4} + \beta_1) q^{20} + (\beta_{5} + 10 \beta_{2}) q^{22} + (2 \beta_{6} - 2 \beta_{2} - 2) q^{23} + (3 \beta_{7} - 3 \beta_{3}) q^{24} + ( - \beta_{6} + \beta_{5} + 1) q^{25} + ( - 2 \beta_{7} + 2 \beta_{4} + \beta_{3} + 2 \beta_1) q^{26} + (\beta_{4} + \beta_{3} + 2 \beta_1) q^{27} + (\beta_{6} + \beta_{2} + 1) q^{29} + (6 \beta_{5} + 3 \beta_{2}) q^{30} + (\beta_{4} - \beta_{3} + 2 \beta_1) q^{31} + ( - \beta_{5} - 4 \beta_{2}) q^{32} + (2 \beta_{7} + 6 \beta_{4} + 3 \beta_1) q^{33} + ( - 2 \beta_{4} + 3 \beta_{3} - 4 \beta_1) q^{34} + ( - 2 \beta_{6} - 3 \beta_{2} - 3) q^{36} + ( - 2 \beta_{6} - 6 \beta_{2} - 6) q^{37} + ( - \beta_{4} - 2 \beta_1) q^{38} + (3 \beta_{6} + \beta_{5} + 2 \beta_{2} - 5) q^{39} + 3 \beta_{3} q^{40} + (2 \beta_{4} - 2 \beta_1) q^{41} + ( - 5 \beta_{5} - 6 \beta_{2}) q^{43} + (4 \beta_{6} - 4 \beta_{5} + 11) q^{44} + ( - 4 \beta_{7} - 2 \beta_{4} - \beta_1) q^{45} + (2 \beta_{5} - 4 \beta_{2}) q^{46} + ( - \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{47} + (2 \beta_{7} + 2 \beta_{4} + \beta_1) q^{48} + ( - \beta_{6} + 2 \beta_{2} + 2) q^{50} + (5 \beta_{6} - 5 \beta_{5} + 9) q^{51} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_1) q^{52} + ( - 2 \beta_{6} + 2 \beta_{5} + 5) q^{53} + ( - 3 \beta_{7} + 3 \beta_{3}) q^{54} + ( - 2 \beta_{7} - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{55} + ( - \beta_{6} + \beta_{5} + 6) q^{57} + ( - \beta_{5} - 4 \beta_{2}) q^{58} + ( - \beta_{7} - 2 \beta_{4} - \beta_1) q^{59} + (5 \beta_{6} - 5 \beta_{5} + 9) q^{60} - 2 \beta_{7} q^{61} + ( - 3 \beta_{7} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_1) q^{62} + ( - 6 \beta_{6} + 6 \beta_{5} - 5) q^{64} + ( - 4 \beta_{6} + 3 \beta_{5} - 7 \beta_{2} - 2) q^{65} + ( - \beta_{4} + 9 \beta_{3} - 2 \beta_1) q^{66} + ( - \beta_{2} - 1) q^{67} + (4 \beta_{7} - 3 \beta_{4} - 4 \beta_{3} + 3 \beta_1) q^{68} + ( - 4 \beta_{7} - 4 \beta_{4} - 2 \beta_1) q^{69} + 4 \beta_{2} q^{71} + ( - 3 \beta_{5} + 9 \beta_{2}) q^{72} - 2 \beta_{3} q^{73} + (6 \beta_{5} + 12 \beta_{2}) q^{74} + (2 \beta_{7} + \beta_{4} - 2 \beta_{3} - \beta_1) q^{75} + (\beta_{7} - \beta_{4} - \beta_{3} + \beta_1) q^{76} + (7 \beta_{6} - 2 \beta_{5} - 4 \beta_{2} + 10) q^{78} + ( - 2 \beta_{6} + 2 \beta_{5} - 4) q^{79} + ( - 2 \beta_{7} - \beta_{4} + 2 \beta_{3} + \beta_1) q^{80} + (4 \beta_{6} + 6 \beta_{2} + 6) q^{81} + (6 \beta_{7} - 4 \beta_{4} - 2 \beta_1) q^{82} + ( - \beta_{4} - \beta_{3} - 2 \beta_1) q^{83} + (5 \beta_{5} + 9 \beta_{2}) q^{85} + ( - 6 \beta_{6} + 6 \beta_{5} - 21) q^{86} + ( - 2 \beta_{4} - \beta_1) q^{87} + ( - 9 \beta_{6} - 3 \beta_{2} - 3) q^{88} + (3 \beta_{7} - 3 \beta_{3}) q^{89} + ( - 3 \beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{90} - 2 q^{92} + (5 \beta_{6} + 9 \beta_{2} + 9) q^{93} + (3 \beta_{7} + 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{94} + ( - \beta_{5} + 6 \beta_{2}) q^{95} + (\beta_{4} - 3 \beta_{3} + 2 \beta_1) q^{96} + (4 \beta_{7} - 2 \beta_{4} - \beta_1) q^{97} + (11 \beta_{6} - 11 \beta_{5} - 6) q^{99}+O(q^{100})$$ q + (-b6 - b2 - 1) * q^2 + (b7 - b3) * q^3 + (b5 + 2*b2) * q^4 + b3 * q^5 + (2*b4 + b1) * q^6 + 3 * q^8 + (-b5 + 3*b2) * q^9 + (-b4 + b1) * q^10 + (-3*b6 - b2 - 1) * q^11 + (-b4 + b3 - 2*b1) * q^12 + (b7 + b1) * q^13 + (b6 - 6*b2 - 6) * q^15 + (-b6 + b2 + 1) * q^16 + (-b7 + 2*b4 + b1) * q^17 + (3*b6 - 3*b5) * q^18 - b7 * q^19 + (-b7 + 2*b4 + b1) * q^20 + (b5 + 10*b2) * q^22 + (2*b6 - 2*b2 - 2) * q^23 + (3*b7 - 3*b3) * q^24 + (-b6 + b5 + 1) * q^25 + (-2*b7 + 2*b4 + b3 + 2*b1) * q^26 + (b4 + b3 + 2*b1) * q^27 + (b6 + b2 + 1) * q^29 + (6*b5 + 3*b2) * q^30 + (b4 - b3 + 2*b1) * q^31 + (-b5 - 4*b2) * q^32 + (2*b7 + 6*b4 + 3*b1) * q^33 + (-2*b4 + 3*b3 - 4*b1) * q^34 + (-2*b6 - 3*b2 - 3) * q^36 + (-2*b6 - 6*b2 - 6) * q^37 + (-b4 - 2*b1) * q^38 + (3*b6 + b5 + 2*b2 - 5) * q^39 + 3*b3 * q^40 + (2*b4 - 2*b1) * q^41 + (-5*b5 - 6*b2) * q^43 + (4*b6 - 4*b5 + 11) * q^44 + (-4*b7 - 2*b4 - b1) * q^45 + (2*b5 - 4*b2) * q^46 + (-b4 - 3*b3 - 2*b1) * q^47 + (2*b7 + 2*b4 + b1) * q^48 + (-b6 + 2*b2 + 2) * q^50 + (5*b6 - 5*b5 + 9) * q^51 + (-b4 + 2*b3 - 3*b1) * q^52 + (-2*b6 + 2*b5 + 5) * q^53 + (-3*b7 + 3*b3) * q^54 + (-2*b7 - 3*b4 + 2*b3 + 3*b1) * q^55 + (-b6 + b5 + 6) * q^57 + (-b5 - 4*b2) * q^58 + (-b7 - 2*b4 - b1) * q^59 + (5*b6 - 5*b5 + 9) * q^60 - 2*b7 * q^61 + (-3*b7 + 2*b4 + 3*b3 - 2*b1) * q^62 + (-6*b6 + 6*b5 - 5) * q^64 + (-4*b6 + 3*b5 - 7*b2 - 2) * q^65 + (-b4 + 9*b3 - 2*b1) * q^66 + (-b2 - 1) * q^67 + (4*b7 - 3*b4 - 4*b3 + 3*b1) * q^68 + (-4*b7 - 4*b4 - 2*b1) * q^69 + 4*b2 * q^71 + (-3*b5 + 9*b2) * q^72 - 2*b3 * q^73 + (6*b5 + 12*b2) * q^74 + (2*b7 + b4 - 2*b3 - b1) * q^75 + (b7 - b4 - b3 + b1) * q^76 + (7*b6 - 2*b5 - 4*b2 + 10) * q^78 + (-2*b6 + 2*b5 - 4) * q^79 + (-2*b7 - b4 + 2*b3 + b1) * q^80 + (4*b6 + 6*b2 + 6) * q^81 + (6*b7 - 4*b4 - 2*b1) * q^82 + (-b4 - b3 - 2*b1) * q^83 + (5*b5 + 9*b2) * q^85 + (-6*b6 + 6*b5 - 21) * q^86 + (-2*b4 - b1) * q^87 + (-9*b6 - 3*b2 - 3) * q^88 + (3*b7 - 3*b3) * q^89 + (-3*b4 - 3*b3 - 6*b1) * q^90 - 2 * q^92 + (5*b6 + 9*b2 + 9) * q^93 + (3*b7 + 2*b4 - 3*b3 - 2*b1) * q^94 + (-b5 + 6*b2) * q^95 + (b4 - 3*b3 + 2*b1) * q^96 + (4*b7 - 2*b4 - b1) * q^97 + (11*b6 - 11*b5 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{2} - 6 q^{4} + 24 q^{8} - 14 q^{9}+O(q^{10})$$ 8 * q - 2 * q^2 - 6 * q^4 + 24 * q^8 - 14 * q^9 $$8 q - 2 q^{2} - 6 q^{4} + 24 q^{8} - 14 q^{9} + 2 q^{11} - 26 q^{15} + 6 q^{16} - 12 q^{18} - 38 q^{22} - 12 q^{23} + 12 q^{25} + 2 q^{29} + 14 q^{32} - 8 q^{36} - 20 q^{37} - 52 q^{39} + 14 q^{43} + 72 q^{44} + 20 q^{46} + 10 q^{50} + 52 q^{51} + 48 q^{53} + 52 q^{57} + 14 q^{58} + 52 q^{60} - 16 q^{64} + 26 q^{65} - 4 q^{67} - 16 q^{71} - 42 q^{72} - 36 q^{74} + 78 q^{78} - 24 q^{79} + 16 q^{81} - 26 q^{85} - 144 q^{86} + 6 q^{88} - 16 q^{92} + 26 q^{93} - 26 q^{95} - 92 q^{99}+O(q^{100})$$ 8 * q - 2 * q^2 - 6 * q^4 + 24 * q^8 - 14 * q^9 + 2 * q^11 - 26 * q^15 + 6 * q^16 - 12 * q^18 - 38 * q^22 - 12 * q^23 + 12 * q^25 + 2 * q^29 + 14 * q^32 - 8 * q^36 - 20 * q^37 - 52 * q^39 + 14 * q^43 + 72 * q^44 + 20 * q^46 + 10 * q^50 + 52 * q^51 + 48 * q^53 + 52 * q^57 + 14 * q^58 + 52 * q^60 - 16 * q^64 + 26 * q^65 - 4 * q^67 - 16 * q^71 - 42 * q^72 - 36 * q^74 + 78 * q^78 - 24 * q^79 + 16 * q^81 - 26 * q^85 - 144 * q^86 + 6 * q^88 - 16 * q^92 + 26 * q^93 - 26 * q^95 - 92 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 13x^{6} + 130x^{4} - 507x^{2} + 1521$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} - 10\nu^{4} + 130\nu^{2} - 507 ) / 390$$ (v^6 - 10*v^4 + 130*v^2 - 507) / 390 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 70\nu^{5} - 520\nu^{3} + 3237\nu ) / 1170$$ (-v^7 + 70*v^5 - 520*v^3 + 3237*v) / 1170 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 10\nu^{5} - 130\nu^{3} + 117\nu ) / 390$$ (-v^7 + 10*v^5 - 130*v^3 + 117*v) / 390 $$\beta_{5}$$ $$=$$ $$( -2\nu^{6} + 35\nu^{4} - 260\nu^{2} + 1014 ) / 195$$ (-2*v^6 + 35*v^4 - 260*v^2 + 1014) / 195 $$\beta_{6}$$ $$=$$ $$( -7\nu^{6} + 70\nu^{4} - 520\nu^{2} + 819 ) / 390$$ (-7*v^6 + 70*v^4 - 520*v^2 + 819) / 390 $$\beta_{7}$$ $$=$$ $$( -11\nu^{7} + 140\nu^{5} - 1040\nu^{3} + 2847\nu ) / 1170$$ (-11*v^7 + 140*v^5 - 1040*v^3 + 2847*v) / 1170
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 7\beta_{2} + 7$$ b6 + 7*b2 + 7 $$\nu^{3}$$ $$=$$ $$2\beta_{7} - 7\beta_{4} - \beta_{3}$$ 2*b7 - 7*b4 - b3 $$\nu^{4}$$ $$=$$ $$13\beta_{5} + 52\beta_{2}$$ 13*b5 + 52*b2 $$\nu^{5}$$ $$=$$ $$13\beta_{7} - 52\beta_{4} + 13\beta_{3} - 52\beta_1$$ 13*b7 - 52*b4 + 13*b3 - 52*b1 $$\nu^{6}$$ $$=$$ $$-130\beta_{6} + 130\beta_{5} - 403$$ -130*b6 + 130*b5 - 403 $$\nu^{7}$$ $$=$$ $$-130\beta_{7} + 260\beta_{3} - 403\beta_1$$ -130*b7 + 260*b3 - 403*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
295.1
 −2.49541 + 1.44073i 2.49541 − 1.44073i 1.87694 − 1.08365i −1.87694 + 1.08365i −2.49541 − 1.44073i 2.49541 + 1.44073i 1.87694 + 1.08365i −1.87694 − 1.08365i
−1.15139 + 1.99426i −1.08365 + 1.87694i −1.65139 2.86029i 2.16731 −2.49541 4.32218i 0 3.00000 −0.848612 1.46984i −2.49541 + 4.32218i
295.2 −1.15139 + 1.99426i 1.08365 1.87694i −1.65139 2.86029i −2.16731 2.49541 + 4.32218i 0 3.00000 −0.848612 1.46984i 2.49541 4.32218i
295.3 0.651388 1.12824i −1.44073 + 2.49541i 0.151388 + 0.262211i 2.88145 1.87694 + 3.25096i 0 3.00000 −2.65139 4.59234i 1.87694 3.25096i
295.4 0.651388 1.12824i 1.44073 2.49541i 0.151388 + 0.262211i −2.88145 −1.87694 3.25096i 0 3.00000 −2.65139 4.59234i −1.87694 + 3.25096i
393.1 −1.15139 1.99426i −1.08365 1.87694i −1.65139 + 2.86029i 2.16731 −2.49541 + 4.32218i 0 3.00000 −0.848612 + 1.46984i −2.49541 4.32218i
393.2 −1.15139 1.99426i 1.08365 + 1.87694i −1.65139 + 2.86029i −2.16731 2.49541 4.32218i 0 3.00000 −0.848612 + 1.46984i 2.49541 + 4.32218i
393.3 0.651388 + 1.12824i −1.44073 2.49541i 0.151388 0.262211i 2.88145 1.87694 3.25096i 0 3.00000 −2.65139 + 4.59234i 1.87694 + 3.25096i
393.4 0.651388 + 1.12824i 1.44073 + 2.49541i 0.151388 0.262211i −2.88145 −1.87694 + 3.25096i 0 3.00000 −2.65139 + 4.59234i −1.87694 3.25096i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 393.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.c even 3 1 inner
91.n odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.f.h 8
7.b odd 2 1 inner 637.2.f.h 8
7.c even 3 1 637.2.g.i 8
7.c even 3 1 637.2.h.j 8
7.d odd 6 1 637.2.g.i 8
7.d odd 6 1 637.2.h.j 8
13.c even 3 1 inner 637.2.f.h 8
13.c even 3 1 8281.2.a.bu 4
13.e even 6 1 8281.2.a.bo 4
91.g even 3 1 637.2.h.j 8
91.h even 3 1 637.2.g.i 8
91.m odd 6 1 637.2.h.j 8
91.n odd 6 1 inner 637.2.f.h 8
91.n odd 6 1 8281.2.a.bu 4
91.t odd 6 1 8281.2.a.bo 4
91.v odd 6 1 637.2.g.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.h 8 1.a even 1 1 trivial
637.2.f.h 8 7.b odd 2 1 inner
637.2.f.h 8 13.c even 3 1 inner
637.2.f.h 8 91.n odd 6 1 inner
637.2.g.i 8 7.c even 3 1
637.2.g.i 8 7.d odd 6 1
637.2.g.i 8 91.h even 3 1
637.2.g.i 8 91.v odd 6 1
637.2.h.j 8 7.c even 3 1
637.2.h.j 8 7.d odd 6 1
637.2.h.j 8 91.g even 3 1
637.2.h.j 8 91.m odd 6 1
8281.2.a.bo 4 13.e even 6 1
8281.2.a.bo 4 91.t odd 6 1
8281.2.a.bu 4 13.c even 3 1
8281.2.a.bu 4 91.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{4} + T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 9$$ T2^4 + T2^3 + 4*T2^2 - 3*T2 + 9 $$T_{3}^{8} + 13T_{3}^{6} + 130T_{3}^{4} + 507T_{3}^{2} + 1521$$ T3^8 + 13*T3^6 + 130*T3^4 + 507*T3^2 + 1521 $$T_{5}^{4} - 13T_{5}^{2} + 39$$ T5^4 - 13*T5^2 + 39

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + T^{3} + 4 T^{2} - 3 T + 9)^{2}$$
$3$ $$T^{8} + 13 T^{6} + 130 T^{4} + \cdots + 1521$$
$5$ $$(T^{4} - 13 T^{2} + 39)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{4} - T^{3} + 30 T^{2} + 29 T + 841)^{2}$$
$13$ $$T^{8} + 13T^{4} + 28561$$
$17$ $$T^{8} + 52 T^{6} + 2665 T^{4} + \cdots + 1521$$
$19$ $$T^{8} + 13 T^{6} + 130 T^{4} + \cdots + 1521$$
$23$ $$(T^{4} + 6 T^{3} + 40 T^{2} - 24 T + 16)^{2}$$
$29$ $$(T^{4} - T^{3} + 4 T^{2} + 3 T + 9)^{2}$$
$31$ $$(T^{4} - 52 T^{2} + 39)^{2}$$
$37$ $$(T^{4} + 10 T^{3} + 88 T^{2} + 120 T + 144)^{2}$$
$41$ $$T^{8} + 156 T^{6} + \cdots + 31539456$$
$43$ $$(T^{4} - 7 T^{3} + 118 T^{2} + 483 T + 4761)^{2}$$
$47$ $$(T^{4} - 156 T^{2} + 351)^{2}$$
$53$ $$(T^{2} - 12 T + 23)^{4}$$
$59$ $$T^{8} + 52 T^{6} + 2353 T^{4} + \cdots + 123201$$
$61$ $$T^{8} + 52 T^{6} + 2080 T^{4} + \cdots + 389376$$
$67$ $$(T^{2} + T + 1)^{4}$$
$71$ $$(T^{2} + 4 T + 16)^{4}$$
$73$ $$(T^{4} - 52 T^{2} + 624)^{2}$$
$79$ $$(T^{2} + 6 T - 4)^{4}$$
$83$ $$(T^{4} - 52 T^{2} + 351)^{2}$$
$89$ $$T^{8} + 117 T^{6} + 10530 T^{4} + \cdots + 9979281$$
$97$ $$T^{8} + 247 T^{6} + \cdots + 127035441$$