# Properties

 Label 125.2.e.a Level $125$ Weight $2$ Character orbit 125.e Analytic conductor $0.998$ 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] = [125,2,Mod(24,125)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(125, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("125.24");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$125 = 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 125.e (of order $$10$$, degree $$4$$, 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: $$0.998130025266$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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 primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}) q^{2} - \zeta_{20} q^{3} + ( - \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{6} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{7} + (\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}) q^{8} - 2 \zeta_{20}^{2} q^{9} +O(q^{10})$$ q + (-z^5 + z^3 - z) * q^2 - z * q^3 + (-z^6 - z^2 + 1) * q^4 + (z^6 - z^4 + z^2) * q^6 + (-z^7 - z^3) * q^7 + (z^7 + z^5 - z^3 - z) * q^8 - 2*z^2 * q^9 $$q + ( - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}) q^{2} - \zeta_{20} q^{3} + ( - \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{6} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{7} + (\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}) q^{8} - 2 \zeta_{20}^{2} q^{9} + ( - 2 \zeta_{20}^{6} + 4 \zeta_{20}^{4} - 4 \zeta_{20}^{2} + 2) q^{11} + (\zeta_{20}^{7} + \zeta_{20}^{3} - \zeta_{20}) q^{12} + ( - 3 \zeta_{20}^{7} - 3 \zeta_{20}^{3} + 3 \zeta_{20}) q^{13} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{14} + ( - 3 \zeta_{20}^{4} + 3 \zeta_{20}^{2} - 3) q^{16} + ( - 2 \zeta_{20}^{7} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{3} + 2 \zeta_{20}) q^{17} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3}) q^{18} + (3 \zeta_{20}^{6} + \zeta_{20}^{4} + 3 \zeta_{20}^{2}) q^{19} + (\zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{21} + (4 \zeta_{20}^{7} - 4 \zeta_{20}^{5} + 2 \zeta_{20}) q^{22} + (2 \zeta_{20}^{5} + 5 \zeta_{20}^{3} + 2 \zeta_{20}) q^{23} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} + 1) q^{24} - 3 q^{26} + 5 \zeta_{20}^{3} q^{27} + (\zeta_{20}^{7} - \zeta_{20}^{5} - 2 \zeta_{20}) q^{28} + ( - 2 \zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{29} - 3 \zeta_{20}^{4} q^{31} + (\zeta_{20}^{7} + 4 \zeta_{20}^{5} + \zeta_{20}^{3}) q^{32} + (2 \zeta_{20}^{7} - 4 \zeta_{20}^{5} + 4 \zeta_{20}^{3} - 2 \zeta_{20}) q^{33} + ( - 4 \zeta_{20}^{4} + 6 \zeta_{20}^{2} - 4) q^{34} + (2 \zeta_{20}^{6} - 2) q^{36} + ( - 3 \zeta_{20}^{7} - 2 \zeta_{20}^{3} + 2 \zeta_{20}) q^{37} + ( - 3 \zeta_{20}^{7} - \zeta_{20}^{3} + \zeta_{20}) q^{38} + (3 \zeta_{20}^{6} - 3) q^{39} + (2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} + 2) q^{41} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}) q^{42} + ( - 3 \zeta_{20}^{7} + 3 \zeta_{20}^{5} - 3 \zeta_{20}^{3}) q^{43} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2}) q^{44} + ( - 2 \zeta_{20}^{6} - 5 \zeta_{20}^{2} + 5) q^{46} + (\zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}) q^{47} + (3 \zeta_{20}^{5} - 3 \zeta_{20}^{3} + 3 \zeta_{20}) q^{48} + (\zeta_{20}^{6} - \zeta_{20}^{4} + 5) q^{49} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 2) q^{51} + ( - 3 \zeta_{20}^{5} - 3 \zeta_{20}^{3} - 3 \zeta_{20}) q^{52} + (4 \zeta_{20}^{7} - 4 \zeta_{20}^{5} - 3 \zeta_{20}) q^{53} + ( - 5 \zeta_{20}^{2} + 5) q^{54} + (\zeta_{20}^{6} + 2 \zeta_{20}^{4} + \zeta_{20}^{2}) q^{56} + ( - 3 \zeta_{20}^{7} - \zeta_{20}^{5} - 3 \zeta_{20}^{3}) q^{57} + ( - \zeta_{20}^{7} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{3} + \zeta_{20}) q^{58} + (3 \zeta_{20}^{4} - 9 \zeta_{20}^{2} + 3) q^{59} + ( - \zeta_{20}^{6} - 5 \zeta_{20}^{4} + 5 \zeta_{20}^{2} + 1) q^{61} + (3 \zeta_{20}^{3} - 3 \zeta_{20}) q^{62} + (2 \zeta_{20}^{7} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{63} + ( - \zeta_{20}^{6} + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{2} + 1) q^{64} + (4 \zeta_{20}^{4} - 6 \zeta_{20}^{2} + 4) q^{66} + (8 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 8 \zeta_{20}) q^{67} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{3}) q^{68} + ( - 2 \zeta_{20}^{6} - 5 \zeta_{20}^{4} - 2 \zeta_{20}^{2}) q^{69} + (5 \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{71} + ( - 4 \zeta_{20}^{7} + 4 \zeta_{20}^{5} + 2 \zeta_{20}) q^{72} - 9 \zeta_{20}^{3} q^{73} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 2) q^{74} + ( - 4 \zeta_{20}^{6} + 4 \zeta_{20}^{4} + 7) q^{76} + ( - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{77} + 3 \zeta_{20} q^{78} + (5 \zeta_{20}^{2} - 5) q^{79} + \zeta_{20}^{4} q^{81} + ( - 2 \zeta_{20}^{7} - 2 \zeta_{20}^{3}) q^{82} + (3 \zeta_{20}^{7} - 5 \zeta_{20}^{5} + 5 \zeta_{20}^{3} - 3 \zeta_{20}) q^{83} + (\zeta_{20}^{4} + \zeta_{20}^{2} + 1) q^{84} + (3 \zeta_{20}^{6} - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{2} - 3) q^{86} + (2 \zeta_{20}^{7} - \zeta_{20}^{3} + \zeta_{20}) q^{87} + ( - 2 \zeta_{20}^{7} + 6 \zeta_{20}^{3} - 6 \zeta_{20}) q^{88} + ( - 4 \zeta_{20}^{6} - 4 \zeta_{20}^{4} + 4 \zeta_{20}^{2} + 4) q^{89} + ( - 3 \zeta_{20}^{4} - 3 \zeta_{20}^{2} - 3) q^{91} + ( - 9 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + 9 \zeta_{20}) q^{92} + 3 \zeta_{20}^{5} q^{93} + \zeta_{20}^{4} q^{94} + ( - 5 \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{96} + ( - 3 \zeta_{20}^{7} + 3 \zeta_{20}^{5} + \zeta_{20}) q^{97} + ( - 6 \zeta_{20}^{5} + 7 \zeta_{20}^{3} - 6 \zeta_{20}) q^{98} + ( - 4 \zeta_{20}^{6} + 4 \zeta_{20}^{4} - 4) q^{99} +O(q^{100})$$ q + (-z^5 + z^3 - z) * q^2 - z * q^3 + (-z^6 - z^2 + 1) * q^4 + (z^6 - z^4 + z^2) * q^6 + (-z^7 - z^3) * q^7 + (z^7 + z^5 - z^3 - z) * q^8 - 2*z^2 * q^9 + (-2*z^6 + 4*z^4 - 4*z^2 + 2) * q^11 + (z^7 + z^3 - z) * q^12 + (-3*z^7 - 3*z^3 + 3*z) * q^13 + (z^6 - z^4 + z^2 - 1) * q^14 + (-3*z^4 + 3*z^2 - 3) * q^16 + (-2*z^7 + 4*z^5 - 4*z^3 + 2*z) * q^17 + (2*z^7 - 2*z^5 + 2*z^3) * q^18 + (3*z^6 + z^4 + 3*z^2) * q^19 + (z^6 + z^2 - 1) * q^21 + (4*z^7 - 4*z^5 + 2*z) * q^22 + (2*z^5 + 5*z^3 + 2*z) * q^23 + (-2*z^6 + 2*z^4 + 1) * q^24 - 3 * q^26 + 5*z^3 * q^27 + (z^7 - z^5 - 2*z) * q^28 + (-2*z^6 + z^2 - 1) * q^29 - 3*z^4 * q^31 + (z^7 + 4*z^5 + z^3) * q^32 + (2*z^7 - 4*z^5 + 4*z^3 - 2*z) * q^33 + (-4*z^4 + 6*z^2 - 4) * q^34 + (2*z^6 - 2) * q^36 + (-3*z^7 - 2*z^3 + 2*z) * q^37 + (-3*z^7 - z^3 + z) * q^38 + (3*z^6 - 3) * q^39 + (2*z^4 + 2*z^2 + 2) * q^41 + (-z^7 + z^5 - z^3 + z) * q^42 + (-3*z^7 + 3*z^5 - 3*z^3) * q^43 + (-2*z^6 + 2*z^4 - 2*z^2) * q^44 + (-2*z^6 - 5*z^2 + 5) * q^46 + (z^7 - z^5 - z) * q^47 + (3*z^5 - 3*z^3 + 3*z) * q^48 + (z^6 - z^4 + 5) * q^49 + (-2*z^6 + 2*z^4 - 2) * q^51 + (-3*z^5 - 3*z^3 - 3*z) * q^52 + (4*z^7 - 4*z^5 - 3*z) * q^53 + (-5*z^2 + 5) * q^54 + (z^6 + 2*z^4 + z^2) * q^56 + (-3*z^7 - z^5 - 3*z^3) * q^57 + (-z^7 + 4*z^5 - 4*z^3 + z) * q^58 + (3*z^4 - 9*z^2 + 3) * q^59 + (-z^6 - 5*z^4 + 5*z^2 + 1) * q^61 + (3*z^3 - 3*z) * q^62 + (2*z^7 + 2*z^3 - 2*z) * q^63 + (-z^6 + 3*z^4 - 3*z^2 + 1) * q^64 + (4*z^4 - 6*z^2 + 4) * q^66 + (8*z^7 - 6*z^5 + 6*z^3 - 8*z) * q^67 + (-2*z^7 + 2*z^5 - 2*z^3) * q^68 + (-2*z^6 - 5*z^4 - 2*z^2) * q^69 + (5*z^6 - z^2 + 1) * q^71 + (-4*z^7 + 4*z^5 + 2*z) * q^72 - 9*z^3 * q^73 + (z^6 - z^4 - 2) * q^74 + (-4*z^6 + 4*z^4 + 7) * q^76 + (-2*z^5 + 2*z^3 - 2*z) * q^77 + 3*z * q^78 + (5*z^2 - 5) * q^79 + z^4 * q^81 + (-2*z^7 - 2*z^3) * q^82 + (3*z^7 - 5*z^5 + 5*z^3 - 3*z) * q^83 + (z^4 + z^2 + 1) * q^84 + (3*z^6 - 6*z^4 + 6*z^2 - 3) * q^86 + (2*z^7 - z^3 + z) * q^87 + (-2*z^7 + 6*z^3 - 6*z) * q^88 + (-4*z^6 - 4*z^4 + 4*z^2 + 4) * q^89 + (-3*z^4 - 3*z^2 - 3) * q^91 + (-9*z^7 + 2*z^5 - 2*z^3 + 9*z) * q^92 + 3*z^5 * q^93 + z^4 * q^94 + (-5*z^6 - z^2 + 1) * q^96 + (-3*z^7 + 3*z^5 + z) * q^97 + (-6*z^5 + 7*z^3 - 6*z) * q^98 + (-4*z^6 + 4*z^4 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} + 6 q^{6} - 4 q^{9}+O(q^{10})$$ 8 * q + 4 * q^4 + 6 * q^6 - 4 * q^9 $$8 q + 4 q^{4} + 6 q^{6} - 4 q^{9} - 4 q^{11} - 2 q^{14} - 12 q^{16} + 10 q^{19} - 4 q^{21} - 24 q^{26} - 10 q^{29} + 6 q^{31} - 12 q^{34} - 12 q^{36} - 18 q^{39} + 16 q^{41} - 12 q^{44} + 26 q^{46} + 44 q^{49} - 24 q^{51} + 30 q^{54} + 26 q^{61} - 6 q^{64} + 12 q^{66} + 2 q^{69} + 16 q^{71} - 12 q^{74} + 40 q^{76} - 30 q^{79} - 2 q^{81} + 8 q^{84} + 6 q^{86} + 40 q^{89} - 24 q^{91} - 2 q^{94} - 4 q^{96} - 48 q^{99}+O(q^{100})$$ 8 * q + 4 * q^4 + 6 * q^6 - 4 * q^9 - 4 * q^11 - 2 * q^14 - 12 * q^16 + 10 * q^19 - 4 * q^21 - 24 * q^26 - 10 * q^29 + 6 * q^31 - 12 * q^34 - 12 * q^36 - 18 * q^39 + 16 * q^41 - 12 * q^44 + 26 * q^46 + 44 * q^49 - 24 * q^51 + 30 * q^54 + 26 * q^61 - 6 * q^64 + 12 * q^66 + 2 * q^69 + 16 * q^71 - 12 * q^74 + 40 * q^76 - 30 * q^79 - 2 * q^81 + 8 * q^84 + 6 * q^86 + 40 * q^89 - 24 * q^91 - 2 * q^94 - 4 * q^96 - 48 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/125\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{20}^{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 0.587785 − 0.809017i −0.587785 + 0.809017i 0.951057 − 0.309017i −0.951057 + 0.309017i 0.951057 + 0.309017i −0.951057 − 0.309017i 0.587785 + 0.809017i −0.587785 − 0.809017i
−1.53884 0.500000i −0.587785 + 0.809017i 0.500000 + 0.363271i 0 1.30902 0.951057i 0.618034i 1.31433 + 1.80902i 0.618034 + 1.90211i 0
24.2 1.53884 + 0.500000i 0.587785 0.809017i 0.500000 + 0.363271i 0 1.30902 0.951057i 0.618034i −1.31433 1.80902i 0.618034 + 1.90211i 0
49.1 −0.363271 + 0.500000i −0.951057 + 0.309017i 0.500000 + 1.53884i 0 0.190983 0.587785i 1.61803i −2.12663 0.690983i −1.61803 + 1.17557i 0
49.2 0.363271 0.500000i 0.951057 0.309017i 0.500000 + 1.53884i 0 0.190983 0.587785i 1.61803i 2.12663 + 0.690983i −1.61803 + 1.17557i 0
74.1 −0.363271 0.500000i −0.951057 0.309017i 0.500000 1.53884i 0 0.190983 + 0.587785i 1.61803i −2.12663 + 0.690983i −1.61803 1.17557i 0
74.2 0.363271 + 0.500000i 0.951057 + 0.309017i 0.500000 1.53884i 0 0.190983 + 0.587785i 1.61803i 2.12663 0.690983i −1.61803 1.17557i 0
99.1 −1.53884 + 0.500000i −0.587785 0.809017i 0.500000 0.363271i 0 1.30902 + 0.951057i 0.618034i 1.31433 1.80902i 0.618034 1.90211i 0
99.2 1.53884 0.500000i 0.587785 + 0.809017i 0.500000 0.363271i 0 1.30902 + 0.951057i 0.618034i −1.31433 + 1.80902i 0.618034 1.90211i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 24.2 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
25.d even 5 1 inner
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.2.e.a 8
5.b even 2 1 inner 125.2.e.a 8
5.c odd 4 1 25.2.d.a 4
5.c odd 4 1 125.2.d.a 4
15.e even 4 1 225.2.h.b 4
20.e even 4 1 400.2.u.b 4
25.d even 5 1 inner 125.2.e.a 8
25.d even 5 1 625.2.b.a 4
25.d even 5 2 625.2.e.c 8
25.e even 10 1 inner 125.2.e.a 8
25.e even 10 1 625.2.b.a 4
25.e even 10 2 625.2.e.c 8
25.f odd 20 1 25.2.d.a 4
25.f odd 20 1 125.2.d.a 4
25.f odd 20 1 625.2.a.b 2
25.f odd 20 1 625.2.a.c 2
25.f odd 20 2 625.2.d.b 4
25.f odd 20 2 625.2.d.h 4
75.l even 20 1 225.2.h.b 4
75.l even 20 1 5625.2.a.d 2
75.l even 20 1 5625.2.a.f 2
100.l even 20 1 400.2.u.b 4
100.l even 20 1 10000.2.a.c 2
100.l even 20 1 10000.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 5.c odd 4 1
25.2.d.a 4 25.f odd 20 1
125.2.d.a 4 5.c odd 4 1
125.2.d.a 4 25.f odd 20 1
125.2.e.a 8 1.a even 1 1 trivial
125.2.e.a 8 5.b even 2 1 inner
125.2.e.a 8 25.d even 5 1 inner
125.2.e.a 8 25.e even 10 1 inner
225.2.h.b 4 15.e even 4 1
225.2.h.b 4 75.l even 20 1
400.2.u.b 4 20.e even 4 1
400.2.u.b 4 100.l even 20 1
625.2.a.b 2 25.f odd 20 1
625.2.a.c 2 25.f odd 20 1
625.2.b.a 4 25.d even 5 1
625.2.b.a 4 25.e even 10 1
625.2.d.b 4 25.f odd 20 2
625.2.d.h 4 25.f odd 20 2
625.2.e.c 8 25.d even 5 2
625.2.e.c 8 25.e even 10 2
5625.2.a.d 2 75.l even 20 1
5625.2.a.f 2 75.l even 20 1
10000.2.a.c 2 100.l even 20 1
10000.2.a.l 2 100.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 4T_{2}^{6} + 6T_{2}^{4} + T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(125, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1$$
$3$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 3 T^{2} + 1)^{2}$$
$11$ $$(T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16)^{2}$$
$13$ $$T^{8} + 9 T^{6} + 486 T^{4} + \cdots + 6561$$
$17$ $$T^{8} + 16 T^{6} + 736 T^{4} + \cdots + 256$$
$19$ $$(T^{4} - 5 T^{3} + 40 T^{2} - 50 T + 25)^{2}$$
$23$ $$T^{8} + 19 T^{6} + 3841 T^{4} + \cdots + 923521$$
$29$ $$(T^{4} + 5 T^{3} + 10 T^{2} + 25)^{2}$$
$31$ $$(T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{2}$$
$37$ $$T^{8} + 11 T^{6} + 321 T^{4} - 29 T^{2} + \cdots + 1$$
$41$ $$(T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16)^{2}$$
$43$ $$(T^{4} + 27 T^{2} + 81)^{2}$$
$47$ $$T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1$$
$53$ $$T^{8} - 41 T^{6} + 681 T^{4} + \cdots + 130321$$
$59$ $$(T^{4} + 90 T^{2} + 675 T + 2025)^{2}$$
$61$ $$(T^{4} - 13 T^{3} + 139 T^{2} - 697 T + 1681)^{2}$$
$67$ $$T^{8} - 124 T^{6} + 5856 T^{4} + \cdots + 3748096$$
$71$ $$(T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841)^{2}$$
$73$ $$T^{8} - 81 T^{6} + 6561 T^{4} + \cdots + 43046721$$
$79$ $$(T^{4} + 15 T^{3} + 100 T^{2} + 250 T + 625)^{2}$$
$83$ $$T^{8} + 19 T^{6} + 1401 T^{4} + \cdots + 14641$$
$89$ $$(T^{4} - 20 T^{3} + 240 T^{2} - 1600 T + 6400)^{2}$$
$97$ $$T^{8} - 4 T^{6} + 166 T^{4} + \cdots + 14641$$