LMFDB - Newform orbit 825.2.bs.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(74,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 3]))

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

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

chi := DirichletCharacter("825.74");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.bs (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:	$6.58765816676$
Analytic rank:	$1$
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 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$\zeta_{20}^{2}$	$-1$	$-1$

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} $

74.1

−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

−0.690983

−

0.951057i

−1.08779

−

1.34786i

0.190983

−

0.587785i

−0.530249

1.96589i

0.951057

−

2.92705i

−2.92705

0.951057i

−0.633446

2.93236i

74.2

−0.690983

−

0.951057i

0.0877853

−

1.72982i

0.190983

−

0.587785i

−1.70582

1.11179i

−0.951057

2.92705i

−2.92705

0.951057i

−2.98459

−

0.303706i

149.1

−1.80902

0.587785i

−1.45106

0.945746i

1.30902

−

0.951057i

2.06909

−

2.56378i

−0.587785

0.427051i

0.427051

−

0.587785i

1.21113

−

2.74466i

149.2

−1.80902

0.587785i

0.451057

−

1.67229i

1.30902

−

0.951057i

0.166977

3.29032i

0.587785

−

0.427051i

0.427051

−

0.587785i

−2.59310

−

1.50859i

299.1

−1.80902

−

0.587785i

−1.45106

−

0.945746i

1.30902

0.951057i

2.06909

2.56378i

−0.587785

−

0.427051i

0.427051

0.587785i

1.21113

2.74466i

299.2

−1.80902

−

0.587785i

0.451057

1.67229i

1.30902

0.951057i

0.166977

−

3.29032i

0.587785

0.427051i

0.427051

0.587785i

−2.59310

1.50859i

524.1

−0.690983

0.951057i

−1.08779

1.34786i

0.190983

0.587785i

−0.530249

−

1.96589i

0.951057

2.92705i

−2.92705

−

0.951057i

−0.633446

−

2.93236i

524.2

−0.690983

0.951057i

0.0877853

1.72982i

0.190983

0.587785i

−1.70582

−

1.11179i

−0.951057

−

2.92705i

−2.92705

−

0.951057i

−2.98459

0.303706i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner
15.d	odd	2	1	inner
165.r	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.bs.a		8
3.b	odd	2	1		825.2.bs.d		8
5.b	even	2	1		825.2.bs.d		8
5.c	odd	4	1		33.2.f.a	✓	8
5.c	odd	4	1		825.2.bi.b		8
11.d	odd	10	1	inner	825.2.bs.a		8
15.d	odd	2	1	inner	825.2.bs.a		8
15.e	even	4	1		33.2.f.a	✓	8
15.e	even	4	1		825.2.bi.b		8
20.e	even	4	1		528.2.bn.c		8
33.f	even	10	1		825.2.bs.d		8
45.k	odd	12	2		891.2.u.a		16
45.l	even	12	2		891.2.u.a		16
55.e	even	4	1		363.2.f.b		8
55.h	odd	10	1		825.2.bs.d		8
55.k	odd	20	1		363.2.d.f		8
55.k	odd	20	1		363.2.f.b		8
55.k	odd	20	1		363.2.f.d		8
55.k	odd	20	1		363.2.f.e		8
55.l	even	20	1		33.2.f.a	✓	8
55.l	even	20	1		363.2.d.f		8
55.l	even	20	1		363.2.f.d		8
55.l	even	20	1		363.2.f.e		8
55.l	even	20	1		825.2.bi.b		8
60.l	odd	4	1		528.2.bn.c		8
165.l	odd	4	1		363.2.f.b		8
165.r	even	10	1	inner	825.2.bs.a		8
165.u	odd	20	1		33.2.f.a	✓	8
165.u	odd	20	1		363.2.d.f		8
165.u	odd	20	1		363.2.f.d		8
165.u	odd	20	1		363.2.f.e		8
165.u	odd	20	1		825.2.bi.b		8
165.v	even	20	1		363.2.d.f		8
165.v	even	20	1		363.2.f.b		8
165.v	even	20	1		363.2.f.d		8
165.v	even	20	1		363.2.f.e		8
220.w	odd	20	1		528.2.bn.c		8
495.bs	even	60	2		891.2.u.a		16
495.bu	odd	60	2		891.2.u.a		16
660.bv	even	20	1		528.2.bn.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.2.f.a	✓	8	5.c	odd	4	1
33.2.f.a	✓	8	15.e	even	4	1
33.2.f.a	✓	8	55.l	even	20	1
33.2.f.a	✓	8	165.u	odd	20	1
363.2.d.f		8	55.k	odd	20	1
363.2.d.f		8	55.l	even	20	1
363.2.d.f		8	165.u	odd	20	1
363.2.d.f		8	165.v	even	20	1
363.2.f.b		8	55.e	even	4	1
363.2.f.b		8	55.k	odd	20	1
363.2.f.b		8	165.l	odd	4	1
363.2.f.b		8	165.v	even	20	1
363.2.f.d		8	55.k	odd	20	1
363.2.f.d		8	55.l	even	20	1
363.2.f.d		8	165.u	odd	20	1
363.2.f.d		8	165.v	even	20	1
363.2.f.e		8	55.k	odd	20	1
363.2.f.e		8	55.l	even	20	1
363.2.f.e		8	165.u	odd	20	1
363.2.f.e		8	165.v	even	20	1
528.2.bn.c		8	20.e	even	4	1
528.2.bn.c		8	60.l	odd	4	1
528.2.bn.c		8	220.w	odd	20	1
528.2.bn.c		8	660.bv	even	20	1
825.2.bi.b		8	5.c	odd	4	1
825.2.bi.b		8	15.e	even	4	1
825.2.bi.b		8	55.l	even	20	1
825.2.bi.b		8	165.u	odd	20	1
825.2.bs.a		8	1.a	even	1	1	trivial
825.2.bs.a		8	11.d	odd	10	1	inner
825.2.bs.a		8	15.d	odd	2	1	inner
825.2.bs.a		8	165.r	even	10	1	inner
825.2.bs.d		8	3.b	odd	2	1
825.2.bs.d		8	5.b	even	2	1
825.2.bs.d		8	33.f	even	10	1
825.2.bs.d		8	55.h	odd	10	1
891.2.u.a		16	45.k	odd	12	2
891.2.u.a		16	45.l	even	12	2
891.2.u.a		16	495.bs	even	60	2
891.2.u.a		16	495.bu	odd	60	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} + 5T_{2}^{3} + 10T_{2}^{2} + 10T_{2} + 5 $ T2^4 + 5*T2^3 + 10*T2^2 + 10*T2 + 5 acting on $S_{2}^{\mathrm{new}}(825, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 5 T^{3} + 10 T^{2} + \cdots + 5)^{2} $ (T^4 + 5T^3 + 10T^2 + 10*T + 5)^2
$3$	$ T^{8} + 4 T^{7} + \cdots + 81 $ T^8 + 4T^7 + 13T^6 + 30T^5 + 61T^4 + 90T^3 + 117T^2 + 108*T + 81
$5$	$ T^{8} $ T^8
$7$	$ T^{8} + 15 T^{6} + \cdots + 25 $ T^8 + 15T^6 + 85T^4 - 25*T^2 + 25
$11$	$ T^{8} + 19 T^{6} + \cdots + 14641 $ T^8 + 19T^6 + 301T^4 + 2299*T^2 + 14641
$13$	$ T^{8} + 15 T^{6} + \cdots + 25 $ T^8 + 15T^6 + 85T^4 - 25*T^2 + 25
$17$	$ (T^{4} + 10 T^{3} + \cdots + 125)^{2} $ (T^4 + 10T^3 + 50T^2 + 125*T + 125)^2
$19$	$ (T^{4} + 10 T^{3} + \cdots + 125)^{2} $ (T^4 + 10T^3 + 50T^2 + 125*T + 125)^2
$23$	$ (T^{2} + 8 T + 11)^{4} $ (T^2 + 8*T + 11)^4
$29$	$ T^{8} + 160 T^{4} + \cdots + 6400 $ T^8 + 160T^4 + 1600T^2 + 6400
$31$	$ (T^{4} + 10 T^{3} + \cdots + 25)^{2} $ (T^4 + 10T^3 + 40T^2 + 25*T + 25)^2
$37$	$ T^{8} - 9 T^{6} + \cdots + 6561 $ T^8 - 9T^6 + 81T^4 - 729*T^2 + 6561
$41$	$ T^{8} - 5 T^{6} + \cdots + 25 $ T^8 - 5T^6 + 85T^4 + 75*T^2 + 25
$43$	$ (T^{4} - 50 T^{2} + 125)^{2} $ (T^4 - 50*T^2 + 125)^2
$47$	$ (T^{4} + 17 T^{3} + \cdots + 3721)^{2} $ (T^4 + 17T^3 + 184T^2 + 1098*T + 3721)^2
$53$	$ (T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 81)^{2} $ (T^4 - 6T^3 + 36T^2 - 81*T + 81)^2
$59$	$ T^{8} + 4 T^{6} + \cdots + 1 $ T^8 + 4T^6 + 46T^4 - 11*T^2 + 1
$61$	$ (T^{4} + 5 T^{3} + 125)^{2} $ (T^4 + 5*T^3 + 125)^2
$67$	$ (T^{4} + 123 T^{2} + 3721)^{2} $ (T^4 + 123*T^2 + 3721)^2
$71$	$ T^{8} - 155 T^{6} + \cdots + 9150625 $ T^8 - 155T^6 + 9150T^4 + 60500*T^2 + 9150625
$73$	$ T^{8} + 40960 T^{4} + \cdots + 419430400 $ T^8 + 40960T^4 + 6553600T^2 + 419430400
$79$	$ (T^{4} + 25 T^{3} + \cdots + 1805)^{2} $ (T^4 + 25T^3 + 225T^2 + 855*T + 1805)^2
$83$	$ (T^{4} + 35 T^{3} + \cdots + 8405)^{2} $ (T^4 + 35T^3 + 455T^2 + 2665*T + 8405)^2
$89$	$ (T^{4} + 90 T^{2} + 25)^{2} $ (T^4 + 90*T^2 + 25)^2
$97$	$ T^{8} - 59 T^{6} + \cdots + 707281 $ T^8 - 59T^6 + 1446T^4 - 3364*T^2 + 707281
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.bs (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{20}^{2} - 1) q^{2} + (\zeta_{20}^{5} + \zeta_{20}^{4} + \cdots + \zeta_{20}) q^{3}+ \cdots + (\zeta_{20}^{6} + 2 \zeta_{20}^{2} + \cdots - 2) q^{9}+O(q^{10}) \) q + (-z^2 - 1) * q^2 + (z^5 + z^4 - z^2 + z) * q^3 + (z^4 + 1) * q^4 + (-z^7 - z^6 - z^5 - z^3 + z^2 - z) * q^6 + (z^7 - 2z^5 + 2z^3) * q^7 + (-z^6 + z^4 + z^2 - 1) * q^8 + (z^6 + 2z^2 - 2z - 2) * q^9	\( q + ( - \zeta_{20}^{2} - 1) q^{2} + (\zeta_{20}^{5} + \zeta_{20}^{4} + \cdots + \zeta_{20}) q^{3}+ \cdots + ( - 2 \zeta_{20}^{7} - 3 \zeta_{20}^{5} + \cdots - 2) q^{99}+O(q^{100}) \) q + (-z^2 - 1) * q^2 + (z^5 + z^4 - z^2 + z) * q^3 + (z^4 + 1) * q^4 + (-z^7 - z^6 - z^5 - z^3 + z^2 - z) * q^6 + (z^7 - 2z^5 + 2z^3) * q^7 + (-z^6 + z^4 + z^2 - 1) * q^8 + (z^6 + 2z^2 - 2z - 2) * q^9 + (-z^7 + z^5 - 3z^3 - z) q^11 + (z^7 + z^5 + z^3 - 1) * q^12 + (2z^7 + 2z^3 - z) * q^13 + (z^5 - 3z^3 + z) q^14 + (-z^6 - 3z^4 - z^2) q^16 + (z^6 - 2z^4 + 3z^2 - 4) * q^17 + (-2z^6 - z^4 + 2z^3 - z^2 + 2z + 3) q^18 + (-2z^6 - z^4 - z^2 - 2) q^19 + (z^7 + z^6 + z^5 - z^4 - 3z^3 + 2z^2 + 2z - 1) q^21 + (z^7 + z^5 + 5z^3) q^22 + (-2z^6 + 2z^4 - 3) * q^23 + (z^7 + 2z^6 - z^5 - 4z^4 + 2z^3 + 3z^2 - z - 1) * q^24 + (-4z^7 - 3z^3 + 3z) q^26 + (3z^7 - z^6 - 4z^5 - 3z^4 + 4z^3 - z^2 - 3z) q^27 + (z^7 + z) * q^28 + (2z^3 + 2z) * q^29 + (-z^6 + 3z^2 - 3) q^31 + (5z^6 - z^4 + 6z^2 - 3) * q^32 + (-2z^7 - 4z^6 + z^4 + 3z^3 - 4z^2 - z + 3) * q^33 + 5 * q^34 + (3z^6 - 2z^5 - 2z^4 + 2z^2 - 2z - 3) q^36 - 3z^7 q^37 + (5z^6 + 5z^2) * q^38 + (3z^6 - z^5 - 2z^4 - z^3 + z^2 - 4) * q^39 + (2z^7 + z^3 - 2z) * q^41 + (-3z^7 - z^6 + 3z^5 - 2z^2 - z + 2) q^42 + (2z^7 + z^5 - 4z^3 + 2z) q^43 + (-3z^7 - z^5 - 2z^3 - z) * q^44 + (2z^6 - 4z^4 + 5z^2 + 1) q^46 + (3z^6 + 4z^4 - 4z^2 - 3) q^47 + (-5z^7 + 3z^4 - 4z^3 - 2z^2 + 4z + 3) q^48 + (-4z^6 - 4z^2) * q^49 + (2z^7 + 2z^6 - 4z^5 - 4z^4 + z^3 + z^2 - 3z + 2) q^51 + (4z^7 - z^5 + 2z^3 - 3z) q^52 + (-3z^6 - 3z^2 + 3) * q^53 + (-2z^7 + 5z^6 + 3z^5 + 3z^4 - 4z^3 + 2z^2 + 6z - 1) q^54 + (-4z^7 + 7z^5 - 4z^3) q^56 + (-4z^7 + z^6 - 2z^5 - 2z^4 - 2z^3 + 3z^2 + z + 1) q^57 + (-2z^5 - 4z^3 - 2z) q^58 + (-2z^7 - z^3 + z) q^59 + (3z^6 - z^4 - z^2 - 2) q^61 + (2z^6 - 4z^4 + z^2 + 2) * q^62 + (-2z^7 + 2z^6 + 4z^5 - 2z^4 - z^3 - 2z^2 - 2z + 2) * q^63 + (-z^6 - 6z^2 + 6) q^64 + (4z^7 + 7z^6 - 5z^5 - z^4 + 5z^2 - z - 7) * q^66 + (-7z^7 + 3z^5 - 7z^3) q^67 + (2z^6 - 4z^4 + z^2 - 3) * q^68 + (2z^6 - 3z^5 - 7z^4 + 2z^3 + 7z^2 - 3z - 2) * q^69 + (-6z^7 - z^5 + z^3 + 6z) * q^71 + (2z^7 + 2z^6 - 2z^5 + z^4 - 2z^3 - 4z^2 + 2z + 2) * q^72 + (-8z^3 - 8z) * q^73 + (6z^7 - 3z^5 + 3z^3 - 3z) * q^74 + (-4z^6 - 2z^4 - 2z^2 + 1) q^76 + (3z^6 - 8z^4 + 4z^2) q^77 + (z^7 - 4z^6 + 2z^5 + 4z^4 + z^3 + 7) q^78 + (z^6 - 2z^4 - 4z^2 - 6) * q^79 + (-4z^7 - 8z^3 - z^2 + 8z) q^81 + (-4z^7 + z^5 - z^3 + 4z) * q^82 + (8z^6 - 7z^4 + 6z^2 - 14) q^83 + (-z^7 + 2z^6 + 2z^5 - z^4 - 2z^3 + z^2 - 1) q^84 + (-5z^7 + 5z^5) * q^86 + (2z^7 + 4z^6 - 2z^3 + 4z^2 - 2) * q^87 + (3z^7 - 8z^5 + 4z^3) q^88 + (-4z^7 - 3z^5 - 4z^3) q^89 + (z^6 + z^4 - z^2 - 1) * q^91 + (-3z^4 + 2z^2 - 3) * q^92 + (2z^7 + 4z^6 - 3z^5 - 7z^4 + 3z^3 + 4z^2 - 2z) q^93 + (-10z^6 + 3z^4 + 4z^2 + 6) q^94 + (10z^7 + z^6 - 3z^5 - 3z^4 + 5z^3 - 3z^2 - 7z + 1) * q^96 + (z^7 - z^5 - 6z) q^97 + (8z^6 + 8z^2 - 4) * q^98 + (-2z^7 - 3z^5 + 4z^4 + 4z^2 + 6z - 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{2} - 4 q^{3} + 6 q^{4} - 10 q^{8} - 10 q^{9}+O(q^{10}) \) 8 * q - 10 * q^2 - 4 * q^3 + 6 * q^4 - 10 * q^8 - 10 * q^9	\( 8 q - 10 q^{2} - 4 q^{3} + 6 q^{4} - 10 q^{8} - 10 q^{9} - 8 q^{12} + 2 q^{16} - 20 q^{17} + 20 q^{18} - 20 q^{19} - 32 q^{23} + 10 q^{24} + 2 q^{27} - 20 q^{31} + 6 q^{33} + 40 q^{34} - 10 q^{36} + 20 q^{38} - 20 q^{39} + 10 q^{42} + 30 q^{46} - 34 q^{47} + 14 q^{48} - 16 q^{49} + 30 q^{51} + 12 q^{53} + 20 q^{57} - 10 q^{61} + 30 q^{62} + 20 q^{63} + 34 q^{64} - 30 q^{66} - 10 q^{68} + 16 q^{69} + 10 q^{72} + 30 q^{77} + 40 q^{78} - 50 q^{79} - 2 q^{81} - 70 q^{83} - 10 q^{91} - 14 q^{92} + 30 q^{93} + 30 q^{94} + 10 q^{96} - 16 q^{99}+O(q^{100}) \) 8 * q - 10 * q^2 - 4 * q^3 + 6 * q^4 - 10 * q^8 - 10 * q^9 - 8 * q^12 + 2 * q^16 - 20 * q^17 + 20 * q^18 - 20 * q^19 - 32 * q^23 + 10 * q^24 + 2 * q^27 - 20 * q^31 + 6 * q^33 + 40 * q^34 - 10 * q^36 + 20 * q^38 - 20 * q^39 + 10 * q^42 + 30 * q^46 - 34 * q^47 + 14 * q^48 - 16 * q^49 + 30 * q^51 + 12 * q^53 + 20 * q^57 - 10 * q^61 + 30 * q^62 + 20 * q^63 + 34 * q^64 - 30 * q^66 - 10 * q^68 + 16 * q^69 + 10 * q^72 + 30 * q^77 + 40 * q^78 - 50 * q^79 - 2 * q^81 - 70 * q^83 - 10 * q^91 - 14 * q^92 + 30 * q^93 + 30 * q^94 + 10 * q^96 - 16 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	825.2.bs.a

Level	$825$
Weight	$2$
Character orbit	825.bs
Analytic conductor	$6.588$
Analytic rank	$1$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(1\)
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 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(n\)	\(376\)	\(551\)	\(727\)
\(\chi(n)\)	\(\zeta_{20}^{2}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 74.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 5 T^{3} + 10 T^{2} + \cdots + 5)^{2} \) (T^4 + 5T^3 + 10T^2 + 10*T + 5)^2
$3$	\( T^{8} + 4 T^{7} + \cdots + 81 \) T^8 + 4T^7 + 13T^6 + 30T^5 + 61T^4 + 90T^3 + 117T^2 + 108*T + 81
$5$	\( T^{8} \) T^8
$7$	\( T^{8} + 15 T^{6} + \cdots + 25 \) T^8 + 15T^6 + 85T^4 - 25*T^2 + 25
$11$	\( T^{8} + 19 T^{6} + \cdots + 14641 \) T^8 + 19T^6 + 301T^4 + 2299*T^2 + 14641
$13$	\( T^{8} + 15 T^{6} + \cdots + 25 \) T^8 + 15T^6 + 85T^4 - 25*T^2 + 25
$17$	\( (T^{4} + 10 T^{3} + \cdots + 125)^{2} \) (T^4 + 10T^3 + 50T^2 + 125*T + 125)^2
$19$	\( (T^{4} + 10 T^{3} + \cdots + 125)^{2} \) (T^4 + 10T^3 + 50T^2 + 125*T + 125)^2
$23$	\( (T^{2} + 8 T + 11)^{4} \) (T^2 + 8*T + 11)^4
$29$	\( T^{8} + 160 T^{4} + \cdots + 6400 \) T^8 + 160T^4 + 1600T^2 + 6400
$31$	\( (T^{4} + 10 T^{3} + \cdots + 25)^{2} \) (T^4 + 10T^3 + 40T^2 + 25*T + 25)^2
$37$	\( T^{8} - 9 T^{6} + \cdots + 6561 \) T^8 - 9T^6 + 81T^4 - 729*T^2 + 6561
$41$	\( T^{8} - 5 T^{6} + \cdots + 25 \) T^8 - 5T^6 + 85T^4 + 75*T^2 + 25
$43$	\( (T^{4} - 50 T^{2} + 125)^{2} \) (T^4 - 50*T^2 + 125)^2
$47$	\( (T^{4} + 17 T^{3} + \cdots + 3721)^{2} \) (T^4 + 17T^3 + 184T^2 + 1098*T + 3721)^2
$53$	\( (T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 81)^{2} \) (T^4 - 6T^3 + 36T^2 - 81*T + 81)^2
$59$	\( T^{8} + 4 T^{6} + \cdots + 1 \) T^8 + 4T^6 + 46T^4 - 11*T^2 + 1
$61$	\( (T^{4} + 5 T^{3} + 125)^{2} \) (T^4 + 5*T^3 + 125)^2
$67$	\( (T^{4} + 123 T^{2} + 3721)^{2} \) (T^4 + 123*T^2 + 3721)^2
$71$	\( T^{8} - 155 T^{6} + \cdots + 9150625 \) T^8 - 155T^6 + 9150T^4 + 60500*T^2 + 9150625
$73$	\( T^{8} + 40960 T^{4} + \cdots + 419430400 \) T^8 + 40960T^4 + 6553600T^2 + 419430400
$79$	\( (T^{4} + 25 T^{3} + \cdots + 1805)^{2} \) (T^4 + 25T^3 + 225T^2 + 855*T + 1805)^2
$83$	\( (T^{4} + 35 T^{3} + \cdots + 8405)^{2} \) (T^4 + 35T^3 + 455T^2 + 2665*T + 8405)^2
$89$	\( (T^{4} + 90 T^{2} + 25)^{2} \) (T^4 + 90*T^2 + 25)^2
$97$	\( T^{8} - 59 T^{6} + \cdots + 707281 \) T^8 - 59T^6 + 1446T^4 - 3364*T^2 + 707281
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