LMFDB - Newform orbit 1521.2.i.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(746,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("1521.746");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.i (of order $4$, 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:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{7} - \beta_{5}) q^{2} + 2 \beta_{3} q^{4} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{5} - \beta_1 q^{7}+O(q^{10}) $ q + (b7 - b5) * q^2 + 2b3 q^4 + (-b7 + b5 + b2) * q^5 - b1 * q^7	$ q + (\beta_{7} - \beta_{5}) q^{2} + 2 \beta_{3} q^{4} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{5} - \beta_1 q^{7} + ( - 2 \beta_{6} - 2 \beta_{3} + \cdots - 2) q^{10}+ \cdots + (4 \beta_{7} + 6 \beta_{5} + 2 \beta_{4}) q^{98}+O(q^{100}) $ q + (b7 - b5) * q^2 + 2b3 q^4 + (-b7 + b5 + b2) * q^5 - b1 * q^7 + (-2b6 - 2b3 + 2b1 - 2) q^10 + (2b7 - b5 - 3b4) * q^11 + (b7 - b4 - b2) * q^14 + 4 * q^16 + (3b7 + b4 - b2) q^17 + (-4b6 + b3 - 1) q^19 + (2b5 + 2b4) * q^20 + (6b6 - 6b3 + 6b1 + 2) q^22 + (b4 - b2) * q^23 + (2b6 - b3 - 2b1 + 2) * q^25 + (2b6 - 2b3 + 2) * q^28 + b5 * q^29 + (b6 - 6b3 + 6) q^31 + (4b7 - 4b5) * q^32 + (8b3 - 4b1 + 8) * q^34 + b5 * q^35 + (3b3 + 2b1 + 3) * q^37 + (2b7 + 4b4 - 4b2) q^38 + 2b2 q^41 + (b6 + 6b3 - b1 + 1) q^43 + (-2b7 + 2b5 + 6b2) q^44 + (2b3 - 4b1 + 2) * q^46 + (-3b7 - 2b5 + b4) * q^47 + (-b6 - 5b3 + b1 - 1) q^49 + (3b7 - b5 - 4b4) * q^50 + (3b7 - 2b5 - 3b4 - 3b2) * q^53 + (-2b6 + 2b3 - 2b1 + 8) q^55 + (-2b3 + 2) q^58 + (-4b7 - b5 + 3b4) * q^59 + (-3b6 + 3b3 - 3b1 + 8) q^61 + (11b7 - b4 + b2) q^62 + 8b3 q^64 + (-7b6 + 4b3 - 4) * q^67 + (2b7 - 8b5 - 2b4 - 2b2) * q^68 + (-2b3 + 2) q^70 + (-6b7 + 6b5 + 5b2) q^71 + (4b3 + 3b1 + 4) * q^73 + (-2b7 - 6b5 + 2b4 + 2b2) * q^74 + (6b3 - 8b1 + 6) * q^76 + (-5b7 + 2b4 - 2b2) q^77 + (-3b6 + 3b3 - 3b1 + 2) q^79 + (-4b7 + 4b5 + 4b2) q^80 + (-4b6 + 4b3 + 4b1 - 4) q^82 + (-3b7 + 3b5 + 2b2) q^83 + (-10b3 + 8b1 - 10) * q^85 + (-5b7 - 7b5 - 2b4) q^86 + (-5b7 - 3b5 + 2b4) q^89 + (2b7 - 2b5 - 2b4 - 2b2) * q^92 + (-2b6 + 2b3 - 2b1 - 10) q^94 + (-7b7 - 3b4 + 3b2) q^95 + (7b6 - 6b3 + 6) * q^97 + (4b7 + 6b5 + 2b4) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7}+O(q^{10}) $ 8 * q - 4 * q^7	$ 8 q - 4 q^{7} + 32 q^{16} + 8 q^{19} + 16 q^{22} + 8 q^{28} + 44 q^{31} + 48 q^{34} + 32 q^{37} + 64 q^{55} + 16 q^{58} + 64 q^{61} - 4 q^{67} + 16 q^{70} + 44 q^{73} + 16 q^{76} + 16 q^{79} - 48 q^{85} - 80 q^{94} + 20 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 + 32 * q^16 + 8 * q^19 + 16 * q^22 + 8 * q^28 + 44 * q^31 + 48 * q^34 + 32 * q^37 + 64 * q^55 + 16 * q^58 + 64 * q^61 - 4 * q^67 + 16 * q^70 + 44 * q^73 + 16 * q^76 + 16 * q^79 - 48 * q^85 - 80 * q^94 + 20 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ \zeta_{24}^{4} + \zeta_{24}^{2} $ v^4 + v^2
$\beta_{2}$	$=$	$ \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} $ v^5 + v^3 + v
$\beta_{3}$	$=$	$ \zeta_{24}^{6} $ v^6
$\beta_{4}$	$=$	$ 2\zeta_{24}^{7} $ 2*v^7
$\beta_{5}$	$=$	$ -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} $ -v^5 - v^3 + v
$\beta_{6}$	$=$	$ -\zeta_{24}^{4} + \zeta_{24}^{2} $ -v^4 + v^2
$\beta_{7}$	$=$	$ -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} $ -v^5 + v^3 + v

Display $\zeta_{24}^j$ in terms of $\beta_i$

$\zeta_{24}$	$=$	$ ( \beta_{5} + \beta_{2} ) / 2 $ (b5 + b2) / 2
$\zeta_{24}^{2}$	$=$	$ ( \beta_{6} + \beta_1 ) / 2 $ (b6 + b1) / 2
$\zeta_{24}^{3}$	$=$	$ ( \beta_{7} - \beta_{5} ) / 2 $ (b7 - b5) / 2
$\zeta_{24}^{4}$	$=$	$ ( -\beta_{6} + \beta_1 ) / 2 $ (-b6 + b1) / 2
$\zeta_{24}^{5}$	$=$	$ ( -\beta_{7} + \beta_{2} ) / 2 $ (-b7 + b2) / 2
$\zeta_{24}^{6}$	$=$	$ \beta_{3} $ b3
$\zeta_{24}^{7}$	$=$	$ ( \beta_{4} ) / 2 $ (b4) / 2

Display $\beta_i$ in terms of $\zeta_{24}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$-1$	$-\beta_{3}$

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} $

746.1

−0.965926	−	0.258819i
0.258819	+	0.965926i
−0.258819	−	0.965926i
0.965926	+	0.258819i
−0.965926	+	0.258819i
0.258819	−	0.965926i
−0.258819	+	0.965926i
0.965926	−	0.258819i

−1.41421

−

1.41421i

2.00000i

−0.517638

−

0.517638i

−1.36603

−

1.36603i

1.46410i

746.2

−1.41421

−

1.41421i

2.00000i

1.93185

1.93185i

0.366025

0.366025i

−

5.46410i

746.3

1.41421

1.41421i

2.00000i

−1.93185

−

1.93185i

0.366025

0.366025i

−

5.46410i

746.4

1.41421

1.41421i

2.00000i

0.517638

0.517638i

−1.36603

−

1.36603i

1.46410i

944.1

−1.41421

1.41421i

−

2.00000i

−0.517638

0.517638i

−1.36603

1.36603i

−

1.46410i

944.2

−1.41421

1.41421i

−

2.00000i

1.93185

−

1.93185i

0.366025

−

0.366025i

5.46410i

944.3

1.41421

−

1.41421i

−

2.00000i

−1.93185

1.93185i

0.366025

−

0.366025i

5.46410i

944.4

1.41421

−

1.41421i

−

2.00000i

0.517638

−

0.517638i

−1.36603

1.36603i

−

1.46410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
13.d	odd	4	1	inner
39.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1521.2.i.d		8
3.b	odd	2	1	inner	1521.2.i.d		8
13.b	even	2	1		1521.2.i.e		8
13.c	even	3	1		117.2.ba.a	✓	8
13.d	odd	4	1	inner	1521.2.i.d		8
13.d	odd	4	1		1521.2.i.e		8
13.f	odd	12	1		117.2.ba.a	✓	8
39.d	odd	2	1		1521.2.i.e		8
39.f	even	4	1	inner	1521.2.i.d		8
39.f	even	4	1		1521.2.i.e		8
39.i	odd	6	1		117.2.ba.a	✓	8
39.k	even	12	1		117.2.ba.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.2.ba.a	✓	8	13.c	even	3	1
117.2.ba.a	✓	8	13.f	odd	12	1
117.2.ba.a	✓	8	39.i	odd	6	1
117.2.ba.a	✓	8	39.k	even	12	1
1521.2.i.d		8	1.a	even	1	1	trivial
1521.2.i.d		8	3.b	odd	2	1	inner
1521.2.i.d		8	13.d	odd	4	1	inner
1521.2.i.d		8	39.f	even	4	1	inner
1521.2.i.e		8	13.b	even	2	1
1521.2.i.e		8	13.d	odd	4	1
1521.2.i.e		8	39.d	odd	2	1
1521.2.i.e		8	39.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1521, [\chi])$:

$ T_{2}^{4} + 16 $

$ T_{5}^{8} + 56T_{5}^{4} + 16 $

$ T_{7}^{4} + 2T_{7}^{3} + 2T_{7}^{2} - 2T_{7} + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 16)^{2} $ (T^4 + 16)^2
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + 56T^{4} + 16 $ T^8 + 56*T^4 + 16
$7$	$ (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} $ (T^4 + 2T^3 + 2T^2 - 2*T + 1)^2
$11$	$ T^{8} + 1784 T^{4} + 456976 $ T^8 + 1784*T^4 + 456976
$13$	$ T^{8} $ T^8
$17$	$ (T^{4} - 48 T^{2} + 144)^{2} $ (T^4 - 48*T^2 + 144)^2
$19$	$ (T^{4} - 4 T^{3} + \cdots + 484)^{2} $ (T^4 - 4T^3 + 8T^2 + 88*T + 484)^2
$23$	$ (T^{2} - 6)^{4} $ (T^2 - 6)^4
$29$	$ (T^{2} + 2)^{4} $ (T^2 + 2)^4
$31$	$ (T^{4} - 22 T^{3} + \cdots + 3481)^{2} $ (T^4 - 22T^3 + 242T^2 - 1298*T + 3481)^2
$37$	$ (T^{4} - 16 T^{3} + \cdots + 676)^{2} $ (T^4 - 16T^3 + 128T^2 - 416*T + 676)^2
$41$	$ T^{8} + 896T^{4} + 4096 $ T^8 + 896*T^4 + 4096
$43$	$ (T^{4} + 78 T^{2} + 1089)^{2} $ (T^4 + 78*T^2 + 1089)^2
$47$	$ T^{8} + 2168 T^{4} + 234256 $ T^8 + 2168*T^4 + 234256
$53$	$ (T^{4} + 112 T^{2} + 2704)^{2} $ (T^4 + 112*T^2 + 2704)^2
$59$	$ T^{8} + 10808T^{4} + 16 $ T^8 + 10808*T^4 + 16
$61$	$ (T^{2} - 16 T + 37)^{4} $ (T^2 - 16*T + 37)^4
$67$	$ (T^{4} + 2 T^{3} + \cdots + 5329)^{2} $ (T^4 + 2T^3 + 2T^2 - 146*T + 5329)^2
$71$	$ T^{8} + 60152 T^{4} + 456976 $ T^8 + 60152*T^4 + 456976
$73$	$ (T^{4} - 22 T^{3} + \cdots + 2209)^{2} $ (T^4 - 22T^3 + 242T^2 - 1034*T + 2209)^2
$79$	$ (T^{2} - 4 T - 23)^{4} $ (T^2 - 4*T - 23)^4
$83$	$ T^{8} + 3104T^{4} + 256 $ T^8 + 3104*T^4 + 256
$89$	$ T^{8} + 17696 T^{4} + 7311616 $ T^8 + 17696*T^4 + 7311616
$97$	$ (T^{4} - 10 T^{3} + \cdots + 3721)^{2} $ (T^4 - 10T^3 + 50T^2 + 610*T + 3721)^2
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Overview	Random
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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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{7} - \beta_{5}) q^{2} + 2 \beta_{3} q^{4} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{5} - \beta_1 q^{7}+O(q^{10}) \) q + (b7 - b5) * q^2 + 2b3 q^4 + (-b7 + b5 + b2) * q^5 - b1 * q^7	\( q + (\beta_{7} - \beta_{5}) q^{2} + 2 \beta_{3} q^{4} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{5} - \beta_1 q^{7} + ( - 2 \beta_{6} - 2 \beta_{3} + \cdots - 2) q^{10}+ \cdots + (4 \beta_{7} + 6 \beta_{5} + 2 \beta_{4}) q^{98}+O(q^{100}) \) q + (b7 - b5) * q^2 + 2b3 q^4 + (-b7 + b5 + b2) * q^5 - b1 * q^7 + (-2b6 - 2b3 + 2b1 - 2) q^10 + (2b7 - b5 - 3b4) * q^11 + (b7 - b4 - b2) * q^14 + 4 * q^16 + (3b7 + b4 - b2) q^17 + (-4b6 + b3 - 1) q^19 + (2b5 + 2b4) * q^20 + (6b6 - 6b3 + 6b1 + 2) q^22 + (b4 - b2) * q^23 + (2b6 - b3 - 2b1 + 2) * q^25 + (2b6 - 2b3 + 2) * q^28 + b5 * q^29 + (b6 - 6b3 + 6) q^31 + (4b7 - 4b5) * q^32 + (8b3 - 4b1 + 8) * q^34 + b5 * q^35 + (3b3 + 2b1 + 3) * q^37 + (2b7 + 4b4 - 4b2) q^38 + 2b2 q^41 + (b6 + 6b3 - b1 + 1) q^43 + (-2b7 + 2b5 + 6b2) q^44 + (2b3 - 4b1 + 2) * q^46 + (-3b7 - 2b5 + b4) * q^47 + (-b6 - 5b3 + b1 - 1) q^49 + (3b7 - b5 - 4b4) * q^50 + (3b7 - 2b5 - 3b4 - 3b2) * q^53 + (-2b6 + 2b3 - 2b1 + 8) q^55 + (-2b3 + 2) q^58 + (-4b7 - b5 + 3b4) * q^59 + (-3b6 + 3b3 - 3b1 + 8) q^61 + (11b7 - b4 + b2) q^62 + 8b3 q^64 + (-7b6 + 4b3 - 4) * q^67 + (2b7 - 8b5 - 2b4 - 2b2) * q^68 + (-2b3 + 2) q^70 + (-6b7 + 6b5 + 5b2) q^71 + (4b3 + 3b1 + 4) * q^73 + (-2b7 - 6b5 + 2b4 + 2b2) * q^74 + (6b3 - 8b1 + 6) * q^76 + (-5b7 + 2b4 - 2b2) q^77 + (-3b6 + 3b3 - 3b1 + 2) q^79 + (-4b7 + 4b5 + 4b2) q^80 + (-4b6 + 4b3 + 4b1 - 4) q^82 + (-3b7 + 3b5 + 2b2) q^83 + (-10b3 + 8b1 - 10) * q^85 + (-5b7 - 7b5 - 2b4) q^86 + (-5b7 - 3b5 + 2b4) q^89 + (2b7 - 2b5 - 2b4 - 2b2) * q^92 + (-2b6 + 2b3 - 2b1 - 10) q^94 + (-7b7 - 3b4 + 3b2) q^95 + (7b6 - 6b3 + 6) * q^97 + (4b7 + 6b5 + 2b4) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7}+O(q^{10}) \) 8 * q - 4 * q^7	\( 8 q - 4 q^{7} + 32 q^{16} + 8 q^{19} + 16 q^{22} + 8 q^{28} + 44 q^{31} + 48 q^{34} + 32 q^{37} + 64 q^{55} + 16 q^{58} + 64 q^{61} - 4 q^{67} + 16 q^{70} + 44 q^{73} + 16 q^{76} + 16 q^{79} - 48 q^{85} - 80 q^{94} + 20 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 + 32 * q^16 + 8 * q^19 + 16 * q^22 + 8 * q^28 + 44 * q^31 + 48 * q^34 + 32 * q^37 + 64 * q^55 + 16 * q^58 + 64 * q^61 - 4 * q^67 + 16 * q^70 + 44 * q^73 + 16 * q^76 + 16 * q^79 - 48 * q^85 - 80 * q^94 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1521.2.i.d

Level	$1521$
Weight	$2$
Character orbit	1521.i
Analytic conductor	$12.145$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(12.1452461474\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( \zeta_{24}^{4} + \zeta_{24}^{2} \) v^4 + v^2
\(\beta_{2}\)	\(=\)	\( \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) v^5 + v^3 + v
\(\beta_{3}\)	\(=\)	\( \zeta_{24}^{6} \) v^6
\(\beta_{4}\)	\(=\)	\( 2\zeta_{24}^{7} \) 2*v^7
\(\beta_{5}\)	\(=\)	\( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) -v^5 - v^3 + v
\(\beta_{6}\)	\(=\)	\( -\zeta_{24}^{4} + \zeta_{24}^{2} \) -v^4 + v^2
\(\beta_{7}\)	\(=\)	\( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) -v^5 + v^3 + v

\(\zeta_{24}\)	\(=\)	\( ( \beta_{5} + \beta_{2} ) / 2 \) (b5 + b2) / 2
\(\zeta_{24}^{2}\)	\(=\)	\( ( \beta_{6} + \beta_1 ) / 2 \) (b6 + b1) / 2
\(\zeta_{24}^{3}\)	\(=\)	\( ( \beta_{7} - \beta_{5} ) / 2 \) (b7 - b5) / 2
\(\zeta_{24}^{4}\)	\(=\)	\( ( -\beta_{6} + \beta_1 ) / 2 \) (-b6 + b1) / 2
\(\zeta_{24}^{5}\)	\(=\)	\( ( -\beta_{7} + \beta_{2} ) / 2 \) (-b7 + b2) / 2
\(\zeta_{24}^{6}\)	\(=\)	\( \beta_{3} \) b3
\(\zeta_{24}^{7}\)	\(=\)	\( ( \beta_{4} ) / 2 \) (b4) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 746.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 16)^{2} \) (T^4 + 16)^2
$3$	\( T^{8} \) T^8
$5$	\( T^{8} + 56T^{4} + 16 \) T^8 + 56*T^4 + 16
$7$	\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) (T^4 + 2T^3 + 2T^2 - 2*T + 1)^2
$11$	\( T^{8} + 1784 T^{4} + 456976 \) T^8 + 1784*T^4 + 456976
$13$	\( T^{8} \) T^8
$17$	\( (T^{4} - 48 T^{2} + 144)^{2} \) (T^4 - 48*T^2 + 144)^2
$19$	\( (T^{4} - 4 T^{3} + \cdots + 484)^{2} \) (T^4 - 4T^3 + 8T^2 + 88*T + 484)^2
$23$	\( (T^{2} - 6)^{4} \) (T^2 - 6)^4
$29$	\( (T^{2} + 2)^{4} \) (T^2 + 2)^4
$31$	\( (T^{4} - 22 T^{3} + \cdots + 3481)^{2} \) (T^4 - 22T^3 + 242T^2 - 1298*T + 3481)^2
$37$	\( (T^{4} - 16 T^{3} + \cdots + 676)^{2} \) (T^4 - 16T^3 + 128T^2 - 416*T + 676)^2
$41$	\( T^{8} + 896T^{4} + 4096 \) T^8 + 896*T^4 + 4096
$43$	\( (T^{4} + 78 T^{2} + 1089)^{2} \) (T^4 + 78*T^2 + 1089)^2
$47$	\( T^{8} + 2168 T^{4} + 234256 \) T^8 + 2168*T^4 + 234256
$53$	\( (T^{4} + 112 T^{2} + 2704)^{2} \) (T^4 + 112*T^2 + 2704)^2
$59$	\( T^{8} + 10808T^{4} + 16 \) T^8 + 10808*T^4 + 16
$61$	\( (T^{2} - 16 T + 37)^{4} \) (T^2 - 16*T + 37)^4
$67$	\( (T^{4} + 2 T^{3} + \cdots + 5329)^{2} \) (T^4 + 2T^3 + 2T^2 - 146*T + 5329)^2
$71$	\( T^{8} + 60152 T^{4} + 456976 \) T^8 + 60152*T^4 + 456976
$73$	\( (T^{4} - 22 T^{3} + \cdots + 2209)^{2} \) (T^4 - 22T^3 + 242T^2 - 1034*T + 2209)^2
$79$	\( (T^{2} - 4 T - 23)^{4} \) (T^2 - 4*T - 23)^4
$83$	\( T^{8} + 3104T^{4} + 256 \) T^8 + 3104*T^4 + 256
$89$	\( T^{8} + 17696 T^{4} + 7311616 \) T^8 + 17696*T^4 + 7311616
$97$	\( (T^{4} - 10 T^{3} + \cdots + 3721)^{2} \) (T^4 - 10T^3 + 50T^2 + 610*T + 3721)^2
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\(n\)	\(677\)	\(847\)
\(\chi(n)\)	\(-1\)	\(-\beta_{3}\)