LMFDB - Newform orbit 578.2.c.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [578,2,Mod(251,578)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("578.251"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 578 = 2 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	578.c (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-8,0,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.61535323683$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 34)
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_{2} q^{2} + \beta_{7} q^{3} - q^{4} - 2 \beta_{7} q^{5} - \beta_{5} q^{6} + 2 \beta_{4} q^{7} - \beta_{2} q^{8} + ( - \beta_{3} - \beta_{2}) q^{9} + 2 \beta_{5} q^{10} + (2 \beta_{5} + \beta_{4}) q^{11}+ \cdots + \beta_{7} q^{99}+O(q^{100}) $ q + b2 * q^2 + b7 * q^3 - q^4 - 2b7 q^5 - b5 * q^6 + 2b4 q^7 - b2 * q^8 + (-b3 - b2) * q^9 + 2b5 q^10 + (2b5 + b4) q^11 - b7 * q^12 + (-2b6 - 2) q^13 - 2b1 q^14 + (2b3 - 4b2) * q^15 + q^16 + (b6 + 1) * q^18 + (-2b3 - 2b2) * q^19 + 2b7 q^20 - 2b6 q^21 + (2b7 - b1) q^22 + 4b4 q^23 + b5 * q^24 + (-4b3 + 3b2) * q^25 + (-2b3 - 2b2) * q^26 + (3b5 - b4) q^27 - 2b4 q^28 - 2b1 q^29 + (-2b6 + 4) q^30 - 2b7 q^31 + b2 * q^32 + (-3b6 + 4) q^33 + 4b6 q^35 + (b3 + b2) * q^36 + (-2b7 - 2b1) * q^37 + (2b6 + 2) q^38 - 2b1 q^39 - 2b5 q^40 + (4b5 + 5b4) * q^41 - 2b3 q^42 + (3b3 + 6b2) * q^43 + (-2b5 - b4) q^44 + 2b4 q^45 - 4b1 q^46 + (4b6 - 4) q^47 + b7 * q^48 + (-4b3 - b2) q^49 + (4b6 - 3) q^50 + (2b6 + 2) q^52 + (-2b3 - 2b2) * q^53 + (3b7 + b1) q^54 + (6b6 - 8) q^55 + 2b1 q^56 - 2b4 q^57 - 2b4 q^58 + (b3 + 2b2) q^59 + (-2b3 + 4b2) * q^60 - 4b5 q^61 + 2b5 q^62 + (2b7 + 4b1) * q^63 - q^64 + 4b1 q^65 + (-3b3 + 4b2) * q^66 + (-b6 + 6) * q^67 - 4b6 q^69 + 4b3 q^70 + (-2b7 - 2b1) * q^71 + (-b6 - 1) * q^72 + 7b1 q^73 + (2b5 - 2b4) * q^74 + (-7b5 - 4b4) * q^75 + (2b3 + 2b2) * q^76 + (2b3 - 4b2) * q^77 - 2b4 q^78 + (-2b5 - 2b4) * q^79 - 2b7 q^80 + (-5b6 + 3) q^81 + (4b7 - 5b1) * q^82 + (3b3 - 2b2) * q^83 + 2b6 q^84 + (-3b6 - 6) q^86 - 2b3 q^87 + (-2b7 + b1) q^88 + (-5b6 + 8) q^89 - 2b1 q^90 + (4b5 - 8b4) * q^91 - 4b4 q^92 + (2b3 - 4b2) * q^93 + (4b3 - 4b2) * q^94 + 4b4 q^95 - b5 * q^96 + 3b7 q^97 + (4b6 + 1) q^98 + b7 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 16 q^{13} + 8 q^{16} + 8 q^{18} + 32 q^{30} + 32 q^{33} + 16 q^{38} - 32 q^{47} - 24 q^{50} + 16 q^{52} - 64 q^{55} - 8 q^{64} + 48 q^{67} - 8 q^{72} + 24 q^{81} - 48 q^{86} + 64 q^{89}+ \cdots + 8 q^{98}+O(q^{100}) $ 8 * q - 8 * q^4 - 16 * q^13 + 8 * q^16 + 8 * q^18 + 32 * q^30 + 32 * q^33 + 16 * q^38 - 32 * q^47 - 24 * q^50 + 16 * q^52 - 64 * q^55 - 8 * q^64 + 48 * q^67 - 8 * q^72 + 24 * q^81 - 48 * q^86 + 64 * q^89 + 8 * q^98

Basis of coefficient ring

$\beta_{1}$	$=$	$ \zeta_{16}^{3} + \zeta_{16} $ v^3 + v
$\beta_{2}$	$=$	$ \zeta_{16}^{4} $ v^4
$\beta_{3}$	$=$	$ \zeta_{16}^{6} + \zeta_{16}^{2} $ v^6 + v^2
$\beta_{4}$	$=$	$ \zeta_{16}^{7} + \zeta_{16}^{5} $ v^7 + v^5
$\beta_{5}$	$=$	$ -\zeta_{16}^{3} + \zeta_{16} $ -v^3 + v
$\beta_{6}$	$=$	$ -\zeta_{16}^{6} + \zeta_{16}^{2} $ -v^6 + v^2
$\beta_{7}$	$=$	$ -\zeta_{16}^{7} + \zeta_{16}^{5} $ -v^7 + v^5

Display $\zeta_{16}^j$ in terms of $\beta_i$

$\zeta_{16}$	$=$	$ ( \beta_{5} + \beta_1 ) / 2 $ (b5 + b1) / 2
$\zeta_{16}^{2}$	$=$	$ ( \beta_{6} + \beta_{3} ) / 2 $ (b6 + b3) / 2
$\zeta_{16}^{3}$	$=$	$ ( -\beta_{5} + \beta_1 ) / 2 $ (-b5 + b1) / 2
$\zeta_{16}^{4}$	$=$	$ \beta_{2} $ b2
$\zeta_{16}^{5}$	$=$	$ ( \beta_{7} + \beta_{4} ) / 2 $ (b7 + b4) / 2
$\zeta_{16}^{6}$	$=$	$ ( -\beta_{6} + \beta_{3} ) / 2 $ (-b6 + b3) / 2
$\zeta_{16}^{7}$	$=$	$ ( -\beta_{7} + \beta_{4} ) / 2 $ (-b7 + b4) / 2

Display $\beta_i$ in terms of $\zeta_{16}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/578\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$\beta_{2}$

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} $

251.1

−0.382683	+	0.923880i
−0.923880	−	0.382683i
0.923880	+	0.382683i
0.382683	−	0.923880i
−0.382683	−	0.923880i
−0.923880	+	0.382683i
0.923880	−	0.382683i
0.382683	+	0.923880i

1.00000i

−1.30656

−

1.30656i

−1.00000

2.61313

2.61313i

1.30656

−

1.30656i

−1.08239

1.08239i

−

1.00000i

0.414214i

−2.61313

2.61313i

251.2

1.00000i

−0.541196

−

0.541196i

−1.00000

1.08239

1.08239i

0.541196

−

0.541196i

2.61313

−

2.61313i

−

1.00000i

−

2.41421i

−1.08239

1.08239i

251.3

1.00000i

0.541196

0.541196i

−1.00000

−1.08239

−

1.08239i

−0.541196

0.541196i

−2.61313

2.61313i

−

1.00000i

−

2.41421i

1.08239

−

1.08239i

251.4

1.00000i

1.30656

1.30656i

−1.00000

−2.61313

−

2.61313i

−1.30656

1.30656i

1.08239

−

1.08239i

−

1.00000i

0.414214i

2.61313

−

2.61313i

327.1

−

1.00000i

−1.30656

1.30656i

−1.00000

2.61313

−

2.61313i

1.30656

1.30656i

−1.08239

−

1.08239i

1.00000i

−

0.414214i

−2.61313

−

2.61313i

327.2

−

1.00000i

−0.541196

0.541196i

−1.00000

1.08239

−

1.08239i

0.541196

0.541196i

2.61313

2.61313i

1.00000i

2.41421i

−1.08239

−

1.08239i

327.3

−

1.00000i

0.541196

−

0.541196i

−1.00000

−1.08239

1.08239i

−0.541196

−

0.541196i

−2.61313

−

2.61313i

1.00000i

2.41421i

1.08239

1.08239i

327.4

−

1.00000i

1.30656

−

1.30656i

−1.00000

−2.61313

2.61313i

−1.30656

−

1.30656i

1.08239

1.08239i

1.00000i

−

0.414214i

2.61313

2.61313i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.b	even	2	1	inner
17.c	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	578.2.c.f		8
17.b	even	2	1	inner	578.2.c.f		8
17.c	even	4	2	inner	578.2.c.f		8
17.d	even	8	2		578.2.a.i		4
17.d	even	8	2		578.2.b.d		4
17.e	odd	16	2		34.2.d.a	✓	4
17.e	odd	16	2		578.2.d.a		4
17.e	odd	16	2		578.2.d.b		4
17.e	odd	16	2		578.2.d.c		4
51.g	odd	8	2		5202.2.a.bw		4
51.i	even	16	2		306.2.l.c		4
68.g	odd	8	2		4624.2.a.bn		4
68.i	even	16	2		272.2.v.b		4
85.o	even	16	2		850.2.o.b		4
85.p	odd	16	2		850.2.l.a		4
85.r	even	16	2		850.2.o.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
34.2.d.a	✓	4	17.e	odd	16	2
272.2.v.b		4	68.i	even	16	2
306.2.l.c		4	51.i	even	16	2
578.2.a.i		4	17.d	even	8	2
578.2.b.d		4	17.d	even	8	2
578.2.c.f		8	1.a	even	1	1	trivial
578.2.c.f		8	17.b	even	2	1	inner
578.2.c.f		8	17.c	even	4	2	inner
578.2.d.a		4	17.e	odd	16	2
578.2.d.b		4	17.e	odd	16	2
578.2.d.c		4	17.e	odd	16	2
850.2.l.a		4	85.p	odd	16	2
850.2.o.a		4	85.r	even	16	2
850.2.o.b		4	85.o	even	16	2
4624.2.a.bn		4	68.g	odd	8	2
5202.2.a.bw		4	51.g	odd	8	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 12T_{3}^{4} + 4 $ T3^8 + 12*T3^4 + 4 acting on $S_{2}^{\mathrm{new}}(578, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 1)^{4} $ (T^2 + 1)^4
$3$	$ T^{8} + 12T^{4} + 4 $ T^8 + 12*T^4 + 4
$5$	$ T^{8} + 192T^{4} + 1024 $ T^8 + 192*T^4 + 1024
$7$	$ T^{8} + 192T^{4} + 1024 $ T^8 + 192*T^4 + 1024
$11$	$ T^{8} + 396T^{4} + 4 $ T^8 + 396*T^4 + 4
$13$	$ (T^{2} + 4 T - 4)^{4} $ (T^2 + 4*T - 4)^4
$17$	$ T^{8} $ T^8
$19$	$ (T^{4} + 24 T^{2} + 16)^{2} $ (T^4 + 24*T^2 + 16)^2
$23$	$ T^{8} + 3072 T^{4} + 262144 $ T^8 + 3072*T^4 + 262144
$29$	$ T^{8} + 192T^{4} + 1024 $ T^8 + 192*T^4 + 1024
$31$	$ T^{8} + 192T^{4} + 1024 $ T^8 + 192*T^4 + 1024
$37$	$ T^{8} + 768 T^{4} + 16384 $ T^8 + 768*T^4 + 16384
$41$	$ T^{8} + 17292 T^{4} + 23059204 $ T^8 + 17292*T^4 + 23059204
$43$	$ (T^{4} + 108 T^{2} + 324)^{2} $ (T^4 + 108*T^2 + 324)^2
$47$	$ (T^{2} + 8 T - 16)^{4} $ (T^2 + 8*T - 16)^4
$53$	$ (T^{4} + 24 T^{2} + 16)^{2} $ (T^4 + 24*T^2 + 16)^2
$59$	$ (T^{4} + 12 T^{2} + 4)^{2} $ (T^4 + 12*T^2 + 4)^2
$61$	$ T^{8} + 3072 T^{4} + 262144 $ T^8 + 3072*T^4 + 262144
$67$	$ (T^{2} - 12 T + 34)^{4} $ (T^2 - 12*T + 34)^4
$71$	$ T^{8} + 768 T^{4} + 16384 $ T^8 + 768*T^4 + 16384
$73$	$ T^{8} + 28812 T^{4} + 23059204 $ T^8 + 28812*T^4 + 23059204
$79$	$ T^{8} + 768 T^{4} + 16384 $ T^8 + 768*T^4 + 16384
$83$	$ (T^{4} + 44 T^{2} + 196)^{2} $ (T^4 + 44*T^2 + 196)^2
$89$	$ (T^{2} - 16 T + 14)^{4} $ (T^2 - 16*T + 14)^4
$97$	$ T^{8} + 972 T^{4} + 26244 $ T^8 + 972*T^4 + 26244
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 578 = 2 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	578.c (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{2} + \beta_{7} q^{3} - q^{4} - 2 \beta_{7} q^{5} - \beta_{5} q^{6} + 2 \beta_{4} q^{7} - \beta_{2} q^{8} + ( - \beta_{3} - \beta_{2}) q^{9} + 2 \beta_{5} q^{10} + (2 \beta_{5} + \beta_{4}) q^{11}+ \cdots + \beta_{7} q^{99}+O(q^{100}) \) q + b2 * q^2 + b7 * q^3 - q^4 - 2b7 q^5 - b5 * q^6 + 2b4 q^7 - b2 * q^8 + (-b3 - b2) * q^9 + 2b5 q^10 + (2b5 + b4) q^11 - b7 * q^12 + (-2b6 - 2) q^13 - 2b1 q^14 + (2b3 - 4b2) * q^15 + q^16 + (b6 + 1) * q^18 + (-2b3 - 2b2) * q^19 + 2b7 q^20 - 2b6 q^21 + (2b7 - b1) q^22 + 4b4 q^23 + b5 * q^24 + (-4b3 + 3b2) * q^25 + (-2b3 - 2b2) * q^26 + (3b5 - b4) q^27 - 2b4 q^28 - 2b1 q^29 + (-2b6 + 4) q^30 - 2b7 q^31 + b2 * q^32 + (-3b6 + 4) q^33 + 4b6 q^35 + (b3 + b2) * q^36 + (-2b7 - 2b1) * q^37 + (2b6 + 2) q^38 - 2b1 q^39 - 2b5 q^40 + (4b5 + 5b4) * q^41 - 2b3 q^42 + (3b3 + 6b2) * q^43 + (-2b5 - b4) q^44 + 2b4 q^45 - 4b1 q^46 + (4b6 - 4) q^47 + b7 * q^48 + (-4b3 - b2) q^49 + (4b6 - 3) q^50 + (2b6 + 2) q^52 + (-2b3 - 2b2) * q^53 + (3b7 + b1) q^54 + (6b6 - 8) q^55 + 2b1 q^56 - 2b4 q^57 - 2b4 q^58 + (b3 + 2b2) q^59 + (-2b3 + 4b2) * q^60 - 4b5 q^61 + 2b5 q^62 + (2b7 + 4b1) * q^63 - q^64 + 4b1 q^65 + (-3b3 + 4b2) * q^66 + (-b6 + 6) * q^67 - 4b6 q^69 + 4b3 q^70 + (-2b7 - 2b1) * q^71 + (-b6 - 1) * q^72 + 7b1 q^73 + (2b5 - 2b4) * q^74 + (-7b5 - 4b4) * q^75 + (2b3 + 2b2) * q^76 + (2b3 - 4b2) * q^77 - 2b4 q^78 + (-2b5 - 2b4) * q^79 - 2b7 q^80 + (-5b6 + 3) q^81 + (4b7 - 5b1) * q^82 + (3b3 - 2b2) * q^83 + 2b6 q^84 + (-3b6 - 6) q^86 - 2b3 q^87 + (-2b7 + b1) q^88 + (-5b6 + 8) q^89 - 2b1 q^90 + (4b5 - 8b4) * q^91 - 4b4 q^92 + (2b3 - 4b2) * q^93 + (4b3 - 4b2) * q^94 + 4b4 q^95 - b5 * q^96 + 3b7 q^97 + (4b6 + 1) q^98 + b7 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 16 q^{13} + 8 q^{16} + 8 q^{18} + 32 q^{30} + 32 q^{33} + 16 q^{38} - 32 q^{47} - 24 q^{50} + 16 q^{52} - 64 q^{55} - 8 q^{64} + 48 q^{67} - 8 q^{72} + 24 q^{81} - 48 q^{86} + 64 q^{89}+ \cdots + 8 q^{98}+O(q^{100}) \) 8 * q - 8 * q^4 - 16 * q^13 + 8 * q^16 + 8 * q^18 + 32 * q^30 + 32 * q^33 + 16 * q^38 - 32 * q^47 - 24 * q^50 + 16 * q^52 - 64 * q^55 - 8 * q^64 + 48 * q^67 - 8 * q^72 + 24 * q^81 - 48 * q^86 + 64 * q^89 + 8 * q^98

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	578.2.c.f

Level	$578$
Weight	$2$
Character orbit	578.c
Analytic conductor	$4.615$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.61535323683\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{16})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 1 \) x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	no (minimal twist has level 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( \zeta_{16}^{3} + \zeta_{16} \) v^3 + v
\(\beta_{2}\)	\(=\)	\( \zeta_{16}^{4} \) v^4
\(\beta_{3}\)	\(=\)	\( \zeta_{16}^{6} + \zeta_{16}^{2} \) v^6 + v^2
\(\beta_{4}\)	\(=\)	\( \zeta_{16}^{7} + \zeta_{16}^{5} \) v^7 + v^5
\(\beta_{5}\)	\(=\)	\( -\zeta_{16}^{3} + \zeta_{16} \) -v^3 + v
\(\beta_{6}\)	\(=\)	\( -\zeta_{16}^{6} + \zeta_{16}^{2} \) -v^6 + v^2
\(\beta_{7}\)	\(=\)	\( -\zeta_{16}^{7} + \zeta_{16}^{5} \) -v^7 + v^5

\(\zeta_{16}\)	\(=\)	\( ( \beta_{5} + \beta_1 ) / 2 \) (b5 + b1) / 2
\(\zeta_{16}^{2}\)	\(=\)	\( ( \beta_{6} + \beta_{3} ) / 2 \) (b6 + b3) / 2
\(\zeta_{16}^{3}\)	\(=\)	\( ( -\beta_{5} + \beta_1 ) / 2 \) (-b5 + b1) / 2
\(\zeta_{16}^{4}\)	\(=\)	\( \beta_{2} \) b2
\(\zeta_{16}^{5}\)	\(=\)	\( ( \beta_{7} + \beta_{4} ) / 2 \) (b7 + b4) / 2
\(\zeta_{16}^{6}\)	\(=\)	\( ( -\beta_{6} + \beta_{3} ) / 2 \) (-b6 + b3) / 2
\(\zeta_{16}^{7}\)	\(=\)	\( ( -\beta_{7} + \beta_{4} ) / 2 \) (-b7 + b4) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 251.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{4} \) (T^2 + 1)^4
$3$	\( T^{8} + 12T^{4} + 4 \) T^8 + 12*T^4 + 4
$5$	\( T^{8} + 192T^{4} + 1024 \) T^8 + 192*T^4 + 1024
$7$	\( T^{8} + 192T^{4} + 1024 \) T^8 + 192*T^4 + 1024
$11$	\( T^{8} + 396T^{4} + 4 \) T^8 + 396*T^4 + 4
$13$	\( (T^{2} + 4 T - 4)^{4} \) (T^2 + 4*T - 4)^4
$17$	\( T^{8} \) T^8
$19$	\( (T^{4} + 24 T^{2} + 16)^{2} \) (T^4 + 24*T^2 + 16)^2
$23$	\( T^{8} + 3072 T^{4} + 262144 \) T^8 + 3072*T^4 + 262144
$29$	\( T^{8} + 192T^{4} + 1024 \) T^8 + 192*T^4 + 1024
$31$	\( T^{8} + 192T^{4} + 1024 \) T^8 + 192*T^4 + 1024
$37$	\( T^{8} + 768 T^{4} + 16384 \) T^8 + 768*T^4 + 16384
$41$	\( T^{8} + 17292 T^{4} + 23059204 \) T^8 + 17292*T^4 + 23059204
$43$	\( (T^{4} + 108 T^{2} + 324)^{2} \) (T^4 + 108*T^2 + 324)^2
$47$	\( (T^{2} + 8 T - 16)^{4} \) (T^2 + 8*T - 16)^4
$53$	\( (T^{4} + 24 T^{2} + 16)^{2} \) (T^4 + 24*T^2 + 16)^2
$59$	\( (T^{4} + 12 T^{2} + 4)^{2} \) (T^4 + 12*T^2 + 4)^2
$61$	\( T^{8} + 3072 T^{4} + 262144 \) T^8 + 3072*T^4 + 262144
$67$	\( (T^{2} - 12 T + 34)^{4} \) (T^2 - 12*T + 34)^4
$71$	\( T^{8} + 768 T^{4} + 16384 \) T^8 + 768*T^4 + 16384
$73$	\( T^{8} + 28812 T^{4} + 23059204 \) T^8 + 28812*T^4 + 23059204
$79$	\( T^{8} + 768 T^{4} + 16384 \) T^8 + 768*T^4 + 16384
$83$	\( (T^{4} + 44 T^{2} + 196)^{2} \) (T^4 + 44*T^2 + 196)^2
$89$	\( (T^{2} - 16 T + 14)^{4} \) (T^2 - 16*T + 14)^4
$97$	\( T^{8} + 972 T^{4} + 26244 \) T^8 + 972*T^4 + 26244
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\(n\)	\(3\)
\(\chi(n)\)	\(\beta_{2}\)