LMFDB - Newform orbit 3267.1.l.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3267,1,Mod(838,3267)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3267, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3267.838");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3267 = 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3267.l (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:	$1.63044539627$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	16.0.6879707136000000000000.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 $ x^16 - 4x^14 + 15x^12 - 56x^10 + 209x^8 - 56x^6 + 15x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} + \cdots)$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{4} - \beta_{12} q^{7}+O(q^{10}) $ q - b4 * q^4 - b12 * q^7	$ q - \beta_{4} q^{4} - \beta_{12} q^{7} + \beta_{3} q^{13} - \beta_{8} q^{16} + (\beta_{7} + \beta_{5}) q^{19} - \beta_{13} q^{25} + (\beta_{12} + \beta_{9} + \beta_{5} - \beta_1) q^{28} + \beta_{14} q^{31} + \beta_{11} q^{43} + ( - \beta_{10} + \beta_{8}) q^{49} + \beta_{7} q^{52} + ( - \beta_{15} - \beta_{11} + \beta_{7} + \beta_{5} + \beta_{3} - \beta_1) q^{61} + ( - \beta_{13} + \beta_{8} + \beta_{4} - 1) q^{64} + \beta_{2} q^{67} + ( - \beta_{15} - \beta_{12}) q^{73} - \beta_{11} q^{76} - \beta_{3} q^{79} - \beta_{13} q^{91} + ( - \beta_{13} + \beta_{8} + \beta_{4} - 1) q^{97}+O(q^{100}) $ q - b4 * q^4 - b12 * q^7 + b3 * q^13 - b8 * q^16 + (b7 + b5) * q^19 - b13 * q^25 + (b12 + b9 + b5 - b1) * q^28 + b14 * q^31 + b11 * q^43 + (-b10 + b8) * q^49 + b7 * q^52 + (-b15 - b11 + b7 + b5 + b3 - b1) * q^61 + (-b13 + b8 + b4 - 1) * q^64 + b2 * q^67 + (-b15 - b12) * q^73 - b11 * q^76 - b3 * q^79 - b13 * q^91 + (-b13 + b8 + b4 - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{4}+O(q^{10}) $ 16 * q - 4 * q^4	$ 16 q - 4 q^{4} - 4 q^{16} + 4 q^{25} + 4 q^{49} - 4 q^{64} + 4 q^{91} - 4 q^{97}+O(q^{100}) $ 16 * q - 4 * q^4 - 4 * q^16 + 4 * q^25 + 4 * q^49 - 4 * q^64 + 4 * q^91 - 4 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 $ x^16 - 4*x^14 + 15*x^12 - 56*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{10} + 362 ) / 209 $ (v^10 + 362) / 209
$\beta_{3}$	$=$	$ ( \nu^{11} + 780\nu ) / 209 $ (v^11 + 780*v) / 209
$\beta_{4}$	$=$	$ ( \nu^{12} + 780\nu^{2} ) / 209 $ (v^12 + 780*v^2) / 209
$\beta_{5}$	$=$	$ ( \nu^{13} + 780\nu^{3} ) / 209 $ (v^13 + 780*v^3) / 209
$\beta_{6}$	$=$	$ ( -2\nu^{12} - 1351\nu^{2} ) / 209 $ (-2v^12 - 1351v^2) / 209
$\beta_{7}$	$=$	$ ( -4\nu^{13} - 2911\nu^{3} ) / 209 $ (-4v^13 - 2911v^3) / 209
$\beta_{8}$	$=$	$ ( -4\nu^{14} - 2911\nu^{4} ) / 209 $ (-4v^14 - 2911v^4) / 209
$\beta_{9}$	$=$	$ ( -4\nu^{15} - 2911\nu^{5} ) / 209 $ (-4v^15 - 2911v^5) / 209
$\beta_{10}$	$=$	$ ( -7\nu^{14} - 5042\nu^{4} ) / 209 $ (-7v^14 - 5042v^4) / 209
$\beta_{11}$	$=$	$ ( -\nu^{15} - 723\nu^{5} ) / 19 $ (-v^15 - 723*v^5) / 19
$\beta_{12}$	$=$	$ ( 60\nu^{15} - 225\nu^{13} + 840\nu^{11} - 3135\nu^{9} + 11704\nu^{7} - 225\nu^{5} + 60\nu^{3} - 15\nu ) / 209 $ (60v^15 - 225v^13 + 840v^11 - 3135v^9 + 11704v^7 - 225v^5 + 60v^3 - 15v) / 209
$\beta_{13}$	$=$	$ ( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209 $ (56v^14 - 224v^12 + 840v^10 - 3135v^8 + 11704v^6 - 3136v^4 + 840*v^2 - 224) / 209
$\beta_{14}$	$=$	$ ( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209 $ (104v^14 - 390v^12 + 1456v^10 - 5434v^8 + 20273v^6 - 390v^4 + 104*v^2 - 26) / 209
$\beta_{15}$	$=$	$ ( 164\nu^{15} - 615\nu^{13} + 2296\nu^{11} - 8569\nu^{9} + 31977\nu^{7} - 615\nu^{5} + 164\nu^{3} - 41\nu ) / 209 $ (164v^15 - 615v^13 + 2296v^11 - 8569v^9 + 31977v^7 - 615v^5 + 164v^3 - 41v) / 209

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{6} + 2\beta_{4} $ b6 + 2*b4
$\nu^{3}$	$=$	$ \beta_{7} + 4\beta_{5} $ b7 + 4*b5
$\nu^{4}$	$=$	$ 4\beta_{10} - 7\beta_{8} $ 4b10 - 7b8
$\nu^{5}$	$=$	$ 4\beta_{11} - 11\beta_{9} $ 4b11 - 11b9
$\nu^{6}$	$=$	$ -15\beta_{14} + 26\beta_{13} - 26\beta_{8} - 26\beta_{4} + 26 $ -15b14 + 26b13 - 26b8 - 26b4 + 26
$\nu^{7}$	$=$	$ -15\beta_{15} + 41\beta_{12} $ -15b15 + 41b12
$\nu^{8}$	$=$	$ -56\beta_{14} + 97\beta_{13} - 56\beta_{10} + 56\beta_{6} + 56\beta_{2} $ -56b14 + 97b13 - 56b10 + 56b6 + 56*b2
$\nu^{9}$	$=$	$ -56\beta_{15} + 153\beta_{12} - 56\beta_{11} + 153\beta_{9} + 56\beta_{7} + 209\beta_{5} + 56\beta_{3} - 209\beta_1 $ -56b15 + 153b12 - 56b11 + 153b9 + 56b7 + 209b5 + 56b3 - 209b1
$\nu^{10}$	$=$	$ 209\beta_{2} - 362 $ 209*b2 - 362
$\nu^{11}$	$=$	$ 209\beta_{3} - 780\beta_1 $ 209b3 - 780b1
$\nu^{12}$	$=$	$ -780\beta_{6} - 1351\beta_{4} $ -780b6 - 1351b4
$\nu^{13}$	$=$	$ -780\beta_{7} - 2911\beta_{5} $ -780b7 - 2911b5
$\nu^{14}$	$=$	$ -2911\beta_{10} + 5042\beta_{8} $ -2911b10 + 5042b8
$\nu^{15}$	$=$	$ -2911\beta_{11} + 7953\beta_{9} $ -2911b11 + 7953b9

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$.

$n$	$244$	$3026$
$\chi(n)$	$\beta_{4}$	$1$

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} $

838.1

−1.83730	−	0.596975i
−0.492303	−	0.159959i
0.492303	+	0.159959i
1.83730	+	0.596975i
1.13551	+	1.56290i
0.304260	+	0.418778i
−0.304260	−	0.418778i
−1.13551	−	1.56290i
1.13551	−	1.56290i
0.304260	−	0.418778i
−0.304260	+	0.418778i
−1.13551	+	1.56290i
−1.83730	+	0.596975i
−0.492303	+	0.159959i
0.492303	−	0.159959i
1.83730	−	0.596975i

−0.809017

−

0.587785i

−1.13551

1.56290i

838.2

−0.809017

−

0.587785i

−0.304260

0.418778i

838.3

−0.809017

−

0.587785i

0.304260

−

0.418778i

838.4

−0.809017

−

0.587785i

1.13551

−

1.56290i

2296.1

0.309017

−

0.951057i

−1.83730

−

0.596975i

2296.2

0.309017

−

0.951057i

−0.492303

−

0.159959i

2296.3

0.309017

−

0.951057i

0.492303

0.159959i

2296.4

0.309017

−

0.951057i

1.83730

0.596975i

2944.1

0.309017

0.951057i

−1.83730

0.596975i

2944.2

0.309017

0.951057i

−0.492303

0.159959i

2944.3

0.309017

0.951057i

0.492303

−

0.159959i

2944.4

0.309017

0.951057i

1.83730

−

0.596975i

2998.1

−0.809017

0.587785i

−1.13551

−

1.56290i

2998.2

−0.809017

0.587785i

−0.304260

−

0.418778i

2998.3

−0.809017

0.587785i

0.304260

0.418778i

2998.4

−0.809017

0.587785i

1.13551

1.56290i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
33.d	even	2	1	inner
33.f	even	10	3	inner
33.h	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3267.1.l.a		16
3.b	odd	2	1	CM	3267.1.l.a		16
11.b	odd	2	1	inner	3267.1.l.a		16
11.c	even	5	1		3267.1.c.a	✓	4
11.c	even	5	3	inner	3267.1.l.a		16
11.d	odd	10	1		3267.1.c.a	✓	4
11.d	odd	10	3	inner	3267.1.l.a		16
33.d	even	2	1	inner	3267.1.l.a		16
33.f	even	10	1		3267.1.c.a	✓	4
33.f	even	10	3	inner	3267.1.l.a		16
33.h	odd	10	1		3267.1.c.a	✓	4
33.h	odd	10	3	inner	3267.1.l.a		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3267.1.c.a	✓	4	11.c	even	5	1
3267.1.c.a	✓	4	11.d	odd	10	1
3267.1.c.a	✓	4	33.f	even	10	1
3267.1.c.a	✓	4	33.h	odd	10	1
3267.1.l.a		16	1.a	even	1	1	trivial
3267.1.l.a		16	3.b	odd	2	1	CM
3267.1.l.a		16	11.b	odd	2	1	inner
3267.1.l.a		16	11.c	even	5	3	inner
3267.1.l.a		16	11.d	odd	10	3	inner
3267.1.l.a		16	33.d	even	2	1	inner
3267.1.l.a		16	33.f	even	10	3	inner
3267.1.l.a		16	33.h	odd	10	3	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3267, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 $ T^16 - 4T^14 + 15T^12 - 56T^10 + 209T^8 - 56T^6 + 15T^4 - 4*T^2 + 1
$11$	$ T^{16} $ T^16
$13$	$ T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 $ T^16 - 4T^14 + 15T^12 - 56T^10 + 209T^8 - 56T^6 + 15T^4 - 4*T^2 + 1
$17$	$ T^{16} $ T^16
$19$	$ (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} $ (T^8 - 2T^6 + 4T^4 - 8*T^2 + 16)^2
$23$	$ T^{16} $ T^16
$29$	$ T^{16} $ T^16
$31$	$ (T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2} $ (T^8 + 3T^6 + 9T^4 + 27*T^2 + 81)^2
$37$	$ T^{16} $ T^16
$41$	$ T^{16} $ T^16
$43$	$ (T^{2} + 2)^{8} $ (T^2 + 2)^8
$47$	$ T^{16} $ T^16
$53$	$ T^{16} $ T^16
$59$	$ T^{16} $ T^16
$61$	$ (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} $ (T^8 - 2T^6 + 4T^4 - 8*T^2 + 16)^2
$67$	$ (T^{2} - 3)^{8} $ (T^2 - 3)^8
$71$	$ T^{16} $ T^16
$73$	$ T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 $ T^16 - 4T^14 + 15T^12 - 56T^10 + 209T^8 - 56T^6 + 15T^4 - 4*T^2 + 1
$79$	$ T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 $ T^16 - 4T^14 + 15T^12 - 56T^10 + 209T^8 - 56T^6 + 15T^4 - 4*T^2 + 1
$83$	$ T^{16} $ T^16
$89$	$ T^{16} $ T^16
$97$	$ (T^{4} + T^{3} + T^{2} + T + 1)^{4} $ (T^4 + T^3 + T^2 + T + 1)^4
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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:	\( N \)	\(=\)	\( 3267 = 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3267.l (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{4} - \beta_{12} q^{7}+O(q^{10}) \) q - b4 * q^4 - b12 * q^7	\( q - \beta_{4} q^{4} - \beta_{12} q^{7} + \beta_{3} q^{13} - \beta_{8} q^{16} + (\beta_{7} + \beta_{5}) q^{19} - \beta_{13} q^{25} + (\beta_{12} + \beta_{9} + \beta_{5} - \beta_1) q^{28} + \beta_{14} q^{31} + \beta_{11} q^{43} + ( - \beta_{10} + \beta_{8}) q^{49} + \beta_{7} q^{52} + ( - \beta_{15} - \beta_{11} + \beta_{7} + \beta_{5} + \beta_{3} - \beta_1) q^{61} + ( - \beta_{13} + \beta_{8} + \beta_{4} - 1) q^{64} + \beta_{2} q^{67} + ( - \beta_{15} - \beta_{12}) q^{73} - \beta_{11} q^{76} - \beta_{3} q^{79} - \beta_{13} q^{91} + ( - \beta_{13} + \beta_{8} + \beta_{4} - 1) q^{97}+O(q^{100}) \) q - b4 * q^4 - b12 * q^7 + b3 * q^13 - b8 * q^16 + (b7 + b5) * q^19 - b13 * q^25 + (b12 + b9 + b5 - b1) * q^28 + b14 * q^31 + b11 * q^43 + (-b10 + b8) * q^49 + b7 * q^52 + (-b15 - b11 + b7 + b5 + b3 - b1) * q^61 + (-b13 + b8 + b4 - 1) * q^64 + b2 * q^67 + (-b15 - b12) * q^73 - b11 * q^76 - b3 * q^79 - b13 * q^91 + (-b13 + b8 + b4 - 1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{4}+O(q^{10}) \) 16 * q - 4 * q^4	\( 16 q - 4 q^{4} - 4 q^{16} + 4 q^{25} + 4 q^{49} - 4 q^{64} + 4 q^{91} - 4 q^{97}+O(q^{100}) \) 16 * q - 4 * q^4 - 4 * q^16 + 4 * q^25 + 4 * q^49 - 4 * q^64 + 4 * q^91 - 4 * 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	3267.1.l.a

Level	$3267$
Weight	$1$
Character orbit	3267.l
Analytic conductor	$1.630$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{12}$
CM discriminant	-3
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(1.63044539627\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	16.0.6879707136000000000000.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) x^16 - 4x^14 + 15x^12 - 56x^10 + 209x^8 - 56x^6 + 15x^4 - 4*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{12}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{10} + 362 ) / 209 \) (v^10 + 362) / 209
\(\beta_{3}\)	\(=\)	\( ( \nu^{11} + 780\nu ) / 209 \) (v^11 + 780*v) / 209
\(\beta_{4}\)	\(=\)	\( ( \nu^{12} + 780\nu^{2} ) / 209 \) (v^12 + 780*v^2) / 209
\(\beta_{5}\)	\(=\)	\( ( \nu^{13} + 780\nu^{3} ) / 209 \) (v^13 + 780*v^3) / 209
\(\beta_{6}\)	\(=\)	\( ( -2\nu^{12} - 1351\nu^{2} ) / 209 \) (-2v^12 - 1351v^2) / 209
\(\beta_{7}\)	\(=\)	\( ( -4\nu^{13} - 2911\nu^{3} ) / 209 \) (-4v^13 - 2911v^3) / 209
\(\beta_{8}\)	\(=\)	\( ( -4\nu^{14} - 2911\nu^{4} ) / 209 \) (-4v^14 - 2911v^4) / 209
\(\beta_{9}\)	\(=\)	\( ( -4\nu^{15} - 2911\nu^{5} ) / 209 \) (-4v^15 - 2911v^5) / 209
\(\beta_{10}\)	\(=\)	\( ( -7\nu^{14} - 5042\nu^{4} ) / 209 \) (-7v^14 - 5042v^4) / 209
\(\beta_{11}\)	\(=\)	\( ( -\nu^{15} - 723\nu^{5} ) / 19 \) (-v^15 - 723*v^5) / 19
\(\beta_{12}\)	\(=\)	\( ( 60\nu^{15} - 225\nu^{13} + 840\nu^{11} - 3135\nu^{9} + 11704\nu^{7} - 225\nu^{5} + 60\nu^{3} - 15\nu ) / 209 \) (60v^15 - 225v^13 + 840v^11 - 3135v^9 + 11704v^7 - 225v^5 + 60v^3 - 15v) / 209
\(\beta_{13}\)	\(=\)	\( ( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209 \) (56v^14 - 224v^12 + 840v^10 - 3135v^8 + 11704v^6 - 3136v^4 + 840*v^2 - 224) / 209
\(\beta_{14}\)	\(=\)	\( ( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209 \) (104v^14 - 390v^12 + 1456v^10 - 5434v^8 + 20273v^6 - 390v^4 + 104*v^2 - 26) / 209
\(\beta_{15}\)	\(=\)	\( ( 164\nu^{15} - 615\nu^{13} + 2296\nu^{11} - 8569\nu^{9} + 31977\nu^{7} - 615\nu^{5} + 164\nu^{3} - 41\nu ) / 209 \) (164v^15 - 615v^13 + 2296v^11 - 8569v^9 + 31977v^7 - 615v^5 + 164v^3 - 41v) / 209

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{6} + 2\beta_{4} \) b6 + 2*b4
\(\nu^{3}\)	\(=\)	\( \beta_{7} + 4\beta_{5} \) b7 + 4*b5
\(\nu^{4}\)	\(=\)	\( 4\beta_{10} - 7\beta_{8} \) 4b10 - 7b8
\(\nu^{5}\)	\(=\)	\( 4\beta_{11} - 11\beta_{9} \) 4b11 - 11b9
\(\nu^{6}\)	\(=\)	\( -15\beta_{14} + 26\beta_{13} - 26\beta_{8} - 26\beta_{4} + 26 \) -15b14 + 26b13 - 26b8 - 26b4 + 26
\(\nu^{7}\)	\(=\)	\( -15\beta_{15} + 41\beta_{12} \) -15b15 + 41b12
\(\nu^{8}\)	\(=\)	\( -56\beta_{14} + 97\beta_{13} - 56\beta_{10} + 56\beta_{6} + 56\beta_{2} \) -56b14 + 97b13 - 56b10 + 56b6 + 56*b2
\(\nu^{9}\)	\(=\)	\( -56\beta_{15} + 153\beta_{12} - 56\beta_{11} + 153\beta_{9} + 56\beta_{7} + 209\beta_{5} + 56\beta_{3} - 209\beta_1 \) -56b15 + 153b12 - 56b11 + 153b9 + 56b7 + 209b5 + 56b3 - 209b1
\(\nu^{10}\)	\(=\)	\( 209\beta_{2} - 362 \) 209*b2 - 362
\(\nu^{11}\)	\(=\)	\( 209\beta_{3} - 780\beta_1 \) 209b3 - 780b1
\(\nu^{12}\)	\(=\)	\( -780\beta_{6} - 1351\beta_{4} \) -780b6 - 1351b4
\(\nu^{13}\)	\(=\)	\( -780\beta_{7} - 2911\beta_{5} \) -780b7 - 2911b5
\(\nu^{14}\)	\(=\)	\( -2911\beta_{10} + 5042\beta_{8} \) -2911b10 + 5042b8
\(\nu^{15}\)	\(=\)	\( -2911\beta_{11} + 7953\beta_{9} \) -2911b11 + 7953b9

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2998.4
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
33.d	even	2	1	inner
33.f	even	10	3	inner
33.h	odd	10	3	inner

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 \) T^16 - 4T^14 + 15T^12 - 56T^10 + 209T^8 - 56T^6 + 15T^4 - 4*T^2 + 1
$11$	\( T^{16} \) T^16
$13$	\( T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 \) T^16 - 4T^14 + 15T^12 - 56T^10 + 209T^8 - 56T^6 + 15T^4 - 4*T^2 + 1
$17$	\( T^{16} \) T^16
$19$	\( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \) (T^8 - 2T^6 + 4T^4 - 8*T^2 + 16)^2
$23$	\( T^{16} \) T^16
$29$	\( T^{16} \) T^16
$31$	\( (T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2} \) (T^8 + 3T^6 + 9T^4 + 27*T^2 + 81)^2
$37$	\( T^{16} \) T^16
$41$	\( T^{16} \) T^16
$43$	\( (T^{2} + 2)^{8} \) (T^2 + 2)^8
$47$	\( T^{16} \) T^16
$53$	\( T^{16} \) T^16
$59$	\( T^{16} \) T^16
$61$	\( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \) (T^8 - 2T^6 + 4T^4 - 8*T^2 + 16)^2
$67$	\( (T^{2} - 3)^{8} \) (T^2 - 3)^8
$71$	\( T^{16} \) T^16
$73$	\( T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 \) T^16 - 4T^14 + 15T^12 - 56T^10 + 209T^8 - 56T^6 + 15T^4 - 4*T^2 + 1
$79$	\( T^{16} - 4 T^{14} + 15 T^{12} - 56 T^{10} + \cdots + 1 \) T^16 - 4T^14 + 15T^12 - 56T^10 + 209T^8 - 56T^6 + 15T^4 - 4*T^2 + 1
$83$	\( T^{16} \) T^16
$89$	\( T^{16} \) T^16
$97$	\( (T^{4} + T^{3} + T^{2} + T + 1)^{4} \) (T^4 + T^3 + T^2 + T + 1)^4
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\(n\)	\(244\)	\(3026\)
\(\chi(n)\)	\(\beta_{4}\)	\(1\)