LMFDB - Newform orbit 867.2.d.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [867,2,Mod(577,867)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(867, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("867.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$8$
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 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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} q^{2} + \beta_1 q^{3} - \beta_{6} q^{4} + (\beta_{4} - \beta_{2} - \beta_1) q^{5} + \beta_{4} q^{6} + ( - \beta_{4} + 2 \beta_1) q^{7} + ( - \beta_{7} - \beta_{5}) q^{8} - q^{9} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{10}+ \cdots + ( - \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{99}+O(q^{100}) $ q + b7 * q^2 + b1 * q^3 - b6 * q^4 + (b4 - b2 - b1) * q^5 + b4 * q^6 + (-b4 + 2b1) q^7 + (-b7 - b5) * q^8 - q^9 + (-b4 - 2b3 + 2b1) * q^10 + (b4 - b3 - b2 + b1) * q^11 - b3 * q^12 + (b6 + 2b5 - 1) q^13 + (2b4 + b3 - 2b1) * q^14 + (-b7 + b5 + 1) * q^15 + (2b6 - 2) q^16 - b7 * q^18 + (2b7 - b6 + 2b5 + 3) * q^19 + (2b4 + b3) q^20 + (b7 - 2) * q^21 + (2b4 - 2b3 - b2 + 2b1) q^22 + (-b4 - 3b2 + 3b1) * q^23 + (-b4 - b2) * q^24 + (2b7 + 2b6 - 2b5) q^25 + (-2b7 + 2b6 + b5) * q^26 - b1 * q^27 + (-b4 - 2b3 + b2) q^28 + (-3b3 - 2b1) * q^29 + (b7 + 2b6 - 2) q^30 + (-b4 - 4b3 - b2 - 2b1) * q^31 + (-2b7 + 4b5) * q^32 + (-b7 + b6 + b5 - 1) * q^33 + (-3b7 - 2b6 + 2b5 + 4) q^35 + b6 * q^36 + (-b4 + 2b3 - 6b1) * q^37 + (4b7 - b5 + 4) q^38 + (b3 + 2b2 - b1) q^39 + (b4 + 2b3 + b2) q^40 + (b4 + 2b3 + 3b2 + 3b1) q^41 + (-2b7 - b6 + 2) q^42 + (-2b7 - b6 + 4b5 - 1) * q^43 + (2b4 - b3 + 2b1) * q^44 + (-b4 + b2 + b1) * q^45 + (3b4 - 2b3 - 2b1) q^46 + (2b7 - 4b6 + 2b5 + 2) q^47 + (2b3 - 2b1) * q^48 + (4b7 + b6 + 1) q^49 + (-2b7 - 4b6 + 2b5 + 4) q^50 + (-2b7 + b6 - 2b5 - 2) * q^52 + (-4b6 + 2b5 - 2) * q^53 - b4 * q^54 + (-2b7 + b6 - 3) q^55 + (-2b4 - 2b2 + 2b1) q^56 + (2b4 - b3 + 2b2 + 3b1) q^57 + (b4 - 3b2) q^58 + (b7 - 2b6 - 6) q^59 + (-2b7 - b6) q^60 + (-b4 - 4b3 + 2b2 + 6b1) q^61 + (2b4 - 4b2 - 2b1) q^62 + (b4 - 2b1) q^63 + 2b6 q^64 + (-3b4 - b3 - b2 - 3b1) * q^65 + (-2b7 + 2b6 + b5 - 2) * q^66 + (2b7 - b6 + 2) q^67 + (b7 + 3b5 - 3) q^69 + (6b7 + 5b6 - 2b5 - 6) q^70 + (-2b4 - 3b3 + 6b1) q^71 + (b7 + b5) * q^72 + (-4b4 - 4b3 + 6b2 - 2b1) * q^73 + (-8b4 + b3 + 2b2 - 2b1) q^74 + (2b4 + 2b3 - 2b2) q^75 + (-3b6 - 4b5 + 2) * q^76 + b5 * q^77 + (-2b4 + 2b3 + b2) * q^78 + (-2b3 + 3b2) * q^79 + (-6b4 - 2b3 + 2b2 + 2b1) * q^80 + q^81 + (b4 + 2b3 + 2b2 + 2b1) q^82 + (-4b7 - 4b6 - 2b5 - 2) q^83 + (b7 + 2b6 - b5) q^84 + (6b6 - b5 - 4) q^86 + (3b6 + 2) q^87 + (-b4 + 2b3 + b2) q^88 + (-2b7 + 4b6 + 3b5 + 4) q^89 + (b4 + 2b3 - 2b1) * q^90 + (2b4 + 3b2 - 2b1) q^91 + (2b4 - 3b3 + 4b2) q^92 + (b7 + 4b6 + b5 + 2) q^93 + (6b7 - 4b5 + 4) * q^94 + (3b4 - 3b3 - 5b2 - 3b1) * q^95 + (-2b4 + 4b2) * q^96 + (-3b4 + 2b3 - b2 + 4b1) q^97 + (-4b6 + b5 + 8) q^98 + (-b4 + b3 + b2 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 8 q^{13} + 8 q^{15} - 16 q^{16} + 24 q^{19} - 16 q^{21} - 16 q^{30} - 8 q^{33} + 32 q^{35} + 32 q^{38} + 16 q^{42} - 8 q^{43} + 16 q^{47} + 8 q^{49} + 32 q^{50} - 16 q^{52} - 16 q^{53} - 24 q^{55}+ \cdots + 64 q^{98}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^13 + 8 * q^15 - 16 * q^16 + 24 * q^19 - 16 * q^21 - 16 * q^30 - 8 * q^33 + 32 * q^35 + 32 * q^38 + 16 * q^42 - 8 * q^43 + 16 * q^47 + 8 * q^49 + 32 * q^50 - 16 * q^52 - 16 * q^53 - 24 * q^55 - 48 * q^59 - 16 * q^66 + 16 * q^67 - 24 * q^69 - 48 * q^70 + 16 * q^76 + 8 * q^81 - 16 * q^83 - 32 * q^86 + 16 * q^87 + 32 * q^89 + 16 * q^93 + 32 * q^94 + 64 * q^98

Basis of coefficient ring

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

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

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

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/867\mathbb{Z}\right)^\times$.

$n$	$290$	$292$
$\chi(n)$	$1$	$-1$

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} $

577.1

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

−1.84776

−

1.00000i

1.41421

3.61313i

1.84776i

−

3.84776i

1.08239

−1.00000

−

6.67619i

577.2

−1.84776

1.00000i

1.41421

−

3.61313i

−

1.84776i

3.84776i

1.08239

−1.00000

6.67619i

577.3

−0.765367

−

1.00000i

−1.41421

−

0.0823922i

0.765367i

−

2.76537i

2.61313

−1.00000

0.0630603i

577.4

−0.765367

1.00000i

−1.41421

0.0823922i

−

0.765367i

2.76537i

2.61313

−1.00000

−

0.0630603i

577.5

0.765367

−

1.00000i

−1.41421

2.08239i

−

0.765367i

−

1.23463i

−2.61313

−1.00000

1.59379i

577.6

0.765367

1.00000i

−1.41421

−

2.08239i

0.765367i

1.23463i

−2.61313

−1.00000

−

1.59379i

577.7

1.84776

−

1.00000i

1.41421

−

1.61313i

−

1.84776i

−

0.152241i

−1.08239

−1.00000

−

2.98067i

577.8

1.84776

1.00000i

1.41421

1.61313i

1.84776i

0.152241i

−1.08239

−1.00000

2.98067i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	867.2.d.e		8
17.b	even	2	1	inner	867.2.d.e		8
17.c	even	4	1		867.2.a.m		4
17.c	even	4	1		867.2.a.n		4
17.d	even	8	2		867.2.e.h		8
17.d	even	8	2		867.2.e.i		8
17.e	odd	16	2		51.2.h.a	✓	8
17.e	odd	16	2		867.2.h.b		8
17.e	odd	16	2		867.2.h.f		8
17.e	odd	16	2		867.2.h.g		8
51.f	odd	4	1		2601.2.a.bc		4
51.f	odd	4	1		2601.2.a.bd		4
51.i	even	16	2		153.2.l.e		8
68.i	even	16	2		816.2.bq.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
51.2.h.a	✓	8	17.e	odd	16	2
153.2.l.e		8	51.i	even	16	2
816.2.bq.a		8	68.i	even	16	2
867.2.a.m		4	17.c	even	4	1
867.2.a.n		4	17.c	even	4	1
867.2.d.e		8	1.a	even	1	1	trivial
867.2.d.e		8	17.b	even	2	1	inner
867.2.e.h		8	17.d	even	8	2
867.2.e.i		8	17.d	even	8	2
867.2.h.b		8	17.e	odd	16	2
867.2.h.f		8	17.e	odd	16	2
867.2.h.g		8	17.e	odd	16	2
2601.2.a.bc		4	51.f	odd	4	1
2601.2.a.bd		4	51.f	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} - 4T_{2}^{2} + 2 $ T2^4 - 4*T2^2 + 2 acting on $S_{2}^{\mathrm{new}}(867, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - 4 T^{2} + 2)^{2} $ (T^4 - 4*T^2 + 2)^2
$3$	$ (T^{2} + 1)^{4} $ (T^2 + 1)^4
$5$	$ T^{8} + 20 T^{6} + \cdots + 1 $ T^8 + 20T^6 + 102T^4 + 148*T^2 + 1
$7$	$ T^{8} + 24 T^{6} + \cdots + 4 $ T^8 + 24T^6 + 148T^4 + 176*T^2 + 4
$11$	$ T^{8} + 28 T^{6} + \cdots + 1 $ T^8 + 28T^6 + 70T^4 + 28*T^2 + 1
$13$	$ (T^{4} + 4 T^{3} - 14 T^{2} + \cdots - 47)^{2} $ (T^4 + 4T^3 - 14T^2 - 68*T - 47)^2
$17$	$ T^{8} $ T^8
$19$	$ (T^{4} - 12 T^{3} + \cdots - 367)^{2} $ (T^4 - 12T^3 + 18T^2 + 172*T - 367)^2
$23$	$ T^{8} + 116 T^{6} + \cdots + 73441 $ T^8 + 116T^6 + 2822T^4 + 25012*T^2 + 73441
$29$	$ (T^{4} + 44 T^{2} + 196)^{2} $ (T^4 + 44*T^2 + 196)^2
$31$	$ T^{8} + 160 T^{6} + \cdots + 399424 $ T^8 + 160T^6 + 6640T^4 + 97536*T^2 + 399424
$37$	$ T^{8} + 184 T^{6} + \cdots + 498436 $ T^8 + 184T^6 + 9108T^4 + 132848*T^2 + 498436
$41$	$ T^{8} + 148 T^{6} + \cdots + 1 $ T^8 + 148T^6 + 102T^4 + 20*T^2 + 1
$43$	$ (T^{4} + 4 T^{3} + \cdots + 1297)^{2} $ (T^4 + 4T^3 - 78T^2 - 196*T + 1297)^2
$47$	$ (T^{4} - 8 T^{3} + \cdots - 752)^{2} $ (T^4 - 8T^3 - 72T^2 + 608*T - 752)^2
$53$	$ (T^{4} + 8 T^{3} + \cdots + 496)^{2} $ (T^4 + 8T^3 - 56T^2 - 160*T + 496)^2
$59$	$ (T^{4} + 24 T^{3} + \cdots + 514)^{2} $ (T^4 + 24T^3 + 196T^2 + 608*T + 514)^2
$61$	$ T^{8} + 312 T^{6} + \cdots + 1110916 $ T^8 + 312T^6 + 20692T^4 + 290800*T^2 + 1110916
$67$	$ (T^{4} - 8 T^{3} + 4 T^{2} + \cdots + 4)^{2} $ (T^4 - 8T^3 + 4T^2 + 16*T + 4)^2
$71$	$ T^{8} + 248 T^{6} + \cdots + 4624 $ T^8 + 248T^6 + 10904T^4 + 90592*T^2 + 4624
$73$	$ T^{8} + 560 T^{6} + \cdots + 565504 $ T^8 + 560T^6 + 94240T^4 + 4183296*T^2 + 565504
$79$	$ T^{8} + 104 T^{6} + \cdots + 3844 $ T^8 + 104T^6 + 2580T^4 + 14288*T^2 + 3844
$83$	$ (T^{4} + 8 T^{3} + \cdots - 272)^{2} $ (T^4 + 8T^3 - 120T^2 - 416*T - 272)^2
$89$	$ (T^{4} - 16 T^{3} + \cdots - 2558)^{2} $ (T^4 - 16T^3 - 20T^2 + 896*T - 2558)^2
$97$	$ T^{8} + 176 T^{6} + \cdots + 399424 $ T^8 + 176T^6 + 7504T^4 + 100736*T^2 + 399424
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Belyi maps

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{7} q^{2} + \beta_1 q^{3} - \beta_{6} q^{4} + (\beta_{4} - \beta_{2} - \beta_1) q^{5} + \beta_{4} q^{6} + ( - \beta_{4} + 2 \beta_1) q^{7} + ( - \beta_{7} - \beta_{5}) q^{8} - q^{9} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{10}+ \cdots + ( - \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{99}+O(q^{100}) \) q + b7 * q^2 + b1 * q^3 - b6 * q^4 + (b4 - b2 - b1) * q^5 + b4 * q^6 + (-b4 + 2b1) q^7 + (-b7 - b5) * q^8 - q^9 + (-b4 - 2b3 + 2b1) * q^10 + (b4 - b3 - b2 + b1) * q^11 - b3 * q^12 + (b6 + 2b5 - 1) q^13 + (2b4 + b3 - 2b1) * q^14 + (-b7 + b5 + 1) * q^15 + (2b6 - 2) q^16 - b7 * q^18 + (2b7 - b6 + 2b5 + 3) * q^19 + (2b4 + b3) q^20 + (b7 - 2) * q^21 + (2b4 - 2b3 - b2 + 2b1) q^22 + (-b4 - 3b2 + 3b1) * q^23 + (-b4 - b2) * q^24 + (2b7 + 2b6 - 2b5) q^25 + (-2b7 + 2b6 + b5) * q^26 - b1 * q^27 + (-b4 - 2b3 + b2) q^28 + (-3b3 - 2b1) * q^29 + (b7 + 2b6 - 2) q^30 + (-b4 - 4b3 - b2 - 2b1) * q^31 + (-2b7 + 4b5) * q^32 + (-b7 + b6 + b5 - 1) * q^33 + (-3b7 - 2b6 + 2b5 + 4) q^35 + b6 * q^36 + (-b4 + 2b3 - 6b1) * q^37 + (4b7 - b5 + 4) q^38 + (b3 + 2b2 - b1) q^39 + (b4 + 2b3 + b2) q^40 + (b4 + 2b3 + 3b2 + 3b1) q^41 + (-2b7 - b6 + 2) q^42 + (-2b7 - b6 + 4b5 - 1) * q^43 + (2b4 - b3 + 2b1) * q^44 + (-b4 + b2 + b1) * q^45 + (3b4 - 2b3 - 2b1) q^46 + (2b7 - 4b6 + 2b5 + 2) q^47 + (2b3 - 2b1) * q^48 + (4b7 + b6 + 1) q^49 + (-2b7 - 4b6 + 2b5 + 4) q^50 + (-2b7 + b6 - 2b5 - 2) * q^52 + (-4b6 + 2b5 - 2) * q^53 - b4 * q^54 + (-2b7 + b6 - 3) q^55 + (-2b4 - 2b2 + 2b1) q^56 + (2b4 - b3 + 2b2 + 3b1) q^57 + (b4 - 3b2) q^58 + (b7 - 2b6 - 6) q^59 + (-2b7 - b6) q^60 + (-b4 - 4b3 + 2b2 + 6b1) q^61 + (2b4 - 4b2 - 2b1) q^62 + (b4 - 2b1) q^63 + 2b6 q^64 + (-3b4 - b3 - b2 - 3b1) * q^65 + (-2b7 + 2b6 + b5 - 2) * q^66 + (2b7 - b6 + 2) q^67 + (b7 + 3b5 - 3) q^69 + (6b7 + 5b6 - 2b5 - 6) q^70 + (-2b4 - 3b3 + 6b1) q^71 + (b7 + b5) * q^72 + (-4b4 - 4b3 + 6b2 - 2b1) * q^73 + (-8b4 + b3 + 2b2 - 2b1) q^74 + (2b4 + 2b3 - 2b2) q^75 + (-3b6 - 4b5 + 2) * q^76 + b5 * q^77 + (-2b4 + 2b3 + b2) * q^78 + (-2b3 + 3b2) * q^79 + (-6b4 - 2b3 + 2b2 + 2b1) * q^80 + q^81 + (b4 + 2b3 + 2b2 + 2b1) q^82 + (-4b7 - 4b6 - 2b5 - 2) q^83 + (b7 + 2b6 - b5) q^84 + (6b6 - b5 - 4) q^86 + (3b6 + 2) q^87 + (-b4 + 2b3 + b2) q^88 + (-2b7 + 4b6 + 3b5 + 4) q^89 + (b4 + 2b3 - 2b1) * q^90 + (2b4 + 3b2 - 2b1) q^91 + (2b4 - 3b3 + 4b2) q^92 + (b7 + 4b6 + b5 + 2) q^93 + (6b7 - 4b5 + 4) * q^94 + (3b4 - 3b3 - 5b2 - 3b1) * q^95 + (-2b4 + 4b2) * q^96 + (-3b4 + 2b3 - b2 + 4b1) q^97 + (-4b6 + b5 + 8) q^98 + (-b4 + b3 + b2 - b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 8 q^{13} + 8 q^{15} - 16 q^{16} + 24 q^{19} - 16 q^{21} - 16 q^{30} - 8 q^{33} + 32 q^{35} + 32 q^{38} + 16 q^{42} - 8 q^{43} + 16 q^{47} + 8 q^{49} + 32 q^{50} - 16 q^{52} - 16 q^{53} - 24 q^{55}+ \cdots + 64 q^{98}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^13 + 8 * q^15 - 16 * q^16 + 24 * q^19 - 16 * q^21 - 16 * q^30 - 8 * q^33 + 32 * q^35 + 32 * q^38 + 16 * q^42 - 8 * q^43 + 16 * q^47 + 8 * q^49 + 32 * q^50 - 16 * q^52 - 16 * q^53 - 24 * q^55 - 48 * q^59 - 16 * q^66 + 16 * q^67 - 24 * q^69 - 48 * q^70 + 16 * q^76 + 8 * q^81 - 16 * q^83 - 32 * q^86 + 16 * q^87 + 32 * q^89 + 16 * q^93 + 32 * q^94 + 64 * q^98

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Newspace parameters

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$q$-expansion

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Hecke kernels

Hecke characteristic polynomials

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Label	867.2.d.e

Level	$867$
Weight	$2$
Character orbit	867.d
Analytic conductor	$6.923$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(6.92302985525\)
Analytic rank:	\(0\)
Dimension:	\(8\)
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 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

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

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 577.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} - 4 T^{2} + 2)^{2} \) (T^4 - 4*T^2 + 2)^2
$3$	\( (T^{2} + 1)^{4} \) (T^2 + 1)^4
$5$	\( T^{8} + 20 T^{6} + \cdots + 1 \) T^8 + 20T^6 + 102T^4 + 148*T^2 + 1
$7$	\( T^{8} + 24 T^{6} + \cdots + 4 \) T^8 + 24T^6 + 148T^4 + 176*T^2 + 4
$11$	\( T^{8} + 28 T^{6} + \cdots + 1 \) T^8 + 28T^6 + 70T^4 + 28*T^2 + 1
$13$	\( (T^{4} + 4 T^{3} - 14 T^{2} + \cdots - 47)^{2} \) (T^4 + 4T^3 - 14T^2 - 68*T - 47)^2
$17$	\( T^{8} \) T^8
$19$	\( (T^{4} - 12 T^{3} + \cdots - 367)^{2} \) (T^4 - 12T^3 + 18T^2 + 172*T - 367)^2
$23$	\( T^{8} + 116 T^{6} + \cdots + 73441 \) T^8 + 116T^6 + 2822T^4 + 25012*T^2 + 73441
$29$	\( (T^{4} + 44 T^{2} + 196)^{2} \) (T^4 + 44*T^2 + 196)^2
$31$	\( T^{8} + 160 T^{6} + \cdots + 399424 \) T^8 + 160T^6 + 6640T^4 + 97536*T^2 + 399424
$37$	\( T^{8} + 184 T^{6} + \cdots + 498436 \) T^8 + 184T^6 + 9108T^4 + 132848*T^2 + 498436
$41$	\( T^{8} + 148 T^{6} + \cdots + 1 \) T^8 + 148T^6 + 102T^4 + 20*T^2 + 1
$43$	\( (T^{4} + 4 T^{3} + \cdots + 1297)^{2} \) (T^4 + 4T^3 - 78T^2 - 196*T + 1297)^2
$47$	\( (T^{4} - 8 T^{3} + \cdots - 752)^{2} \) (T^4 - 8T^3 - 72T^2 + 608*T - 752)^2
$53$	\( (T^{4} + 8 T^{3} + \cdots + 496)^{2} \) (T^4 + 8T^3 - 56T^2 - 160*T + 496)^2
$59$	\( (T^{4} + 24 T^{3} + \cdots + 514)^{2} \) (T^4 + 24T^3 + 196T^2 + 608*T + 514)^2
$61$	\( T^{8} + 312 T^{6} + \cdots + 1110916 \) T^8 + 312T^6 + 20692T^4 + 290800*T^2 + 1110916
$67$	\( (T^{4} - 8 T^{3} + 4 T^{2} + \cdots + 4)^{2} \) (T^4 - 8T^3 + 4T^2 + 16*T + 4)^2
$71$	\( T^{8} + 248 T^{6} + \cdots + 4624 \) T^8 + 248T^6 + 10904T^4 + 90592*T^2 + 4624
$73$	\( T^{8} + 560 T^{6} + \cdots + 565504 \) T^8 + 560T^6 + 94240T^4 + 4183296*T^2 + 565504
$79$	\( T^{8} + 104 T^{6} + \cdots + 3844 \) T^8 + 104T^6 + 2580T^4 + 14288*T^2 + 3844
$83$	\( (T^{4} + 8 T^{3} + \cdots - 272)^{2} \) (T^4 + 8T^3 - 120T^2 - 416*T - 272)^2
$89$	\( (T^{4} - 16 T^{3} + \cdots - 2558)^{2} \) (T^4 - 16T^3 - 20T^2 + 896*T - 2558)^2
$97$	\( T^{8} + 176 T^{6} + \cdots + 399424 \) T^8 + 176T^6 + 7504T^4 + 100736*T^2 + 399424
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\(n\)	\(290\)	\(292\)
\(\chi(n)\)	\(1\)	\(-1\)