LMFDB - Newform orbit 1584.2.cd.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,2,Mod(17,1584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1584, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 5, 9]))

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

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

chi := DirichletCharacter("1584.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1584.cd (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:	$12.6483036802$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.64000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 198)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-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_{6} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{7} + 2 \beta_{6} - \beta_{4} + \cdots - 1) q^{7}+O(q^{10}) $ q + (-b7 + b6 - b2 + b1) * q^5 + (b7 + 2b6 - b4 + 2b3 + b2 - 2b1 - 1) q^7	$ q + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{7} - 9 \beta_{6} + 2 \beta_{5} + \cdots + 1) q^{97}+O(q^{100}) $ q + (-b7 + b6 - b2 + b1) * q^5 + (b7 + 2b6 - b4 + 2b3 + b2 - 2b1 - 1) q^7 + (b7 - 2b6 - b5 - b3 + 1) q^11 + (-b7 + 2b6 + b5 - 4b4 + 2b2 + b1 - 2) q^13 + (-b7 + 2b6 + 2b2 - b1) * q^17 + (-b5 - 2b4 + b3 - 2b2 - b1) * q^19 + (4b6 + b5 + 4b2 - 2) * q^23 + (4b7 - b6 - 2b5 + 2b3 - 2b1 + 1) * q^25 + (b7 - 2b5 - 3b4 + b3 + 4b2 - b1 - 3) q^29 + (2b7 - 2b6 + 4b5 - 2b3 - 3b2 + b1 + 3) q^31 + (b7 + 5b6 - 3b5 + 4b3 - 2b1) * q^35 + (-2b7 + 4b5 - 3b3 + b1) q^37 + (6b7 - 2b5 - 2b4 + 3b3 + 2b2 - 4b1) * q^41 + (b7 - 4b6 - 4b5 + b3 - 4b2 + 2) q^43 + (2b6 + b5 - 2b4 - 2b2 + b1 + 2) q^47 + (-4b7 - b6 - 2b5 - 5b4 - 2b3 - b2 - 4b1) q^49 + (3b7 - 3b5 - 3b2 + 2b1 - 3) * q^53 + (-4b7 + 3b6 + 3b5 - 4b4 - 5b3 + 3b2 + 4b1 - 6) q^55 + (5b7 - 2b6 + 3b4 + 3b3 - b2 - 3b1 - 1) q^59 + (6b7 + 2b6 - 2b5 + 2b4 + 2b3 - 6b2 - 6b1 + 4) q^61 + (3b7 - 3b5 + 3b3 - 6b1 + 8) * q^65 + (4b7 + 2b6 - b5 - 2b4 - 2b3 - 2b1 + 6) q^67 + (-6b6 - 3b5 + 2b4 + 3b3 + 2b2 + 4) q^71 + (4b7 + 4b6 - b4 - 4b3 + 2b2 + 4b1 - 3) q^73 + (-6b6 + 4b5 + 11b4 - b3 - 3b2 + 5b1 + 5) q^77 + (3b7 + b6 - 3b5 - 2b4 + 3b2 - 8b1 + 1) q^79 + (b7 - 3b6 - 3b5 - 3b3 - 3b2 + b1) * q^83 + (b5 - 4b4 + 5b3 - 4b2 + b1) q^85 + (6b6 + 6b4) * q^89 + (-6b7 + 4b6 + 6b5 - 6b4 - 3b3 + 6b2 - 4) * q^91 + (3b7 - 6b5 + 4b4 + 2b3 - 4b2 - 4b1 + 4) * q^95 + (2b7 - 9b6 + 2b5 - b2 + 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 4 q^{11} + 8 q^{17} + 6 q^{25} - 10 q^{29} + 14 q^{31} + 10 q^{35} + 8 q^{41} + 20 q^{47} + 6 q^{49} - 30 q^{53} - 28 q^{55} - 20 q^{59} + 20 q^{61} + 64 q^{65} + 56 q^{67} + 20 q^{71} - 10 q^{73} + 20 q^{79} - 12 q^{83} + 16 q^{95} - 12 q^{97}+O(q^{100}) $ 8 * q + 4 * q^11 + 8 * q^17 + 6 * q^25 - 10 * q^29 + 14 * q^31 + 10 * q^35 + 8 * q^41 + 20 * q^47 + 6 * q^49 - 30 * q^53 - 28 * q^55 - 20 * q^59 + 20 * q^61 + 64 * q^65 + 56 * q^67 + 20 * q^71 - 10 * q^73 + 20 * q^79 - 12 * q^83 + 16 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2*x^6 + 4*x^4 - 8*x^2 + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{3}$	$=$	$ ( \nu^{3} ) / 2 $ (v^3) / 2
$\beta_{4}$	$=$	$ ( \nu^{4} ) / 4 $ (v^4) / 4
$\beta_{5}$	$=$	$ ( \nu^{5} ) / 4 $ (v^5) / 4
$\beta_{6}$	$=$	$ ( \nu^{6} ) / 8 $ (v^6) / 8
$\beta_{7}$	$=$	$ ( \nu^{7} ) / 8 $ (v^7) / 8

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{2} $ 2*b2
$\nu^{3}$	$=$	$ 2\beta_{3} $ 2*b3
$\nu^{4}$	$=$	$ 4\beta_{4} $ 4*b4
$\nu^{5}$	$=$	$ 4\beta_{5} $ 4*b5
$\nu^{6}$	$=$	$ 8\beta_{6} $ 8*b6
$\nu^{7}$	$=$	$ 8\beta_{7} $ 8*b7

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$\beta_{2}$	$-1$	$1$	$1$

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} $

17.1

−1.34500	+	0.437016i
1.34500	−	0.437016i
0.831254	−	1.14412i
−0.831254	+	1.14412i
0.831254	+	1.14412i
−0.831254	−	1.14412i
−1.34500	−	0.437016i
1.34500	+	0.437016i

−3.29428

−

1.07038i

0.740706

1.01949i

17.2

1.05822

0.343836i

−2.97677

−

4.09718i

161.1

0.604291

0.831735i

−1.88947

0.613926i

161.2

1.63178

2.24595i

4.12554

−

1.34047i

305.1

0.604291

−

0.831735i

−1.88947

−

0.613926i

305.2

1.63178

−

2.24595i

4.12554

1.34047i

1025.1

−3.29428

1.07038i

0.740706

−

1.01949i

1025.2

1.05822

−

0.343836i

−2.97677

4.09718i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
33.f	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1584.2.cd.b		8
3.b	odd	2	1		1584.2.cd.a		8
4.b	odd	2	1		198.2.l.b	yes	8
11.d	odd	10	1		1584.2.cd.a		8
12.b	even	2	1		198.2.l.a	✓	8
33.f	even	10	1	inner	1584.2.cd.b		8
44.g	even	10	1		198.2.l.a	✓	8
44.g	even	10	1		2178.2.b.j		8
44.h	odd	10	1		2178.2.b.i		8
132.n	odd	10	1		198.2.l.b	yes	8
132.n	odd	10	1		2178.2.b.i		8
132.o	even	10	1		2178.2.b.j		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
198.2.l.a	✓	8	12.b	even	2	1
198.2.l.a	✓	8	44.g	even	10	1
198.2.l.b	yes	8	4.b	odd	2	1
198.2.l.b	yes	8	132.n	odd	10	1
1584.2.cd.a		8	3.b	odd	2	1
1584.2.cd.a		8	11.d	odd	10	1
1584.2.cd.b		8	1.a	even	1	1	trivial
1584.2.cd.b		8	33.f	even	10	1	inner
2178.2.b.i		8	44.h	odd	10	1
2178.2.b.i		8	132.n	odd	10	1
2178.2.b.j		8	44.g	even	10	1
2178.2.b.j		8	132.o	even	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 8T_{5}^{6} + 30T_{5}^{5} + 34T_{5}^{4} - 240T_{5}^{3} + 403T_{5}^{2} - 330T_{5} + 121 $ T5^8 - 8*T5^6 + 30*T5^5 + 34*T5^4 - 240*T5^3 + 403*T5^2 - 330*T5 + 121 acting on $S_{2}^{\mathrm{new}}(1584, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - 8 T^{6} + \cdots + 121 $ T^8 - 8T^6 + 30T^5 + 34T^4 - 240T^3 + 403T^2 - 330T + 121
$7$	$ T^{8} - 10 T^{6} + \cdots + 3025 $ T^8 - 10T^6 - 110T^5 + 260T^4 + 1100T^3 - 75T^2 - 550T + 3025
$11$	$ T^{8} - 4 T^{7} + \cdots + 14641 $ T^8 - 4T^7 + 44T^5 - 226T^4 + 484T^3 - 5324*T + 14641
$13$	$ T^{8} - 8 T^{6} + \cdots + 16 $ T^8 - 8T^6 + 120T^5 + 184T^4 - 960T^3 + 3928T^2 + 480T + 16
$17$	$ T^{8} - 8 T^{7} + \cdots + 1936 $ T^8 - 8T^7 + 48T^6 - 136T^5 + 200T^4 + 224T^3 + 408T^2 + 352*T + 1936
$19$	$ T^{8} - 8 T^{6} + \cdots + 16 $ T^8 - 8T^6 - 120T^5 + 184T^4 + 960T^3 + 3928T^2 - 480T + 16
$23$	$ T^{8} + 88 T^{6} + \cdots + 16 $ T^8 + 88T^6 + 1944T^4 + 10592*T^2 + 16
$29$	$ T^{8} + 10 T^{7} + \cdots + 3025 $ T^8 + 10T^7 + 115T^6 + 460T^5 + 1390T^4 + 1850T^3 + 500T^2 - 3300*T + 3025
$31$	$ T^{8} - 14 T^{7} + \cdots + 1771561 $ T^8 - 14T^7 + 147T^6 - 352T^5 + 29282T^3 + 322102T^2 + 644204T + 1771561
$37$	$ T^{8} + 80 T^{6} + \cdots + 400 $ T^8 + 80T^6 + 2440T^4 - 600*T^2 + 400
$41$	$ T^{8} - 8 T^{7} + \cdots + 234256 $ T^8 - 8T^7 - 2T^6 + 424T^5 + 5580T^4 - 21296T^3 + 281688T^2 - 170368*T + 234256
$43$	$ T^{8} + 188 T^{6} + \cdots + 1157776 $ T^8 + 188T^6 + 11164T^4 + 224912*T^2 + 1157776
$47$	$ T^{8} - 20 T^{7} + \cdots + 13456 $ T^8 - 20T^7 + 142T^6 - 400T^5 + 864T^4 - 6800T^3 + 25648T^2 - 23200*T + 13456
$53$	$ T^{8} + 30 T^{7} + \cdots + 7027801 $ T^8 + 30T^7 + 463T^6 + 3720T^5 + 17194T^4 + 100110T^3 + 843952T^2 + 3976500*T + 7027801
$59$	$ T^{8} + 20 T^{7} + \cdots + 5387041 $ T^8 + 20T^7 + 218T^6 + 1630T^5 + 16204T^4 + 180940T^3 + 1213607T^2 + 4061750*T + 5387041
$61$	$ T^{8} - 20 T^{7} + \cdots + 198697216 $ T^8 - 20T^7 + 28T^6 + 3680T^5 - 42816T^4 - 80960T^3 + 5748352T^2 - 51873280*T + 198697216
$67$	$ (T^{4} - 28 T^{3} + \cdots - 1604)^{2} $ (T^4 - 28T^3 + 184T^2 + 248*T - 1604)^2
$71$	$ T^{8} - 20 T^{7} + \cdots + 1628176 $ T^8 - 20T^7 + 118T^6 - 1440T^5 + 34744T^4 - 314320T^3 + 1186272T^2 - 1837440*T + 1628176
$73$	$ T^{8} + 10 T^{7} + \cdots + 19456921 $ T^8 + 10T^7 - 287T^6 - 3680T^5 + 21714T^4 + 510610T^3 + 4110552T^2 + 11468600*T + 19456921
$79$	$ T^{8} - 20 T^{7} + \cdots + 101761 $ T^8 - 20T^7 - 102T^6 + 3910T^5 - 4036T^4 - 218020T^3 + 2606447T^2 + 602910*T + 101761
$83$	$ T^{8} + 12 T^{7} + \cdots + 942841 $ T^8 + 12T^7 + 138T^6 + 174T^5 - 1760T^4 - 18336T^3 + 406253T^2 - 425298*T + 942841
$89$	$ (T^{4} + 180 T^{2} + 6480)^{2} $ (T^4 + 180*T^2 + 6480)^2
$97$	$ T^{8} + 12 T^{7} + \cdots + 7623121 $ T^8 + 12T^7 + 228T^6 + 3754T^5 + 40950T^4 + 166324T^3 + 709113T^2 + 2766522*T + 7623121
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1584.cd (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{7} + 2 \beta_{6} - \beta_{4} + \cdots - 1) q^{7}+O(q^{10}) \) q + (-b7 + b6 - b2 + b1) * q^5 + (b7 + 2b6 - b4 + 2b3 + b2 - 2b1 - 1) q^7	\( q + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{7} - 9 \beta_{6} + 2 \beta_{5} + \cdots + 1) q^{97}+O(q^{100}) \) q + (-b7 + b6 - b2 + b1) * q^5 + (b7 + 2b6 - b4 + 2b3 + b2 - 2b1 - 1) q^7 + (b7 - 2b6 - b5 - b3 + 1) q^11 + (-b7 + 2b6 + b5 - 4b4 + 2b2 + b1 - 2) q^13 + (-b7 + 2b6 + 2b2 - b1) * q^17 + (-b5 - 2b4 + b3 - 2b2 - b1) * q^19 + (4b6 + b5 + 4b2 - 2) * q^23 + (4b7 - b6 - 2b5 + 2b3 - 2b1 + 1) * q^25 + (b7 - 2b5 - 3b4 + b3 + 4b2 - b1 - 3) q^29 + (2b7 - 2b6 + 4b5 - 2b3 - 3b2 + b1 + 3) q^31 + (b7 + 5b6 - 3b5 + 4b3 - 2b1) * q^35 + (-2b7 + 4b5 - 3b3 + b1) q^37 + (6b7 - 2b5 - 2b4 + 3b3 + 2b2 - 4b1) * q^41 + (b7 - 4b6 - 4b5 + b3 - 4b2 + 2) q^43 + (2b6 + b5 - 2b4 - 2b2 + b1 + 2) q^47 + (-4b7 - b6 - 2b5 - 5b4 - 2b3 - b2 - 4b1) q^49 + (3b7 - 3b5 - 3b2 + 2b1 - 3) * q^53 + (-4b7 + 3b6 + 3b5 - 4b4 - 5b3 + 3b2 + 4b1 - 6) q^55 + (5b7 - 2b6 + 3b4 + 3b3 - b2 - 3b1 - 1) q^59 + (6b7 + 2b6 - 2b5 + 2b4 + 2b3 - 6b2 - 6b1 + 4) q^61 + (3b7 - 3b5 + 3b3 - 6b1 + 8) * q^65 + (4b7 + 2b6 - b5 - 2b4 - 2b3 - 2b1 + 6) q^67 + (-6b6 - 3b5 + 2b4 + 3b3 + 2b2 + 4) q^71 + (4b7 + 4b6 - b4 - 4b3 + 2b2 + 4b1 - 3) q^73 + (-6b6 + 4b5 + 11b4 - b3 - 3b2 + 5b1 + 5) q^77 + (3b7 + b6 - 3b5 - 2b4 + 3b2 - 8b1 + 1) q^79 + (b7 - 3b6 - 3b5 - 3b3 - 3b2 + b1) * q^83 + (b5 - 4b4 + 5b3 - 4b2 + b1) q^85 + (6b6 + 6b4) * q^89 + (-6b7 + 4b6 + 6b5 - 6b4 - 3b3 + 6b2 - 4) * q^91 + (3b7 - 6b5 + 4b4 + 2b3 - 4b2 - 4b1 + 4) * q^95 + (2b7 - 9b6 + 2b5 - b2 + 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 4 q^{11} + 8 q^{17} + 6 q^{25} - 10 q^{29} + 14 q^{31} + 10 q^{35} + 8 q^{41} + 20 q^{47} + 6 q^{49} - 30 q^{53} - 28 q^{55} - 20 q^{59} + 20 q^{61} + 64 q^{65} + 56 q^{67} + 20 q^{71} - 10 q^{73} + 20 q^{79} - 12 q^{83} + 16 q^{95} - 12 q^{97}+O(q^{100}) \) 8 * q + 4 * q^11 + 8 * q^17 + 6 * q^25 - 10 * q^29 + 14 * q^31 + 10 * q^35 + 8 * q^41 + 20 * q^47 + 6 * q^49 - 30 * q^53 - 28 * q^55 - 20 * q^59 + 20 * q^61 + 64 * q^65 + 56 * q^67 + 20 * q^71 - 10 * q^73 + 20 * q^79 - 12 * q^83 + 16 * q^95 - 12 * q^97

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