LMFDB - Newform orbit 1472.2.c.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,2,Mod(1471,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1472.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1472.c (of order $2$, degree $1$, 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:	$11.7539791775$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8869743.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 92)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{3} + ( - \beta_{2} - 3) q^{9} + ( - \beta_{4} - \beta_{2}) q^{13} + \beta_{3} q^{23} + 5 q^{25} + (2 \beta_{3} + 3 \beta_1) q^{27} + (\beta_{4} - \beta_{2}) q^{29} - \beta_{5} q^{31} + (\beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{39}+ \cdots + ( - 3 \beta_{4} - \beta_{2} + 2) q^{93}+O(q^{100}) $ q - b1 * q^3 + (-b2 - 3) * q^9 + (-b4 - b2) * q^13 + b3 * q^23 + 5 * q^25 + (2b3 + 3b1) * q^27 + (b4 - b2) * q^29 - b5 * q^31 + (b5 + 2b3 + 2b1) * q^39 + (-2b4 - b2) q^41 + (b5 - 2b1) q^47 - 7 * q^49 + 2b3 q^59 + (b4 + 3b2) q^69 + (-b5 - 4b1) q^71 + (2b4 - b2) q^73 - 5b1 q^75 + (2b4 + 6b2 + 9) * q^81 + (-b5 + 2b3 + 4b1) * q^87 + (-3b4 - b2 + 2) q^93
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 18 q^{9} + 30 q^{25} - 42 q^{49} + 54 q^{81} + 12 q^{93}+O(q^{100}) $ 6 * q - 18 * q^9 + 30 * q^25 - 42 * q^49 + 54 * q^81 + 12 * q^93

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8 :

$\beta_{1}$	$=$	$ ( \nu^{5} + 2\nu^{4} + \nu^{2} - 2\nu ) / 4 $ (v^5 + 2v^4 + v^2 - 2v) / 4
$\beta_{2}$	$=$	$ ( -\nu^{5} - 2\nu^{4} + 7\nu^{2} + 10\nu ) / 4 $ (-v^5 - 2v^4 + 7v^2 + 10*v) / 4
$\beta_{3}$	$=$	$ 2\nu^{3} - 3 $ 2*v^3 - 3
$\beta_{4}$	$=$	$ ( -\nu^{5} + 2\nu^{4} - \nu^{2} - 2\nu ) / 2 $ (-v^5 + 2v^4 - v^2 - 2v) / 2
$\beta_{5}$	$=$	$ ( 3\nu^{5} - 2\nu^{4} - 13\nu^{2} + 18\nu ) / 4 $ (3v^5 - 2v^4 - 13v^2 + 18v) / 4

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

$\nu$	$=$	$ ( \beta_{5} + \beta_{4} + 2\beta_{2} + \beta_1 ) / 8 $ (b5 + b4 + 2*b2 + b1) / 8
$\nu^{2}$	$=$	$ ( -\beta_{5} - \beta_{4} + 2\beta_{2} + 3\beta_1 ) / 8 $ (-b5 - b4 + 2b2 + 3b1) / 8
$\nu^{3}$	$=$	$ ( \beta_{3} + 3 ) / 2 $ (b3 + 3) / 2
$\nu^{4}$	$=$	$ ( \beta_{5} + 5\beta_{4} + 2\beta_{2} + 9\beta_1 ) / 8 $ (b5 + 5b4 + 2b2 + 9*b1) / 8
$\nu^{5}$	$=$	$ ( \beta_{5} - 7\beta_{4} - 2\beta_{2} + 13\beta_1 ) / 8 $ (b5 - 7b4 - 2b2 + 13*b1) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times$.

$n$	$645$	$833$	$1151$
$\chi(n)$	$1$	$-1$	$-1$

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} $

1471.1

1.33454	+	0.467979i
−1.07255	−	0.921756i
−0.261988	+	1.38973i
−0.261988	−	1.38973i
−1.07255	+	0.921756i
1.33454	−	0.467979i

−

3.43410i

−8.79306

1471.2

−

2.11101i

−1.45636

1471.3

−

1.32309i

1.24943

1471.4

1.32309i

1.24943

1471.5

2.11101i

−1.45636

1471.6

3.43410i

−8.79306

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.b	odd	2	1	CM by $\Q(\sqrt{-23}) $
4.b	odd	2	1	inner
92.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1472.2.c.c		6
4.b	odd	2	1	inner	1472.2.c.c		6
8.b	even	2	1		92.2.b.b	✓	6
8.d	odd	2	1		92.2.b.b	✓	6
23.b	odd	2	1	CM	1472.2.c.c		6
24.f	even	2	1		828.2.e.b		6
24.h	odd	2	1		828.2.e.b		6
92.b	even	2	1	inner	1472.2.c.c		6
184.e	odd	2	1		92.2.b.b	✓	6
184.h	even	2	1		92.2.b.b	✓	6
552.b	even	2	1		828.2.e.b		6
552.h	odd	2	1		828.2.e.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
92.2.b.b	✓	6	8.b	even	2	1
92.2.b.b	✓	6	8.d	odd	2	1
92.2.b.b	✓	6	184.e	odd	2	1
92.2.b.b	✓	6	184.h	even	2	1
828.2.e.b		6	24.f	even	2	1
828.2.e.b		6	24.h	odd	2	1
828.2.e.b		6	552.b	even	2	1
828.2.e.b		6	552.h	odd	2	1
1472.2.c.c		6	1.a	even	1	1	trivial
1472.2.c.c		6	4.b	odd	2	1	inner
1472.2.c.c		6	23.b	odd	2	1	CM
1472.2.c.c		6	92.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{6} + 18T_{3}^{4} + 81T_{3}^{2} + 92 $ T3^6 + 18*T3^4 + 81*T3^2 + 92 acting on $S_{2}^{\mathrm{new}}(1472, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + 18 T^{4} + \cdots + 92 $ T^6 + 18T^4 + 81T^2 + 92
$5$	$ T^{6} $ T^6
$7$	$ T^{6} $ T^6
$11$	$ T^{6} $ T^6
$13$	$ (T^{3} - 39 T - 74)^{2} $ (T^3 - 39*T - 74)^2
$17$	$ T^{6} $ T^6
$19$	$ T^{6} $ T^6
$23$	$ (T^{2} + 23)^{3} $ (T^2 + 23)^3
$29$	$ (T^{3} - 87 T - 282)^{2} $ (T^3 - 87*T - 282)^2
$31$	$ T^{6} + 186 T^{4} + \cdots + 828 $ T^6 + 186T^4 + 8649T^2 + 828
$37$	$ T^{6} $ T^6
$41$	$ (T^{3} - 123 T - 426)^{2} $ (T^3 - 123*T - 426)^2
$43$	$ T^{6} $ T^6
$47$	$ T^{6} + 282 T^{4} + \cdots + 412988 $ T^6 + 282T^4 + 19881T^2 + 412988
$53$	$ T^{6} $ T^6
$59$	$ (T^{2} + 92)^{3} $ (T^2 + 92)^3
$61$	$ T^{6} $ T^6
$67$	$ T^{6} $ T^6
$71$	$ T^{6} + 426 T^{4} + \cdots + 48668 $ T^6 + 426T^4 + 45369T^2 + 48668
$73$	$ (T^{3} - 219 T - 1226)^{2} $ (T^3 - 219*T - 1226)^2
$79$	$ T^{6} $ T^6
$83$	$ T^{6} $ T^6
$89$	$ T^{6} $ T^6
$97$	$ T^{6} $ T^6
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Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1472.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} + ( - \beta_{2} - 3) q^{9} + ( - \beta_{4} - \beta_{2}) q^{13} + \beta_{3} q^{23} + 5 q^{25} + (2 \beta_{3} + 3 \beta_1) q^{27} + (\beta_{4} - \beta_{2}) q^{29} - \beta_{5} q^{31} + (\beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{39}+ \cdots + ( - 3 \beta_{4} - \beta_{2} + 2) q^{93}+O(q^{100}) \) q - b1 * q^3 + (-b2 - 3) * q^9 + (-b4 - b2) * q^13 + b3 * q^23 + 5 * q^25 + (2b3 + 3b1) * q^27 + (b4 - b2) * q^29 - b5 * q^31 + (b5 + 2b3 + 2b1) * q^39 + (-2b4 - b2) q^41 + (b5 - 2b1) q^47 - 7 * q^49 + 2b3 q^59 + (b4 + 3b2) q^69 + (-b5 - 4b1) q^71 + (2b4 - b2) q^73 - 5b1 q^75 + (2b4 + 6b2 + 9) * q^81 + (-b5 + 2b3 + 4b1) * q^87 + (-3b4 - b2 + 2) q^93
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 18 q^{9} + 30 q^{25} - 42 q^{49} + 54 q^{81} + 12 q^{93}+O(q^{100}) \) 6 * q - 18 * q^9 + 30 * q^25 - 42 * q^49 + 54 * q^81 + 12 * q^93

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