LMFDB - Newform orbit 3024.2.b.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3024,2,Mod(1567,3024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 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("3024.1567"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-14,0,0,0,0,0,10,0,0,0, 0,0,8,0,0,0,0,0,-22,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(47)] == traces)

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

Self dual:	no
Analytic conductor:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
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 $\beta = \frac{1}{2}(1 + 3\sqrt{-3})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta - 1) q^{7} + ( - 2 \beta + 1) q^{13} - 7 q^{19} + 5 q^{25} + 4 q^{31} - 11 q^{37} + ( - 4 \beta + 2) q^{43} + ( - \beta - 6) q^{49} + ( - 6 \beta + 3) q^{61} + (6 \beta - 3) q^{67} + ( - 6 \beta + 3) q^{73}+ \cdots + ( - 2 \beta + 1) q^{97}+O(q^{100}) $ q + (b - 1) * q^7 + (-2b + 1) q^13 - 7 * q^19 + 5 * q^25 + 4 * q^31 - 11 * q^37 + (-4b + 2) q^43 + (-b - 6) * q^49 + (-6b + 3) q^61 + (6b - 3) q^67 + (-6b + 3) q^73 + (-2b + 1) q^79 + (b + 13) * q^91 + (-2b + 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{7} - 14 q^{19} + 10 q^{25} + 8 q^{31} - 22 q^{37} - 13 q^{49} + 27 q^{91}+O(q^{100}) $ 2 * q - q^7 - 14 * q^19 + 10 * q^25 + 8 * q^31 - 22 * q^37 - 13 * q^49 + 27 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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} $

1567.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

−

2.59808i

1567.2

−0.500000

2.59808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
28.d	even	2	1	inner
84.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3024.2.b.f	✓	2
3.b	odd	2	1	CM	3024.2.b.f	✓	2
4.b	odd	2	1		3024.2.b.k	yes	2
7.b	odd	2	1		3024.2.b.k	yes	2
12.b	even	2	1		3024.2.b.k	yes	2
21.c	even	2	1		3024.2.b.k	yes	2
28.d	even	2	1	inner	3024.2.b.f	✓	2
84.h	odd	2	1	inner	3024.2.b.f	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3024.2.b.f	✓	2	1.a	even	1	1	trivial
3024.2.b.f	✓	2	3.b	odd	2	1	CM
3024.2.b.f	✓	2	28.d	even	2	1	inner
3024.2.b.f	✓	2	84.h	odd	2	1	inner
3024.2.b.k	yes	2	4.b	odd	2	1
3024.2.b.k	yes	2	7.b	odd	2	1
3024.2.b.k	yes	2	12.b	even	2	1
3024.2.b.k	yes	2	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3024, [\chi])$:

$ T_{5} $

T5

$ T_{11} $

T11

$ T_{13}^{2} + 27 $

T13^2 + 27

$ T_{17} $

T17

$ T_{19} + 7 $

T19 + 7

$ T_{29} $

T29

$ T_{47} $

T47

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + T + 7 $ T^2 + T + 7
$11$	$ T^{2} $ T^2
$13$	$ T^{2} + 27 $ T^2 + 27
$17$	$ T^{2} $ T^2
$19$	$ (T + 7)^{2} $ (T + 7)^2
$23$	$ T^{2} $ T^2
$29$	$ T^{2} $ T^2
$31$	$ (T - 4)^{2} $ (T - 4)^2
$37$	$ (T + 11)^{2} $ (T + 11)^2
$41$	$ T^{2} $ T^2
$43$	$ T^{2} + 108 $ T^2 + 108
$47$	$ T^{2} $ T^2
$53$	$ T^{2} $ T^2
$59$	$ T^{2} $ T^2
$61$	$ T^{2} + 243 $ T^2 + 243
$67$	$ T^{2} + 243 $ T^2 + 243
$71$	$ T^{2} $ T^2
$73$	$ T^{2} + 243 $ T^2 + 243
$79$	$ T^{2} + 27 $ T^2 + 27
$83$	$ T^{2} $ T^2
$89$	$ T^{2} $ T^2
$97$	$ T^{2} + 27 $ T^2 + 27
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Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{7} + ( - 2 \beta + 1) q^{13} - 7 q^{19} + 5 q^{25} + 4 q^{31} - 11 q^{37} + ( - 4 \beta + 2) q^{43} + ( - \beta - 6) q^{49} + ( - 6 \beta + 3) q^{61} + (6 \beta - 3) q^{67} + ( - 6 \beta + 3) q^{73}+ \cdots + ( - 2 \beta + 1) q^{97}+O(q^{100}) \) q + (b - 1) * q^7 + (-2b + 1) q^13 - 7 * q^19 + 5 * q^25 + 4 * q^31 - 11 * q^37 + (-4b + 2) q^43 + (-b - 6) * q^49 + (-6b + 3) q^61 + (6b - 3) q^67 + (-6b + 3) q^73 + (-2b + 1) q^79 + (b + 13) * q^91 + (-2b + 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{7} - 14 q^{19} + 10 q^{25} + 8 q^{31} - 22 q^{37} - 13 q^{49} + 27 q^{91}+O(q^{100}) \) 2 * q - q^7 - 14 * q^19 + 10 * q^25 + 8 * q^31 - 22 * q^37 - 13 * q^49 + 27 * q^91

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