LMFDB - Newform orbit 1152.1.h.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,1,Mod(449,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1152.449"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1152.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$0.574922894553$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.6912.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + (\zeta_{8}^{3} - \zeta_{8}) q^{5} + 2 \zeta_{8}^{2} q^{13} + (\zeta_{8}^{3} + \zeta_{8}) q^{17} + q^{25} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{29} + (\zeta_{8}^{3} + \zeta_{8}) q^{41} - q^{49} + (\zeta_{8}^{3} - \zeta_{8}) q^{53}+ \cdots + ( - \zeta_{8}^{3} - \zeta_{8}) q^{89}+O(q^{100}) $ q + (z^3 - z) * q^5 + 2z^2 q^13 + (z^3 + z) * q^17 + q^25 + (-z^3 + z) * q^29 + (z^3 + z) * q^41 - q^49 + (z^3 - z) * q^53 + (-2z^3 - 2z) * q^65 - 2z^2 q^85 + (-z^3 - z) * q^89
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{25} - 4 q^{49}+O(q^{100}) $ 4 * q + 4 * q^25 - 4 * q^49

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$-1$	$-1$

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} $

449.1

0.707107	−	0.707107i
0.707107	+	0.707107i
−0.707107	+	0.707107i
−0.707107	−	0.707107i

−1.41421

449.2

−1.41421

449.3

1.41421

449.4

1.41421

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1152.1.h.a	✓	4
3.b	odd	2	1	inner	1152.1.h.a	✓	4
4.b	odd	2	1	CM	1152.1.h.a	✓	4
8.b	even	2	1	inner	1152.1.h.a	✓	4
8.d	odd	2	1	inner	1152.1.h.a	✓	4
12.b	even	2	1	inner	1152.1.h.a	✓	4
16.e	even	4	1		2304.1.e.a		2
16.e	even	4	1		2304.1.e.b		2
16.f	odd	4	1		2304.1.e.a		2
16.f	odd	4	1		2304.1.e.b		2
24.f	even	2	1	inner	1152.1.h.a	✓	4
24.h	odd	2	1	inner	1152.1.h.a	✓	4
48.i	odd	4	1		2304.1.e.a		2
48.i	odd	4	1		2304.1.e.b		2
48.k	even	4	1		2304.1.e.a		2
48.k	even	4	1		2304.1.e.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1152.1.h.a	✓	4	1.a	even	1	1	trivial
1152.1.h.a	✓	4	3.b	odd	2	1	inner
1152.1.h.a	✓	4	4.b	odd	2	1	CM
1152.1.h.a	✓	4	8.b	even	2	1	inner
1152.1.h.a	✓	4	8.d	odd	2	1	inner
1152.1.h.a	✓	4	12.b	even	2	1	inner
1152.1.h.a	✓	4	24.f	even	2	1	inner
1152.1.h.a	✓	4	24.h	odd	2	1	inner
2304.1.e.a		2	16.e	even	4	1
2304.1.e.a		2	16.f	odd	4	1
2304.1.e.a		2	48.i	odd	4	1
2304.1.e.a		2	48.k	even	4	1
2304.1.e.b		2	16.e	even	4	1
2304.1.e.b		2	16.f	odd	4	1
2304.1.e.b		2	48.i	odd	4	1
2304.1.e.b		2	48.k	even	4	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1152, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$7$	$ T^{4} $ T^4
$11$	$ T^{4} $ T^4
$13$	$ (T^{2} + 4)^{2} $ (T^2 + 4)^2
$17$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$19$	$ T^{4} $ T^4
$23$	$ T^{4} $ T^4
$29$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$31$	$ T^{4} $ T^4
$37$	$ T^{4} $ T^4
$41$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$43$	$ T^{4} $ T^4
$47$	$ T^{4} $ T^4
$53$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$59$	$ T^{4} $ T^4
$61$	$ T^{4} $ T^4
$67$	$ T^{4} $ T^4
$71$	$ T^{4} $ T^4
$73$	$ T^{4} $ T^4
$79$	$ T^{4} $ T^4
$83$	$ T^{4} $ T^4
$89$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$97$	$ T^{4} $ T^4
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Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1152.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{8}^{3} - \zeta_{8}) q^{5} + 2 \zeta_{8}^{2} q^{13} + (\zeta_{8}^{3} + \zeta_{8}) q^{17} + q^{25} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{29} + (\zeta_{8}^{3} + \zeta_{8}) q^{41} - q^{49} + (\zeta_{8}^{3} - \zeta_{8}) q^{53}+ \cdots + ( - \zeta_{8}^{3} - \zeta_{8}) q^{89}+O(q^{100}) \) q + (z^3 - z) * q^5 + 2z^2 q^13 + (z^3 + z) * q^17 + q^25 + (-z^3 + z) * q^29 + (z^3 + z) * q^41 - q^49 + (z^3 - z) * q^53 + (-2z^3 - 2z) * q^65 - 2z^2 q^85 + (-z^3 - z) * q^89
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{25} - 4 q^{49}+O(q^{100}) \) 4 * q + 4 * q^25 - 4 * q^49

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