LMFDB - Newform orbit 384.2.c.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,2,Mod(383,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 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("384.383"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$384 = 2^{7} \cdot 3$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	384.c (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,0,0,0,0,0,0,-12,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$3.06625543762$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 3x^{2} + 1$ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_1 + 1) q^{3} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{3} + \beta_1 - 2) q^{11} + (2 \beta_{3} - 2 \beta_1 - 2) q^{13}+ \cdots + (5 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{99}+O(q^{100})$ q + (b1 + 1) * q^3 + (b3 + b1) * q^5 + (b3 + b2 + b1) * q^7 + (-b3 + b2 + b1 + 1) * q^9 + (-b3 + b1 - 2) * q^11 + (2b3 - 2b1 - 2) * q^13 + (-b3 + b2 + b1 - 2) * q^15 + (-2b3 - 2b2 - 2b1) q^17 + (b3 + 2b2 + b1) q^19 + (b3 + 2b2 + b1 - 2) q^21 + (-2b3 + 2b1 + 4) * q^23 + (-2b3 + 2b1 + 1) * q^25 + (b3 + 2b2 + 3) q^27 + (-b3 - 4b2 - b1) q^29 + (b3 - b2 + b1) * q^31 + (-b3 + b2 - 3b1) q^33 - 4 * q^35 + (-2b3 + 2b1 - 2) * q^37 + (2b3 - 2b2 - 6) * q^39 + (2b3 + 2b1) * q^41 + (b3 - 2b2 + b1) q^43 + (b3 + 2b2 - 3b1) * q^45 + 8 * q^47 + (2b3 - 2b1 - 1) * q^49 + (-2b3 - 4b2 - 2b1 + 4) q^51 + (b3 + b1) * q^53 + (-4b3 + 2b2 - 4b1) q^55 + (3b3 + 3b2 + b1 - 2) * q^57 + (b3 - b1 - 2) * q^59 + (-2b3 + 2b1 + 6) * q^61 + (3b3 + 3b2 - b1 - 4) * q^63 + (2b3 - 4b2 + 2b1) q^65 + (-3b3 - 3b1) * q^67 + (-2b3 + 2b2 + 2b1 + 8) q^69 + (6b3 - 6b1 - 4) * q^71 - 2 * q^73 + (-2b3 + 2b2 - b1 + 5) * q^75 + (-2b3 - 2b1) * q^77 + (-3b3 - 5b2 - 3b1) q^79 + (4b3 + 2b2 + 4b1 + 1) q^81 + (b3 - b1 - 6) * q^83 + 8 * q^85 + (-7b3 - 5b2 - b1 + 2) * q^87 - 2b2 q^89 + (-2b3 - 6b2 - 2b1) q^91 + (-3b3 + b1 - 2) q^93 + (2b3 - 2b1 - 4) * q^95 + (2b3 - 2b1 - 6) * q^97 + (5b3 - 2b2 - b1 + 2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{3} - 12 q^{11} - 12 q^{15} - 8 q^{21} + 8 q^{23} - 4 q^{25} + 14 q^{27} + 4 q^{33} - 16 q^{35} - 16 q^{37} - 20 q^{39} + 8 q^{45} + 32 q^{47} + 4 q^{49} + 16 q^{51} - 4 q^{57} - 4 q^{59} + 16 q^{61}+ \cdots + 20 q^{99}+O(q^{100})$ 4 * q + 2 * q^3 - 12 * q^11 - 12 * q^15 - 8 * q^21 + 8 * q^23 - 4 * q^25 + 14 * q^27 + 4 * q^33 - 16 * q^35 - 16 * q^37 - 20 * q^39 + 8 * q^45 + 32 * q^47 + 4 * q^49 + 16 * q^51 - 4 * q^57 - 4 * q^59 + 16 * q^61 - 8 * q^63 + 24 * q^69 + 8 * q^71 - 8 * q^73 + 18 * q^75 + 4 * q^81 - 20 * q^83 + 32 * q^85 - 4 * q^87 - 16 * q^93 - 8 * q^95 - 16 * q^97 + 20 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} + 3x^{2} + 1$ x^4 + 3*x^2 + 1 :

$\beta_{1}$	$=$	$\nu^{2} + \nu + 1$ v^2 + v + 1
$\beta_{2}$	$=$	$2\nu^{3} + 4\nu$ 2v^3 + 4v
$\beta_{3}$	$=$	$-\nu^{2} + \nu - 1$ -v^2 + v - 1

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

$\nu$	$=$	$( \beta_{3} + \beta_1 ) / 2$ (b3 + b1) / 2
$\nu^{2}$	$=$	$( -\beta_{3} + \beta _1 - 2 ) / 2$ (-b3 + b1 - 2) / 2
$\nu^{3}$	$=$	$( -2\beta_{3} + \beta_{2} - 2\beta_1 ) / 2$ (-2b3 + b2 - 2b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$ .

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$-1$

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}

383.1

	−	1.61803i
		1.61803i
	−	0.618034i
		0.618034i

−0.618034

−

1.61803i

−

3.23607i

−

1.23607i

−2.23607

2.00000i

383.2

−0.618034

1.61803i

3.23607i

1.23607i

−2.23607

−

2.00000i

383.3

1.61803

−

0.618034i

−

1.23607i

−

3.23607i

2.23607

−

2.00000i

383.4

1.61803

0.618034i

1.23607i

3.23607i

2.23607

2.00000i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	384.2.c.c	yes	4
3.b	odd	2	1		384.2.c.b	yes	4
4.b	odd	2	1		384.2.c.b	yes	4
8.b	even	2	1		384.2.c.a	✓	4
8.d	odd	2	1		384.2.c.d	yes	4
12.b	even	2	1	inner	384.2.c.c	yes	4
16.e	even	4	1		768.2.f.c		4
16.e	even	4	1		768.2.f.e		4
16.f	odd	4	1		768.2.f.b		4
16.f	odd	4	1		768.2.f.f		4
24.f	even	2	1		384.2.c.a	✓	4
24.h	odd	2	1		384.2.c.d	yes	4
48.i	odd	4	1		768.2.f.b		4
48.i	odd	4	1		768.2.f.f		4
48.k	even	4	1		768.2.f.c		4
48.k	even	4	1		768.2.f.e		4

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(384, [\chi])$ :

T_{11}^{2} + 6T_{11} + 4

T11^2 + 6*T11 + 4

T_{23}^{2} - 4T_{23} - 16

T23^2 - 4*T23 - 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} - 2 T^{3} + \cdots + 9$ T^4 - 2T^3 + 2T^2 - 6*T + 9
$5$	$T^{4} + 12T^{2} + 16$ T^4 + 12*T^2 + 16
$7$	$T^{4} + 12T^{2} + 16$ T^4 + 12*T^2 + 16
$11$	$(T^{2} + 6 T + 4)^{2}$ (T^2 + 6*T + 4)^2
$13$	$(T^{2} - 20)^{2}$ (T^2 - 20)^2
$17$	$T^{4} + 48T^{2} + 256$ T^4 + 48*T^2 + 256
$19$	$T^{4} + 28T^{2} + 16$ T^4 + 28*T^2 + 16
$23$	$(T^{2} - 4 T - 16)^{2}$ (T^2 - 4*T - 16)^2
$29$	$T^{4} + 108T^{2} + 1936$ T^4 + 108*T^2 + 1936
$31$	$T^{4} + 28T^{2} + 16$ T^4 + 28*T^2 + 16
$37$	$(T^{2} + 8 T - 4)^{2}$ (T^2 + 8*T - 4)^2
$41$	$T^{4} + 48T^{2} + 256$ T^4 + 48*T^2 + 256
$43$	$T^{4} + 60T^{2} + 400$ T^4 + 60*T^2 + 400
$47$	$(T - 8)^{4}$ (T - 8)^4
$53$	$T^{4} + 12T^{2} + 16$ T^4 + 12*T^2 + 16
$59$	$(T^{2} + 2 T - 4)^{2}$ (T^2 + 2*T - 4)^2
$61$	$(T^{2} - 8 T - 4)^{2}$ (T^2 - 8*T - 4)^2
$67$	$T^{4} + 108T^{2} + 1296$ T^4 + 108*T^2 + 1296
$71$	$(T^{2} - 4 T - 176)^{2}$ (T^2 - 4*T - 176)^2
$73$	$(T + 2)^{4}$ (T + 2)^4
$79$	$T^{4} + 188T^{2} + 16$ T^4 + 188*T^2 + 16
$83$	$(T^{2} + 10 T + 20)^{2}$ (T^2 + 10*T + 20)^2
$89$	$(T^{2} + 16)^{2}$ (T^2 + 16)^2
$97$	$(T^{2} + 8 T - 4)^{2}$ (T^2 + 8*T - 4)^2
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Newspace parameters

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Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	384.2.c.c

Level	$384$
Weight	$2$
Character orbit	384.c
Analytic conductor	$3.066$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$