# Properties

 Label 384.2.a.f Level $384$ Weight $2$ Character orbit 384.a Self dual yes Analytic conductor $3.066$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,2,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.a (trivial)

## 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: yes Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 2 * q^7 + q^9 $$q + q^{3} - 2 q^{7} + q^{9} + 4 q^{11} + 6 q^{13} + 6 q^{17} - 2 q^{21} - 4 q^{23} - 5 q^{25} + q^{27} + 4 q^{29} - 10 q^{31} + 4 q^{33} + 2 q^{37} + 6 q^{39} - 2 q^{41} - 8 q^{43} + 12 q^{47} - 3 q^{49} + 6 q^{51} - 12 q^{53} + 4 q^{59} + 2 q^{61} - 2 q^{63} - 4 q^{67} - 4 q^{69} + 4 q^{71} - 10 q^{73} - 5 q^{75} - 8 q^{77} + 6 q^{79} + q^{81} - 12 q^{83} + 4 q^{87} + 2 q^{89} - 12 q^{91} - 10 q^{93} - 6 q^{97} + 4 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^7 + q^9 + 4 * q^11 + 6 * q^13 + 6 * q^17 - 2 * q^21 - 4 * q^23 - 5 * q^25 + q^27 + 4 * q^29 - 10 * q^31 + 4 * q^33 + 2 * q^37 + 6 * q^39 - 2 * q^41 - 8 * q^43 + 12 * q^47 - 3 * q^49 + 6 * q^51 - 12 * q^53 + 4 * q^59 + 2 * q^61 - 2 * q^63 - 4 * q^67 - 4 * q^69 + 4 * q^71 - 10 * q^73 - 5 * q^75 - 8 * q^77 + 6 * q^79 + q^81 - 12 * q^83 + 4 * q^87 + 2 * q^89 - 12 * q^91 - 10 * q^93 - 6 * q^97 + 4 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 1.00000 0 0 0 −2.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.a.f yes 1
3.b odd 2 1 1152.2.a.i 1
4.b odd 2 1 384.2.a.c yes 1
5.b even 2 1 9600.2.a.w 1
8.b even 2 1 384.2.a.b 1
8.d odd 2 1 384.2.a.g yes 1
12.b even 2 1 1152.2.a.l 1
16.e even 4 2 768.2.d.g 2
16.f odd 4 2 768.2.d.b 2
20.d odd 2 1 9600.2.a.bh 1
24.f even 2 1 1152.2.a.k 1
24.h odd 2 1 1152.2.a.j 1
40.e odd 2 1 9600.2.a.h 1
40.f even 2 1 9600.2.a.bw 1
48.i odd 4 2 2304.2.d.m 2
48.k even 4 2 2304.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.b 1 8.b even 2 1
384.2.a.c yes 1 4.b odd 2 1
384.2.a.f yes 1 1.a even 1 1 trivial
384.2.a.g yes 1 8.d odd 2 1
768.2.d.b 2 16.f odd 4 2
768.2.d.g 2 16.e even 4 2
1152.2.a.i 1 3.b odd 2 1
1152.2.a.j 1 24.h odd 2 1
1152.2.a.k 1 24.f even 2 1
1152.2.a.l 1 12.b even 2 1
2304.2.d.d 2 48.k even 4 2
2304.2.d.m 2 48.i odd 4 2
9600.2.a.h 1 40.e odd 2 1
9600.2.a.w 1 5.b even 2 1
9600.2.a.bh 1 20.d odd 2 1
9600.2.a.bw 1 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(384))$$:

 $$T_{5}$$ T5 $$T_{7} + 2$$ T7 + 2 $$T_{11} - 4$$ T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T - 4$$
$13$ $$T - 6$$
$17$ $$T - 6$$
$19$ $$T$$
$23$ $$T + 4$$
$29$ $$T - 4$$
$31$ $$T + 10$$
$37$ $$T - 2$$
$41$ $$T + 2$$
$43$ $$T + 8$$
$47$ $$T - 12$$
$53$ $$T + 12$$
$59$ $$T - 4$$
$61$ $$T - 2$$
$67$ $$T + 4$$
$71$ $$T - 4$$
$73$ $$T + 10$$
$79$ $$T - 6$$
$83$ $$T + 12$$
$89$ $$T - 2$$
$97$ $$T + 6$$