# Properties

 Label 384.2.c.d Level $384$ Weight $2$ Character orbit 384.c Analytic conductor $3.066$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,2,Mod(383,384)]

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([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("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: f = N[0] # Warning: the index may be different

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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_1) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^3 + (-b3 - b1) * q^5 + (-b3 - b2 - b1) * q^7 + (-b3 + b2 + b1 + 1) * q^9 $$q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_1) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{3} + \beta_1 - 2) q^{11} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{13} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{15} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{17} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{19} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{21} + (2 \beta_{3} - 2 \beta_1 - 4) q^{23} + ( - 2 \beta_{3} + 2 \beta_1 + 1) q^{25} + (\beta_{3} + 2 \beta_{2} + 3) q^{27} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{31} + ( - \beta_{3} + \beta_{2} - 3 \beta_1) q^{33} - 4 q^{35} + (2 \beta_{3} - 2 \beta_1 + 2) q^{37} + ( - 2 \beta_{3} + 2 \beta_{2} + 6) q^{39} + (2 \beta_{3} + 2 \beta_1) q^{41} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{43} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{45} - 8 q^{47} + (2 \beta_{3} - 2 \beta_1 - 1) q^{49} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{51}+ \cdots + (5 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{99}+O(q^{100})$$ 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 + (-2*b3 + 2*b1 + 2) * q^13 + (b3 - b2 - b1 + 2) * q^15 + (-2*b3 - 2*b2 - 2*b1) * q^17 + (b3 + 2*b2 + b1) * q^19 + (-b3 - 2*b2 - b1 + 2) * q^21 + (2*b3 - 2*b1 - 4) * q^23 + (-2*b3 + 2*b1 + 1) * q^25 + (b3 + 2*b2 + 3) * q^27 + (b3 + 4*b2 + b1) * q^29 + (-b3 + b2 - b1) * q^31 + (-b3 + b2 - 3*b1) * q^33 - 4 * q^35 + (2*b3 - 2*b1 + 2) * q^37 + (-2*b3 + 2*b2 + 6) * q^39 + (2*b3 + 2*b1) * q^41 + (b3 - 2*b2 + b1) * q^43 + (-b3 - 2*b2 + 3*b1) * q^45 - 8 * q^47 + (2*b3 - 2*b1 - 1) * q^49 + (-2*b3 - 4*b2 - 2*b1 + 4) * q^51 + (-b3 - b1) * q^53 + (4*b3 - 2*b2 + 4*b1) * q^55 + (3*b3 + 3*b2 + b1 - 2) * q^57 + (b3 - b1 - 2) * q^59 + (2*b3 - 2*b1 - 6) * q^61 + (-3*b3 - 3*b2 + b1 + 4) * q^63 + (2*b3 - 4*b2 + 2*b1) * q^65 + (-3*b3 - 3*b1) * q^67 + (2*b3 - 2*b2 - 2*b1 - 8) * q^69 + (-6*b3 + 6*b1 + 4) * q^71 - 2 * q^73 + (-2*b3 + 2*b2 - b1 + 5) * q^75 + (2*b3 + 2*b1) * q^77 + (3*b3 + 5*b2 + 3*b1) * q^79 + (4*b3 + 2*b2 + 4*b1 + 1) * q^81 + (b3 - b1 - 6) * q^83 - 8 * q^85 + (7*b3 + 5*b2 + b1 - 2) * q^87 - 2*b2 * q^89 + (-2*b3 - 6*b2 - 2*b1) * q^91 + (3*b3 - b1 + 2) * q^93 + (-2*b3 + 2*b1 + 4) * q^95 + (2*b3 - 2*b1 - 6) * q^97 + (5*b3 - 2*b2 - b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3}+O(q^{10})$$ 4 * q + 2 * q^3 $$4 q + 2 q^{3} - 12 q^{11} + 12 q^{15} + 8 q^{21} - 8 q^{23} - 4 q^{25} + 14 q^{27} + 4 q^{33} - 16 q^{35} + 16 q^{37} + 20 q^{39} - 8 q^{45} - 32 q^{47} + 4 q^{49} + 16 q^{51} - 4 q^{57} - 4 q^{59} - 16 q^{61} + 8 q^{63} - 24 q^{69} - 8 q^{71} - 8 q^{73} + 18 q^{75} + 4 q^{81} - 20 q^{83} - 32 q^{85} + 4 q^{87} + 16 q^{93} + 8 q^{95} - 16 q^{97} + 20 q^{99}+O(q^{100})$$ 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 $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu + 1$$ v^2 + v + 1 $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 4\nu$$ 2*v^3 + 4*v $$\beta_{3}$$ $$=$$ $$-\nu^{2} + \nu - 1$$ -v^2 + v - 1
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + \beta _1 - 2 ) / 2$$ (-b3 + b1 - 2) / 2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + \beta_{2} - 2\beta_1 ) / 2$$ (-2*b3 + b2 - 2*b1) / 2

## 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 −0.618034 1.61803i 0 3.23607i 0 1.23607i 0 −2.23607 + 2.00000i 0
383.2 0 −0.618034 + 1.61803i 0 3.23607i 0 1.23607i 0 −2.23607 2.00000i 0
383.3 0 1.61803 0.618034i 0 1.23607i 0 3.23607i 0 2.23607 2.00000i 0
383.4 0 1.61803 + 0.618034i 0 1.23607i 0 3.23607i 0 2.23607 + 2.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.d yes 4
3.b odd 2 1 384.2.c.a 4
4.b odd 2 1 384.2.c.a 4
8.b even 2 1 384.2.c.b yes 4
8.d odd 2 1 384.2.c.c yes 4
12.b even 2 1 inner 384.2.c.d yes 4
16.e even 4 1 768.2.f.b 4
16.e even 4 1 768.2.f.f 4
16.f odd 4 1 768.2.f.c 4
16.f odd 4 1 768.2.f.e 4
24.f even 2 1 384.2.c.b yes 4
24.h odd 2 1 384.2.c.c yes 4
48.i odd 4 1 768.2.f.c 4
48.i odd 4 1 768.2.f.e 4
48.k even 4 1 768.2.f.b 4
48.k even 4 1 768.2.f.f 4

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

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