Properties

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

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,2,Mod(193,384)]

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, 1, 0]))

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

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 4 q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + 4 * q^7 - q^9 $$q + i q^{3} + 4 q^{7} - q^{9} + 4 i q^{11} - 4 i q^{13} - 2 q^{17} + 4 i q^{19} + 4 i q^{21} + 8 q^{23} + 5 q^{25} - i q^{27} + 8 i q^{29} - 4 q^{31} - 4 q^{33} - 4 i q^{37} + 4 q^{39} - 6 q^{41} + 4 i q^{43} - 8 q^{47} + 9 q^{49} - 2 i q^{51} - 8 i q^{53} - 4 q^{57} - 12 i q^{59} - 12 i q^{61} - 4 q^{63} - 12 i q^{67} + 8 i q^{69} - 8 q^{71} + 6 q^{73} + 5 i q^{75} + 16 i q^{77} - 4 q^{79} + q^{81} + 4 i q^{83} - 8 q^{87} + 6 q^{89} - 16 i q^{91} - 4 i q^{93} - 2 q^{97} - 4 i q^{99} +O(q^{100})$$ q + i * q^3 + 4 * q^7 - q^9 + 4*i * q^11 - 4*i * q^13 - 2 * q^17 + 4*i * q^19 + 4*i * q^21 + 8 * q^23 + 5 * q^25 - i * q^27 + 8*i * q^29 - 4 * q^31 - 4 * q^33 - 4*i * q^37 + 4 * q^39 - 6 * q^41 + 4*i * q^43 - 8 * q^47 + 9 * q^49 - 2*i * q^51 - 8*i * q^53 - 4 * q^57 - 12*i * q^59 - 12*i * q^61 - 4 * q^63 - 12*i * q^67 + 8*i * q^69 - 8 * q^71 + 6 * q^73 + 5*i * q^75 + 16*i * q^77 - 4 * q^79 + q^81 + 4*i * q^83 - 8 * q^87 + 6 * q^89 - 16*i * q^91 - 4*i * q^93 - 2 * q^97 - 4*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 8 * q^7 - 2 * q^9 $$2 q + 8 q^{7} - 2 q^{9} - 4 q^{17} + 16 q^{23} + 10 q^{25} - 8 q^{31} - 8 q^{33} + 8 q^{39} - 12 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{57} - 8 q^{63} - 16 q^{71} + 12 q^{73} - 8 q^{79} + 2 q^{81} - 16 q^{87} + 12 q^{89} - 4 q^{97}+O(q^{100})$$ 2 * q + 8 * q^7 - 2 * q^9 - 4 * q^17 + 16 * q^23 + 10 * q^25 - 8 * q^31 - 8 * q^33 + 8 * q^39 - 12 * q^41 - 16 * q^47 + 18 * q^49 - 8 * q^57 - 8 * q^63 - 16 * q^71 + 12 * q^73 - 8 * q^79 + 2 * q^81 - 16 * q^87 + 12 * q^89 - 4 * q^97

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 4.00000 0 −1.00000 0
193.2 0 1.00000i 0 0 0 4.00000 0 −1.00000 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
8.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.d.b yes 2
3.b odd 2 1 1152.2.d.f 2
4.b odd 2 1 384.2.d.a 2
8.b even 2 1 inner 384.2.d.b yes 2
8.d odd 2 1 384.2.d.a 2
12.b even 2 1 1152.2.d.a 2
16.e even 4 1 768.2.a.b 1
16.e even 4 1 768.2.a.f 1
16.f odd 4 1 768.2.a.c 1
16.f odd 4 1 768.2.a.g 1
24.f even 2 1 1152.2.d.a 2
24.h odd 2 1 1152.2.d.f 2
48.i odd 4 1 2304.2.a.f 1
48.i odd 4 1 2304.2.a.g 1
48.k even 4 1 2304.2.a.j 1
48.k even 4 1 2304.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.a 2 4.b odd 2 1
384.2.d.a 2 8.d odd 2 1
384.2.d.b yes 2 1.a even 1 1 trivial
384.2.d.b yes 2 8.b even 2 1 inner
768.2.a.b 1 16.e even 4 1
768.2.a.c 1 16.f odd 4 1
768.2.a.f 1 16.e even 4 1
768.2.a.g 1 16.f odd 4 1
1152.2.d.a 2 12.b even 2 1
1152.2.d.a 2 24.f even 2 1
1152.2.d.f 2 3.b odd 2 1
1152.2.d.f 2 24.h odd 2 1
2304.2.a.f 1 48.i odd 4 1
2304.2.a.g 1 48.i odd 4 1
2304.2.a.j 1 48.k even 4 1
2304.2.a.k 1 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 $$S_{2}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7} - 4$$ T7 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2} + 16$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T - 8)^{2}$$
$29$ $$T^{2} + 64$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 64$$
$59$ $$T^{2} + 144$$
$61$ $$T^{2} + 144$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T + 2)^{2}$$