# Properties

 Label 3888.2.a.z Level $3888$ Weight $2$ Character orbit 3888.a Self dual yes Analytic conductor $31.046$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3888,2,Mod(1,3888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3888, 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("3888.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3888 = 2^{4} \cdot 3^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3888.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: $$31.0458363059$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 243) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - 2 q^{7}+O(q^{10})$$ q + b * q^5 - 2 * q^7 $$q + \beta q^{5} - 2 q^{7} + \beta q^{11} - q^{13} + 3 \beta q^{17} + q^{19} - \beta q^{23} + q^{25} + 2 \beta q^{29} + q^{31} - 2 \beta q^{35} + 8 q^{37} - 2 \beta q^{41} - 11 q^{43} + 4 \beta q^{47} - 3 q^{49} - 3 \beta q^{53} + 6 q^{55} - \beta q^{59} + 5 q^{61} - \beta q^{65} + 7 q^{67} + 3 \beta q^{71} + 11 q^{73} - 2 \beta q^{77} + 7 q^{79} - 5 \beta q^{83} + 18 q^{85} + 2 q^{91} + \beta q^{95} - 7 q^{97} +O(q^{100})$$ q + b * q^5 - 2 * q^7 + b * q^11 - q^13 + 3*b * q^17 + q^19 - b * q^23 + q^25 + 2*b * q^29 + q^31 - 2*b * q^35 + 8 * q^37 - 2*b * q^41 - 11 * q^43 + 4*b * q^47 - 3 * q^49 - 3*b * q^53 + 6 * q^55 - b * q^59 + 5 * q^61 - b * q^65 + 7 * q^67 + 3*b * q^71 + 11 * q^73 - 2*b * q^77 + 7 * q^79 - 5*b * q^83 + 18 * q^85 + 2 * q^91 + b * q^95 - 7 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7}+O(q^{10})$$ 2 * q - 4 * q^7 $$2 q - 4 q^{7} - 2 q^{13} + 2 q^{19} + 2 q^{25} + 2 q^{31} + 16 q^{37} - 22 q^{43} - 6 q^{49} + 12 q^{55} + 10 q^{61} + 14 q^{67} + 22 q^{73} + 14 q^{79} + 36 q^{85} + 4 q^{91} - 14 q^{97}+O(q^{100})$$ 2 * q - 4 * q^7 - 2 * q^13 + 2 * q^19 + 2 * q^25 + 2 * q^31 + 16 * q^37 - 22 * q^43 - 6 * q^49 + 12 * q^55 + 10 * q^61 + 14 * q^67 + 22 * q^73 + 14 * q^79 + 36 * q^85 + 4 * q^91 - 14 * q^97

## 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
 −2.44949 2.44949
0 0 0 −2.44949 0 −2.00000 0 0 0
1.2 0 0 0 2.44949 0 −2.00000 0 0 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

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3888.2.a.z 2
3.b odd 2 1 inner 3888.2.a.z 2
4.b odd 2 1 243.2.a.d 2
12.b even 2 1 243.2.a.d 2
20.d odd 2 1 6075.2.a.bn 2
36.f odd 6 2 243.2.c.c 4
36.h even 6 2 243.2.c.c 4
60.h even 2 1 6075.2.a.bn 2
108.j odd 18 6 729.2.e.p 12
108.l even 18 6 729.2.e.p 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.d 2 4.b odd 2 1
243.2.a.d 2 12.b even 2 1
243.2.c.c 4 36.f odd 6 2
243.2.c.c 4 36.h even 6 2
729.2.e.p 12 108.j odd 18 6
729.2.e.p 12 108.l even 18 6
3888.2.a.z 2 1.a even 1 1 trivial
3888.2.a.z 2 3.b odd 2 1 inner
6075.2.a.bn 2 20.d odd 2 1
6075.2.a.bn 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3888))$$:

 $$T_{5}^{2} - 6$$ T5^2 - 6 $$T_{7} + 2$$ T7 + 2 $$T_{11}^{2} - 6$$ T11^2 - 6 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} - 6$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 54$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 6$$
$29$ $$T^{2} - 24$$
$31$ $$(T - 1)^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 24$$
$43$ $$(T + 11)^{2}$$
$47$ $$T^{2} - 96$$
$53$ $$T^{2} - 54$$
$59$ $$T^{2} - 6$$
$61$ $$(T - 5)^{2}$$
$67$ $$(T - 7)^{2}$$
$71$ $$T^{2} - 54$$
$73$ $$(T - 11)^{2}$$
$79$ $$(T - 7)^{2}$$
$83$ $$T^{2} - 150$$
$89$ $$T^{2}$$
$97$ $$(T + 7)^{2}$$