# Properties

 Label 3888.2.a.bd Level $3888$ Weight $2$ Character orbit 3888.a Self dual yes Analytic conductor $31.046$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 2) q^{5} + (2 \beta_{2} - \beta_1 + 1) q^{7}+O(q^{10})$$ q + (-b1 - 2) * q^5 + (2*b2 - b1 + 1) * q^7 $$q + ( - \beta_1 - 2) q^{5} + (2 \beta_{2} - \beta_1 + 1) q^{7} + ( - 3 \beta_{2} + 2 \beta_1 + 1) q^{11} + (\beta_{2} + \beta_1 - 1) q^{13} - 3 q^{17} + ( - 3 \beta_{2} + 1) q^{19} + (3 \beta_{2} - 2 \beta_1 + 2) q^{23} + (\beta_{2} + 4 \beta_1 + 1) q^{25} + ( - 3 \beta_{2} + \beta_1 - 4) q^{29} + (2 \beta_{2} - \beta_1 + 4) q^{31} + ( - 3 \beta_{2} - \beta_1 - 2) q^{35} + ( - 3 \beta_1 - 1) q^{37} + ( - 3 \beta_{2} + 5 \beta_1 + 1) q^{41} + ( - \beta_{2} + 2 \beta_1 + 4) q^{43} + (5 \beta_1 - 2) q^{47} + (\beta_{2} - 2 \beta_1) q^{49} + (3 \beta_{2} - 6) q^{53} + (4 \beta_{2} - 2 \beta_1 - 3) q^{55} + (\beta_1 - 7) q^{59} + ( - 5 \beta_{2} + \beta_1 + 2) q^{61} + ( - 3 \beta_{2} - 2 \beta_1 - 1) q^{65} + ( - \beta_{2} + 5 \beta_1 - 2) q^{67} + (3 \beta_{2} - 9 \beta_1 - 3) q^{71} + ( - 3 \beta_{2} + 6 \beta_1 + 2) q^{73} + (3 \beta_{2} + 2 \beta_1 - 8) q^{77} + ( - 4 \beta_{2} + 5 \beta_1 - 2) q^{79} + ( - 3 \beta_{2} - \beta_1 - 2) q^{83} + (3 \beta_1 + 6) q^{85} + (6 \beta_{2} - 9 \beta_1) q^{89} + ( - 4 \beta_{2} + 5 \beta_1 + 2) q^{91} + (6 \beta_{2} + 2 \beta_1 + 1) q^{95} + (4 \beta_{2} - 8 \beta_1 + 5) q^{97}+O(q^{100})$$ q + (-b1 - 2) * q^5 + (2*b2 - b1 + 1) * q^7 + (-3*b2 + 2*b1 + 1) * q^11 + (b2 + b1 - 1) * q^13 - 3 * q^17 + (-3*b2 + 1) * q^19 + (3*b2 - 2*b1 + 2) * q^23 + (b2 + 4*b1 + 1) * q^25 + (-3*b2 + b1 - 4) * q^29 + (2*b2 - b1 + 4) * q^31 + (-3*b2 - b1 - 2) * q^35 + (-3*b1 - 1) * q^37 + (-3*b2 + 5*b1 + 1) * q^41 + (-b2 + 2*b1 + 4) * q^43 + (5*b1 - 2) * q^47 + (b2 - 2*b1) * q^49 + (3*b2 - 6) * q^53 + (4*b2 - 2*b1 - 3) * q^55 + (b1 - 7) * q^59 + (-5*b2 + b1 + 2) * q^61 + (-3*b2 - 2*b1 - 1) * q^65 + (-b2 + 5*b1 - 2) * q^67 + (3*b2 - 9*b1 - 3) * q^71 + (-3*b2 + 6*b1 + 2) * q^73 + (3*b2 + 2*b1 - 8) * q^77 + (-4*b2 + 5*b1 - 2) * q^79 + (-3*b2 - b1 - 2) * q^83 + (3*b1 + 6) * q^85 + (6*b2 - 9*b1) * q^89 + (-4*b2 + 5*b1 + 2) * q^91 + (6*b2 + 2*b1 + 1) * q^95 + (4*b2 - 8*b1 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{5} + 3 q^{7}+O(q^{10})$$ 3 * q - 6 * q^5 + 3 * q^7 $$3 q - 6 q^{5} + 3 q^{7} + 3 q^{11} - 3 q^{13} - 9 q^{17} + 3 q^{19} + 6 q^{23} + 3 q^{25} - 12 q^{29} + 12 q^{31} - 6 q^{35} - 3 q^{37} + 3 q^{41} + 12 q^{43} - 6 q^{47} - 18 q^{53} - 9 q^{55} - 21 q^{59} + 6 q^{61} - 3 q^{65} - 6 q^{67} - 9 q^{71} + 6 q^{73} - 24 q^{77} - 6 q^{79} - 6 q^{83} + 18 q^{85} + 6 q^{91} + 3 q^{95} + 15 q^{97}+O(q^{100})$$ 3 * q - 6 * q^5 + 3 * q^7 + 3 * q^11 - 3 * q^13 - 9 * q^17 + 3 * q^19 + 6 * q^23 + 3 * q^25 - 12 * q^29 + 12 * q^31 - 6 * q^35 - 3 * q^37 + 3 * q^41 + 12 * q^43 - 6 * q^47 - 18 * q^53 - 9 * q^55 - 21 * q^59 + 6 * q^61 - 3 * q^65 - 6 * q^67 - 9 * q^71 + 6 * q^73 - 24 * q^77 - 6 * q^79 - 6 * q^83 + 18 * q^85 + 6 * q^91 + 3 * q^95 + 15 * q^97

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 1.87939 −0.347296 −1.53209
0 0 0 −3.87939 0 2.18479 0 0 0
1.2 0 0 0 −1.65270 0 −2.41147 0 0 0
1.3 0 0 0 −0.467911 0 3.22668 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

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 3888.2.a.bd 3
3.b odd 2 1 3888.2.a.bk 3
4.b odd 2 1 243.2.a.e 3
12.b even 2 1 243.2.a.f yes 3
20.d odd 2 1 6075.2.a.bv 3
36.f odd 6 2 243.2.c.f 6
36.h even 6 2 243.2.c.e 6
60.h even 2 1 6075.2.a.bq 3
108.j odd 18 2 729.2.e.a 6
108.j odd 18 2 729.2.e.g 6
108.j odd 18 2 729.2.e.h 6
108.l even 18 2 729.2.e.b 6
108.l even 18 2 729.2.e.c 6
108.l even 18 2 729.2.e.i 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.e 3 4.b odd 2 1
243.2.a.f yes 3 12.b even 2 1
243.2.c.e 6 36.h even 6 2
243.2.c.f 6 36.f odd 6 2
729.2.e.a 6 108.j odd 18 2
729.2.e.b 6 108.l even 18 2
729.2.e.c 6 108.l even 18 2
729.2.e.g 6 108.j odd 18 2
729.2.e.h 6 108.j odd 18 2
729.2.e.i 6 108.l even 18 2
3888.2.a.bd 3 1.a even 1 1 trivial
3888.2.a.bk 3 3.b odd 2 1
6075.2.a.bq 3 60.h even 2 1
6075.2.a.bv 3 20.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{3} + 6T_{5}^{2} + 9T_{5} + 3$$ T5^3 + 6*T5^2 + 9*T5 + 3 $$T_{7}^{3} - 3T_{7}^{2} - 6T_{7} + 17$$ T7^3 - 3*T7^2 - 6*T7 + 17 $$T_{11}^{3} - 3T_{11}^{2} - 18T_{11} + 3$$ T11^3 - 3*T11^2 - 18*T11 + 3 $$T_{13}^{3} + 3T_{13}^{2} - 6T_{13} - 17$$ T13^3 + 3*T13^2 - 6*T13 - 17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 6 T^{2} + 9 T + 3$$
$7$ $$T^{3} - 3 T^{2} - 6 T + 17$$
$11$ $$T^{3} - 3 T^{2} - 18 T + 3$$
$13$ $$T^{3} + 3 T^{2} - 6 T - 17$$
$17$ $$(T + 3)^{3}$$
$19$ $$T^{3} - 3 T^{2} - 24 T - 1$$
$23$ $$T^{3} - 6 T^{2} - 9 T + 51$$
$29$ $$T^{3} + 12 T^{2} + 27 T - 57$$
$31$ $$T^{3} - 12 T^{2} + 39 T - 19$$
$37$ $$T^{3} + 3 T^{2} - 24 T + 1$$
$41$ $$T^{3} - 3 T^{2} - 54 T + 219$$
$43$ $$T^{3} - 12 T^{2} + 39 T - 19$$
$47$ $$T^{3} + 6 T^{2} - 63 T - 267$$
$53$ $$T^{3} + 18 T^{2} + 81 T + 81$$
$59$ $$T^{3} + 21 T^{2} + 144 T + 321$$
$61$ $$T^{3} - 6 T^{2} - 51 T - 53$$
$67$ $$T^{3} + 6 T^{2} - 51 T - 109$$
$71$ $$T^{3} + 9 T^{2} - 162 T - 999$$
$73$ $$T^{3} - 6 T^{2} - 69 T + 397$$
$79$ $$T^{3} + 6 T^{2} - 51 T + 53$$
$83$ $$T^{3} + 6 T^{2} - 27 T - 51$$
$89$ $$T^{3} - 189T - 999$$
$97$ $$T^{3} - 15 T^{2} - 69 T + 19$$