# Properties

 Label 6384.2.a.bv Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ 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] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6384.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: $$50.9764966504$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 3192) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 6$$ 2*v^2 - 6
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 6 ) / 2$$ (b2 + 6) / 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
 2.34292 0.470683 −1.81361
0 −1.00000 0 0 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 0 0 1.00000 0 1.00000 0
1.3 0 −1.00000 0 0 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$19$$ $$-1$$

## 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 6384.2.a.bv 3
4.b odd 2 1 3192.2.a.v 3
12.b even 2 1 9576.2.a.cc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.v 3 4.b odd 2 1
6384.2.a.bv 3 1.a even 1 1 trivial
9576.2.a.cc 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6384))$$:

 $$T_{5}$$ T5 $$T_{11}^{3} - 4T_{11}^{2} - 12T_{11} + 16$$ T11^3 - 4*T11^2 - 12*T11 + 16 $$T_{13}^{3} + 6T_{13}^{2} - 16T_{13} - 64$$ T13^3 + 6*T13^2 - 16*T13 - 64 $$T_{17}^{3} - 4T_{17}^{2} - 24T_{17} + 64$$ T17^3 - 4*T17^2 - 24*T17 + 64 $$T_{23}^{3} - 4T_{23}^{2} - 60T_{23} + 256$$ T23^3 - 4*T23^2 - 60*T23 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$(T - 1)^{3}$$
$11$ $$T^{3} - 4 T^{2} - 12 T + 16$$
$13$ $$T^{3} + 6 T^{2} - 16 T - 64$$
$17$ $$T^{3} - 4 T^{2} - 24 T + 64$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 4 T^{2} - 60 T + 256$$
$29$ $$T^{3} + 12 T^{2} + 20 T - 32$$
$31$ $$T^{3} + 4 T^{2} - 56 T - 256$$
$37$ $$T^{3} + 10 T^{2} - 28 T - 344$$
$41$ $$T^{3} + 2 T^{2} - 68 T - 8$$
$43$ $$T^{3} - 4 T^{2} - 64 T + 128$$
$47$ $$T^{3} - 2 T^{2} - 16 T + 16$$
$53$ $$T^{3} + 24 T^{2} + 164 T + 272$$
$59$ $$T^{3} + 4 T^{2} - 112 T + 64$$
$61$ $$T^{3} + 14 T^{2} - 52 T - 664$$
$67$ $$T^{3} - 156T - 128$$
$71$ $$T^{3} - 14T^{2} + 32$$
$73$ $$T^{3} - 2 T^{2} - 68 T + 8$$
$79$ $$T^{3} + 4 T^{2} - 124 T + 352$$
$83$ $$T^{3} - 14 T^{2} - 64 T + 1024$$
$89$ $$T^{3} + 18 T^{2} - 4 T - 584$$
$97$ $$T^{3} + 2 T^{2} - 96 T + 304$$