# Properties

 Label 6384.2.a.bi Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
0 −1.00000 0 −2.73205 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 0.732051 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.bi 2
4.b odd 2 1 3192.2.a.s 2
12.b even 2 1 9576.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.s 2 4.b odd 2 1
6384.2.a.bi 2 1.a even 1 1 trivial
9576.2.a.bw 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6384))$$:

 $$T_{5}^{2} + 2T_{5} - 2$$ T5^2 + 2*T5 - 2 $$T_{11} - 4$$ T11 - 4 $$T_{13}^{2} - 4T_{13} - 8$$ T13^2 - 4*T13 - 8 $$T_{17}^{2} + 10T_{17} + 22$$ T17^2 + 10*T17 + 22 $$T_{23}^{2} - 48$$ T23^2 - 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 2T - 2$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} - 4T - 8$$
$17$ $$T^{2} + 10T + 22$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 48$$
$29$ $$T^{2} + 10T + 22$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} - 4T - 44$$
$41$ $$T^{2} + 4T - 44$$
$43$ $$T^{2} - 4T - 44$$
$47$ $$T^{2} - 10T - 2$$
$53$ $$T^{2} + 2T - 74$$
$59$ $$T^{2} - 8T - 32$$
$61$ $$T^{2} + 12T - 12$$
$67$ $$T^{2} + 4T - 8$$
$71$ $$T^{2} - 2T - 26$$
$73$ $$T^{2} - 108$$
$79$ $$T^{2} - 8T - 32$$
$83$ $$T^{2} + 6T - 18$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 4T - 188$$