# Properties

 Label 6080.2.a.bk Level $6080$ Weight $2$ Character orbit 6080.a Self dual yes Analytic conductor $48.549$ Analytic rank $0$ 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] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + q^{5} + (2 \beta + 1) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + q^5 + (2*b + 1) * q^9 $$q + (\beta + 1) q^{3} + q^{5} + (2 \beta + 1) q^{9} + 2 q^{11} + (\beta - 1) q^{13} + (\beta + 1) q^{15} + (2 \beta + 4) q^{17} + q^{19} + (2 \beta - 2) q^{23} + q^{25} + 4 q^{27} + 2 \beta q^{29} - 4 q^{31} + (2 \beta + 2) q^{33} + ( - 5 \beta + 1) q^{37} + 2 q^{39} + ( - 2 \beta + 4) q^{41} + ( - 4 \beta + 4) q^{43} + (2 \beta + 1) q^{45} + (4 \beta - 4) q^{47} - 7 q^{49} + (6 \beta + 10) q^{51} + (5 \beta + 3) q^{53} + 2 q^{55} + (\beta + 1) q^{57} + (2 \beta - 2) q^{59} + ( - 2 \beta + 10) q^{61} + (\beta - 1) q^{65} + ( - 3 \beta + 1) q^{67} + 4 q^{69} + ( - 4 \beta - 4) q^{71} - 6 \beta q^{73} + (\beta + 1) q^{75} + (6 \beta - 2) q^{79} + ( - 2 \beta + 1) q^{81} + ( - 2 \beta - 2) q^{83} + (2 \beta + 4) q^{85} + (2 \beta + 6) q^{87} - 2 \beta q^{89} + ( - 4 \beta - 4) q^{93} + q^{95} + ( - 5 \beta - 7) q^{97} + (4 \beta + 2) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + q^5 + (2*b + 1) * q^9 + 2 * q^11 + (b - 1) * q^13 + (b + 1) * q^15 + (2*b + 4) * q^17 + q^19 + (2*b - 2) * q^23 + q^25 + 4 * q^27 + 2*b * q^29 - 4 * q^31 + (2*b + 2) * q^33 + (-5*b + 1) * q^37 + 2 * q^39 + (-2*b + 4) * q^41 + (-4*b + 4) * q^43 + (2*b + 1) * q^45 + (4*b - 4) * q^47 - 7 * q^49 + (6*b + 10) * q^51 + (5*b + 3) * q^53 + 2 * q^55 + (b + 1) * q^57 + (2*b - 2) * q^59 + (-2*b + 10) * q^61 + (b - 1) * q^65 + (-3*b + 1) * q^67 + 4 * q^69 + (-4*b - 4) * q^71 - 6*b * q^73 + (b + 1) * q^75 + (6*b - 2) * q^79 + (-2*b + 1) * q^81 + (-2*b - 2) * q^83 + (2*b + 4) * q^85 + (2*b + 6) * q^87 - 2*b * q^89 + (-4*b - 4) * q^93 + q^95 + (-5*b - 7) * q^97 + (4*b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 8 q^{17} + 2 q^{19} - 4 q^{23} + 2 q^{25} + 8 q^{27} - 8 q^{31} + 4 q^{33} + 2 q^{37} + 4 q^{39} + 8 q^{41} + 8 q^{43} + 2 q^{45} - 8 q^{47} - 14 q^{49} + 20 q^{51} + 6 q^{53} + 4 q^{55} + 2 q^{57} - 4 q^{59} + 20 q^{61} - 2 q^{65} + 2 q^{67} + 8 q^{69} - 8 q^{71} + 2 q^{75} - 4 q^{79} + 2 q^{81} - 4 q^{83} + 8 q^{85} + 12 q^{87} - 8 q^{93} + 2 q^{95} - 14 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^15 + 8 * q^17 + 2 * q^19 - 4 * q^23 + 2 * q^25 + 8 * q^27 - 8 * q^31 + 4 * q^33 + 2 * q^37 + 4 * q^39 + 8 * q^41 + 8 * q^43 + 2 * q^45 - 8 * q^47 - 14 * q^49 + 20 * q^51 + 6 * q^53 + 4 * q^55 + 2 * q^57 - 4 * q^59 + 20 * q^61 - 2 * q^65 + 2 * q^67 + 8 * q^69 - 8 * q^71 + 2 * q^75 - 4 * q^79 + 2 * q^81 - 4 * q^83 + 8 * q^85 + 12 * q^87 - 8 * q^93 + 2 * q^95 - 14 * q^97 + 4 * 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 −0.732051 0 1.00000 0 0 0 −2.46410 0
1.2 0 2.73205 0 1.00000 0 0 0 4.46410 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 6080.2.a.bk 2
4.b odd 2 1 6080.2.a.ba 2
8.b even 2 1 1520.2.a.k 2
8.d odd 2 1 760.2.a.g 2
24.f even 2 1 6840.2.a.y 2
40.e odd 2 1 3800.2.a.j 2
40.f even 2 1 7600.2.a.bd 2
40.k even 4 2 3800.2.d.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.g 2 8.d odd 2 1
1520.2.a.k 2 8.b even 2 1
3800.2.a.j 2 40.e odd 2 1
3800.2.d.h 4 40.k even 4 2
6080.2.a.ba 2 4.b odd 2 1
6080.2.a.bk 2 1.a even 1 1 trivial
6840.2.a.y 2 24.f even 2 1
7600.2.a.bd 2 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6080))$$:

 $$T_{3}^{2} - 2T_{3} - 2$$ T3^2 - 2*T3 - 2 $$T_{7}$$ T7 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 2T - 2$$
$17$ $$T^{2} - 8T + 4$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 4T - 8$$
$29$ $$T^{2} - 12$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - 2T - 74$$
$41$ $$T^{2} - 8T + 4$$
$43$ $$T^{2} - 8T - 32$$
$47$ $$T^{2} + 8T - 32$$
$53$ $$T^{2} - 6T - 66$$
$59$ $$T^{2} + 4T - 8$$
$61$ $$T^{2} - 20T + 88$$
$67$ $$T^{2} - 2T - 26$$
$71$ $$T^{2} + 8T - 32$$
$73$ $$T^{2} - 108$$
$79$ $$T^{2} + 4T - 104$$
$83$ $$T^{2} + 4T - 8$$
$89$ $$T^{2} - 12$$
$97$ $$T^{2} + 14T - 26$$