# Properties

 Label 6080.2.a.bs Level $6080$ Weight $2$ Character orbit 6080.a Self dual yes Analytic conductor $48.549$ 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] = [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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{5} + (\beta_{2} + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^3 + q^5 + (b2 + 2) * q^7 + (b2 + 2*b1 + 1) * q^9 $$q - \beta_1 q^{3} + q^{5} + (\beta_{2} + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_1 - 2) q^{11} + ( - 2 \beta_{2} - \beta_1 - 2) q^{13} - \beta_1 q^{15} + ( - \beta_{2} - 4) q^{17} - q^{19} + (\beta_{2} - 2 \beta_1 + 2) q^{21} + ( - \beta_{2} + 2 \beta_1 + 2) q^{23} + q^{25} + ( - \beta_{2} - 2 \beta_1 - 6) q^{27} + 3 \beta_{2} q^{29} + ( - 2 \beta_{2} + 4) q^{31} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{33} + (\beta_{2} + 2) q^{35} + (\beta_{2} - \beta_1 - 4) q^{37} + ( - \beta_{2} + 4 \beta_1) q^{39} + (2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{43} + (\beta_{2} + 2 \beta_1 + 1) q^{45} + ( - 2 \beta_{2} - 2 \beta_1) q^{47} + (3 \beta_{2} - 2 \beta_1 + 3) q^{49} + ( - \beta_{2} + 4 \beta_1 - 2) q^{51} + ( - 2 \beta_{2} - \beta_1 + 2) q^{53} + (2 \beta_1 - 2) q^{55} + \beta_1 q^{57} + (\beta_{2} - 2) q^{59} + (4 \beta_1 - 8) q^{61} + (2 \beta_1 + 4) q^{63} + ( - 2 \beta_{2} - \beta_1 - 2) q^{65} + (4 \beta_{2} + 3 \beta_1) q^{67} + ( - 3 \beta_{2} - 6 \beta_1 - 10) q^{69} + ( - 4 \beta_1 + 4) q^{71} + ( - 3 \beta_{2} - 4 \beta_1 - 4) q^{73} - \beta_1 q^{75} + ( - 4 \beta_{2} + 4 \beta_1 - 8) q^{77} + (4 \beta_{2} + 2 \beta_1 - 4) q^{79} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{81} + (2 \beta_{2} - 8) q^{83} + ( - \beta_{2} - 4) q^{85} + (3 \beta_{2} + 6) q^{87} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{89} + ( - 3 \beta_{2} + 2 \beta_1 - 14) q^{91} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{93} - q^{95} + (\beta_{2} + 7 \beta_1 - 4) q^{97} + (6 \beta_1 + 10) q^{99}+O(q^{100})$$ q - b1 * q^3 + q^5 + (b2 + 2) * q^7 + (b2 + 2*b1 + 1) * q^9 + (2*b1 - 2) * q^11 + (-2*b2 - b1 - 2) * q^13 - b1 * q^15 + (-b2 - 4) * q^17 - q^19 + (b2 - 2*b1 + 2) * q^21 + (-b2 + 2*b1 + 2) * q^23 + q^25 + (-b2 - 2*b1 - 6) * q^27 + 3*b2 * q^29 + (-2*b2 + 4) * q^31 + (-2*b2 - 2*b1 - 8) * q^33 + (b2 + 2) * q^35 + (b2 - b1 - 4) * q^37 + (-b2 + 4*b1) * q^39 + (2*b2 - 2*b1 - 2) * q^41 + (-4*b2 - 2*b1 - 4) * q^43 + (b2 + 2*b1 + 1) * q^45 + (-2*b2 - 2*b1) * q^47 + (3*b2 - 2*b1 + 3) * q^49 + (-b2 + 4*b1 - 2) * q^51 + (-2*b2 - b1 + 2) * q^53 + (2*b1 - 2) * q^55 + b1 * q^57 + (b2 - 2) * q^59 + (4*b1 - 8) * q^61 + (2*b1 + 4) * q^63 + (-2*b2 - b1 - 2) * q^65 + (4*b2 + 3*b1) * q^67 + (-3*b2 - 6*b1 - 10) * q^69 + (-4*b1 + 4) * q^71 + (-3*b2 - 4*b1 - 4) * q^73 - b1 * q^75 + (-4*b2 + 4*b1 - 8) * q^77 + (4*b2 + 2*b1 - 4) * q^79 + (-2*b2 + 4*b1 + 3) * q^81 + (2*b2 - 8) * q^83 + (-b2 - 4) * q^85 + (3*b2 + 6) * q^87 + (-2*b2 - 2*b1 + 6) * q^89 + (-3*b2 + 2*b1 - 14) * q^91 + (-2*b2 - 4*b1 - 4) * q^93 - q^95 + (b2 + 7*b1 - 4) * q^97 + (6*b1 + 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} + 3 q^{5} + 5 q^{7} + 4 q^{9}+O(q^{10})$$ 3 * q - q^3 + 3 * q^5 + 5 * q^7 + 4 * q^9 $$3 q - q^{3} + 3 q^{5} + 5 q^{7} + 4 q^{9} - 4 q^{11} - 5 q^{13} - q^{15} - 11 q^{17} - 3 q^{19} + 3 q^{21} + 9 q^{23} + 3 q^{25} - 19 q^{27} - 3 q^{29} + 14 q^{31} - 24 q^{33} + 5 q^{35} - 14 q^{37} + 5 q^{39} - 10 q^{41} - 10 q^{43} + 4 q^{45} + 4 q^{49} - q^{51} + 7 q^{53} - 4 q^{55} + q^{57} - 7 q^{59} - 20 q^{61} + 14 q^{63} - 5 q^{65} - q^{67} - 33 q^{69} + 8 q^{71} - 13 q^{73} - q^{75} - 16 q^{77} - 14 q^{79} + 15 q^{81} - 26 q^{83} - 11 q^{85} + 15 q^{87} + 18 q^{89} - 37 q^{91} - 14 q^{93} - 3 q^{95} - 6 q^{97} + 36 q^{99}+O(q^{100})$$ 3 * q - q^3 + 3 * q^5 + 5 * q^7 + 4 * q^9 - 4 * q^11 - 5 * q^13 - q^15 - 11 * q^17 - 3 * q^19 + 3 * q^21 + 9 * q^23 + 3 * q^25 - 19 * q^27 - 3 * q^29 + 14 * q^31 - 24 * q^33 + 5 * q^35 - 14 * q^37 + 5 * q^39 - 10 * q^41 - 10 * q^43 + 4 * q^45 + 4 * q^49 - q^51 + 7 * q^53 - 4 * q^55 + q^57 - 7 * q^59 - 20 * q^61 + 14 * q^63 - 5 * q^65 - q^67 - 33 * q^69 + 8 * q^71 - 13 * q^73 - q^75 - 16 * q^77 - 14 * q^79 + 15 * q^81 - 26 * q^83 - 11 * q^85 + 15 * q^87 + 18 * q^89 - 37 * q^91 - 14 * q^93 - 3 * q^95 - 6 * q^97 + 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 4

## 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
 3.12489 −0.363328 −1.76156
0 −3.12489 0 1.00000 0 1.51514 0 6.76491 0
1.2 0 0.363328 0 1.00000 0 −1.14134 0 −2.86799 0
1.3 0 1.76156 0 1.00000 0 4.62620 0 0.103084 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.bs 3
4.b odd 2 1 6080.2.a.bw 3
8.b even 2 1 1520.2.a.r 3
8.d odd 2 1 760.2.a.h 3
24.f even 2 1 6840.2.a.bj 3
40.e odd 2 1 3800.2.a.y 3
40.f even 2 1 7600.2.a.bo 3
40.k even 4 2 3800.2.d.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.h 3 8.d odd 2 1
1520.2.a.r 3 8.b even 2 1
3800.2.a.y 3 40.e odd 2 1
3800.2.d.k 6 40.k even 4 2
6080.2.a.bs 3 1.a even 1 1 trivial
6080.2.a.bw 3 4.b odd 2 1
6840.2.a.bj 3 24.f even 2 1
7600.2.a.bo 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6080))$$:

 $$T_{3}^{3} + T_{3}^{2} - 6T_{3} + 2$$ T3^3 + T3^2 - 6*T3 + 2 $$T_{7}^{3} - 5T_{7}^{2} + 8$$ T7^3 - 5*T7^2 + 8 $$T_{11}^{3} + 4T_{11}^{2} - 20T_{11} - 64$$ T11^3 + 4*T11^2 - 20*T11 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + T^{2} - 6T + 2$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 5T^{2} + 8$$
$11$ $$T^{3} + 4 T^{2} - 20 T - 64$$
$13$ $$T^{3} + 5 T^{2} - 22 T - 106$$
$17$ $$T^{3} + 11 T^{2} + 32 T + 20$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} - 9 T^{2} - 16 T + 160$$
$29$ $$T^{3} + 3 T^{2} - 72 T - 108$$
$31$ $$T^{3} - 14 T^{2} + 32 T + 64$$
$37$ $$T^{3} + 14 T^{2} + 46 T - 20$$
$41$ $$T^{3} + 10 T^{2} - 44 T - 472$$
$43$ $$T^{3} + 10 T^{2} - 88 T - 848$$
$47$ $$T^{3} - 40T - 64$$
$53$ $$T^{3} - 7 T^{2} - 14 T - 2$$
$59$ $$T^{3} + 7 T^{2} + 8 T - 8$$
$61$ $$T^{3} + 20 T^{2} + 32 T - 640$$
$67$ $$T^{3} + T^{2} - 134 T + 530$$
$71$ $$T^{3} - 8 T^{2} - 80 T + 512$$
$73$ $$T^{3} + 13 T^{2} - 64 T - 500$$
$79$ $$T^{3} + 14 T^{2} - 56 T + 16$$
$83$ $$T^{3} + 26 T^{2} + 192 T + 352$$
$89$ $$T^{3} - 18 T^{2} + 68 T - 40$$
$97$ $$T^{3} + 6 T^{2} - 274 T - 2308$$