# Properties

 Label 6080.2.a.b Level $6080$ Weight $2$ Character orbit 6080.a Self dual yes Analytic conductor $48.549$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10})$$ q - 3 * q^3 + q^5 + 5 * q^7 + 6 * q^9 $$q - 3 q^{3} + q^{5} + 5 q^{7} + 6 q^{9} - 4 q^{11} + q^{13} - 3 q^{15} - 3 q^{17} + q^{19} - 15 q^{21} - 7 q^{23} + q^{25} - 9 q^{27} + 3 q^{29} + 2 q^{31} + 12 q^{33} + 5 q^{35} + 2 q^{37} - 3 q^{39} - 6 q^{41} + 6 q^{43} + 6 q^{45} + 18 q^{49} + 9 q^{51} + 13 q^{53} - 4 q^{55} - 3 q^{57} - 9 q^{59} + 12 q^{61} + 30 q^{63} + q^{65} - 3 q^{67} + 21 q^{69} + 11 q^{73} - 3 q^{75} - 20 q^{77} + 2 q^{79} + 9 q^{81} - 10 q^{83} - 3 q^{85} - 9 q^{87} + 2 q^{89} + 5 q^{91} - 6 q^{93} + q^{95} - 2 q^{97} - 24 q^{99}+O(q^{100})$$ q - 3 * q^3 + q^5 + 5 * q^7 + 6 * q^9 - 4 * q^11 + q^13 - 3 * q^15 - 3 * q^17 + q^19 - 15 * q^21 - 7 * q^23 + q^25 - 9 * q^27 + 3 * q^29 + 2 * q^31 + 12 * q^33 + 5 * q^35 + 2 * q^37 - 3 * q^39 - 6 * q^41 + 6 * q^43 + 6 * q^45 + 18 * q^49 + 9 * q^51 + 13 * q^53 - 4 * q^55 - 3 * q^57 - 9 * q^59 + 12 * q^61 + 30 * q^63 + q^65 - 3 * q^67 + 21 * q^69 + 11 * q^73 - 3 * q^75 - 20 * q^77 + 2 * q^79 + 9 * q^81 - 10 * q^83 - 3 * q^85 - 9 * q^87 + 2 * q^89 + 5 * q^91 - 6 * q^93 + q^95 - 2 * q^97 - 24 * 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
 0
0 −3.00000 0 1.00000 0 5.00000 0 6.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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.b 1
4.b odd 2 1 6080.2.a.x 1
8.b even 2 1 1520.2.a.j 1
8.d odd 2 1 190.2.a.b 1
24.f even 2 1 1710.2.a.g 1
40.e odd 2 1 950.2.a.c 1
40.f even 2 1 7600.2.a.a 1
40.k even 4 2 950.2.b.a 2
56.e even 2 1 9310.2.a.u 1
120.m even 2 1 8550.2.a.bm 1
152.b even 2 1 3610.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.b 1 8.d odd 2 1
950.2.a.c 1 40.e odd 2 1
950.2.b.a 2 40.k even 4 2
1520.2.a.j 1 8.b even 2 1
1710.2.a.g 1 24.f even 2 1
3610.2.a.e 1 152.b even 2 1
6080.2.a.b 1 1.a even 1 1 trivial
6080.2.a.x 1 4.b odd 2 1
7600.2.a.a 1 40.f even 2 1
8550.2.a.bm 1 120.m even 2 1
9310.2.a.u 1 56.e even 2 1

## Hecke kernels

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

 $$T_{3} + 3$$ T3 + 3 $$T_{7} - 5$$ T7 - 5 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 1$$
$7$ $$T - 5$$
$11$ $$T + 4$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T - 1$$
$23$ $$T + 7$$
$29$ $$T - 3$$
$31$ $$T - 2$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T - 6$$
$47$ $$T$$
$53$ $$T - 13$$
$59$ $$T + 9$$
$61$ $$T - 12$$
$67$ $$T + 3$$
$71$ $$T$$
$73$ $$T - 11$$
$79$ $$T - 2$$
$83$ $$T + 10$$
$89$ $$T - 2$$
$97$ $$T + 2$$