# Properties

 Label 6080.2.a.p Level $6080$ Weight $2$ Character orbit 6080.a Self dual yes Analytic conductor $48.549$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - q^5 + q^7 - 2 * q^9 $$q + q^{3} - q^{5} + q^{7} - 2 q^{9} + q^{13} - q^{15} - 3 q^{17} + q^{19} + q^{21} - 3 q^{23} + q^{25} - 5 q^{27} + 3 q^{29} - 2 q^{31} - q^{35} + 10 q^{37} + q^{39} + 6 q^{41} + 2 q^{43} + 2 q^{45} - 6 q^{49} - 3 q^{51} - 3 q^{53} + q^{57} + 3 q^{59} - 8 q^{61} - 2 q^{63} - q^{65} - 7 q^{67} - 3 q^{69} - 12 q^{71} - 13 q^{73} + q^{75} - 14 q^{79} + q^{81} + 6 q^{83} + 3 q^{85} + 3 q^{87} + 6 q^{89} + q^{91} - 2 q^{93} - q^{95} - 10 q^{97}+O(q^{100})$$ q + q^3 - q^5 + q^7 - 2 * q^9 + q^13 - q^15 - 3 * q^17 + q^19 + q^21 - 3 * q^23 + q^25 - 5 * q^27 + 3 * q^29 - 2 * q^31 - q^35 + 10 * q^37 + q^39 + 6 * q^41 + 2 * q^43 + 2 * q^45 - 6 * q^49 - 3 * q^51 - 3 * q^53 + q^57 + 3 * q^59 - 8 * q^61 - 2 * q^63 - q^65 - 7 * q^67 - 3 * q^69 - 12 * q^71 - 13 * q^73 + q^75 - 14 * q^79 + q^81 + 6 * q^83 + 3 * q^85 + 3 * q^87 + 6 * q^89 + q^91 - 2 * q^93 - q^95 - 10 * q^97

## 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 1.00000 0 −1.00000 0 1.00000 0 −2.00000 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.p 1
4.b odd 2 1 6080.2.a.h 1
8.b even 2 1 1520.2.a.d 1
8.d odd 2 1 190.2.a.c 1
24.f even 2 1 1710.2.a.d 1
40.e odd 2 1 950.2.a.a 1
40.f even 2 1 7600.2.a.m 1
40.k even 4 2 950.2.b.e 2
56.e even 2 1 9310.2.a.o 1
120.m even 2 1 8550.2.a.bd 1
152.b even 2 1 3610.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 8.d odd 2 1
950.2.a.a 1 40.e odd 2 1
950.2.b.e 2 40.k even 4 2
1520.2.a.d 1 8.b even 2 1
1710.2.a.d 1 24.f even 2 1
3610.2.a.b 1 152.b even 2 1
6080.2.a.h 1 4.b odd 2 1
6080.2.a.p 1 1.a even 1 1 trivial
7600.2.a.m 1 40.f even 2 1
8550.2.a.bd 1 120.m even 2 1
9310.2.a.o 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6080))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} - 1$$ T7 - 1 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T - 1$$
$23$ $$T + 3$$
$29$ $$T - 3$$
$31$ $$T + 2$$
$37$ $$T - 10$$
$41$ $$T - 6$$
$43$ $$T - 2$$
$47$ $$T$$
$53$ $$T + 3$$
$59$ $$T - 3$$
$61$ $$T + 8$$
$67$ $$T + 7$$
$71$ $$T + 12$$
$73$ $$T + 13$$
$79$ $$T + 14$$
$83$ $$T - 6$$
$89$ $$T - 6$$
$97$ $$T + 10$$