# Properties

 Label 6840.2.a.o Level $6840$ Weight $2$ Character orbit 6840.a Self dual yes Analytic conductor $54.618$ 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] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6840.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: $$54.6176749826$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{5} + 4 q^{7}+O(q^{10})$$ q - q^5 + 4 * q^7 $$q - q^{5} + 4 q^{7} + 4 q^{11} - 6 q^{17} - q^{19} - 8 q^{23} + q^{25} + 6 q^{29} - 8 q^{31} - 4 q^{35} - 8 q^{37} + 2 q^{41} - 12 q^{47} + 9 q^{49} - 4 q^{53} - 4 q^{55} - 8 q^{59} - 14 q^{61} - 2 q^{67} + 8 q^{71} - 2 q^{73} + 16 q^{77} + 4 q^{79} - 12 q^{83} + 6 q^{85} - 6 q^{89} + q^{95}+O(q^{100})$$ q - q^5 + 4 * q^7 + 4 * q^11 - 6 * q^17 - q^19 - 8 * q^23 + q^25 + 6 * q^29 - 8 * q^31 - 4 * q^35 - 8 * q^37 + 2 * q^41 - 12 * q^47 + 9 * q^49 - 4 * q^53 - 4 * q^55 - 8 * q^59 - 14 * q^61 - 2 * q^67 + 8 * q^71 - 2 * q^73 + 16 * q^77 + 4 * q^79 - 12 * q^83 + 6 * q^85 - 6 * q^89 + q^95

## 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 0 0 −1.00000 0 4.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$19$$ $$+1$$

## 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 6840.2.a.o 1
3.b odd 2 1 760.2.a.d 1
12.b even 2 1 1520.2.a.c 1
15.d odd 2 1 3800.2.a.b 1
15.e even 4 2 3800.2.d.c 2
24.f even 2 1 6080.2.a.s 1
24.h odd 2 1 6080.2.a.e 1
60.h even 2 1 7600.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.d 1 3.b odd 2 1
1520.2.a.c 1 12.b even 2 1
3800.2.a.b 1 15.d odd 2 1
3800.2.d.c 2 15.e even 4 2
6080.2.a.e 1 24.h odd 2 1
6080.2.a.s 1 24.f even 2 1
6840.2.a.o 1 1.a even 1 1 trivial
7600.2.a.s 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6840))$$:

 $$T_{7} - 4$$ T7 - 4 $$T_{11} - 4$$ T11 - 4 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 4$$
$11$ $$T - 4$$
$13$ $$T$$
$17$ $$T + 6$$
$19$ $$T + 1$$
$23$ $$T + 8$$
$29$ $$T - 6$$
$31$ $$T + 8$$
$37$ $$T + 8$$
$41$ $$T - 2$$
$43$ $$T$$
$47$ $$T + 12$$
$53$ $$T + 4$$
$59$ $$T + 8$$
$61$ $$T + 14$$
$67$ $$T + 2$$
$71$ $$T - 8$$
$73$ $$T + 2$$
$79$ $$T - 4$$
$83$ $$T + 12$$
$89$ $$T + 6$$
$97$ $$T$$