# Properties

 Label 1824.2.a.f Level $1824$ Weight $2$ Character orbit 1824.a Self dual yes Analytic conductor $14.565$ 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] = [1824,2,Mod(1,1824)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1824, 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("1824.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1824 = 2^{5} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1824.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: $$14.5647133287$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + 3 q^{5} - 3 q^{7} + q^{9}+O(q^{10})$$ q - q^3 + 3 * q^5 - 3 * q^7 + q^9 $$q - q^{3} + 3 q^{5} - 3 q^{7} + q^{9} - 3 q^{11} - 3 q^{15} + q^{17} - q^{19} + 3 q^{21} + 4 q^{25} - q^{27} - 8 q^{29} + 2 q^{31} + 3 q^{33} - 9 q^{35} - 4 q^{37} - 12 q^{41} + q^{43} + 3 q^{45} - 9 q^{47} + 2 q^{49} - q^{51} + 6 q^{53} - 9 q^{55} + q^{57} + 6 q^{59} - q^{61} - 3 q^{63} + 10 q^{67} + 10 q^{71} - 11 q^{73} - 4 q^{75} + 9 q^{77} - 6 q^{79} + q^{81} + 3 q^{85} + 8 q^{87} - 6 q^{89} - 2 q^{93} - 3 q^{95} - 8 q^{97} - 3 q^{99}+O(q^{100})$$ q - q^3 + 3 * q^5 - 3 * q^7 + q^9 - 3 * q^11 - 3 * q^15 + q^17 - q^19 + 3 * q^21 + 4 * q^25 - q^27 - 8 * q^29 + 2 * q^31 + 3 * q^33 - 9 * q^35 - 4 * q^37 - 12 * q^41 + q^43 + 3 * q^45 - 9 * q^47 + 2 * q^49 - q^51 + 6 * q^53 - 9 * q^55 + q^57 + 6 * q^59 - q^61 - 3 * q^63 + 10 * q^67 + 10 * q^71 - 11 * q^73 - 4 * q^75 + 9 * q^77 - 6 * q^79 + q^81 + 3 * q^85 + 8 * q^87 - 6 * q^89 - 2 * q^93 - 3 * q^95 - 8 * q^97 - 3 * 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 −1.00000 0 3.00000 0 −3.00000 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$19$$ $$1$$

## 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 1824.2.a.f 1
3.b odd 2 1 5472.2.a.b 1
4.b odd 2 1 1824.2.a.l yes 1
8.b even 2 1 3648.2.a.t 1
8.d odd 2 1 3648.2.a.b 1
12.b even 2 1 5472.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.f 1 1.a even 1 1 trivial
1824.2.a.l yes 1 4.b odd 2 1
3648.2.a.b 1 8.d odd 2 1
3648.2.a.t 1 8.b even 2 1
5472.2.a.b 1 3.b odd 2 1
5472.2.a.c 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1824))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{7} + 3$$ T7 + 3 $$T_{11} + 3$$ T11 + 3 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 3$$
$7$ $$T + 3$$
$11$ $$T + 3$$
$13$ $$T$$
$17$ $$T - 1$$
$19$ $$T + 1$$
$23$ $$T$$
$29$ $$T + 8$$
$31$ $$T - 2$$
$37$ $$T + 4$$
$41$ $$T + 12$$
$43$ $$T - 1$$
$47$ $$T + 9$$
$53$ $$T - 6$$
$59$ $$T - 6$$
$61$ $$T + 1$$
$67$ $$T - 10$$
$71$ $$T - 10$$
$73$ $$T + 11$$
$79$ $$T + 6$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T + 8$$