# Properties

 Label 1824.2.a.h 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 - q^5 - q^7 + q^9 $$q + q^{3} - q^{5} - q^{7} + q^{9} - 5 q^{11} + 4 q^{13} - q^{15} - 3 q^{17} + q^{19} - q^{21} - 4 q^{25} + q^{27} - 4 q^{29} - 2 q^{31} - 5 q^{33} + q^{35} + 4 q^{37} + 4 q^{39} - 4 q^{41} - 5 q^{43} - q^{45} - 7 q^{47} - 6 q^{49} - 3 q^{51} - 2 q^{53} + 5 q^{55} + q^{57} - 10 q^{59} + 15 q^{61} - q^{63} - 4 q^{65} - 14 q^{67} + 2 q^{71} - 3 q^{73} - 4 q^{75} + 5 q^{77} - 14 q^{79} + q^{81} + 3 q^{85} - 4 q^{87} + 10 q^{89} - 4 q^{91} - 2 q^{93} - q^{95} + 4 q^{97} - 5 q^{99}+O(q^{100})$$ q + q^3 - q^5 - q^7 + q^9 - 5 * q^11 + 4 * q^13 - q^15 - 3 * q^17 + q^19 - q^21 - 4 * q^25 + q^27 - 4 * q^29 - 2 * q^31 - 5 * q^33 + q^35 + 4 * q^37 + 4 * q^39 - 4 * q^41 - 5 * q^43 - q^45 - 7 * q^47 - 6 * q^49 - 3 * q^51 - 2 * q^53 + 5 * q^55 + q^57 - 10 * q^59 + 15 * q^61 - q^63 - 4 * q^65 - 14 * q^67 + 2 * q^71 - 3 * q^73 - 4 * q^75 + 5 * q^77 - 14 * q^79 + q^81 + 3 * q^85 - 4 * q^87 + 10 * q^89 - 4 * q^91 - 2 * q^93 - q^95 + 4 * q^97 - 5 * 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 −1.00000 0 −1.00000 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.h yes 1
3.b odd 2 1 5472.2.a.o 1
4.b odd 2 1 1824.2.a.c 1
8.b even 2 1 3648.2.a.l 1
8.d odd 2 1 3648.2.a.be 1
12.b even 2 1 5472.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.c 1 4.b odd 2 1
1824.2.a.h yes 1 1.a even 1 1 trivial
3648.2.a.l 1 8.b even 2 1
3648.2.a.be 1 8.d odd 2 1
5472.2.a.o 1 3.b odd 2 1
5472.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1824))$$:

 $$T_{5} + 1$$ T5 + 1 $$T_{7} + 1$$ T7 + 1 $$T_{11} + 5$$ T11 + 5 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T + 5$$
$13$ $$T - 4$$
$17$ $$T + 3$$
$19$ $$T - 1$$
$23$ $$T$$
$29$ $$T + 4$$
$31$ $$T + 2$$
$37$ $$T - 4$$
$41$ $$T + 4$$
$43$ $$T + 5$$
$47$ $$T + 7$$
$53$ $$T + 2$$
$59$ $$T + 10$$
$61$ $$T - 15$$
$67$ $$T + 14$$
$71$ $$T - 2$$
$73$ $$T + 3$$
$79$ $$T + 14$$
$83$ $$T$$
$89$ $$T - 10$$
$97$ $$T - 4$$