# Properties

 Label 3024.2.a.a Level $3024$ Weight $2$ Character orbit 3024.a Self dual yes Analytic conductor $24.147$ 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] = [3024,2,Mod(1,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, 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("3024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.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: $$24.1467615712$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 378) 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{5} + q^{7}+O(q^{10})$$ q - 4 * q^5 + q^7 $$q - 4 q^{5} + q^{7} - 4 q^{11} + 3 q^{13} + 7 q^{17} - 2 q^{19} - q^{23} + 11 q^{25} - q^{29} + 9 q^{31} - 4 q^{35} + 2 q^{37} - 6 q^{41} - 11 q^{43} - 6 q^{47} + q^{49} + 9 q^{53} + 16 q^{55} - 5 q^{59} - 6 q^{61} - 12 q^{65} - 7 q^{67} - 7 q^{71} - 14 q^{73} - 4 q^{77} + 6 q^{79} - 4 q^{83} - 28 q^{85} + 3 q^{89} + 3 q^{91} + 8 q^{95} - 8 q^{97}+O(q^{100})$$ q - 4 * q^5 + q^7 - 4 * q^11 + 3 * q^13 + 7 * q^17 - 2 * q^19 - q^23 + 11 * q^25 - q^29 + 9 * q^31 - 4 * q^35 + 2 * q^37 - 6 * q^41 - 11 * q^43 - 6 * q^47 + q^49 + 9 * q^53 + 16 * q^55 - 5 * q^59 - 6 * q^61 - 12 * q^65 - 7 * q^67 - 7 * q^71 - 14 * q^73 - 4 * q^77 + 6 * q^79 - 4 * q^83 - 28 * q^85 + 3 * q^89 + 3 * q^91 + 8 * q^95 - 8 * 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 0 0 −4.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## 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 3024.2.a.a 1
3.b odd 2 1 3024.2.a.bd 1
4.b odd 2 1 378.2.a.a 1
12.b even 2 1 378.2.a.h yes 1
20.d odd 2 1 9450.2.a.dv 1
28.d even 2 1 2646.2.a.o 1
36.f odd 6 2 1134.2.f.p 2
36.h even 6 2 1134.2.f.a 2
60.h even 2 1 9450.2.a.bc 1
84.h odd 2 1 2646.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.a 1 4.b odd 2 1
378.2.a.h yes 1 12.b even 2 1
1134.2.f.a 2 36.h even 6 2
1134.2.f.p 2 36.f odd 6 2
2646.2.a.o 1 28.d even 2 1
2646.2.a.p 1 84.h odd 2 1
3024.2.a.a 1 1.a even 1 1 trivial
3024.2.a.bd 1 3.b odd 2 1
9450.2.a.bc 1 60.h even 2 1
9450.2.a.dv 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3024))$$:

 $$T_{5} + 4$$ T5 + 4 $$T_{11} + 4$$ T11 + 4 $$T_{17} - 7$$ T17 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 4$$
$7$ $$T - 1$$
$11$ $$T + 4$$
$13$ $$T - 3$$
$17$ $$T - 7$$
$19$ $$T + 2$$
$23$ $$T + 1$$
$29$ $$T + 1$$
$31$ $$T - 9$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T + 11$$
$47$ $$T + 6$$
$53$ $$T - 9$$
$59$ $$T + 5$$
$61$ $$T + 6$$
$67$ $$T + 7$$
$71$ $$T + 7$$
$73$ $$T + 14$$
$79$ $$T - 6$$
$83$ $$T + 4$$
$89$ $$T - 3$$
$97$ $$T + 8$$