# Properties

 Label 3024.2.a.c Level $3024$ Weight $2$ Character orbit 3024.a Self dual yes Analytic conductor $24.147$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{5} - q^{7}+O(q^{10})$$ q - 3 * q^5 - q^7 $$q - 3 q^{5} - q^{7} - 3 q^{11} - 4 q^{13} - 6 q^{17} + 7 q^{19} + 3 q^{23} + 4 q^{25} - 5 q^{31} + 3 q^{35} - 7 q^{37} - 9 q^{41} + 10 q^{43} - 6 q^{47} + q^{49} + 12 q^{53} + 9 q^{55} + 6 q^{59} + 8 q^{61} + 12 q^{65} + 4 q^{67} - 9 q^{71} + 2 q^{73} + 3 q^{77} + 10 q^{79} + 18 q^{85} + 15 q^{89} + 4 q^{91} - 21 q^{95} + 8 q^{97}+O(q^{100})$$ q - 3 * q^5 - q^7 - 3 * q^11 - 4 * q^13 - 6 * q^17 + 7 * q^19 + 3 * q^23 + 4 * q^25 - 5 * q^31 + 3 * q^35 - 7 * q^37 - 9 * q^41 + 10 * q^43 - 6 * q^47 + q^49 + 12 * q^53 + 9 * q^55 + 6 * q^59 + 8 * q^61 + 12 * q^65 + 4 * q^67 - 9 * q^71 + 2 * q^73 + 3 * q^77 + 10 * q^79 + 18 * q^85 + 15 * q^89 + 4 * q^91 - 21 * 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 −3.00000 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.c 1
3.b odd 2 1 3024.2.a.bb 1
4.b odd 2 1 378.2.a.b 1
12.b even 2 1 378.2.a.g yes 1
20.d odd 2 1 9450.2.a.cu 1
28.d even 2 1 2646.2.a.n 1
36.f odd 6 2 1134.2.f.o 2
36.h even 6 2 1134.2.f.b 2
60.h even 2 1 9450.2.a.h 1
84.h odd 2 1 2646.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.b 1 4.b odd 2 1
378.2.a.g yes 1 12.b even 2 1
1134.2.f.b 2 36.h even 6 2
1134.2.f.o 2 36.f odd 6 2
2646.2.a.n 1 28.d even 2 1
2646.2.a.q 1 84.h odd 2 1
3024.2.a.c 1 1.a even 1 1 trivial
3024.2.a.bb 1 3.b odd 2 1
9450.2.a.h 1 60.h even 2 1
9450.2.a.cu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3024))$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{11} + 3$$ T11 + 3 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T + 1$$
$11$ $$T + 3$$
$13$ $$T + 4$$
$17$ $$T + 6$$
$19$ $$T - 7$$
$23$ $$T - 3$$
$29$ $$T$$
$31$ $$T + 5$$
$37$ $$T + 7$$
$41$ $$T + 9$$
$43$ $$T - 10$$
$47$ $$T + 6$$
$53$ $$T - 12$$
$59$ $$T - 6$$
$61$ $$T - 8$$
$67$ $$T - 4$$
$71$ $$T + 9$$
$73$ $$T - 2$$
$79$ $$T - 10$$
$83$ $$T$$
$89$ $$T - 15$$
$97$ $$T - 8$$