# Properties

 Label 2304.2.a.n Level $2304$ Weight $2$ Character orbit 2304.a Self dual yes Analytic conductor $18.398$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(1,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.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: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1152) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 2 q^{5}+O(q^{10})$$ q + 2 * q^5 $$q + 2 q^{5} + 4 q^{13} + 8 q^{17} - q^{25} - 10 q^{29} + 12 q^{37} - 8 q^{41} - 7 q^{49} + 14 q^{53} + 12 q^{61} + 8 q^{65} + 6 q^{73} + 16 q^{85} - 16 q^{89} + 18 q^{97}+O(q^{100})$$ q + 2 * q^5 + 4 * q^13 + 8 * q^17 - q^25 - 10 * q^29 + 12 * q^37 - 8 * q^41 - 7 * q^49 + 14 * q^53 + 12 * q^61 + 8 * q^65 + 6 * q^73 + 16 * q^85 - 16 * q^89 + 18 * 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 2.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.n 1
3.b odd 2 1 2304.2.a.d 1
4.b odd 2 1 CM 2304.2.a.n 1
8.b even 2 1 2304.2.a.c 1
8.d odd 2 1 2304.2.a.c 1
12.b even 2 1 2304.2.a.d 1
16.e even 4 2 1152.2.d.e yes 2
16.f odd 4 2 1152.2.d.e yes 2
24.f even 2 1 2304.2.a.m 1
24.h odd 2 1 2304.2.a.m 1
48.i odd 4 2 1152.2.d.b 2
48.k even 4 2 1152.2.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.d.b 2 48.i odd 4 2
1152.2.d.b 2 48.k even 4 2
1152.2.d.e yes 2 16.e even 4 2
1152.2.d.e yes 2 16.f odd 4 2
2304.2.a.c 1 8.b even 2 1
2304.2.a.c 1 8.d odd 2 1
2304.2.a.d 1 3.b odd 2 1
2304.2.a.d 1 12.b even 2 1
2304.2.a.m 1 24.f even 2 1
2304.2.a.m 1 24.h odd 2 1
2304.2.a.n 1 1.a even 1 1 trivial
2304.2.a.n 1 4.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2304))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13} - 4$$ T13 - 4 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 4$$
$17$ $$T - 8$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 10$$
$31$ $$T$$
$37$ $$T - 12$$
$41$ $$T + 8$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T - 14$$
$59$ $$T$$
$61$ $$T - 12$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 16$$
$97$ $$T - 18$$