# Properties

 Label 2304.2.a.h Level $2304$ Weight $2$ Character orbit 2304.a Self dual yes Analytic conductor $18.398$ Analytic rank $1$ Dimension $1$ CM discriminant -8 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 64) 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)

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q - 6 q^{11} + 6 q^{17} + 2 q^{19} - 5 q^{25} - 6 q^{41} - 10 q^{43} - 7 q^{49} - 6 q^{59} - 14 q^{67} - 2 q^{73} - 18 q^{83} + 18 q^{89} + 10 q^{97}+O(q^{100})$$ q - 6 * q^11 + 6 * q^17 + 2 * q^19 - 5 * q^25 - 6 * q^41 - 10 * q^43 - 7 * q^49 - 6 * q^59 - 14 * q^67 - 2 * q^73 - 18 * q^83 + 18 * q^89 + 10 * 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 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.a.h 1
3.b odd 2 1 256.2.a.d 1
4.b odd 2 1 2304.2.a.i 1
8.b even 2 1 2304.2.a.i 1
8.d odd 2 1 CM 2304.2.a.h 1
12.b even 2 1 256.2.a.a 1
15.d odd 2 1 6400.2.a.a 1
16.e even 4 2 576.2.d.a 2
16.f odd 4 2 576.2.d.a 2
24.f even 2 1 256.2.a.d 1
24.h odd 2 1 256.2.a.a 1
48.i odd 4 2 64.2.b.a 2
48.k even 4 2 64.2.b.a 2
60.h even 2 1 6400.2.a.x 1
96.o even 8 4 1024.2.e.l 4
96.p odd 8 4 1024.2.e.l 4
120.i odd 2 1 6400.2.a.x 1
120.m even 2 1 6400.2.a.a 1
240.t even 4 2 1600.2.d.a 2
240.z odd 4 2 1600.2.f.b 2
240.bb even 4 2 1600.2.f.a 2
240.bd odd 4 2 1600.2.f.a 2
240.bf even 4 2 1600.2.f.b 2
240.bm odd 4 2 1600.2.d.a 2
336.v odd 4 2 3136.2.b.b 2
336.y even 4 2 3136.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 48.i odd 4 2
64.2.b.a 2 48.k even 4 2
256.2.a.a 1 12.b even 2 1
256.2.a.a 1 24.h odd 2 1
256.2.a.d 1 3.b odd 2 1
256.2.a.d 1 24.f even 2 1
576.2.d.a 2 16.e even 4 2
576.2.d.a 2 16.f odd 4 2
1024.2.e.l 4 96.o even 8 4
1024.2.e.l 4 96.p odd 8 4
1600.2.d.a 2 240.t even 4 2
1600.2.d.a 2 240.bm odd 4 2
1600.2.f.a 2 240.bb even 4 2
1600.2.f.a 2 240.bd odd 4 2
1600.2.f.b 2 240.z odd 4 2
1600.2.f.b 2 240.bf even 4 2
2304.2.a.h 1 1.a even 1 1 trivial
2304.2.a.h 1 8.d odd 2 1 CM
2304.2.a.i 1 4.b odd 2 1
2304.2.a.i 1 8.b even 2 1
3136.2.b.b 2 336.v odd 4 2
3136.2.b.b 2 336.y even 4 2
6400.2.a.a 1 15.d odd 2 1
6400.2.a.a 1 120.m even 2 1
6400.2.a.x 1 60.h even 2 1
6400.2.a.x 1 120.i odd 2 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{7}$$ T7 $$T_{11} + 6$$ T11 + 6 $$T_{13}$$ T13 $$T_{19} - 2$$ T19 - 2

## Hecke characteristic polynomials

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