# Properties

 Label 2304.2.a.a Level $2304$ Weight $2$ Character orbit 2304.a Self dual yes Analytic conductor $18.398$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 128) 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{5}+O(q^{10})$$ q - 4 * q^5 $$q - 4 q^{5} + 4 q^{13} + 2 q^{17} + 11 q^{25} - 4 q^{29} - 12 q^{37} + 10 q^{41} - 7 q^{49} - 4 q^{53} - 12 q^{61} - 16 q^{65} - 6 q^{73} - 8 q^{85} - 10 q^{89} - 18 q^{97}+O(q^{100})$$ q - 4 * q^5 + 4 * q^13 + 2 * q^17 + 11 * q^25 - 4 * q^29 - 12 * q^37 + 10 * q^41 - 7 * q^49 - 4 * q^53 - 12 * q^61 - 16 * q^65 - 6 * q^73 - 8 * q^85 - 10 * 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 −4.00000 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.a 1
3.b odd 2 1 256.2.a.c 1
4.b odd 2 1 CM 2304.2.a.a 1
8.b even 2 1 2304.2.a.p 1
8.d odd 2 1 2304.2.a.p 1
12.b even 2 1 256.2.a.c 1
15.d odd 2 1 6400.2.a.l 1
16.e even 4 2 1152.2.d.d 2
16.f odd 4 2 1152.2.d.d 2
24.f even 2 1 256.2.a.b 1
24.h odd 2 1 256.2.a.b 1
48.i odd 4 2 128.2.b.b 2
48.k even 4 2 128.2.b.b 2
60.h even 2 1 6400.2.a.l 1
96.o even 8 4 1024.2.e.k 4
96.p odd 8 4 1024.2.e.k 4
120.i odd 2 1 6400.2.a.m 1
120.m even 2 1 6400.2.a.m 1
240.t even 4 2 3200.2.d.e 2
240.z odd 4 2 3200.2.f.c 2
240.bb even 4 2 3200.2.f.c 2
240.bd odd 4 2 3200.2.f.d 2
240.bf even 4 2 3200.2.f.d 2
240.bm odd 4 2 3200.2.d.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.b 2 48.i odd 4 2
128.2.b.b 2 48.k even 4 2
256.2.a.b 1 24.f even 2 1
256.2.a.b 1 24.h odd 2 1
256.2.a.c 1 3.b odd 2 1
256.2.a.c 1 12.b even 2 1
1024.2.e.k 4 96.o even 8 4
1024.2.e.k 4 96.p odd 8 4
1152.2.d.d 2 16.e even 4 2
1152.2.d.d 2 16.f odd 4 2
2304.2.a.a 1 1.a even 1 1 trivial
2304.2.a.a 1 4.b odd 2 1 CM
2304.2.a.p 1 8.b even 2 1
2304.2.a.p 1 8.d odd 2 1
3200.2.d.e 2 240.t even 4 2
3200.2.d.e 2 240.bm odd 4 2
3200.2.f.c 2 240.z odd 4 2
3200.2.f.c 2 240.bb even 4 2
3200.2.f.d 2 240.bd odd 4 2
3200.2.f.d 2 240.bf even 4 2
6400.2.a.l 1 15.d odd 2 1
6400.2.a.l 1 60.h even 2 1
6400.2.a.m 1 120.i odd 2 1
6400.2.a.m 1 120.m even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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