# Properties

 Label 2304.4.a.a Level $2304$ Weight $4$ Character orbit 2304.a Self dual yes Analytic conductor $135.940$ 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] = [2304,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$135.940400653$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 22 q^{5}+O(q^{10})$$ q - 22 * q^5 $$q - 22 q^{5} - 92 q^{13} + 104 q^{17} + 359 q^{25} - 130 q^{29} + 396 q^{37} + 472 q^{41} - 343 q^{49} + 518 q^{53} - 468 q^{61} + 2024 q^{65} - 1098 q^{73} - 2288 q^{85} + 176 q^{89} + 594 q^{97}+O(q^{100})$$ q - 22 * q^5 - 92 * q^13 + 104 * q^17 + 359 * q^25 - 130 * q^29 + 396 * q^37 + 472 * q^41 - 343 * q^49 + 518 * q^53 - 468 * q^61 + 2024 * q^65 - 1098 * q^73 - 2288 * q^85 + 176 * q^89 + 594 * 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 −22.0000 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 2304.4.a.a 1
3.b odd 2 1 2304.4.a.o 1
4.b odd 2 1 CM 2304.4.a.a 1
8.b even 2 1 2304.4.a.p 1
8.d odd 2 1 2304.4.a.p 1
12.b even 2 1 2304.4.a.o 1
16.e even 4 2 1152.4.d.f yes 2
16.f odd 4 2 1152.4.d.f yes 2
24.f even 2 1 2304.4.a.b 1
24.h odd 2 1 2304.4.a.b 1
48.i odd 4 2 1152.4.d.c 2
48.k even 4 2 1152.4.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.4.d.c 2 48.i odd 4 2
1152.4.d.c 2 48.k even 4 2
1152.4.d.f yes 2 16.e even 4 2
1152.4.d.f yes 2 16.f odd 4 2
2304.4.a.a 1 1.a even 1 1 trivial
2304.4.a.a 1 4.b odd 2 1 CM
2304.4.a.b 1 24.f even 2 1
2304.4.a.b 1 24.h odd 2 1
2304.4.a.o 1 3.b odd 2 1
2304.4.a.o 1 12.b even 2 1
2304.4.a.p 1 8.b even 2 1
2304.4.a.p 1 8.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2304))$$:

 $$T_{5} + 22$$ T5 + 22 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13} + 92$$ T13 + 92 $$T_{17} - 104$$ T17 - 104 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 22$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T + 92$$
$17$ $$T - 104$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 130$$
$31$ $$T$$
$37$ $$T - 396$$
$41$ $$T - 472$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T - 518$$
$59$ $$T$$
$61$ $$T + 468$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T + 1098$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 176$$
$97$ $$T - 594$$