# Properties

 Label 2304.2.d.h Level $2304$ Weight $2$ Character orbit 2304.d Analytic conductor $18.398$ Analytic rank $1$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(1153,2304)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2304.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5}+O(q^{10})$$ q + 2*b * q^5 $$q + 2 \beta q^{5} - 3 \beta q^{13} - 8 q^{17} - 11 q^{25} + 2 \beta q^{29} + \beta q^{37} - 8 q^{41} - 7 q^{49} - 2 \beta q^{53} - 5 \beta q^{61} + 24 q^{65} - 6 q^{73} - 16 \beta q^{85} + 16 q^{89} - 18 q^{97} +O(q^{100})$$ q + 2*b * q^5 - 3*b * q^13 - 8 * q^17 - 11 * q^25 + 2*b * q^29 + b * q^37 - 8 * q^41 - 7 * q^49 - 2*b * q^53 - 5*b * q^61 + 24 * q^65 - 6 * q^73 - 16*b * q^85 + 16 * q^89 - 18 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 16 q^{17} - 22 q^{25} - 16 q^{41} - 14 q^{49} + 48 q^{65} - 12 q^{73} + 32 q^{89} - 36 q^{97}+O(q^{100})$$ 2 * q - 16 * q^17 - 22 * q^25 - 16 * q^41 - 14 * q^49 + 48 * q^65 - 12 * q^73 + 32 * q^89 - 36 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1153.1
 − 1.00000i 1.00000i
0 0 0 4.00000i 0 0 0 0 0
1153.2 0 0 0 4.00000i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.d.h 2
3.b odd 2 1 2304.2.d.l 2
4.b odd 2 1 CM 2304.2.d.h 2
8.b even 2 1 inner 2304.2.d.h 2
8.d odd 2 1 inner 2304.2.d.h 2
12.b even 2 1 2304.2.d.l 2
16.e even 4 1 288.2.a.a 1
16.e even 4 1 576.2.a.i 1
16.f odd 4 1 288.2.a.a 1
16.f odd 4 1 576.2.a.i 1
24.f even 2 1 2304.2.d.l 2
24.h odd 2 1 2304.2.d.l 2
48.i odd 4 1 288.2.a.e yes 1
48.i odd 4 1 576.2.a.a 1
48.k even 4 1 288.2.a.e yes 1
48.k even 4 1 576.2.a.a 1
80.i odd 4 1 7200.2.f.n 2
80.j even 4 1 7200.2.f.n 2
80.k odd 4 1 7200.2.a.bf 1
80.q even 4 1 7200.2.a.bf 1
80.s even 4 1 7200.2.f.n 2
80.t odd 4 1 7200.2.f.n 2
144.u even 12 2 2592.2.i.a 2
144.v odd 12 2 2592.2.i.x 2
144.w odd 12 2 2592.2.i.a 2
144.x even 12 2 2592.2.i.x 2
240.t even 4 1 7200.2.a.be 1
240.z odd 4 1 7200.2.f.q 2
240.bb even 4 1 7200.2.f.q 2
240.bd odd 4 1 7200.2.f.q 2
240.bf even 4 1 7200.2.f.q 2
240.bm odd 4 1 7200.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.a.a 1 16.e even 4 1
288.2.a.a 1 16.f odd 4 1
288.2.a.e yes 1 48.i odd 4 1
288.2.a.e yes 1 48.k even 4 1
576.2.a.a 1 48.i odd 4 1
576.2.a.a 1 48.k even 4 1
576.2.a.i 1 16.e even 4 1
576.2.a.i 1 16.f odd 4 1
2304.2.d.h 2 1.a even 1 1 trivial
2304.2.d.h 2 4.b odd 2 1 CM
2304.2.d.h 2 8.b even 2 1 inner
2304.2.d.h 2 8.d odd 2 1 inner
2304.2.d.l 2 3.b odd 2 1
2304.2.d.l 2 12.b even 2 1
2304.2.d.l 2 24.f even 2 1
2304.2.d.l 2 24.h odd 2 1
2592.2.i.a 2 144.u even 12 2
2592.2.i.a 2 144.w odd 12 2
2592.2.i.x 2 144.v odd 12 2
2592.2.i.x 2 144.x even 12 2
7200.2.a.be 1 240.t even 4 1
7200.2.a.be 1 240.bm odd 4 1
7200.2.a.bf 1 80.k odd 4 1
7200.2.a.bf 1 80.q even 4 1
7200.2.f.n 2 80.i odd 4 1
7200.2.f.n 2 80.j even 4 1
7200.2.f.n 2 80.s even 4 1
7200.2.f.n 2 80.t odd 4 1
7200.2.f.q 2 240.z odd 4 1
7200.2.f.q 2 240.bb even 4 1
7200.2.f.q 2 240.bd odd 4 1
7200.2.f.q 2 240.bf even 4 1

## Hecke kernels

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

 $$T_{5}^{2} + 16$$ T5^2 + 16 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{17} + 8$$ T17 + 8 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 16$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$(T + 8)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 16$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 100$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T + 6)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 16)^{2}$$
$97$ $$(T + 18)^{2}$$