Properties

 Label 2304.4.a.bd Level $2304$ Weight $4$ Character orbit 2304.a Self dual yes Analytic conductor $135.940$ Analytic rank $1$ Dimension $2$ CM discriminant -3 Inner twists $4$

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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 576) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 18\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{7} +O(q^{10})$$ q - b * q^7 $$q - \beta q^{7} + 2 \beta q^{13} + 56 q^{19} - 125 q^{25} - 5 \beta q^{31} + 14 \beta q^{37} - 520 q^{43} + 629 q^{49} + 30 \beta q^{61} + 880 q^{67} - 1190 q^{73} + 35 \beta q^{79} - 1944 q^{91} + 1330 q^{97} +O(q^{100})$$ q - b * q^7 + 2*b * q^13 + 56 * q^19 - 125 * q^25 - 5*b * q^31 + 14*b * q^37 - 520 * q^43 + 629 * q^49 + 30*b * q^61 + 880 * q^67 - 1190 * q^73 + 35*b * q^79 - 1944 * q^91 + 1330 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 112 q^{19} - 250 q^{25} - 1040 q^{43} + 1258 q^{49} + 1760 q^{67} - 2380 q^{73} - 3888 q^{91} + 2660 q^{97}+O(q^{100})$$ 2 * q + 112 * q^19 - 250 * q^25 - 1040 * q^43 + 1258 * q^49 + 1760 * q^67 - 2380 * q^73 - 3888 * q^91 + 2660 * 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
 1.73205 −1.73205
0 0 0 0 0 −31.1769 0 0 0
1.2 0 0 0 0 0 31.1769 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.4.a.bd 2
3.b odd 2 1 CM 2304.4.a.bd 2
4.b odd 2 1 2304.4.a.bc 2
8.b even 2 1 2304.4.a.bc 2
8.d odd 2 1 inner 2304.4.a.bd 2
12.b even 2 1 2304.4.a.bc 2
16.e even 4 2 576.4.d.d 4
16.f odd 4 2 576.4.d.d 4
24.f even 2 1 inner 2304.4.a.bd 2
24.h odd 2 1 2304.4.a.bc 2
48.i odd 4 2 576.4.d.d 4
48.k even 4 2 576.4.d.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.4.d.d 4 16.e even 4 2
576.4.d.d 4 16.f odd 4 2
576.4.d.d 4 48.i odd 4 2
576.4.d.d 4 48.k even 4 2
2304.4.a.bc 2 4.b odd 2 1
2304.4.a.bc 2 8.b even 2 1
2304.4.a.bc 2 12.b even 2 1
2304.4.a.bc 2 24.h odd 2 1
2304.4.a.bd 2 1.a even 1 1 trivial
2304.4.a.bd 2 3.b odd 2 1 CM
2304.4.a.bd 2 8.d odd 2 1 inner
2304.4.a.bd 2 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2304))$$:

 $$T_{5}$$ T5 $$T_{7}^{2} - 972$$ T7^2 - 972 $$T_{11}$$ T11 $$T_{13}^{2} - 3888$$ T13^2 - 3888 $$T_{17}$$ T17 $$T_{19} - 56$$ T19 - 56

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 972$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3888$$
$17$ $$T^{2}$$
$19$ $$(T - 56)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 24300$$
$37$ $$T^{2} - 190512$$
$41$ $$T^{2}$$
$43$ $$(T + 520)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 874800$$
$67$ $$(T - 880)^{2}$$
$71$ $$T^{2}$$
$73$ $$(T + 1190)^{2}$$
$79$ $$T^{2} - 1190700$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 1330)^{2}$$