# Properties

 Label 2304.4.a.bm Level $2304$ Weight $4$ Character orbit 2304.a Self dual yes Analytic conductor $135.940$ Analytic rank $0$ Dimension $2$ CM no 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: $$0$$ 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: $$2$$ Twist minimal: no (minimal twist has level 576) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \beta q^{5} + \beta q^{7}+O(q^{10})$$ q + 4*b * q^5 + b * q^7 $$q + 4 \beta q^{5} + \beta q^{7} + 48 q^{11} - 6 \beta q^{13} + 96 q^{17} + 40 q^{19} + 32 \beta q^{23} + 67 q^{25} - 4 \beta q^{29} - 59 \beta q^{31} + 48 q^{35} + 86 \beta q^{37} + 288 q^{41} - 152 q^{43} + 160 \beta q^{47} - 331 q^{49} - 52 \beta q^{53} + 192 \beta q^{55} - 480 q^{59} - 218 \beta q^{61} - 288 q^{65} + 848 q^{67} + 256 \beta q^{71} + 538 q^{73} + 48 \beta q^{77} - 291 \beta q^{79} - 432 q^{83} + 384 \beta q^{85} - 1344 q^{89} - 72 q^{91} + 160 \beta q^{95} - 590 q^{97} +O(q^{100})$$ q + 4*b * q^5 + b * q^7 + 48 * q^11 - 6*b * q^13 + 96 * q^17 + 40 * q^19 + 32*b * q^23 + 67 * q^25 - 4*b * q^29 - 59*b * q^31 + 48 * q^35 + 86*b * q^37 + 288 * q^41 - 152 * q^43 + 160*b * q^47 - 331 * q^49 - 52*b * q^53 + 192*b * q^55 - 480 * q^59 - 218*b * q^61 - 288 * q^65 + 848 * q^67 + 256*b * q^71 + 538 * q^73 + 48*b * q^77 - 291*b * q^79 - 432 * q^83 + 384*b * q^85 - 1344 * q^89 - 72 * q^91 + 160*b * q^95 - 590 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 96 q^{11} + 192 q^{17} + 80 q^{19} + 134 q^{25} + 96 q^{35} + 576 q^{41} - 304 q^{43} - 662 q^{49} - 960 q^{59} - 576 q^{65} + 1696 q^{67} + 1076 q^{73} - 864 q^{83} - 2688 q^{89} - 144 q^{91} - 1180 q^{97}+O(q^{100})$$ 2 * q + 96 * q^11 + 192 * q^17 + 80 * q^19 + 134 * q^25 + 96 * q^35 + 576 * q^41 - 304 * q^43 - 662 * q^49 - 960 * q^59 - 576 * q^65 + 1696 * q^67 + 1076 * q^73 - 864 * q^83 - 2688 * q^89 - 144 * q^91 - 1180 * 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 −13.8564 0 −3.46410 0 0 0
1.2 0 0 0 13.8564 0 3.46410 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 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{2} - 192$$ T5^2 - 192 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{11} - 48$$ T11 - 48 $$T_{13}^{2} - 432$$ T13^2 - 432 $$T_{17} - 96$$ T17 - 96 $$T_{19} - 40$$ T19 - 40

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 192$$
$7$ $$T^{2} - 12$$
$11$ $$(T - 48)^{2}$$
$13$ $$T^{2} - 432$$
$17$ $$(T - 96)^{2}$$
$19$ $$(T - 40)^{2}$$
$23$ $$T^{2} - 12288$$
$29$ $$T^{2} - 192$$
$31$ $$T^{2} - 41772$$
$37$ $$T^{2} - 88752$$
$41$ $$(T - 288)^{2}$$
$43$ $$(T + 152)^{2}$$
$47$ $$T^{2} - 307200$$
$53$ $$T^{2} - 32448$$
$59$ $$(T + 480)^{2}$$
$61$ $$T^{2} - 570288$$
$67$ $$(T - 848)^{2}$$
$71$ $$T^{2} - 786432$$
$73$ $$(T - 538)^{2}$$
$79$ $$T^{2} - 1016172$$
$83$ $$(T + 432)^{2}$$
$89$ $$(T + 1344)^{2}$$
$97$ $$(T + 590)^{2}$$