# Properties

 Label 576.4.d.f Level $576$ Weight $4$ Character orbit 576.d Analytic conductor $33.985$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(289,576)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("576.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 576.d (of order $$2$$, degree $$1$$, 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: $$33.9851001633$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 192) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{5} + \beta_{2} q^{7}+O(q^{10})$$ q - b3 * q^5 + b2 * q^7 $$q - \beta_{3} q^{5} + \beta_{2} q^{7} + 12 \beta_1 q^{11} - 4 \beta_{3} q^{13} + 42 q^{17} - 23 \beta_1 q^{19} + 2 \beta_{2} q^{23} - 271 q^{25} - \beta_{3} q^{29} + 7 \beta_{2} q^{31} + 99 \beta_1 q^{35} + 10 \beta_{3} q^{37} + 6 q^{41} + 23 \beta_1 q^{43} + 2 \beta_{2} q^{47} + 53 q^{49} + 25 \beta_{3} q^{53} - 48 \beta_{2} q^{55} + 129 \beta_1 q^{59} + 18 \beta_{3} q^{61} - 1584 q^{65} + 131 \beta_1 q^{67} - 50 \beta_{2} q^{71} - 430 q^{73} - 48 \beta_{3} q^{77} + 59 \beta_{2} q^{79} - 108 \beta_1 q^{83} - 42 \beta_{3} q^{85} - 630 q^{89} + 396 \beta_1 q^{91} + 92 \beta_{2} q^{95} + 862 q^{97}+O(q^{100})$$ q - b3 * q^5 + b2 * q^7 + 12*b1 * q^11 - 4*b3 * q^13 + 42 * q^17 - 23*b1 * q^19 + 2*b2 * q^23 - 271 * q^25 - b3 * q^29 + 7*b2 * q^31 + 99*b1 * q^35 + 10*b3 * q^37 + 6 * q^41 + 23*b1 * q^43 + 2*b2 * q^47 + 53 * q^49 + 25*b3 * q^53 - 48*b2 * q^55 + 129*b1 * q^59 + 18*b3 * q^61 - 1584 * q^65 + 131*b1 * q^67 - 50*b2 * q^71 - 430 * q^73 - 48*b3 * q^77 + 59*b2 * q^79 - 108*b1 * q^83 - 42*b3 * q^85 - 630 * q^89 + 396*b1 * q^91 + 92*b2 * q^95 + 862 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 168 q^{17} - 1084 q^{25} + 24 q^{41} + 212 q^{49} - 6336 q^{65} - 1720 q^{73} - 2520 q^{89} + 3448 q^{97}+O(q^{100})$$ 4 * q + 168 * q^17 - 1084 * q^25 + 24 * q^41 + 212 * q^49 - 6336 * q^65 - 1720 * q^73 - 2520 * q^89 + 3448 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( -4\nu^{3} + 8\nu ) / 3$$ (-4*v^3 + 8*v) / 3 $$\beta_{2}$$ $$=$$ $$-2\nu^{3} + 16\nu$$ -2*v^3 + 16*v $$\beta_{3}$$ $$=$$ $$12\nu^{2} - 30$$ 12*v^2 - 30
 $$\nu$$ $$=$$ $$( 2\beta_{2} - 3\beta_1 ) / 24$$ (2*b2 - 3*b1) / 24 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 30 ) / 12$$ (b3 + 30) / 12 $$\nu^{3}$$ $$=$$ $$( \beta_{2} - 6\beta_1 ) / 6$$ (b2 - 6*b1) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −1.65831 − 0.500000i 1.65831 + 0.500000i −1.65831 + 0.500000i 1.65831 − 0.500000i
0 0 0 19.8997i 0 −19.8997 0 0 0
289.2 0 0 0 19.8997i 0 19.8997 0 0 0
289.3 0 0 0 19.8997i 0 −19.8997 0 0 0
289.4 0 0 0 19.8997i 0 19.8997 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 inner
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 576.4.d.f 4
3.b odd 2 1 192.4.d.b 4
4.b odd 2 1 inner 576.4.d.f 4
8.b even 2 1 inner 576.4.d.f 4
8.d odd 2 1 inner 576.4.d.f 4
12.b even 2 1 192.4.d.b 4
16.e even 4 1 2304.4.a.y 2
16.e even 4 1 2304.4.a.bl 2
16.f odd 4 1 2304.4.a.y 2
16.f odd 4 1 2304.4.a.bl 2
24.f even 2 1 192.4.d.b 4
24.h odd 2 1 192.4.d.b 4
48.i odd 4 1 768.4.a.i 2
48.i odd 4 1 768.4.a.l 2
48.k even 4 1 768.4.a.i 2
48.k even 4 1 768.4.a.l 2

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(576, [\chi])$$:

 $$T_{5}^{2} + 396$$ T5^2 + 396 $$T_{7}^{2} - 396$$ T7^2 - 396 $$T_{17} - 42$$ T17 - 42

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 396)^{2}$$
$7$ $$(T^{2} - 396)^{2}$$
$11$ $$(T^{2} + 2304)^{2}$$
$13$ $$(T^{2} + 6336)^{2}$$
$17$ $$(T - 42)^{4}$$
$19$ $$(T^{2} + 8464)^{2}$$
$23$ $$(T^{2} - 1584)^{2}$$
$29$ $$(T^{2} + 396)^{2}$$
$31$ $$(T^{2} - 19404)^{2}$$
$37$ $$(T^{2} + 39600)^{2}$$
$41$ $$(T - 6)^{4}$$
$43$ $$(T^{2} + 8464)^{2}$$
$47$ $$(T^{2} - 1584)^{2}$$
$53$ $$(T^{2} + 247500)^{2}$$
$59$ $$(T^{2} + 266256)^{2}$$
$61$ $$(T^{2} + 128304)^{2}$$
$67$ $$(T^{2} + 274576)^{2}$$
$71$ $$(T^{2} - 990000)^{2}$$
$73$ $$(T + 430)^{4}$$
$79$ $$(T^{2} - 1378476)^{2}$$
$83$ $$(T^{2} + 186624)^{2}$$
$89$ $$(T + 630)^{4}$$
$97$ $$(T - 862)^{4}$$