# Properties

 Label 576.4.d.b 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: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{2} q^{5} + \beta_1 q^{7}+O(q^{10})$$ q - 2*b2 * q^5 + b1 * q^7 $$q - 2 \beta_{2} q^{5} + \beta_1 q^{7} + 6 \beta_{3} q^{11} + 3 \beta_{2} q^{13} - 96 q^{17} + 5 \beta_{3} q^{19} - 32 \beta_1 q^{23} - 67 q^{25} - 2 \beta_{2} q^{29} + 59 \beta_1 q^{31} - 6 \beta_{3} q^{35} + 43 \beta_{2} q^{37} + 288 q^{41} + 19 \beta_{3} q^{43} + 160 \beta_1 q^{47} - 331 q^{49} + 26 \beta_{2} q^{53} + 192 \beta_1 q^{55} - 60 \beta_{3} q^{59} + 109 \beta_{2} q^{61} + 288 q^{65} + 106 \beta_{3} q^{67} - 256 \beta_1 q^{71} - 538 q^{73} + 24 \beta_{2} q^{77} + 291 \beta_1 q^{79} + 54 \beta_{3} q^{83} + 192 \beta_{2} q^{85} - 1344 q^{89} + 9 \beta_{3} q^{91} + 160 \beta_1 q^{95} - 590 q^{97}+O(q^{100})$$ q - 2*b2 * q^5 + b1 * q^7 + 6*b3 * q^11 + 3*b2 * q^13 - 96 * q^17 + 5*b3 * q^19 - 32*b1 * q^23 - 67 * q^25 - 2*b2 * q^29 + 59*b1 * q^31 - 6*b3 * q^35 + 43*b2 * q^37 + 288 * q^41 + 19*b3 * q^43 + 160*b1 * q^47 - 331 * q^49 + 26*b2 * q^53 + 192*b1 * q^55 - 60*b3 * q^59 + 109*b2 * q^61 + 288 * q^65 + 106*b3 * q^67 - 256*b1 * q^71 - 538 * q^73 + 24*b2 * q^77 + 291*b1 * q^79 + 54*b3 * q^83 + 192*b2 * q^85 - 1344 * q^89 + 9*b3 * q^91 + 160*b1 * q^95 - 590 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 384 q^{17} - 268 q^{25} + 1152 q^{41} - 1324 q^{49} + 1152 q^{65} - 2152 q^{73} - 5376 q^{89} - 2360 q^{97}+O(q^{100})$$ 4 * q - 384 * q^17 - 268 * q^25 + 1152 * q^41 - 1324 * q^49 + 1152 * q^65 - 2152 * q^73 - 5376 * q^89 - 2360 * q^97

Basis of coefficient ring

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

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 13.8564i 0 −3.46410 0 0 0
289.2 0 0 0 13.8564i 0 3.46410 0 0 0
289.3 0 0 0 13.8564i 0 −3.46410 0 0 0
289.4 0 0 0 13.8564i 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

## 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.b 4
3.b odd 2 1 576.4.d.h yes 4
4.b odd 2 1 inner 576.4.d.b 4
8.b even 2 1 inner 576.4.d.b 4
8.d odd 2 1 inner 576.4.d.b 4
12.b even 2 1 576.4.d.h yes 4
16.e even 4 1 2304.4.a.w 2
16.e even 4 1 2304.4.a.bj 2
16.f odd 4 1 2304.4.a.w 2
16.f odd 4 1 2304.4.a.bj 2
24.f even 2 1 576.4.d.h yes 4
24.h odd 2 1 576.4.d.h yes 4
48.i odd 4 1 2304.4.a.z 2
48.i odd 4 1 2304.4.a.bm 2
48.k even 4 1 2304.4.a.z 2
48.k even 4 1 2304.4.a.bm 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 192$$ T5^2 + 192 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{17} + 96$$ T17 + 96

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 192)^{2}$$
$7$ $$(T^{2} - 12)^{2}$$
$11$ $$(T^{2} + 2304)^{2}$$
$13$ $$(T^{2} + 432)^{2}$$
$17$ $$(T + 96)^{4}$$
$19$ $$(T^{2} + 1600)^{2}$$
$23$ $$(T^{2} - 12288)^{2}$$
$29$ $$(T^{2} + 192)^{2}$$
$31$ $$(T^{2} - 41772)^{2}$$
$37$ $$(T^{2} + 88752)^{2}$$
$41$ $$(T - 288)^{4}$$
$43$ $$(T^{2} + 23104)^{2}$$
$47$ $$(T^{2} - 307200)^{2}$$
$53$ $$(T^{2} + 32448)^{2}$$
$59$ $$(T^{2} + 230400)^{2}$$
$61$ $$(T^{2} + 570288)^{2}$$
$67$ $$(T^{2} + 719104)^{2}$$
$71$ $$(T^{2} - 786432)^{2}$$
$73$ $$(T + 538)^{4}$$
$79$ $$(T^{2} - 1016172)^{2}$$
$83$ $$(T^{2} + 186624)^{2}$$
$89$ $$(T + 1344)^{4}$$
$97$ $$(T + 590)^{4}$$