# Properties

 Label 576.4.d.g 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: $$2^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q - 3 \beta_{2} q^{5} + \beta_1 q^{7}+O(q^{10})$$ q - 3*b2 * q^5 + b1 * q^7 $$q - 3 \beta_{2} q^{5} + \beta_1 q^{7} + 16 \beta_{2} q^{13} + 90 q^{17} + 29 \beta_{3} q^{19} - 30 \beta_1 q^{23} + 17 q^{25} - 75 \beta_{2} q^{29} + 87 \beta_1 q^{31} - 9 \beta_{3} q^{35} + 10 \beta_{2} q^{37} + 54 q^{41} - 5 \beta_{3} q^{43} + 114 \beta_1 q^{47} - 331 q^{49} - 141 \beta_{2} q^{53} + 81 \beta_{3} q^{59} - 166 \beta_{2} q^{61} + 576 q^{65} - 29 \beta_{3} q^{67} + 318 \beta_1 q^{71} + 1106 q^{73} + 43 \beta_1 q^{79} + 288 \beta_{3} q^{83} - 270 \beta_{2} q^{85} - 918 q^{89} + 48 \beta_{3} q^{91} + 348 \beta_1 q^{95} + 190 q^{97}+O(q^{100})$$ q - 3*b2 * q^5 + b1 * q^7 + 16*b2 * q^13 + 90 * q^17 + 29*b3 * q^19 - 30*b1 * q^23 + 17 * q^25 - 75*b2 * q^29 + 87*b1 * q^31 - 9*b3 * q^35 + 10*b2 * q^37 + 54 * q^41 - 5*b3 * q^43 + 114*b1 * q^47 - 331 * q^49 - 141*b2 * q^53 + 81*b3 * q^59 - 166*b2 * q^61 + 576 * q^65 - 29*b3 * q^67 + 318*b1 * q^71 + 1106 * q^73 + 43*b1 * q^79 + 288*b3 * q^83 - 270*b2 * q^85 - 918 * q^89 + 48*b3 * q^91 + 348*b1 * q^95 + 190 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 360 q^{17} + 68 q^{25} + 216 q^{41} - 1324 q^{49} + 2304 q^{65} + 4424 q^{73} - 3672 q^{89} + 760 q^{97}+O(q^{100})$$ 4 * q + 360 * q^17 + 68 * q^25 + 216 * q^41 - 1324 * q^49 + 2304 * q^65 + 4424 * q^73 - 3672 * q^89 + 760 * q^97

Basis of coefficient ring

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

## 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 10.3923i 0 −3.46410 0 0 0
289.2 0 0 0 10.3923i 0 3.46410 0 0 0
289.3 0 0 0 10.3923i 0 −3.46410 0 0 0
289.4 0 0 0 10.3923i 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.g 4
3.b odd 2 1 192.4.d.a 4
4.b odd 2 1 inner 576.4.d.g 4
8.b even 2 1 inner 576.4.d.g 4
8.d odd 2 1 inner 576.4.d.g 4
12.b even 2 1 192.4.d.a 4
16.e even 4 1 2304.4.a.be 2
16.e even 4 1 2304.4.a.bg 2
16.f odd 4 1 2304.4.a.be 2
16.f odd 4 1 2304.4.a.bg 2
24.f even 2 1 192.4.d.a 4
24.h odd 2 1 192.4.d.a 4
48.i odd 4 1 768.4.a.g 2
48.i odd 4 1 768.4.a.n 2
48.k even 4 1 768.4.a.g 2
48.k even 4 1 768.4.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.4.d.a 4 3.b odd 2 1
192.4.d.a 4 12.b even 2 1
192.4.d.a 4 24.f even 2 1
192.4.d.a 4 24.h odd 2 1
576.4.d.g 4 1.a even 1 1 trivial
576.4.d.g 4 4.b odd 2 1 inner
576.4.d.g 4 8.b even 2 1 inner
576.4.d.g 4 8.d odd 2 1 inner
768.4.a.g 2 48.i odd 4 1
768.4.a.g 2 48.k even 4 1
768.4.a.n 2 48.i odd 4 1
768.4.a.n 2 48.k even 4 1
2304.4.a.be 2 16.e even 4 1
2304.4.a.be 2 16.f odd 4 1
2304.4.a.bg 2 16.e even 4 1
2304.4.a.bg 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 $$S_{4}^{\mathrm{new}}(576, [\chi])$$:

 $$T_{5}^{2} + 108$$ T5^2 + 108 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{17} - 90$$ T17 - 90

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 108)^{2}$$
$7$ $$(T^{2} - 12)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 3072)^{2}$$
$17$ $$(T - 90)^{4}$$
$19$ $$(T^{2} + 13456)^{2}$$
$23$ $$(T^{2} - 10800)^{2}$$
$29$ $$(T^{2} + 67500)^{2}$$
$31$ $$(T^{2} - 90828)^{2}$$
$37$ $$(T^{2} + 1200)^{2}$$
$41$ $$(T - 54)^{4}$$
$43$ $$(T^{2} + 400)^{2}$$
$47$ $$(T^{2} - 155952)^{2}$$
$53$ $$(T^{2} + 238572)^{2}$$
$59$ $$(T^{2} + 104976)^{2}$$
$61$ $$(T^{2} + 330672)^{2}$$
$67$ $$(T^{2} + 13456)^{2}$$
$71$ $$(T^{2} - 1213488)^{2}$$
$73$ $$(T - 1106)^{4}$$
$79$ $$(T^{2} - 22188)^{2}$$
$83$ $$(T^{2} + 1327104)^{2}$$
$89$ $$(T + 918)^{4}$$
$97$ $$(T - 190)^{4}$$