Properties

 Label 1152.4.d.n Level $1152$ Weight $4$ Character orbit 1152.d Analytic conductor $67.970$ 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] = [1152,4,Mod(577,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, 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("1152.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1152.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: $$67.9702003266$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 384) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + 5 \beta_{3} q^{7}+O(q^{10})$$ q + b2 * q^5 + 5*b3 * q^7 $$q + \beta_{2} q^{5} + 5 \beta_{3} q^{7} - 5 \beta_1 q^{11} + 14 \beta_{2} q^{13} + 34 q^{17} - 13 \beta_1 q^{19} - 22 \beta_{3} q^{23} + 117 q^{25} - 71 \beta_{2} q^{29} + 39 \beta_{3} q^{31} + 10 \beta_1 q^{35} + 96 \beta_{2} q^{37} - 26 q^{41} + 63 \beta_1 q^{43} + 122 \beta_{3} q^{47} - 143 q^{49} - 241 \beta_{2} q^{53} + 20 \beta_{3} q^{55} - 91 \beta_1 q^{59} + 260 \beta_{2} q^{61} - 112 q^{65} - 157 \beta_1 q^{67} + 118 \beta_{3} q^{71} - 338 q^{73} - 100 \beta_{2} q^{77} + 279 \beta_{3} q^{79} + 259 \beta_1 q^{83} + 34 \beta_{2} q^{85} + 234 q^{89} + 140 \beta_1 q^{91} + 52 \beta_{3} q^{95} - 178 q^{97}+O(q^{100})$$ q + b2 * q^5 + 5*b3 * q^7 - 5*b1 * q^11 + 14*b2 * q^13 + 34 * q^17 - 13*b1 * q^19 - 22*b3 * q^23 + 117 * q^25 - 71*b2 * q^29 + 39*b3 * q^31 + 10*b1 * q^35 + 96*b2 * q^37 - 26 * q^41 + 63*b1 * q^43 + 122*b3 * q^47 - 143 * q^49 - 241*b2 * q^53 + 20*b3 * q^55 - 91*b1 * q^59 + 260*b2 * q^61 - 112 * q^65 - 157*b1 * q^67 + 118*b3 * q^71 - 338 * q^73 - 100*b2 * q^77 + 279*b3 * q^79 + 259*b1 * q^83 + 34*b2 * q^85 + 234 * q^89 + 140*b1 * q^91 + 52*b3 * q^95 - 178 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 136 q^{17} + 468 q^{25} - 104 q^{41} - 572 q^{49} - 448 q^{65} - 1352 q^{73} + 936 q^{89} - 712 q^{97}+O(q^{100})$$ 4 * q + 136 * q^17 + 468 * q^25 - 104 * q^41 - 572 * q^49 - 448 * q^65 - 1352 * q^73 + 936 * q^89 - 712 * q^97

Basis of coefficient ring

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 0 0 2.82843i 0 −14.1421 0 0 0
577.2 0 0 0 2.82843i 0 14.1421 0 0 0
577.3 0 0 0 2.82843i 0 −14.1421 0 0 0
577.4 0 0 0 2.82843i 0 14.1421 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 1152.4.d.n 4
3.b odd 2 1 384.4.d.d 4
4.b odd 2 1 inner 1152.4.d.n 4
8.b even 2 1 inner 1152.4.d.n 4
8.d odd 2 1 inner 1152.4.d.n 4
12.b even 2 1 384.4.d.d 4
16.e even 4 1 2304.4.a.bb 2
16.e even 4 1 2304.4.a.bh 2
16.f odd 4 1 2304.4.a.bb 2
16.f odd 4 1 2304.4.a.bh 2
24.f even 2 1 384.4.d.d 4
24.h odd 2 1 384.4.d.d 4
48.i odd 4 1 768.4.a.h 2
48.i odd 4 1 768.4.a.m 2
48.k even 4 1 768.4.a.h 2
48.k even 4 1 768.4.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.d 4 3.b odd 2 1
384.4.d.d 4 12.b even 2 1
384.4.d.d 4 24.f even 2 1
384.4.d.d 4 24.h odd 2 1
768.4.a.h 2 48.i odd 4 1
768.4.a.h 2 48.k even 4 1
768.4.a.m 2 48.i odd 4 1
768.4.a.m 2 48.k even 4 1
1152.4.d.n 4 1.a even 1 1 trivial
1152.4.d.n 4 4.b odd 2 1 inner
1152.4.d.n 4 8.b even 2 1 inner
1152.4.d.n 4 8.d odd 2 1 inner
2304.4.a.bb 2 16.e even 4 1
2304.4.a.bb 2 16.f odd 4 1
2304.4.a.bh 2 16.e even 4 1
2304.4.a.bh 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 $$S_{4}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{2} + 8$$ T5^2 + 8 $$T_{7}^{2} - 200$$ T7^2 - 200 $$T_{17} - 34$$ T17 - 34

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 8)^{2}$$
$7$ $$(T^{2} - 200)^{2}$$
$11$ $$(T^{2} + 400)^{2}$$
$13$ $$(T^{2} + 1568)^{2}$$
$17$ $$(T - 34)^{4}$$
$19$ $$(T^{2} + 2704)^{2}$$
$23$ $$(T^{2} - 3872)^{2}$$
$29$ $$(T^{2} + 40328)^{2}$$
$31$ $$(T^{2} - 12168)^{2}$$
$37$ $$(T^{2} + 73728)^{2}$$
$41$ $$(T + 26)^{4}$$
$43$ $$(T^{2} + 63504)^{2}$$
$47$ $$(T^{2} - 119072)^{2}$$
$53$ $$(T^{2} + 464648)^{2}$$
$59$ $$(T^{2} + 132496)^{2}$$
$61$ $$(T^{2} + 540800)^{2}$$
$67$ $$(T^{2} + 394384)^{2}$$
$71$ $$(T^{2} - 111392)^{2}$$
$73$ $$(T + 338)^{4}$$
$79$ $$(T^{2} - 622728)^{2}$$
$83$ $$(T^{2} + 1073296)^{2}$$
$89$ $$(T - 234)^{4}$$
$97$ $$(T + 178)^{4}$$