# Properties

 Label 1152.2.f.a Level $1152$ Weight $2$ Character orbit 1152.f Analytic conductor $9.199$ 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] = [1152,2,Mod(575,1152)]

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([1, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1152.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.f (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: $$9.19876631285$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{5} - 2 \beta_{2} q^{7}+O(q^{10})$$ q - b3 * q^5 - 2*b2 * q^7 $$q - \beta_{3} q^{5} - 2 \beta_{2} q^{7} + 2 \beta_1 q^{11} - \beta_1 q^{13} - \beta_{2} q^{17} - 4 \beta_{3} q^{19} - 4 q^{23} - 3 q^{25} + 5 \beta_{3} q^{29} - 6 \beta_{2} q^{31} + 2 \beta_1 q^{35} + 4 \beta_1 q^{37} + 3 \beta_{2} q^{41} - 8 \beta_{3} q^{43} - 12 q^{47} - q^{49} - 9 \beta_{3} q^{53} - 4 \beta_{2} q^{55} - 4 \beta_1 q^{61} + 2 \beta_{2} q^{65} - 4 \beta_{3} q^{67} - 4 q^{71} + 8 q^{73} + 8 \beta_{3} q^{77} - 2 \beta_{2} q^{79} + 6 \beta_1 q^{83} + \beta_1 q^{85} - 11 \beta_{2} q^{89} - 4 \beta_{3} q^{91} + 8 q^{95}+O(q^{100})$$ q - b3 * q^5 - 2*b2 * q^7 + 2*b1 * q^11 - b1 * q^13 - b2 * q^17 - 4*b3 * q^19 - 4 * q^23 - 3 * q^25 + 5*b3 * q^29 - 6*b2 * q^31 + 2*b1 * q^35 + 4*b1 * q^37 + 3*b2 * q^41 - 8*b3 * q^43 - 12 * q^47 - q^49 - 9*b3 * q^53 - 4*b2 * q^55 - 4*b1 * q^61 + 2*b2 * q^65 - 4*b3 * q^67 - 4 * q^71 + 8 * q^73 + 8*b3 * q^77 - 2*b2 * q^79 + 6*b1 * q^83 + b1 * q^85 - 11*b2 * q^89 - 4*b3 * q^91 + 8 * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 16 q^{23} - 12 q^{25} - 48 q^{47} - 4 q^{49} - 16 q^{71} + 32 q^{73} + 32 q^{95}+O(q^{100})$$ 4 * q - 16 * q^23 - 12 * q^25 - 48 * q^47 - 4 * q^49 - 16 * q^71 + 32 * q^73 + 32 * q^95

Basis of coefficient ring

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 −1.41421 0 2.82843i 0 0 0
575.2 0 0 0 −1.41421 0 2.82843i 0 0 0
575.3 0 0 0 1.41421 0 2.82843i 0 0 0
575.4 0 0 0 1.41421 0 2.82843i 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
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.f.a 4
3.b odd 2 1 1152.2.f.d yes 4
4.b odd 2 1 1152.2.f.d yes 4
8.b even 2 1 inner 1152.2.f.a 4
8.d odd 2 1 1152.2.f.d yes 4
12.b even 2 1 inner 1152.2.f.a 4
16.e even 4 1 2304.2.c.a 2
16.e even 4 1 2304.2.c.h 2
16.f odd 4 1 2304.2.c.b 2
16.f odd 4 1 2304.2.c.g 2
24.f even 2 1 inner 1152.2.f.a 4
24.h odd 2 1 1152.2.f.d yes 4
48.i odd 4 1 2304.2.c.b 2
48.i odd 4 1 2304.2.c.g 2
48.k even 4 1 2304.2.c.a 2
48.k even 4 1 2304.2.c.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.f.a 4 1.a even 1 1 trivial
1152.2.f.a 4 8.b even 2 1 inner
1152.2.f.a 4 12.b even 2 1 inner
1152.2.f.a 4 24.f even 2 1 inner
1152.2.f.d yes 4 3.b odd 2 1
1152.2.f.d yes 4 4.b odd 2 1
1152.2.f.d yes 4 8.d odd 2 1
1152.2.f.d yes 4 24.h odd 2 1
2304.2.c.a 2 16.e even 4 1
2304.2.c.a 2 48.k even 4 1
2304.2.c.b 2 16.f odd 4 1
2304.2.c.b 2 48.i odd 4 1
2304.2.c.g 2 16.f odd 4 1
2304.2.c.g 2 48.i odd 4 1
2304.2.c.h 2 16.e even 4 1
2304.2.c.h 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 $$S_{2}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{2} - 2$$ T5^2 - 2 $$T_{7}^{2} + 8$$ T7^2 + 8 $$T_{23} + 4$$ T23 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 2)^{2}$$
$7$ $$(T^{2} + 8)^{2}$$
$11$ $$(T^{2} + 16)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$(T^{2} + 2)^{2}$$
$19$ $$(T^{2} - 32)^{2}$$
$23$ $$(T + 4)^{4}$$
$29$ $$(T^{2} - 50)^{2}$$
$31$ $$(T^{2} + 72)^{2}$$
$37$ $$(T^{2} + 64)^{2}$$
$41$ $$(T^{2} + 18)^{2}$$
$43$ $$(T^{2} - 128)^{2}$$
$47$ $$(T + 12)^{4}$$
$53$ $$(T^{2} - 162)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 64)^{2}$$
$67$ $$(T^{2} - 32)^{2}$$
$71$ $$(T + 4)^{4}$$
$73$ $$(T - 8)^{4}$$
$79$ $$(T^{2} + 8)^{2}$$
$83$ $$(T^{2} + 144)^{2}$$
$89$ $$(T^{2} + 242)^{2}$$
$97$ $$T^{4}$$