# Properties

 Label 1152.4.d.g Level $1152$ Weight $4$ Character orbit 1152.d Analytic conductor $67.970$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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 $$\beta = 4i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} + 12 q^{7}+O(q^{10})$$ q + 2*b * q^5 + 12 * q^7 $$q + 2 \beta q^{5} + 12 q^{7} + 3 \beta q^{11} - 5 \beta q^{13} - 62 q^{17} - 27 \beta q^{19} + 72 q^{23} + 61 q^{25} - 32 \beta q^{29} - 204 q^{31} + 24 \beta q^{35} - 57 \beta q^{37} + 22 q^{41} - 51 \beta q^{43} + 600 q^{47} - 199 q^{49} - 64 \beta q^{53} - 96 q^{55} + 207 \beta q^{59} + 21 \beta q^{61} + 160 q^{65} - 87 \beta q^{67} - 456 q^{71} + 822 q^{73} + 36 \beta q^{77} - 1356 q^{79} + 27 \beta q^{83} - 124 \beta q^{85} + 938 q^{89} - 60 \beta q^{91} + 864 q^{95} + 1278 q^{97} +O(q^{100})$$ q + 2*b * q^5 + 12 * q^7 + 3*b * q^11 - 5*b * q^13 - 62 * q^17 - 27*b * q^19 + 72 * q^23 + 61 * q^25 - 32*b * q^29 - 204 * q^31 + 24*b * q^35 - 57*b * q^37 + 22 * q^41 - 51*b * q^43 + 600 * q^47 - 199 * q^49 - 64*b * q^53 - 96 * q^55 + 207*b * q^59 + 21*b * q^61 + 160 * q^65 - 87*b * q^67 - 456 * q^71 + 822 * q^73 + 36*b * q^77 - 1356 * q^79 + 27*b * q^83 - 124*b * q^85 + 938 * q^89 - 60*b * q^91 + 864 * q^95 + 1278 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 24 q^{7}+O(q^{10})$$ 2 * q + 24 * q^7 $$2 q + 24 q^{7} - 124 q^{17} + 144 q^{23} + 122 q^{25} - 408 q^{31} + 44 q^{41} + 1200 q^{47} - 398 q^{49} - 192 q^{55} + 320 q^{65} - 912 q^{71} + 1644 q^{73} - 2712 q^{79} + 1876 q^{89} + 1728 q^{95} + 2556 q^{97}+O(q^{100})$$ 2 * q + 24 * q^7 - 124 * q^17 + 144 * q^23 + 122 * q^25 - 408 * q^31 + 44 * q^41 + 1200 * q^47 - 398 * q^49 - 192 * q^55 + 320 * q^65 - 912 * q^71 + 1644 * q^73 - 2712 * q^79 + 1876 * q^89 + 1728 * q^95 + 2556 * q^97

## 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
 − 1.00000i 1.00000i
0 0 0 8.00000i 0 12.0000 0 0 0
577.2 0 0 0 8.00000i 0 12.0000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.4.d.g 2
3.b odd 2 1 384.4.d.b yes 2
4.b odd 2 1 1152.4.d.b 2
8.b even 2 1 inner 1152.4.d.g 2
8.d odd 2 1 1152.4.d.b 2
12.b even 2 1 384.4.d.a 2
16.e even 4 1 2304.4.a.e 1
16.e even 4 1 2304.4.a.k 1
16.f odd 4 1 2304.4.a.f 1
16.f odd 4 1 2304.4.a.l 1
24.f even 2 1 384.4.d.a 2
24.h odd 2 1 384.4.d.b yes 2
48.i odd 4 1 768.4.a.b 1
48.i odd 4 1 768.4.a.c 1
48.k even 4 1 768.4.a.a 1
48.k even 4 1 768.4.a.d 1

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

 $$T_{5}^{2} + 64$$ T5^2 + 64 $$T_{7} - 12$$ T7 - 12 $$T_{17} + 62$$ T17 + 62

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 64$$
$7$ $$(T - 12)^{2}$$
$11$ $$T^{2} + 144$$
$13$ $$T^{2} + 400$$
$17$ $$(T + 62)^{2}$$
$19$ $$T^{2} + 11664$$
$23$ $$(T - 72)^{2}$$
$29$ $$T^{2} + 16384$$
$31$ $$(T + 204)^{2}$$
$37$ $$T^{2} + 51984$$
$41$ $$(T - 22)^{2}$$
$43$ $$T^{2} + 41616$$
$47$ $$(T - 600)^{2}$$
$53$ $$T^{2} + 65536$$
$59$ $$T^{2} + 685584$$
$61$ $$T^{2} + 7056$$
$67$ $$T^{2} + 121104$$
$71$ $$(T + 456)^{2}$$
$73$ $$(T - 822)^{2}$$
$79$ $$(T + 1356)^{2}$$
$83$ $$T^{2} + 11664$$
$89$ $$(T - 938)^{2}$$
$97$ $$(T - 1278)^{2}$$