# Properties

 Label 1152.4.d.i Level $1152$ Weight $4$ Character orbit 1152.d Analytic conductor $67.970$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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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: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1) q^{5} + (\beta_{3} - 8) q^{7}+O(q^{10})$$ q + (b2 - b1) * q^5 + (b3 - 8) * q^7 $$q + (\beta_{2} - \beta_1) q^{5} + (\beta_{3} - 8) q^{7} + (4 \beta_{2} + \beta_1) q^{11} + (2 \beta_{2} + 9 \beta_1) q^{13} + (4 \beta_{3} + 18) q^{17} + (4 \beta_{2} - 17 \beta_1) q^{19} + (2 \beta_{3} + 128) q^{23} + (8 \beta_{3} - 99) q^{25} + (9 \beta_{2} - 19 \beta_1) q^{29} + ( - 13 \beta_{3} - 40) q^{31} + ( - 12 \beta_{2} + 60 \beta_1) q^{35} + (4 \beta_{2} - 17 \beta_1) q^{37} + (20 \beta_{3} - 218) q^{41} + (4 \beta_{2} - 89 \beta_1) q^{43} + ( - 14 \beta_{3} + 112) q^{47} + ( - 16 \beta_{3} - 71) q^{49} + (15 \beta_{2} + 43 \beta_1) q^{53} + (12 \beta_{3} - 816) q^{55} + 81 \beta_1 q^{59} + 81 \beta_1 q^{61} + ( - 28 \beta_{3} - 272) q^{65} + ( - 48 \beta_{2} - 57 \beta_1) q^{67} + ( - 2 \beta_{3} + 1024) q^{71} + (48 \beta_{3} - 330) q^{73} + ( - 28 \beta_{2} + 200 \beta_1) q^{77} + (59 \beta_{3} + 248) q^{79} + (28 \beta_{2} + 97 \beta_1) q^{83} + (2 \beta_{2} + 190 \beta_1) q^{85} + ( - 8 \beta_{3} + 266) q^{89} + (20 \beta_{2} + 32 \beta_1) q^{91} + (84 \beta_{3} - 1104) q^{95} + ( - 88 \beta_{3} - 610) q^{97}+O(q^{100})$$ q + (b2 - b1) * q^5 + (b3 - 8) * q^7 + (4*b2 + b1) * q^11 + (2*b2 + 9*b1) * q^13 + (4*b3 + 18) * q^17 + (4*b2 - 17*b1) * q^19 + (2*b3 + 128) * q^23 + (8*b3 - 99) * q^25 + (9*b2 - 19*b1) * q^29 + (-13*b3 - 40) * q^31 + (-12*b2 + 60*b1) * q^35 + (4*b2 - 17*b1) * q^37 + (20*b3 - 218) * q^41 + (4*b2 - 89*b1) * q^43 + (-14*b3 + 112) * q^47 + (-16*b3 - 71) * q^49 + (15*b2 + 43*b1) * q^53 + (12*b3 - 816) * q^55 + 81*b1 * q^59 + 81*b1 * q^61 + (-28*b3 - 272) * q^65 + (-48*b2 - 57*b1) * q^67 + (-2*b3 + 1024) * q^71 + (48*b3 - 330) * q^73 + (-28*b2 + 200*b1) * q^77 + (59*b3 + 248) * q^79 + (28*b2 + 97*b1) * q^83 + (2*b2 + 190*b1) * q^85 + (-8*b3 + 266) * q^89 + (20*b2 + 32*b1) * q^91 + (84*b3 - 1104) * q^95 + (-88*b3 - 610) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 32 q^{7}+O(q^{10})$$ 4 * q - 32 * q^7 $$4 q - 32 q^{7} + 72 q^{17} + 512 q^{23} - 396 q^{25} - 160 q^{31} - 872 q^{41} + 448 q^{47} - 284 q^{49} - 3264 q^{55} - 1088 q^{65} + 4096 q^{71} - 1320 q^{73} + 992 q^{79} + 1064 q^{89} - 4416 q^{95} - 2440 q^{97}+O(q^{100})$$ 4 * q - 32 * q^7 + 72 * q^17 + 512 * q^23 - 396 * q^25 - 160 * q^31 - 872 * q^41 + 448 * q^47 - 284 * q^49 - 3264 * q^55 - 1088 * q^65 + 4096 * q^71 - 1320 * q^73 + 992 * q^79 + 1064 * q^89 - 4416 * q^95 - 2440 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 4\nu^{3} + 16\nu ) / 3$$ (4*v^3 + 16*v) / 3 $$\beta_{2}$$ $$=$$ $$( 4\nu^{3} + 40\nu ) / 3$$ (4*v^3 + 40*v) / 3 $$\beta_{3}$$ $$=$$ $$8\nu^{2} + 28$$ 8*v^2 + 28
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 8$$ (b2 - b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 28 ) / 8$$ (b3 - 28) / 8 $$\nu^{3}$$ $$=$$ $$( -2\beta_{2} + 5\beta_1 ) / 4$$ (-2*b2 + 5*b1) / 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
 − 2.30278i − 1.30278i 1.30278i 2.30278i
0 0 0 18.4222i 0 −22.4222 0 0 0
577.2 0 0 0 10.4222i 0 6.42221 0 0 0
577.3 0 0 0 10.4222i 0 6.42221 0 0 0
577.4 0 0 0 18.4222i 0 −22.4222 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.i 4
3.b odd 2 1 384.4.d.c 4
4.b odd 2 1 1152.4.d.o 4
8.b even 2 1 inner 1152.4.d.i 4
8.d odd 2 1 1152.4.d.o 4
12.b even 2 1 384.4.d.e yes 4
16.e even 4 1 2304.4.a.t 2
16.e even 4 1 2304.4.a.bq 2
16.f odd 4 1 2304.4.a.s 2
16.f odd 4 1 2304.4.a.bp 2
24.f even 2 1 384.4.d.e yes 4
24.h odd 2 1 384.4.d.c 4
48.i odd 4 1 768.4.a.j 2
48.i odd 4 1 768.4.a.k 2
48.k even 4 1 768.4.a.e 2
48.k even 4 1 768.4.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.c 4 3.b odd 2 1
384.4.d.c 4 24.h odd 2 1
384.4.d.e yes 4 12.b even 2 1
384.4.d.e yes 4 24.f even 2 1
768.4.a.e 2 48.k even 4 1
768.4.a.j 2 48.i odd 4 1
768.4.a.k 2 48.i odd 4 1
768.4.a.p 2 48.k even 4 1
1152.4.d.i 4 1.a even 1 1 trivial
1152.4.d.i 4 8.b even 2 1 inner
1152.4.d.o 4 4.b odd 2 1
1152.4.d.o 4 8.d odd 2 1
2304.4.a.s 2 16.f odd 4 1
2304.4.a.t 2 16.e even 4 1
2304.4.a.bp 2 16.f odd 4 1
2304.4.a.bq 2 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{4} + 448T_{5}^{2} + 36864$$ T5^4 + 448*T5^2 + 36864 $$T_{7}^{2} + 16T_{7} - 144$$ T7^2 + 16*T7 - 144 $$T_{17}^{2} - 36T_{17} - 3004$$ T17^2 - 36*T17 - 3004

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 448 T^{2} + 36864$$
$7$ $$(T^{2} + 16 T - 144)^{2}$$
$11$ $$T^{4} + 6688 T^{2} + \cdots + 10969344$$
$13$ $$T^{4} + 4256 T^{2} + 215296$$
$17$ $$(T^{2} - 36 T - 3004)^{2}$$
$19$ $$T^{4} + 15904 T^{2} + \cdots + 1679616$$
$23$ $$(T^{2} - 256 T + 15552)^{2}$$
$29$ $$T^{4} + 45248 T^{2} + \cdots + 122589184$$
$31$ $$(T^{2} + 80 T - 33552)^{2}$$
$37$ $$T^{4} + 15904 T^{2} + \cdots + 1679616$$
$41$ $$(T^{2} + 436 T - 35676)^{2}$$
$43$ $$T^{4} + 260128 T^{2} + \cdots + 15229534464$$
$47$ $$(T^{2} - 224 T - 28224)^{2}$$
$53$ $$T^{4} + 152768 T^{2} + \cdots + 296390656$$
$59$ $$(T^{2} + 104976)^{2}$$
$61$ $$(T^{2} + 104976)^{2}$$
$67$ $$T^{4} + 1062432 T^{2} + \cdots + 182540853504$$
$71$ $$(T^{2} - 2048 T + 1047744)^{2}$$
$73$ $$(T^{2} + 660 T - 370332)^{2}$$
$79$ $$(T^{2} - 496 T - 662544)^{2}$$
$83$ $$T^{4} + 627232 T^{2} + \cdots + 156950784$$
$89$ $$(T^{2} - 532 T + 57444)^{2}$$
$97$ $$(T^{2} + 1220 T - 1238652)^{2}$$
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