# Properties

 Label 1152.4.d.p Level $1152$ Weight $4$ Character orbit 1152.d Analytic conductor $67.970$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.1534132224.8 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1$$ x^8 + 18*x^6 + 107*x^4 + 210*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{26}\cdot 3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{5} + \beta_{2} q^{7}+O(q^{10})$$ q - b4 * q^5 + b2 * q^7 $$q - \beta_{4} q^{5} + \beta_{2} q^{7} + ( - \beta_{3} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{4}) q^{13} + (\beta_{5} - 30) q^{17} + (3 \beta_{3} + \beta_1) q^{19} + (\beta_{7} + 3 \beta_{2}) q^{23} + ( - 2 \beta_{5} - 43) q^{25} + ( - 2 \beta_{6} + 5 \beta_{4}) q^{29} + (\beta_{7} - 4 \beta_{2}) q^{31} + (3 \beta_{3} - 10 \beta_1) q^{35} + ( - \beta_{6} - 17 \beta_{4}) q^{37} + (5 \beta_{5} + 102) q^{41} + (3 \beta_{3} - 11 \beta_1) q^{43} + (2 \beta_{7} - 12 \beta_{2}) q^{47} + ( - 8 \beta_{5} + 209) q^{49} + (2 \beta_{6} - 13 \beta_{4}) q^{53} + (\beta_{7} + 5 \beta_{2}) q^{55} + (16 \beta_{3} + 39 \beta_1) q^{59} + (5 \beta_{6} + 49 \beta_{4}) q^{61} + (9 \beta_{5} + 144) q^{65} + (12 \beta_{3} + 49 \beta_1) q^{67} + ( - 3 \beta_{7} + 27 \beta_{2}) q^{71} + ( - 8 \beta_{5} - 242) q^{73} + (12 \beta_{6} + 16 \beta_{4}) q^{77} + ( - 5 \beta_{7} + 6 \beta_{2}) q^{79} + (17 \beta_{3} - 71 \beta_1) q^{83} + (8 \beta_{6} - 26 \beta_{4}) q^{85} + (6 \beta_{5} - 918) q^{89} + (63 \beta_{3} - 108 \beta_1) q^{91} + ( - 7 \beta_{7} - 3 \beta_{2}) q^{95} + (2 \beta_{5} - 370) q^{97}+O(q^{100})$$ q - b4 * q^5 + b2 * q^7 + (-b3 + b1) * q^11 + (-b6 + b4) * q^13 + (b5 - 30) * q^17 + (3*b3 + b1) * q^19 + (b7 + 3*b2) * q^23 + (-2*b5 - 43) * q^25 + (-2*b6 + 5*b4) * q^29 + (b7 - 4*b2) * q^31 + (3*b3 - 10*b1) * q^35 + (-b6 - 17*b4) * q^37 + (5*b5 + 102) * q^41 + (3*b3 - 11*b1) * q^43 + (2*b7 - 12*b2) * q^47 + (-8*b5 + 209) * q^49 + (2*b6 - 13*b4) * q^53 + (b7 + 5*b2) * q^55 + (16*b3 + 39*b1) * q^59 + (5*b6 + 49*b4) * q^61 + (9*b5 + 144) * q^65 + (12*b3 + 49*b1) * q^67 + (-3*b7 + 27*b2) * q^71 + (-8*b5 - 242) * q^73 + (12*b6 + 16*b4) * q^77 + (-5*b7 + 6*b2) * q^79 + (17*b3 - 71*b1) * q^83 + (8*b6 - 26*b4) * q^85 + (6*b5 - 918) * q^89 + (63*b3 - 108*b1) * q^91 + (-7*b7 - 3*b2) * q^95 + (2*b5 - 370) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 240 q^{17} - 344 q^{25} + 816 q^{41} + 1672 q^{49} + 1152 q^{65} - 1936 q^{73} - 7344 q^{89} - 2960 q^{97}+O(q^{100})$$ 8 * q - 240 * q^17 - 344 * q^25 + 816 * q^41 + 1672 * q^49 + 1152 * q^65 - 1936 * q^73 - 7344 * q^89 - 2960 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$6\nu^{5} + 66\nu^{3} + 174\nu$$ 6*v^5 + 66*v^3 + 174*v $$\beta_{2}$$ $$=$$ $$-\nu^{6} - 6\nu^{4} + 2\nu^{2} + 3$$ -v^6 - 6*v^4 + 2*v^2 + 3 $$\beta_{3}$$ $$=$$ $$8\nu^{5} + 104\nu^{3} + 328\nu$$ 8*v^5 + 104*v^3 + 328*v $$\beta_{4}$$ $$=$$ $$2\nu^{7} + 31\nu^{5} + 155\nu^{3} + 253\nu$$ 2*v^7 + 31*v^5 + 155*v^3 + 253*v $$\beta_{5}$$ $$=$$ $$24\nu^{6} + 288\nu^{4} + 864\nu^{2} + 72$$ 24*v^6 + 288*v^4 + 864*v^2 + 72 $$\beta_{6}$$ $$=$$ $$-10\nu^{7} - 131\nu^{5} - 487\nu^{3} - 377\nu$$ -10*v^7 - 131*v^5 - 487*v^3 - 377*v $$\beta_{7}$$ $$=$$ $$33\nu^{6} + 390\nu^{4} + 1086\nu^{2} - 195$$ 33*v^6 + 390*v^4 + 1086*v^2 - 195
 $$\nu$$ $$=$$ $$( 2\beta_{6} + 10\beta_{4} - 3\beta_{3} - 4\beta_1 ) / 96$$ (2*b6 + 10*b4 - 3*b3 - 4*b1) / 96 $$\nu^{2}$$ $$=$$ $$( -3\beta_{7} + 4\beta_{5} - 3\beta_{2} - 864 ) / 192$$ (-3*b7 + 4*b5 - 3*b2 - 864) / 192 $$\nu^{3}$$ $$=$$ $$( -3\beta_{6} - 15\beta_{4} + 6\beta_{3} + 4\beta_1 ) / 24$$ (-3*b6 - 15*b4 + 6*b3 + 4*b1) / 24 $$\nu^{4}$$ $$=$$ $$( 19\beta_{7} - 24\beta_{5} + 51\beta_{2} + 5280 ) / 192$$ (19*b7 - 24*b5 + 51*b2 + 5280) / 192 $$\nu^{5}$$ $$=$$ $$( 74\beta_{6} + 370\beta_{4} - 177\beta_{3} - 44\beta_1 ) / 96$$ (74*b6 + 370*b4 - 177*b3 - 44*b1) / 96 $$\nu^{6}$$ $$=$$ $$( -15\beta_{7} + 19\beta_{5} - 63\beta_{2} - 4104 ) / 24$$ (-15*b7 + 19*b5 - 63*b2 - 4104) / 24 $$\nu^{7}$$ $$=$$ $$( -470\beta_{6} - 2302\beta_{4} + 1263\beta_{3} - 52\beta_1 ) / 96$$ (-470*b6 - 2302*b4 + 1263*b3 - 52*b1) / 96

## 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
 2.48330i 0.0690906i 2.21597i 2.63019i − 2.21597i − 2.63019i − 2.48330i − 0.0690906i
0 0 0 17.4288i 0 −2.99032 0 0 0
577.2 0 0 0 17.4288i 0 2.99032 0 0 0
577.3 0 0 0 5.67763i 0 −33.0917 0 0 0
577.4 0 0 0 5.67763i 0 33.0917 0 0 0
577.5 0 0 0 5.67763i 0 −33.0917 0 0 0
577.6 0 0 0 5.67763i 0 33.0917 0 0 0
577.7 0 0 0 17.4288i 0 −2.99032 0 0 0
577.8 0 0 0 17.4288i 0 2.99032 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 577.8 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.p 8
3.b odd 2 1 384.4.d.f 8
4.b odd 2 1 inner 1152.4.d.p 8
8.b even 2 1 inner 1152.4.d.p 8
8.d odd 2 1 inner 1152.4.d.p 8
12.b even 2 1 384.4.d.f 8
16.e even 4 1 2304.4.a.by 4
16.e even 4 1 2304.4.a.cb 4
16.f odd 4 1 2304.4.a.by 4
16.f odd 4 1 2304.4.a.cb 4
24.f even 2 1 384.4.d.f 8
24.h odd 2 1 384.4.d.f 8
48.i odd 4 1 768.4.a.u 4
48.i odd 4 1 768.4.a.v 4
48.k even 4 1 768.4.a.u 4
48.k even 4 1 768.4.a.v 4

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

 $$T_{5}^{4} + 336T_{5}^{2} + 9792$$ T5^4 + 336*T5^2 + 9792 $$T_{7}^{4} - 1104T_{7}^{2} + 9792$$ T7^4 - 1104*T7^2 + 9792 $$T_{17}^{2} + 60T_{17} - 3708$$ T17^2 + 60*T17 - 3708

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 336 T^{2} + 9792)^{2}$$
$7$ $$(T^{4} - 1104 T^{2} + 9792)^{2}$$
$11$ $$(T^{4} + 1312 T^{2} + 135424)^{2}$$
$13$ $$(T^{4} + 8640 T^{2} + 12690432)^{2}$$
$17$ $$(T^{2} + 60 T - 3708)^{4}$$
$19$ $$(T^{4} + 9504 T^{2} + 19927296)^{2}$$
$23$ $$(T^{4} - 53568 T^{2} + \cdots + 621831168)^{2}$$
$29$ $$(T^{4} + 41040 T^{2} + \cdots + 419577408)^{2}$$
$31$ $$(T^{4} - 55248 T^{2} + \cdots + 461095488)^{2}$$
$37$ $$(T^{4} + 107136 T^{2} + \cdots + 2487324672)^{2}$$
$41$ $$(T^{2} - 204 T - 104796)^{4}$$
$43$ $$(T^{4} + 44064 T^{2} + \cdots + 164249856)^{2}$$
$47$ $$(T^{4} - 302400 T^{2} + \cdots + 21332616192)^{2}$$
$53$ $$(T^{4} + 87888 T^{2} + \cdots + 806557248)^{2}$$
$59$ $$(T^{4} + 700192 T^{2} + \cdots + 7735554304)^{2}$$
$61$ $$(T^{4} + 1040256 T^{2} + \cdots + 270509248512)^{2}$$
$67$ $$(T^{4} + 838944 T^{2} + \cdots + 73992704256)^{2}$$
$71$ $$(T^{4} - 1104192 T^{2} + \cdots + 297070322688)^{2}$$
$73$ $$(T^{2} + 484 T - 236348)^{4}$$
$79$ $$(T^{4} - 1039824 T^{2} + \cdots + 1904357952)^{2}$$
$83$ $$(T^{4} + 1747744 T^{2} + \cdots + 334010020096)^{2}$$
$89$ $$(T^{2} + 1836 T + 676836)^{4}$$
$97$ $$(T^{2} + 740 T + 118468)^{4}$$