Properties

 Label 1584.4.a.bc Level $1584$ Weight $4$ Character orbit 1584.a Self dual yes Analytic conductor $93.459$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,4,Mod(1,1584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1584.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1584 = 2^{4} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1584.a (trivial)

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: yes Analytic conductor: $$93.4590254491$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 $$\beta = 4\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \beta - 1) q^{5} + ( - \beta - 10) q^{7}+O(q^{10})$$ q + (2*b - 1) * q^5 + (-b - 10) * q^7 $$q + (2 \beta - 1) q^{5} + ( - \beta - 10) q^{7} - 11 q^{11} + (5 \beta + 40) q^{13} + (3 \beta + 62) q^{17} + (15 \beta - 36) q^{19} + (9 \beta - 49) q^{23} + ( - 4 \beta + 68) q^{25} + ( - 14 \beta - 72) q^{29} + (7 \beta + 17) q^{31} + ( - 19 \beta - 86) q^{35} + (2 \beta + 27) q^{37} + ( - \beta - 268) q^{41} + ( - 4 \beta + 30) q^{43} + (30 \beta - 136) q^{47} + (20 \beta - 195) q^{49} + ( - 14 \beta + 246) q^{53} + ( - 22 \beta + 11) q^{55} + (33 \beta + 317) q^{59} + ( - 46 \beta + 420) q^{61} + (75 \beta + 440) q^{65} + ( - 5 \beta - 377) q^{67} + ( - 19 \beta - 339) q^{71} + (117 \beta - 200) q^{73} + (11 \beta + 110) q^{77} + (164 \beta - 158) q^{79} + ( - 30 \beta + 234) q^{83} + (121 \beta + 226) q^{85} + ( - 82 \beta + 921) q^{89} + ( - 90 \beta - 640) q^{91} + ( - 87 \beta + 1476) q^{95} + ( - 36 \beta + 1097) q^{97}+O(q^{100})$$ q + (2*b - 1) * q^5 + (-b - 10) * q^7 - 11 * q^11 + (5*b + 40) * q^13 + (3*b + 62) * q^17 + (15*b - 36) * q^19 + (9*b - 49) * q^23 + (-4*b + 68) * q^25 + (-14*b - 72) * q^29 + (7*b + 17) * q^31 + (-19*b - 86) * q^35 + (2*b + 27) * q^37 + (-b - 268) * q^41 + (-4*b + 30) * q^43 + (30*b - 136) * q^47 + (20*b - 195) * q^49 + (-14*b + 246) * q^53 + (-22*b + 11) * q^55 + (33*b + 317) * q^59 + (-46*b + 420) * q^61 + (75*b + 440) * q^65 + (-5*b - 377) * q^67 + (-19*b - 339) * q^71 + (117*b - 200) * q^73 + (11*b + 110) * q^77 + (164*b - 158) * q^79 + (-30*b + 234) * q^83 + (121*b + 226) * q^85 + (-82*b + 921) * q^89 + (-90*b - 640) * q^91 + (-87*b + 1476) * q^95 + (-36*b + 1097) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 20 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 20 * q^7 $$2 q - 2 q^{5} - 20 q^{7} - 22 q^{11} + 80 q^{13} + 124 q^{17} - 72 q^{19} - 98 q^{23} + 136 q^{25} - 144 q^{29} + 34 q^{31} - 172 q^{35} + 54 q^{37} - 536 q^{41} + 60 q^{43} - 272 q^{47} - 390 q^{49} + 492 q^{53} + 22 q^{55} + 634 q^{59} + 840 q^{61} + 880 q^{65} - 754 q^{67} - 678 q^{71} - 400 q^{73} + 220 q^{77} - 316 q^{79} + 468 q^{83} + 452 q^{85} + 1842 q^{89} - 1280 q^{91} + 2952 q^{95} + 2194 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 20 * q^7 - 22 * q^11 + 80 * q^13 + 124 * q^17 - 72 * q^19 - 98 * q^23 + 136 * q^25 - 144 * q^29 + 34 * q^31 - 172 * q^35 + 54 * q^37 - 536 * q^41 + 60 * q^43 - 272 * q^47 - 390 * q^49 + 492 * q^53 + 22 * q^55 + 634 * q^59 + 840 * q^61 + 880 * q^65 - 754 * q^67 - 678 * q^71 - 400 * q^73 + 220 * q^77 - 316 * q^79 + 468 * q^83 + 452 * q^85 + 1842 * q^89 - 1280 * q^91 + 2952 * q^95 + 2194 * q^97

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
0 0 0 −14.8564 0 −3.07180 0 0 0
1.2 0 0 0 12.8564 0 −16.9282 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$+1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.4.a.bc 2
3.b odd 2 1 176.4.a.i 2
4.b odd 2 1 99.4.a.c 2
12.b even 2 1 11.4.a.a 2
20.d odd 2 1 2475.4.a.q 2
24.f even 2 1 704.4.a.p 2
24.h odd 2 1 704.4.a.n 2
33.d even 2 1 1936.4.a.w 2
44.c even 2 1 1089.4.a.v 2
60.h even 2 1 275.4.a.b 2
60.l odd 4 2 275.4.b.c 4
84.h odd 2 1 539.4.a.e 2
132.d odd 2 1 121.4.a.c 2
132.n odd 10 4 121.4.c.f 8
132.o even 10 4 121.4.c.c 8
156.h even 2 1 1859.4.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 12.b even 2 1
99.4.a.c 2 4.b odd 2 1
121.4.a.c 2 132.d odd 2 1
121.4.c.c 8 132.o even 10 4
121.4.c.f 8 132.n odd 10 4
176.4.a.i 2 3.b odd 2 1
275.4.a.b 2 60.h even 2 1
275.4.b.c 4 60.l odd 4 2
539.4.a.e 2 84.h odd 2 1
704.4.a.n 2 24.h odd 2 1
704.4.a.p 2 24.f even 2 1
1089.4.a.v 2 44.c even 2 1
1584.4.a.bc 2 1.a even 1 1 trivial
1859.4.a.a 2 156.h even 2 1
1936.4.a.w 2 33.d even 2 1
2475.4.a.q 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1584))$$:

 $$T_{5}^{2} + 2T_{5} - 191$$ T5^2 + 2*T5 - 191 $$T_{7}^{2} + 20T_{7} + 52$$ T7^2 + 20*T7 + 52

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T - 191$$
$7$ $$T^{2} + 20T + 52$$
$11$ $$(T + 11)^{2}$$
$13$ $$T^{2} - 80T + 400$$
$17$ $$T^{2} - 124T + 3412$$
$19$ $$T^{2} + 72T - 9504$$
$23$ $$T^{2} + 98T - 1487$$
$29$ $$T^{2} + 144T - 4224$$
$31$ $$T^{2} - 34T - 2063$$
$37$ $$T^{2} - 54T + 537$$
$41$ $$T^{2} + 536T + 71776$$
$43$ $$T^{2} - 60T + 132$$
$47$ $$T^{2} + 272T - 24704$$
$53$ $$T^{2} - 492T + 51108$$
$59$ $$T^{2} - 634T + 48217$$
$61$ $$T^{2} - 840T + 74832$$
$67$ $$T^{2} + 754T + 140929$$
$71$ $$T^{2} + 678T + 97593$$
$73$ $$T^{2} + 400T - 617072$$
$79$ $$T^{2} + 316 T - 1266044$$
$83$ $$T^{2} - 468T + 11556$$
$89$ $$T^{2} - 1842 T + 525489$$
$97$ $$T^{2} - 2194 T + 1141201$$