# Properties

 Label 5328.2.a.bc Level $5328$ Weight $2$ Character orbit 5328.a Self dual yes Analytic conductor $42.544$ Analytic rank $1$ 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] = [5328,2,Mod(1,5328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5328, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5328.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5328 = 2^{4} \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5328.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: $$42.5442941969$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 3 \beta + 1) q^{5} + 2 \beta q^{7}+O(q^{10})$$ q + (-3*b + 1) * q^5 + 2*b * q^7 $$q + ( - 3 \beta + 1) q^{5} + 2 \beta q^{7} + (\beta - 3) q^{11} + ( - 3 \beta + 2) q^{13} + (4 \beta - 2) q^{17} + ( - 4 \beta + 2) q^{19} + (3 \beta - 2) q^{23} + (3 \beta + 5) q^{25} + (7 \beta - 2) q^{29} + (\beta - 9) q^{31} + ( - 4 \beta - 6) q^{35} - q^{37} + ( - \beta - 8) q^{41} + (2 \beta + 2) q^{43} + ( - 2 \beta + 2) q^{47} + (4 \beta - 3) q^{49} + ( - 4 \beta + 6) q^{53} + (7 \beta - 6) q^{55} + (2 \beta - 8) q^{59} + (\beta + 9) q^{61} + 11 q^{65} + ( - 5 \beta + 7) q^{67} + (8 \beta - 10) q^{71} + (5 \beta - 1) q^{73} + ( - 4 \beta + 2) q^{77} + (9 \beta - 6) q^{79} + ( - 4 \beta - 8) q^{83} + ( - 2 \beta - 14) q^{85} + ( - 4 \beta + 8) q^{89} + ( - 2 \beta - 6) q^{91} + (2 \beta + 14) q^{95} + ( - 4 \beta + 6) q^{97}+O(q^{100})$$ q + (-3*b + 1) * q^5 + 2*b * q^7 + (b - 3) * q^11 + (-3*b + 2) * q^13 + (4*b - 2) * q^17 + (-4*b + 2) * q^19 + (3*b - 2) * q^23 + (3*b + 5) * q^25 + (7*b - 2) * q^29 + (b - 9) * q^31 + (-4*b - 6) * q^35 - q^37 + (-b - 8) * q^41 + (2*b + 2) * q^43 + (-2*b + 2) * q^47 + (4*b - 3) * q^49 + (-4*b + 6) * q^53 + (7*b - 6) * q^55 + (2*b - 8) * q^59 + (b + 9) * q^61 + 11 * q^65 + (-5*b + 7) * q^67 + (8*b - 10) * q^71 + (5*b - 1) * q^73 + (-4*b + 2) * q^77 + (9*b - 6) * q^79 + (-4*b - 8) * q^83 + (-2*b - 14) * q^85 + (-4*b + 8) * q^89 + (-2*b - 6) * q^91 + (2*b + 14) * q^95 + (-4*b + 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q - q^5 + 2 * q^7 $$2 q - q^{5} + 2 q^{7} - 5 q^{11} + q^{13} - q^{23} + 13 q^{25} + 3 q^{29} - 17 q^{31} - 16 q^{35} - 2 q^{37} - 17 q^{41} + 6 q^{43} + 2 q^{47} - 2 q^{49} + 8 q^{53} - 5 q^{55} - 14 q^{59} + 19 q^{61} + 22 q^{65} + 9 q^{67} - 12 q^{71} + 3 q^{73} - 3 q^{79} - 20 q^{83} - 30 q^{85} + 12 q^{89} - 14 q^{91} + 30 q^{95} + 8 q^{97}+O(q^{100})$$ 2 * q - q^5 + 2 * q^7 - 5 * q^11 + q^13 - q^23 + 13 * q^25 + 3 * q^29 - 17 * q^31 - 16 * q^35 - 2 * q^37 - 17 * q^41 + 6 * q^43 + 2 * q^47 - 2 * q^49 + 8 * q^53 - 5 * q^55 - 14 * q^59 + 19 * q^61 + 22 * q^65 + 9 * q^67 - 12 * q^71 + 3 * q^73 - 3 * q^79 - 20 * q^83 - 30 * q^85 + 12 * q^89 - 14 * q^91 + 30 * q^95 + 8 * 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
 1.61803 −0.618034
0 0 0 −3.85410 0 3.23607 0 0 0
1.2 0 0 0 2.85410 0 −1.23607 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$37$$ $$+1$$

## 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 5328.2.a.bc 2
3.b odd 2 1 592.2.a.g 2
4.b odd 2 1 666.2.a.i 2
12.b even 2 1 74.2.a.b 2
24.f even 2 1 2368.2.a.y 2
24.h odd 2 1 2368.2.a.u 2
60.h even 2 1 1850.2.a.t 2
60.l odd 4 2 1850.2.b.j 4
84.h odd 2 1 3626.2.a.s 2
132.d odd 2 1 8954.2.a.j 2
444.g even 2 1 2738.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 12.b even 2 1
592.2.a.g 2 3.b odd 2 1
666.2.a.i 2 4.b odd 2 1
1850.2.a.t 2 60.h even 2 1
1850.2.b.j 4 60.l odd 4 2
2368.2.a.u 2 24.h odd 2 1
2368.2.a.y 2 24.f even 2 1
2738.2.a.g 2 444.g even 2 1
3626.2.a.s 2 84.h odd 2 1
5328.2.a.bc 2 1.a even 1 1 trivial
8954.2.a.j 2 132.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5328))$$:

 $$T_{5}^{2} + T_{5} - 11$$ T5^2 + T5 - 11 $$T_{7}^{2} - 2T_{7} - 4$$ T7^2 - 2*T7 - 4 $$T_{11}^{2} + 5T_{11} + 5$$ T11^2 + 5*T11 + 5 $$T_{13}^{2} - T_{13} - 11$$ T13^2 - T13 - 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T - 11$$
$7$ $$T^{2} - 2T - 4$$
$11$ $$T^{2} + 5T + 5$$
$13$ $$T^{2} - T - 11$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} - 20$$
$23$ $$T^{2} + T - 11$$
$29$ $$T^{2} - 3T - 59$$
$31$ $$T^{2} + 17T + 71$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} + 17T + 71$$
$43$ $$T^{2} - 6T + 4$$
$47$ $$T^{2} - 2T - 4$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} + 14T + 44$$
$61$ $$T^{2} - 19T + 89$$
$67$ $$T^{2} - 9T - 11$$
$71$ $$T^{2} + 12T - 44$$
$73$ $$T^{2} - 3T - 29$$
$79$ $$T^{2} + 3T - 99$$
$83$ $$T^{2} + 20T + 80$$
$89$ $$T^{2} - 12T + 16$$
$97$ $$T^{2} - 8T - 4$$