# Properties

 Label 5328.2.a.r Level $5328$ Weight $2$ Character orbit 5328.a Self dual yes Analytic conductor $42.544$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 37) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{5} + q^{7}+O(q^{10})$$ q + 2 * q^5 + q^7 $$q + 2 q^{5} + q^{7} - 5 q^{11} - 2 q^{13} + 2 q^{23} - q^{25} - 6 q^{29} + 4 q^{31} + 2 q^{35} - q^{37} + 9 q^{41} - 2 q^{43} - 9 q^{47} - 6 q^{49} - q^{53} - 10 q^{55} + 8 q^{59} - 8 q^{61} - 4 q^{65} - 8 q^{67} + 9 q^{71} - q^{73} - 5 q^{77} - 4 q^{79} - 15 q^{83} - 4 q^{89} - 2 q^{91} + 4 q^{97}+O(q^{100})$$ q + 2 * q^5 + q^7 - 5 * q^11 - 2 * q^13 + 2 * q^23 - q^25 - 6 * q^29 + 4 * q^31 + 2 * q^35 - q^37 + 9 * q^41 - 2 * q^43 - 9 * q^47 - 6 * q^49 - q^53 - 10 * q^55 + 8 * q^59 - 8 * q^61 - 4 * q^65 - 8 * q^67 + 9 * q^71 - q^73 - 5 * q^77 - 4 * q^79 - 15 * q^83 - 4 * q^89 - 2 * q^91 + 4 * 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
 0
0 0 0 2.00000 0 1.00000 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.r 1
3.b odd 2 1 592.2.a.e 1
4.b odd 2 1 333.2.a.d 1
12.b even 2 1 37.2.a.a 1
20.d odd 2 1 8325.2.a.e 1
24.f even 2 1 2368.2.a.q 1
24.h odd 2 1 2368.2.a.b 1
60.h even 2 1 925.2.a.e 1
60.l odd 4 2 925.2.b.b 2
84.h odd 2 1 1813.2.a.a 1
132.d odd 2 1 4477.2.a.b 1
156.h even 2 1 6253.2.a.c 1
444.g even 2 1 1369.2.a.e 1
444.j odd 4 2 1369.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.a 1 12.b even 2 1
333.2.a.d 1 4.b odd 2 1
592.2.a.e 1 3.b odd 2 1
925.2.a.e 1 60.h even 2 1
925.2.b.b 2 60.l odd 4 2
1369.2.a.e 1 444.g even 2 1
1369.2.b.c 2 444.j odd 4 2
1813.2.a.a 1 84.h odd 2 1
2368.2.a.b 1 24.h odd 2 1
2368.2.a.q 1 24.f even 2 1
4477.2.a.b 1 132.d odd 2 1
5328.2.a.r 1 1.a even 1 1 trivial
6253.2.a.c 1 156.h even 2 1
8325.2.a.e 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5328))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7} - 1$$ T7 - 1 $$T_{11} + 5$$ T11 + 5 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T - 1$$
$11$ $$T + 5$$
$13$ $$T + 2$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 2$$
$29$ $$T + 6$$
$31$ $$T - 4$$
$37$ $$T + 1$$
$41$ $$T - 9$$
$43$ $$T + 2$$
$47$ $$T + 9$$
$53$ $$T + 1$$
$59$ $$T - 8$$
$61$ $$T + 8$$
$67$ $$T + 8$$
$71$ $$T - 9$$
$73$ $$T + 1$$
$79$ $$T + 4$$
$83$ $$T + 15$$
$89$ $$T + 4$$
$97$ $$T - 4$$