# Properties

 Label 3528.2.a.p Level $3528$ Weight $2$ Character orbit 3528.a Self dual yes Analytic conductor $28.171$ 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] = [3528,2,Mod(1,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, 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("3528.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3528.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: $$28.1712218331$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) 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)

 $$f(q)$$ $$=$$ $$q + q^{5}+O(q^{10})$$ q + q^5 $$q + q^{5} - 3 q^{11} - 6 q^{13} + 5 q^{17} + q^{19} + 7 q^{23} - 4 q^{25} - 2 q^{29} - 5 q^{31} + 3 q^{37} + 2 q^{41} - 4 q^{43} - 5 q^{47} + q^{53} - 3 q^{55} - 15 q^{59} - 5 q^{61} - 6 q^{65} - 9 q^{67} + 7 q^{73} + q^{79} - 12 q^{83} + 5 q^{85} - 7 q^{89} + q^{95} - 2 q^{97}+O(q^{100})$$ q + q^5 - 3 * q^11 - 6 * q^13 + 5 * q^17 + q^19 + 7 * q^23 - 4 * q^25 - 2 * q^29 - 5 * q^31 + 3 * q^37 + 2 * q^41 - 4 * q^43 - 5 * q^47 + q^53 - 3 * q^55 - 15 * q^59 - 5 * q^61 - 6 * q^65 - 9 * q^67 + 7 * q^73 + q^79 - 12 * q^83 + 5 * q^85 - 7 * q^89 + q^95 - 2 * 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 1.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## 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 3528.2.a.p 1
3.b odd 2 1 392.2.a.c 1
4.b odd 2 1 7056.2.a.bj 1
7.b odd 2 1 3528.2.a.j 1
7.c even 3 2 504.2.s.c 2
7.d odd 6 2 3528.2.s.q 2
12.b even 2 1 784.2.a.h 1
15.d odd 2 1 9800.2.a.be 1
21.c even 2 1 392.2.a.e 1
21.g even 6 2 392.2.i.b 2
21.h odd 6 2 56.2.i.b 2
24.f even 2 1 3136.2.a.j 1
24.h odd 2 1 3136.2.a.u 1
28.d even 2 1 7056.2.a.u 1
28.g odd 6 2 1008.2.s.g 2
84.h odd 2 1 784.2.a.c 1
84.j odd 6 2 784.2.i.h 2
84.n even 6 2 112.2.i.a 2
105.g even 2 1 9800.2.a.s 1
105.o odd 6 2 1400.2.q.d 2
105.x even 12 4 1400.2.bh.a 4
168.e odd 2 1 3136.2.a.t 1
168.i even 2 1 3136.2.a.i 1
168.s odd 6 2 448.2.i.b 2
168.v even 6 2 448.2.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.b 2 21.h odd 6 2
112.2.i.a 2 84.n even 6 2
392.2.a.c 1 3.b odd 2 1
392.2.a.e 1 21.c even 2 1
392.2.i.b 2 21.g even 6 2
448.2.i.b 2 168.s odd 6 2
448.2.i.d 2 168.v even 6 2
504.2.s.c 2 7.c even 3 2
784.2.a.c 1 84.h odd 2 1
784.2.a.h 1 12.b even 2 1
784.2.i.h 2 84.j odd 6 2
1008.2.s.g 2 28.g odd 6 2
1400.2.q.d 2 105.o odd 6 2
1400.2.bh.a 4 105.x even 12 4
3136.2.a.i 1 168.i even 2 1
3136.2.a.j 1 24.f even 2 1
3136.2.a.t 1 168.e odd 2 1
3136.2.a.u 1 24.h odd 2 1
3528.2.a.j 1 7.b odd 2 1
3528.2.a.p 1 1.a even 1 1 trivial
3528.2.s.q 2 7.d odd 6 2
7056.2.a.u 1 28.d even 2 1
7056.2.a.bj 1 4.b odd 2 1
9800.2.a.s 1 105.g even 2 1
9800.2.a.be 1 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3528))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{11} + 3$$ T11 + 3 $$T_{13} + 6$$ T13 + 6 $$T_{23} - 7$$ T23 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 3$$
$13$ $$T + 6$$
$17$ $$T - 5$$
$19$ $$T - 1$$
$23$ $$T - 7$$
$29$ $$T + 2$$
$31$ $$T + 5$$
$37$ $$T - 3$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T + 5$$
$53$ $$T - 1$$
$59$ $$T + 15$$
$61$ $$T + 5$$
$67$ $$T + 9$$
$71$ $$T$$
$73$ $$T - 7$$
$79$ $$T - 1$$
$83$ $$T + 12$$
$89$ $$T + 7$$
$97$ $$T + 2$$