# Properties

 Label 3200.2.a.q Level $3200$ Weight $2$ Character orbit 3200.a Self dual yes Analytic conductor $25.552$ 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] = [3200,2,Mod(1,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.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: $$25.5521286468$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{9}+O(q^{10})$$ q + q^3 - 2 * q^9 $$q + q^{3} - 2 q^{9} - q^{11} - 2 q^{13} + 3 q^{17} + 3 q^{19} - 6 q^{23} - 5 q^{27} - 2 q^{29} - 2 q^{31} - q^{33} + 4 q^{37} - 2 q^{39} - 3 q^{41} - 4 q^{43} + 6 q^{47} - 7 q^{49} + 3 q^{51} - 10 q^{53} + 3 q^{57} + 12 q^{59} - 12 q^{61} - q^{67} - 6 q^{69} - 10 q^{71} + q^{73} - 16 q^{79} + q^{81} + 11 q^{83} - 2 q^{87} - 13 q^{89} - 2 q^{93} + 6 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^9 - q^11 - 2 * q^13 + 3 * q^17 + 3 * q^19 - 6 * q^23 - 5 * q^27 - 2 * q^29 - 2 * q^31 - q^33 + 4 * q^37 - 2 * q^39 - 3 * q^41 - 4 * q^43 + 6 * q^47 - 7 * q^49 + 3 * q^51 - 10 * q^53 + 3 * q^57 + 12 * q^59 - 12 * q^61 - q^67 - 6 * q^69 - 10 * q^71 + q^73 - 16 * q^79 + q^81 + 11 * q^83 - 2 * q^87 - 13 * q^89 - 2 * q^93 + 6 * q^97 + 2 * q^99

## 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 1.00000 0 0 0 0 0 −2.00000 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
$$2$$ $$1$$
$$5$$ $$1$$

## 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 3200.2.a.q yes 1
4.b odd 2 1 3200.2.a.k yes 1
5.b even 2 1 3200.2.a.j yes 1
5.c odd 4 2 3200.2.c.n 2
8.b even 2 1 3200.2.a.l yes 1
8.d odd 2 1 3200.2.a.r yes 1
20.d odd 2 1 3200.2.a.t yes 1
20.e even 4 2 3200.2.c.p 2
40.e odd 2 1 3200.2.a.i 1
40.f even 2 1 3200.2.a.s yes 1
40.i odd 4 2 3200.2.c.o 2
40.k even 4 2 3200.2.c.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.a.i 1 40.e odd 2 1
3200.2.a.j yes 1 5.b even 2 1
3200.2.a.k yes 1 4.b odd 2 1
3200.2.a.l yes 1 8.b even 2 1
3200.2.a.q yes 1 1.a even 1 1 trivial
3200.2.a.r yes 1 8.d odd 2 1
3200.2.a.s yes 1 40.f even 2 1
3200.2.a.t yes 1 20.d odd 2 1
3200.2.c.m 2 40.k even 4 2
3200.2.c.n 2 5.c odd 4 2
3200.2.c.o 2 40.i odd 4 2
3200.2.c.p 2 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3200))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7}$$ T7 $$T_{11} + 1$$ T11 + 1 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T - 3$$
$19$ $$T - 3$$
$23$ $$T + 6$$
$29$ $$T + 2$$
$31$ $$T + 2$$
$37$ $$T - 4$$
$41$ $$T + 3$$
$43$ $$T + 4$$
$47$ $$T - 6$$
$53$ $$T + 10$$
$59$ $$T - 12$$
$61$ $$T + 12$$
$67$ $$T + 1$$
$71$ $$T + 10$$
$73$ $$T - 1$$
$79$ $$T + 16$$
$83$ $$T - 11$$
$89$ $$T + 13$$
$97$ $$T - 6$$