# Properties

 Label 3920.2.a.x Level $3920$ Weight $2$ Character orbit 3920.a Self dual yes Analytic conductor $31.301$ 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] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.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: $$31.3013575923$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1960) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} - 2 q^{9}+O(q^{10})$$ q + q^3 + q^5 - 2 * q^9 $$q + q^{3} + q^{5} - 2 q^{9} - 3 q^{11} + q^{13} + q^{15} - 5 q^{17} + 6 q^{19} + q^{25} - 5 q^{27} - 5 q^{29} - 2 q^{31} - 3 q^{33} - 4 q^{37} + q^{39} - 2 q^{41} - 10 q^{43} - 2 q^{45} + 9 q^{47} - 5 q^{51} + 6 q^{53} - 3 q^{55} + 6 q^{57} + 6 q^{59} - 12 q^{61} + q^{65} + 2 q^{67} - 14 q^{73} + q^{75} - q^{79} + q^{81} - 12 q^{83} - 5 q^{85} - 5 q^{87} - 2 q^{93} + 6 q^{95} - 9 q^{97} + 6 q^{99}+O(q^{100})$$ q + q^3 + q^5 - 2 * q^9 - 3 * q^11 + q^13 + q^15 - 5 * q^17 + 6 * q^19 + q^25 - 5 * q^27 - 5 * q^29 - 2 * q^31 - 3 * q^33 - 4 * q^37 + q^39 - 2 * q^41 - 10 * q^43 - 2 * q^45 + 9 * q^47 - 5 * q^51 + 6 * q^53 - 3 * q^55 + 6 * q^57 + 6 * q^59 - 12 * q^61 + q^65 + 2 * q^67 - 14 * q^73 + q^75 - q^79 + q^81 - 12 * q^83 - 5 * q^85 - 5 * q^87 - 2 * q^93 + 6 * q^95 - 9 * q^97 + 6 * 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 1.00000 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
$$2$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 3920.2.a.x 1
4.b odd 2 1 1960.2.a.f 1
7.b odd 2 1 3920.2.a.l 1
20.d odd 2 1 9800.2.a.bd 1
28.d even 2 1 1960.2.a.j yes 1
28.f even 6 2 1960.2.q.f 2
28.g odd 6 2 1960.2.q.j 2
140.c even 2 1 9800.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.f 1 4.b odd 2 1
1960.2.a.j yes 1 28.d even 2 1
1960.2.q.f 2 28.f even 6 2
1960.2.q.j 2 28.g odd 6 2
3920.2.a.l 1 7.b odd 2 1
3920.2.a.x 1 1.a even 1 1 trivial
9800.2.a.t 1 140.c even 2 1
9800.2.a.bd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3920))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{11} + 3$$ T11 + 3 $$T_{13} - 1$$ T13 - 1 $$T_{17} + 5$$ T17 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 3$$
$13$ $$T - 1$$
$17$ $$T + 5$$
$19$ $$T - 6$$
$23$ $$T$$
$29$ $$T + 5$$
$31$ $$T + 2$$
$37$ $$T + 4$$
$41$ $$T + 2$$
$43$ $$T + 10$$
$47$ $$T - 9$$
$53$ $$T - 6$$
$59$ $$T - 6$$
$61$ $$T + 12$$
$67$ $$T - 2$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T + 1$$
$83$ $$T + 12$$
$89$ $$T$$
$97$ $$T + 9$$