# Properties

 Label 3920.2.a.f 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{5} + q^{9}+O(q^{10})$$ q - 2 * q^3 + q^5 + q^9 $$q - 2 q^{3} + q^{5} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} - 2 q^{19} + 4 q^{23} + q^{25} + 4 q^{27} + 10 q^{29} - 4 q^{31} + 8 q^{33} - 2 q^{37} - 4 q^{39} - 12 q^{41} + 4 q^{43} + q^{45} - 4 q^{47} + 2 q^{53} - 4 q^{55} + 4 q^{57} - 10 q^{59} + 6 q^{61} + 2 q^{65} - 4 q^{67} - 8 q^{69} + 12 q^{71} - 4 q^{73} - 2 q^{75} + 4 q^{79} - 11 q^{81} - 14 q^{83} - 20 q^{87} + 8 q^{89} + 8 q^{93} - 2 q^{95} - 8 q^{97} - 4 q^{99}+O(q^{100})$$ q - 2 * q^3 + q^5 + q^9 - 4 * q^11 + 2 * q^13 - 2 * q^15 - 2 * q^19 + 4 * q^23 + q^25 + 4 * q^27 + 10 * q^29 - 4 * q^31 + 8 * q^33 - 2 * q^37 - 4 * q^39 - 12 * q^41 + 4 * q^43 + q^45 - 4 * q^47 + 2 * q^53 - 4 * q^55 + 4 * q^57 - 10 * q^59 + 6 * q^61 + 2 * q^65 - 4 * q^67 - 8 * q^69 + 12 * q^71 - 4 * q^73 - 2 * q^75 + 4 * q^79 - 11 * q^81 - 14 * q^83 - 20 * q^87 + 8 * q^89 + 8 * q^93 - 2 * q^95 - 8 * q^97 - 4 * 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 −2.00000 0 1.00000 0 0 0 1.00000 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.f 1
4.b odd 2 1 1960.2.a.n yes 1
7.b odd 2 1 3920.2.a.bd 1
20.d odd 2 1 9800.2.a.i 1
28.d even 2 1 1960.2.a.b 1
28.f even 6 2 1960.2.q.n 2
28.g odd 6 2 1960.2.q.b 2
140.c even 2 1 9800.2.a.bn 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.b 1 28.d even 2 1
1960.2.a.n yes 1 4.b odd 2 1
1960.2.q.b 2 28.g odd 6 2
1960.2.q.n 2 28.f even 6 2
3920.2.a.f 1 1.a even 1 1 trivial
3920.2.a.bd 1 7.b odd 2 1
9800.2.a.i 1 20.d odd 2 1
9800.2.a.bn 1 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3920))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 2$$ T13 - 2 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T - 2$$
$17$ $$T$$
$19$ $$T + 2$$
$23$ $$T - 4$$
$29$ $$T - 10$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T + 12$$
$43$ $$T - 4$$
$47$ $$T + 4$$
$53$ $$T - 2$$
$59$ $$T + 10$$
$61$ $$T - 6$$
$67$ $$T + 4$$
$71$ $$T - 12$$
$73$ $$T + 4$$
$79$ $$T - 4$$
$83$ $$T + 14$$
$89$ $$T - 8$$
$97$ $$T + 8$$