# Properties

 Label 3920.2.a.b Level $3920$ Weight $2$ Character orbit 3920.a Self dual yes Analytic conductor $31.301$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - q^{5} + 6 q^{9}+O(q^{10})$$ q - 3 * q^3 - q^5 + 6 * q^9 $$q - 3 q^{3} - q^{5} + 6 q^{9} + 2 q^{11} + 3 q^{15} - 4 q^{17} + 6 q^{19} - 3 q^{23} + q^{25} - 9 q^{27} + 9 q^{29} + 4 q^{31} - 6 q^{33} - 4 q^{37} - 7 q^{41} + 5 q^{43} - 6 q^{45} - 8 q^{47} + 12 q^{51} - 2 q^{53} - 2 q^{55} - 18 q^{57} - 10 q^{59} + q^{61} + 9 q^{67} + 9 q^{69} - 2 q^{71} - 4 q^{73} - 3 q^{75} - 10 q^{79} + 9 q^{81} + 7 q^{83} + 4 q^{85} - 27 q^{87} + q^{89} - 12 q^{93} - 6 q^{95} + 14 q^{97} + 12 q^{99}+O(q^{100})$$ q - 3 * q^3 - q^5 + 6 * q^9 + 2 * q^11 + 3 * q^15 - 4 * q^17 + 6 * q^19 - 3 * q^23 + q^25 - 9 * q^27 + 9 * q^29 + 4 * q^31 - 6 * q^33 - 4 * q^37 - 7 * q^41 + 5 * q^43 - 6 * q^45 - 8 * q^47 + 12 * q^51 - 2 * q^53 - 2 * q^55 - 18 * q^57 - 10 * q^59 + q^61 + 9 * q^67 + 9 * q^69 - 2 * q^71 - 4 * q^73 - 3 * q^75 - 10 * q^79 + 9 * q^81 + 7 * q^83 + 4 * q^85 - 27 * q^87 + q^89 - 12 * q^93 - 6 * q^95 + 14 * q^97 + 12 * 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 −3.00000 0 −1.00000 0 0 0 6.00000 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.b 1
4.b odd 2 1 490.2.a.k 1
7.b odd 2 1 3920.2.a.bk 1
7.c even 3 2 560.2.q.i 2
12.b even 2 1 4410.2.a.r 1
20.d odd 2 1 2450.2.a.b 1
20.e even 4 2 2450.2.c.s 2
28.d even 2 1 490.2.a.e 1
28.f even 6 2 490.2.e.f 2
28.g odd 6 2 70.2.e.a 2
84.h odd 2 1 4410.2.a.h 1
84.n even 6 2 630.2.k.f 2
140.c even 2 1 2450.2.a.q 1
140.j odd 4 2 2450.2.c.a 2
140.p odd 6 2 350.2.e.l 2
140.w even 12 4 350.2.j.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 28.g odd 6 2
350.2.e.l 2 140.p odd 6 2
350.2.j.f 4 140.w even 12 4
490.2.a.e 1 28.d even 2 1
490.2.a.k 1 4.b odd 2 1
490.2.e.f 2 28.f even 6 2
560.2.q.i 2 7.c even 3 2
630.2.k.f 2 84.n even 6 2
2450.2.a.b 1 20.d odd 2 1
2450.2.a.q 1 140.c even 2 1
2450.2.c.a 2 140.j odd 4 2
2450.2.c.s 2 20.e even 4 2
3920.2.a.b 1 1.a even 1 1 trivial
3920.2.a.bk 1 7.b odd 2 1
4410.2.a.h 1 84.h odd 2 1
4410.2.a.r 1 12.b even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T - 2$$
$13$ $$T$$
$17$ $$T + 4$$
$19$ $$T - 6$$
$23$ $$T + 3$$
$29$ $$T - 9$$
$31$ $$T - 4$$
$37$ $$T + 4$$
$41$ $$T + 7$$
$43$ $$T - 5$$
$47$ $$T + 8$$
$53$ $$T + 2$$
$59$ $$T + 10$$
$61$ $$T - 1$$
$67$ $$T - 9$$
$71$ $$T + 2$$
$73$ $$T + 4$$
$79$ $$T + 10$$
$83$ $$T - 7$$
$89$ $$T - 1$$
$97$ $$T - 14$$