# Properties

 Label 3920.2.a.r 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 280) 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 $$q - q^{3} + q^{5} - 2 q^{9} + 5 q^{11} - q^{13} - q^{15} - 3 q^{17} - 6 q^{19} + 6 q^{23} + q^{25} + 5 q^{27} - 9 q^{29} - 5 q^{33} + 6 q^{37} + q^{39} - 8 q^{41} - 6 q^{43} - 2 q^{45} + 3 q^{47} + 3 q^{51} - 12 q^{53} + 5 q^{55} + 6 q^{57} + 8 q^{59} + 4 q^{61} - q^{65} + 4 q^{67} - 6 q^{69} - 8 q^{71} - 10 q^{73} - q^{75} + 3 q^{79} + q^{81} - 12 q^{83} - 3 q^{85} + 9 q^{87} + 16 q^{89} - 6 q^{95} - 7 q^{97} - 10 q^{99}+O(q^{100})$$ q - q^3 + q^5 - 2 * q^9 + 5 * q^11 - q^13 - q^15 - 3 * q^17 - 6 * q^19 + 6 * q^23 + q^25 + 5 * q^27 - 9 * q^29 - 5 * q^33 + 6 * q^37 + q^39 - 8 * q^41 - 6 * q^43 - 2 * q^45 + 3 * q^47 + 3 * q^51 - 12 * q^53 + 5 * q^55 + 6 * q^57 + 8 * q^59 + 4 * q^61 - q^65 + 4 * q^67 - 6 * q^69 - 8 * q^71 - 10 * q^73 - q^75 + 3 * q^79 + q^81 - 12 * q^83 - 3 * q^85 + 9 * q^87 + 16 * q^89 - 6 * q^95 - 7 * q^97 - 10 * 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.r 1
4.b odd 2 1 1960.2.a.k 1
7.b odd 2 1 560.2.a.e 1
20.d odd 2 1 9800.2.a.n 1
21.c even 2 1 5040.2.a.be 1
28.d even 2 1 280.2.a.b 1
28.f even 6 2 1960.2.q.m 2
28.g odd 6 2 1960.2.q.e 2
35.c odd 2 1 2800.2.a.i 1
35.f even 4 2 2800.2.g.m 2
56.e even 2 1 2240.2.a.v 1
56.h odd 2 1 2240.2.a.j 1
84.h odd 2 1 2520.2.a.p 1
140.c even 2 1 1400.2.a.k 1
140.j odd 4 2 1400.2.g.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.b 1 28.d even 2 1
560.2.a.e 1 7.b odd 2 1
1400.2.a.k 1 140.c even 2 1
1400.2.g.e 2 140.j odd 4 2
1960.2.a.k 1 4.b odd 2 1
1960.2.q.e 2 28.g odd 6 2
1960.2.q.m 2 28.f even 6 2
2240.2.a.j 1 56.h odd 2 1
2240.2.a.v 1 56.e even 2 1
2520.2.a.p 1 84.h odd 2 1
2800.2.a.i 1 35.c odd 2 1
2800.2.g.m 2 35.f even 4 2
3920.2.a.r 1 1.a even 1 1 trivial
5040.2.a.be 1 21.c even 2 1
9800.2.a.n 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 $$T_{11} - 5$$ T11 - 5 $$T_{13} + 1$$ T13 + 1 $$T_{17} + 3$$ T17 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T - 5$$
$13$ $$T + 1$$
$17$ $$T + 3$$
$19$ $$T + 6$$
$23$ $$T - 6$$
$29$ $$T + 9$$
$31$ $$T$$
$37$ $$T - 6$$
$41$ $$T + 8$$
$43$ $$T + 6$$
$47$ $$T - 3$$
$53$ $$T + 12$$
$59$ $$T - 8$$
$61$ $$T - 4$$
$67$ $$T - 4$$
$71$ $$T + 8$$
$73$ $$T + 10$$
$79$ $$T - 3$$
$83$ $$T + 12$$
$89$ $$T - 16$$
$97$ $$T + 7$$