# Properties

 Label 3920.2.a.t 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - 3 q^{9}+O(q^{10})$$ q + q^5 - 3 * q^9 $$q + q^{5} - 3 q^{9} - 4 q^{11} + 6 q^{13} - 2 q^{17} + q^{25} + 6 q^{29} + 8 q^{31} - 10 q^{37} - 2 q^{41} - 4 q^{43} - 3 q^{45} + 8 q^{47} - 2 q^{53} - 4 q^{55} - 8 q^{59} + 14 q^{61} + 6 q^{65} + 12 q^{67} + 16 q^{71} - 2 q^{73} + 8 q^{79} + 9 q^{81} + 8 q^{83} - 2 q^{85} - 10 q^{89} - 2 q^{97} + 12 q^{99}+O(q^{100})$$ q + q^5 - 3 * q^9 - 4 * q^11 + 6 * q^13 - 2 * q^17 + q^25 + 6 * q^29 + 8 * q^31 - 10 * q^37 - 2 * q^41 - 4 * q^43 - 3 * q^45 + 8 * q^47 - 2 * q^53 - 4 * q^55 - 8 * q^59 + 14 * q^61 + 6 * q^65 + 12 * q^67 + 16 * q^71 - 2 * q^73 + 8 * q^79 + 9 * q^81 + 8 * q^83 - 2 * q^85 - 10 * q^89 - 2 * 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 0 0 1.00000 0 0 0 −3.00000 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.t 1
4.b odd 2 1 490.2.a.h 1
7.b odd 2 1 560.2.a.d 1
12.b even 2 1 4410.2.a.b 1
20.d odd 2 1 2450.2.a.l 1
20.e even 4 2 2450.2.c.k 2
21.c even 2 1 5040.2.a.bm 1
28.d even 2 1 70.2.a.a 1
28.f even 6 2 490.2.e.d 2
28.g odd 6 2 490.2.e.c 2
35.c odd 2 1 2800.2.a.m 1
35.f even 4 2 2800.2.g.n 2
56.e even 2 1 2240.2.a.n 1
56.h odd 2 1 2240.2.a.q 1
84.h odd 2 1 630.2.a.d 1
140.c even 2 1 350.2.a.b 1
140.j odd 4 2 350.2.c.b 2
308.g odd 2 1 8470.2.a.j 1
420.o odd 2 1 3150.2.a.bj 1
420.w even 4 2 3150.2.g.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 28.d even 2 1
350.2.a.b 1 140.c even 2 1
350.2.c.b 2 140.j odd 4 2
490.2.a.h 1 4.b odd 2 1
490.2.e.c 2 28.g odd 6 2
490.2.e.d 2 28.f even 6 2
560.2.a.d 1 7.b odd 2 1
630.2.a.d 1 84.h odd 2 1
2240.2.a.n 1 56.e even 2 1
2240.2.a.q 1 56.h odd 2 1
2450.2.a.l 1 20.d odd 2 1
2450.2.c.k 2 20.e even 4 2
2800.2.a.m 1 35.c odd 2 1
2800.2.g.n 2 35.f even 4 2
3150.2.a.bj 1 420.o odd 2 1
3150.2.g.c 2 420.w even 4 2
3920.2.a.t 1 1.a even 1 1 trivial
4410.2.a.b 1 12.b even 2 1
5040.2.a.bm 1 21.c even 2 1
8470.2.a.j 1 308.g odd 2 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 6$$ T13 - 6 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T - 6$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 8$$
$37$ $$T + 10$$
$41$ $$T + 2$$
$43$ $$T + 4$$
$47$ $$T - 8$$
$53$ $$T + 2$$
$59$ $$T + 8$$
$61$ $$T - 14$$
$67$ $$T - 12$$
$71$ $$T - 16$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T - 8$$
$89$ $$T + 10$$
$97$ $$T + 2$$