# Properties

 Label 380.2.a.b Level $380$ Weight $2$ Character orbit 380.a Self dual yes Analytic conductor $3.034$ 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] = [380,2,Mod(1,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 - q^5 + 2 * q^7 + q^9 $$q + 2 q^{3} - q^{5} + 2 q^{7} + q^{9} + 6 q^{13} - 2 q^{15} + 2 q^{17} - q^{19} + 4 q^{21} - 2 q^{23} + q^{25} - 4 q^{27} - 2 q^{29} + 4 q^{31} - 2 q^{35} - 10 q^{37} + 12 q^{39} - 10 q^{41} + 6 q^{43} - q^{45} - 6 q^{47} - 3 q^{49} + 4 q^{51} + 6 q^{53} - 2 q^{57} - 4 q^{59} + 2 q^{61} + 2 q^{63} - 6 q^{65} - 2 q^{67} - 4 q^{69} + 12 q^{71} - 6 q^{73} + 2 q^{75} + 8 q^{79} - 11 q^{81} - 2 q^{83} - 2 q^{85} - 4 q^{87} + 2 q^{89} + 12 q^{91} + 8 q^{93} + q^{95} - 18 q^{97}+O(q^{100})$$ q + 2 * q^3 - q^5 + 2 * q^7 + q^9 + 6 * q^13 - 2 * q^15 + 2 * q^17 - q^19 + 4 * q^21 - 2 * q^23 + q^25 - 4 * q^27 - 2 * q^29 + 4 * q^31 - 2 * q^35 - 10 * q^37 + 12 * q^39 - 10 * q^41 + 6 * q^43 - q^45 - 6 * q^47 - 3 * q^49 + 4 * q^51 + 6 * q^53 - 2 * q^57 - 4 * q^59 + 2 * q^61 + 2 * q^63 - 6 * q^65 - 2 * q^67 - 4 * q^69 + 12 * q^71 - 6 * q^73 + 2 * q^75 + 8 * q^79 - 11 * q^81 - 2 * q^83 - 2 * q^85 - 4 * q^87 + 2 * q^89 + 12 * q^91 + 8 * q^93 + q^95 - 18 * q^97

## 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 2.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$1$$

## 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 380.2.a.b 1
3.b odd 2 1 3420.2.a.f 1
4.b odd 2 1 1520.2.a.b 1
5.b even 2 1 1900.2.a.a 1
5.c odd 4 2 1900.2.c.a 2
8.b even 2 1 6080.2.a.f 1
8.d odd 2 1 6080.2.a.v 1
19.b odd 2 1 7220.2.a.b 1
20.d odd 2 1 7600.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.b 1 1.a even 1 1 trivial
1520.2.a.b 1 4.b odd 2 1
1900.2.a.a 1 5.b even 2 1
1900.2.c.a 2 5.c odd 4 2
3420.2.a.f 1 3.b odd 2 1
6080.2.a.f 1 8.b even 2 1
6080.2.a.v 1 8.d odd 2 1
7220.2.a.b 1 19.b odd 2 1
7600.2.a.r 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(380))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T + 1$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T - 2$$
$19$ $$T + 1$$
$23$ $$T + 2$$
$29$ $$T + 2$$
$31$ $$T - 4$$
$37$ $$T + 10$$
$41$ $$T + 10$$
$43$ $$T - 6$$
$47$ $$T + 6$$
$53$ $$T - 6$$
$59$ $$T + 4$$
$61$ $$T - 2$$
$67$ $$T + 2$$
$71$ $$T - 12$$
$73$ $$T + 6$$
$79$ $$T - 8$$
$83$ $$T + 2$$
$89$ $$T - 2$$
$97$ $$T + 18$$