# Properties

 Label 380.2.a.d Level $380$ Weight $2$ Character orbit 380.a Self dual yes Analytic conductor $3.034$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + q^{5} + 2 q^{7} + (2 \beta + 1) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + q^5 + 2 * q^7 + (2*b + 1) * q^9 $$q + (\beta + 1) q^{3} + q^{5} + 2 q^{7} + (2 \beta + 1) q^{9} - 2 \beta q^{11} + ( - \beta - 1) q^{13} + (\beta + 1) q^{15} - 2 \beta q^{17} + q^{19} + (2 \beta + 2) q^{21} - 2 \beta q^{23} + q^{25} + 4 q^{27} + 2 \beta q^{29} + ( - 2 \beta + 2) q^{31} + ( - 2 \beta - 6) q^{33} + 2 q^{35} + (\beta + 5) q^{37} + ( - 2 \beta - 4) q^{39} - 6 q^{41} + ( - 4 \beta + 2) q^{43} + (2 \beta + 1) q^{45} + (4 \beta + 6) q^{47} - 3 q^{49} + ( - 2 \beta - 6) q^{51} + ( - \beta - 9) q^{53} - 2 \beta q^{55} + (\beta + 1) q^{57} + 4 \beta q^{59} + (6 \beta + 2) q^{61} + (4 \beta + 2) q^{63} + ( - \beta - 1) q^{65} + (\beta + 5) q^{67} + ( - 2 \beta - 6) q^{69} + (2 \beta - 6) q^{71} + (2 \beta - 4) q^{73} + (\beta + 1) q^{75} - 4 \beta q^{77} + (4 \beta - 4) q^{79} + ( - 2 \beta + 1) q^{81} + 2 \beta q^{83} - 2 \beta q^{85} + (2 \beta + 6) q^{87} + ( - 2 \beta - 12) q^{89} + ( - 2 \beta - 2) q^{91} - 4 q^{93} + q^{95} + ( - 9 \beta - 1) q^{97} + ( - 2 \beta - 12) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + q^5 + 2 * q^7 + (2*b + 1) * q^9 - 2*b * q^11 + (-b - 1) * q^13 + (b + 1) * q^15 - 2*b * q^17 + q^19 + (2*b + 2) * q^21 - 2*b * q^23 + q^25 + 4 * q^27 + 2*b * q^29 + (-2*b + 2) * q^31 + (-2*b - 6) * q^33 + 2 * q^35 + (b + 5) * q^37 + (-2*b - 4) * q^39 - 6 * q^41 + (-4*b + 2) * q^43 + (2*b + 1) * q^45 + (4*b + 6) * q^47 - 3 * q^49 + (-2*b - 6) * q^51 + (-b - 9) * q^53 - 2*b * q^55 + (b + 1) * q^57 + 4*b * q^59 + (6*b + 2) * q^61 + (4*b + 2) * q^63 + (-b - 1) * q^65 + (b + 5) * q^67 + (-2*b - 6) * q^69 + (2*b - 6) * q^71 + (2*b - 4) * q^73 + (b + 1) * q^75 - 4*b * q^77 + (4*b - 4) * q^79 + (-2*b + 1) * q^81 + 2*b * q^83 - 2*b * q^85 + (2*b + 6) * q^87 + (-2*b - 12) * q^89 + (-2*b - 2) * q^91 - 4 * q^93 + q^95 + (-9*b - 1) * q^97 + (-2*b - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 2 q^{19} + 4 q^{21} + 2 q^{25} + 8 q^{27} + 4 q^{31} - 12 q^{33} + 4 q^{35} + 10 q^{37} - 8 q^{39} - 12 q^{41} + 4 q^{43} + 2 q^{45} + 12 q^{47} - 6 q^{49} - 12 q^{51} - 18 q^{53} + 2 q^{57} + 4 q^{61} + 4 q^{63} - 2 q^{65} + 10 q^{67} - 12 q^{69} - 12 q^{71} - 8 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 12 q^{87} - 24 q^{89} - 4 q^{91} - 8 q^{93} + 2 q^{95} - 2 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 2 * q^19 + 4 * q^21 + 2 * q^25 + 8 * q^27 + 4 * q^31 - 12 * q^33 + 4 * q^35 + 10 * q^37 - 8 * q^39 - 12 * q^41 + 4 * q^43 + 2 * q^45 + 12 * q^47 - 6 * q^49 - 12 * q^51 - 18 * q^53 + 2 * q^57 + 4 * q^61 + 4 * q^63 - 2 * q^65 + 10 * q^67 - 12 * q^69 - 12 * q^71 - 8 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^89 - 4 * q^91 - 8 * q^93 + 2 * q^95 - 2 * q^97 - 24 * 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
 −1.73205 1.73205
0 −0.732051 0 1.00000 0 2.00000 0 −2.46410 0
1.2 0 2.73205 0 1.00000 0 2.00000 0 4.46410 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.d 2
3.b odd 2 1 3420.2.a.h 2
4.b odd 2 1 1520.2.a.l 2
5.b even 2 1 1900.2.a.d 2
5.c odd 4 2 1900.2.c.e 4
8.b even 2 1 6080.2.a.z 2
8.d odd 2 1 6080.2.a.bj 2
19.b odd 2 1 7220.2.a.h 2
20.d odd 2 1 7600.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.d 2 1.a even 1 1 trivial
1520.2.a.l 2 4.b odd 2 1
1900.2.a.d 2 5.b even 2 1
1900.2.c.e 4 5.c odd 4 2
3420.2.a.h 2 3.b odd 2 1
6080.2.a.z 2 8.b even 2 1
6080.2.a.bj 2 8.d odd 2 1
7220.2.a.h 2 19.b odd 2 1
7600.2.a.bf 2 20.d odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} - 12$$
$13$ $$T^{2} + 2T - 2$$
$17$ $$T^{2} - 12$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 12$$
$29$ $$T^{2} - 12$$
$31$ $$T^{2} - 4T - 8$$
$37$ $$T^{2} - 10T + 22$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} - 4T - 44$$
$47$ $$T^{2} - 12T - 12$$
$53$ $$T^{2} + 18T + 78$$
$59$ $$T^{2} - 48$$
$61$ $$T^{2} - 4T - 104$$
$67$ $$T^{2} - 10T + 22$$
$71$ $$T^{2} + 12T + 24$$
$73$ $$T^{2} + 8T + 4$$
$79$ $$T^{2} + 8T - 32$$
$83$ $$T^{2} - 12$$
$89$ $$T^{2} + 24T + 132$$
$97$ $$T^{2} + 2T - 242$$