# Properties

 Label 380.2.y.a Level $380$ Weight $2$ Character orbit 380.y Analytic conductor $3.034$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(217,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("380.217");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.y (of order $$12$$, degree $$4$$, minimal)

## 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: no Analytic conductor: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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 a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots + 1) q^{3}+ \cdots + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{9}+O(q^{10})$$ q + (-2*z^3 + z^2 + z + 1) * q^3 + (-z^2 - 2*z + 1) * q^5 + (-2*z^3 - 2) * q^7 + (-3*z^3 + 3*z) * q^9 $$q + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots + 1) q^{3}+ \cdots + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{99}+O(q^{100})$$ q + (-2*z^3 + z^2 + z + 1) * q^3 + (-z^2 - 2*z + 1) * q^5 + (-2*z^3 - 2) * q^7 + (-3*z^3 + 3*z) * q^9 + q^11 + (-z^3 - z^2 - z + 2) * q^13 + (-3*z^3 + z^2 - 3*z - 2) * q^15 + (-z^2 + z + 1) * q^17 + (-3*z^3 + 5*z) * q^19 + (-4*z^2 - 4) * q^21 + (4*z^3 + 3*z^2 - 4*z) * q^25 + (5*z^3 + 5*z) * q^29 + (-2*z^2 + 1) * q^31 + (-2*z^3 + z^2 + z + 1) * q^33 + (6*z^2 + 2*z - 6) * q^35 + (2*z^3 + 4*z^2 - 4*z - 2) * q^37 - 6*z^3 * q^39 + (2*z^2 + 2) * q^41 + (4*z^2 + 4*z - 4) * q^43 + (-3*z^3 - 6) * q^45 + (3*z^3 + 3*z^2 - 3*z) * q^47 + z^3 * q^49 + (-2*z^2 + 4) * q^51 + (-3*z^3 - 3*z^2 - 3*z + 6) * q^53 + (-z^2 - 2*z + 1) * q^55 + (-z^3 - 8*z^2 + 8*z + 7) * q^57 + (2*z^3 - z) * q^59 - 13*z^2 * q^61 + (6*z^3 - 6*z^2 - 6*z) * q^63 + (3*z^3 + 2*z^2 - 6*z - 1) * q^65 + (5*z^3 - 5*z^2 + 5*z + 10) * q^67 + (7*z^2 + 7) * q^71 + (-10*z^2 - 10*z + 10) * q^73 + (z^3 + 14*z^2 - 2*z - 7) * q^75 + (-2*z^3 - 2) * q^77 + (14*z^3 - 7*z) * q^79 + (9*z^2 - 9) * q^81 + (4*z^3 - 4) * q^83 + (z^3 - 3*z^2 - z) * q^85 + (15*z^3 + 15) * q^87 + (9*z^3 + 9*z) * q^89 + (4*z^2 - 8) * q^91 + (-3*z^2 - 3*z + 3) * q^93 + (-5*z^3 - 4*z^2 + 2*z - 6) * q^95 + (-16*z^3 - 8*z^2 + 8*z - 8) * q^97 + (-3*z^3 + 3*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} + 2 q^{5} - 8 q^{7}+O(q^{10})$$ 4 * q + 6 * q^3 + 2 * q^5 - 8 * q^7 $$4 q + 6 q^{3} + 2 q^{5} - 8 q^{7} + 4 q^{11} + 6 q^{13} - 6 q^{15} + 2 q^{17} - 24 q^{21} + 6 q^{25} + 6 q^{33} - 12 q^{35} + 12 q^{41} - 8 q^{43} - 24 q^{45} + 6 q^{47} + 12 q^{51} + 18 q^{53} + 2 q^{55} + 12 q^{57} - 26 q^{61} - 12 q^{63} + 30 q^{67} + 42 q^{71} + 20 q^{73} - 8 q^{77} - 18 q^{81} - 16 q^{83} - 6 q^{85} + 60 q^{87} - 24 q^{91} + 6 q^{93} - 32 q^{95} - 48 q^{97}+O(q^{100})$$ 4 * q + 6 * q^3 + 2 * q^5 - 8 * q^7 + 4 * q^11 + 6 * q^13 - 6 * q^15 + 2 * q^17 - 24 * q^21 + 6 * q^25 + 6 * q^33 - 12 * q^35 + 12 * q^41 - 8 * q^43 - 24 * q^45 + 6 * q^47 + 12 * q^51 + 18 * q^53 + 2 * q^55 + 12 * q^57 - 26 * q^61 - 12 * q^63 + 30 * q^67 + 42 * q^71 + 20 * q^73 - 8 * q^77 - 18 * q^81 - 16 * q^83 - 6 * q^85 + 60 * q^87 - 24 * q^91 + 6 * q^93 - 32 * q^95 - 48 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$\zeta_{12}^{3}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
217.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 2.36603 0.633975i 0 −1.23205 1.86603i 0 −2.00000 2.00000i 0 2.59808 1.50000i 0
293.1 0 0.633975 + 2.36603i 0 2.23205 + 0.133975i 0 −2.00000 + 2.00000i 0 −2.59808 + 1.50000i 0
297.1 0 0.633975 2.36603i 0 2.23205 0.133975i 0 −2.00000 2.00000i 0 −2.59808 1.50000i 0
373.1 0 2.36603 + 0.633975i 0 −1.23205 + 1.86603i 0 −2.00000 + 2.00000i 0 2.59808 + 1.50000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.y.a 4
5.c odd 4 1 inner 380.2.y.a 4
19.d odd 6 1 inner 380.2.y.a 4
95.l even 12 1 inner 380.2.y.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.y.a 4 1.a even 1 1 trivial
380.2.y.a 4 5.c odd 4 1 inner
380.2.y.a 4 19.d odd 6 1 inner
380.2.y.a 4 95.l even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 6T_{3}^{3} + 18T_{3}^{2} - 36T_{3} + 36$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 6 T^{3} + \cdots + 36$$
$5$ $$T^{4} - 2 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 4 T + 8)^{2}$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4} - 6 T^{3} + \cdots + 36$$
$17$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$19$ $$T^{4} - 37T^{2} + 361$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 75T^{2} + 5625$$
$31$ $$(T^{2} + 3)^{2}$$
$37$ $$T^{4} + 576$$
$41$ $$(T^{2} - 6 T + 12)^{2}$$
$43$ $$T^{4} + 8 T^{3} + \cdots + 1024$$
$47$ $$T^{4} - 6 T^{3} + \cdots + 324$$
$53$ $$T^{4} - 18 T^{3} + \cdots + 2916$$
$59$ $$T^{4} + 3T^{2} + 9$$
$61$ $$(T^{2} + 13 T + 169)^{2}$$
$67$ $$T^{4} - 30 T^{3} + \cdots + 22500$$
$71$ $$(T^{2} - 21 T + 147)^{2}$$
$73$ $$T^{4} - 20 T^{3} + \cdots + 40000$$
$79$ $$T^{4} + 147 T^{2} + 21609$$
$83$ $$(T^{2} + 8 T + 32)^{2}$$
$89$ $$T^{4} + 243 T^{2} + 59049$$
$97$ $$T^{4} + 48 T^{3} + \cdots + 147456$$