# Properties

 Label 380.2.v.b Level $380$ Weight $2$ Character orbit 380.v 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(7,380)]

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([6, 3, 4]))

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

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

chi := DirichletCharacter("380.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.v (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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{3}+ \cdots - 3 \zeta_{12} q^{9} +O(q^{10})$$ q + (-z^2 + z + 1) * q^2 + (z^3 + z^2 + z - 2) * q^3 + (-2*z^3 + 2*z) * q^4 + (2*z^3 - z^2 - 2*z) * q^5 + (4*z^2 - 2) * q^6 + (2*z^3 + 4*z^2 - 4*z - 2) * q^7 + (-2*z^3 + 2) * q^8 - 3*z * q^9 $$q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{3}+ \cdots + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{99}+O(q^{100})$$ q + (-z^2 + z + 1) * q^2 + (z^3 + z^2 + z - 2) * q^3 + (-2*z^3 + 2*z) * q^4 + (2*z^3 - z^2 - 2*z) * q^5 + (4*z^2 - 2) * q^6 + (2*z^3 + 4*z^2 - 4*z - 2) * q^7 + (-2*z^3 + 2) * q^8 - 3*z * q^9 + (z^3 - 3) * q^10 + (2*z^2 - 1) * q^11 + (4*z^3 + 2*z^2 - 2*z + 2) * q^12 + (-z^2 - z + 1) * q^13 + (8*z^3 - 4*z) * q^14 + (-6*z^3 - z^2 + 3*z - 1) * q^15 + (-4*z^2 + 4) * q^16 + (3*z^3 + 3*z^2 - 3*z) * q^17 + (3*z^3 - 3*z^2 - 3*z) * q^18 + (2*z^3 + 3*z) * q^19 + (4*z^2 - 2*z - 4) * q^20 - 12*z^2 * q^21 + (2*z^3 + z^2 - z + 1) * q^22 + (-4*z^3 + 2*z^2 + 2*z + 2) * q^23 + (4*z^3 + 4*z) * q^24 + (-3*z^2 + 4*z + 3) * q^25 - 2*z^2 * q^26 + (4*z^3 + 4*z^2 + 4*z - 8) * q^28 - z * q^29 + (-4*z^3 - 2*z^2 - 4*z + 4) * q^30 + (-10*z^2 + 5) * q^31 + (-4*z^3 - 4*z^2 + 4*z) * q^32 + (3*z^3 - 3*z^2 - 3*z) * q^33 + 6*z^3 * q^34 + (-2*z^3 - 6*z^2 - 2*z + 12) * q^35 - 6 * q^36 + (2*z^3 + 2) * q^37 + (-3*z^3 + 5*z^2 + 5*z - 2) * q^38 + (-2*z^3 + 4*z) * q^39 + (6*z^3 + 2*z^2 - 6*z) * q^40 + 8*z^2 * q^41 + (-12*z^3 - 12) * q^42 + (2*z^3 + 2*z^2 + 2*z - 4) * q^43 + (2*z^3 + 2*z) * q^44 + (3*z^3 + 6) * q^45 + (-4*z^2 + 8) * q^46 + (10*z^3 + 5*z^2 - 5*z + 5) * q^47 + (-4*z^3 + 8*z^2 + 8*z - 4) * q^48 - 17*z^3 * q^49 + (-7*z^3 + z^2 + 7*z) * q^50 + (-6*z^2 - 6) * q^51 + (-2*z^3 - 2) * q^52 + (-3*z^2 - 3*z + 3) * q^53 + (-2*z^3 - z^2 - 2*z + 2) * q^55 + (16*z^2 - 8) * q^56 + (z^3 + 8*z^2 - 8*z - 7) * q^57 + (z^3 - z^2 - z) * q^58 + (-z^3 - z) * q^59 + (2*z^3 - 12*z^2 - 4*z + 6) * q^60 + (-3*z^2 + 3) * q^61 + (-10*z^3 - 5*z^2 + 5*z - 5) * q^62 + (-12*z^3 + 6*z^2 + 6*z + 6) * q^63 - 8*z^3 * q^64 + (3*z^3 + 1) * q^65 - 6 * q^66 + (-2*z^3 - z^2 + z - 1) * q^67 + (6*z^2 + 6*z - 6) * q^68 + 12*z^3 * q^69 + (-4*z^3 - 16*z^2 + 8*z + 8) * q^70 + (-z^2 + 2) * q^71 + (6*z^2 - 6*z - 6) * q^72 + (-8*z^3 + 8*z^2 + 8*z) * q^73 + 4*z * q^74 + (z^3 + 14*z^2 - 2*z - 7) * q^75 + (4*z^2 + 6) * q^76 + (-6*z^3 - 6) * q^77 + (-4*z^3 + 2*z^2 + 2*z + 2) * q^78 + (5*z^3 + 5*z) * q^79 + (8*z^3 - 4) * q^80 - 9*z^2 * q^81 + (8*z^3 + 8) * q^82 + (4*z^3 - 8*z^2 - 8*z + 4) * q^83 - 24*z * q^84 + (-9*z^2 - 3*z + 9) * q^85 + (8*z^2 - 4) * q^86 + (-z^3 - 2*z^2 + 2*z + 1) * q^87 + (-2*z^3 + 4*z^2 + 4*z - 2) * q^88 + z * q^89 + (-3*z^2 + 9*z + 3) * q^90 + (4*z^2 + 4) * q^91 + (-4*z^3 - 8*z^2 + 8*z + 4) * q^92 + (-15*z^3 + 15*z^2 + 15*z) * q^93 + (10*z^3 + 10*z) * q^94 + (-5*z^3 - 4*z^2 + 2*z - 6) * q^95 + (8*z^2 + 8) * q^96 + (-4*z^3 - 4*z^2 + 4*z) * q^97 + (-17*z^2 - 17*z + 17) * q^98 + (-6*z^3 + 3*z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 6 q^{3} - 2 q^{5} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 6 * q^3 - 2 * q^5 + 8 * q^8 $$4 q + 2 q^{2} - 6 q^{3} - 2 q^{5} + 8 q^{8} - 12 q^{10} + 12 q^{12} + 2 q^{13} - 6 q^{15} + 8 q^{16} + 6 q^{17} - 6 q^{18} - 8 q^{20} - 24 q^{21} + 6 q^{22} + 12 q^{23} + 6 q^{25} - 4 q^{26} - 24 q^{28} + 12 q^{30} - 8 q^{32} - 6 q^{33} + 36 q^{35} - 24 q^{36} + 8 q^{37} + 2 q^{38} + 4 q^{40} + 16 q^{41} - 48 q^{42} - 12 q^{43} + 24 q^{45} + 24 q^{46} + 30 q^{47} + 2 q^{50} - 36 q^{51} - 8 q^{52} + 6 q^{53} + 6 q^{55} - 12 q^{57} - 2 q^{58} + 6 q^{61} - 30 q^{62} + 36 q^{63} + 4 q^{65} - 24 q^{66} - 6 q^{67} - 12 q^{68} + 6 q^{71} - 12 q^{72} + 16 q^{73} + 32 q^{76} - 24 q^{77} + 12 q^{78} - 16 q^{80} - 18 q^{81} + 32 q^{82} + 18 q^{85} + 6 q^{90} + 24 q^{91} + 30 q^{93} - 32 q^{95} + 48 q^{96} - 8 q^{97} + 34 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 6 * q^3 - 2 * q^5 + 8 * q^8 - 12 * q^10 + 12 * q^12 + 2 * q^13 - 6 * q^15 + 8 * q^16 + 6 * q^17 - 6 * q^18 - 8 * q^20 - 24 * q^21 + 6 * q^22 + 12 * q^23 + 6 * q^25 - 4 * q^26 - 24 * q^28 + 12 * q^30 - 8 * q^32 - 6 * q^33 + 36 * q^35 - 24 * q^36 + 8 * q^37 + 2 * q^38 + 4 * q^40 + 16 * q^41 - 48 * q^42 - 12 * q^43 + 24 * q^45 + 24 * q^46 + 30 * q^47 + 2 * q^50 - 36 * q^51 - 8 * q^52 + 6 * q^53 + 6 * q^55 - 12 * q^57 - 2 * q^58 + 6 * q^61 - 30 * q^62 + 36 * q^63 + 4 * q^65 - 24 * q^66 - 6 * q^67 - 12 * q^68 + 6 * q^71 - 12 * q^72 + 16 * q^73 + 32 * q^76 - 24 * q^77 + 12 * q^78 - 16 * q^80 - 18 * q^81 + 32 * q^82 + 18 * q^85 + 6 * q^90 + 24 * q^91 + 30 * q^93 - 32 * q^95 + 48 * q^96 - 8 * q^97 + 34 * q^98

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
1.36603 0.366025i −0.633975 + 2.36603i 1.73205 1.00000i −2.23205 + 0.133975i 3.46410i −3.46410 + 3.46410i 2.00000 2.00000i −2.59808 1.50000i −3.00000 + 1.00000i
83.1 −0.366025 1.36603i −2.36603 0.633975i −1.73205 + 1.00000i 1.23205 1.86603i 3.46410i 3.46410 + 3.46410i 2.00000 + 2.00000i 2.59808 + 1.50000i −3.00000 1.00000i
87.1 −0.366025 + 1.36603i −2.36603 + 0.633975i −1.73205 1.00000i 1.23205 + 1.86603i 3.46410i 3.46410 3.46410i 2.00000 2.00000i 2.59808 1.50000i −3.00000 + 1.00000i
163.1 1.36603 + 0.366025i −0.633975 2.36603i 1.73205 + 1.00000i −2.23205 0.133975i 3.46410i −3.46410 3.46410i 2.00000 + 2.00000i −2.59808 + 1.50000i −3.00000 1.00000i
 $$n$$: 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
76.g odd 6 1 inner
380.v 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.v.b yes 4
4.b odd 2 1 380.2.v.a 4
5.c odd 4 1 inner 380.2.v.b yes 4
19.c even 3 1 380.2.v.a 4
20.e even 4 1 380.2.v.a 4
76.g odd 6 1 inner 380.2.v.b yes 4
95.m odd 12 1 380.2.v.a 4
380.v even 12 1 inner 380.2.v.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.v.a 4 4.b odd 2 1
380.2.v.a 4 19.c even 3 1
380.2.v.a 4 20.e even 4 1
380.2.v.a 4 95.m odd 12 1
380.2.v.b yes 4 1.a even 1 1 trivial
380.2.v.b yes 4 5.c odd 4 1 inner
380.2.v.b yes 4 76.g odd 6 1 inner
380.2.v.b yes 4 380.v 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

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$3$ $$T^{4} + 6 T^{3} + \cdots + 36$$
$5$ $$T^{4} + 2 T^{3} + \cdots + 25$$
$7$ $$T^{4} + 576$$
$11$ $$(T^{2} + 3)^{2}$$
$13$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$17$ $$T^{4} - 6 T^{3} + \cdots + 324$$
$19$ $$T^{4} + 11T^{2} + 361$$
$23$ $$T^{4} - 12 T^{3} + \cdots + 576$$
$29$ $$T^{4} - T^{2} + 1$$
$31$ $$(T^{2} + 75)^{2}$$
$37$ $$(T^{2} - 4 T + 8)^{2}$$
$41$ $$(T^{2} - 8 T + 64)^{2}$$
$43$ $$T^{4} + 12 T^{3} + \cdots + 576$$
$47$ $$T^{4} - 30 T^{3} + \cdots + 22500$$
$53$ $$T^{4} - 6 T^{3} + \cdots + 324$$
$59$ $$T^{4} + 3T^{2} + 9$$
$61$ $$(T^{2} - 3 T + 9)^{2}$$
$67$ $$T^{4} + 6 T^{3} + \cdots + 36$$
$71$ $$(T^{2} - 3 T + 3)^{2}$$
$73$ $$T^{4} - 16 T^{3} + \cdots + 16384$$
$79$ $$T^{4} + 75T^{2} + 5625$$
$83$ $$T^{4} + 9216$$
$89$ $$T^{4} - T^{2} + 1$$
$97$ $$T^{4} + 8 T^{3} + \cdots + 1024$$