# Properties

 Label 380.2.bd.a Level $380$ Weight $2$ Character orbit 380.bd Analytic conductor $3.034$ Analytic rank $0$ Dimension $60$ 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(9,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 9, 8]))

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

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

chi := DirichletCharacter("380.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.bd (of order $$18$$, degree $$6$$, 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: $$60$$ Relative dimension: $$10$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$60 q+O(q^{10})$$ 60 * q $$\operatorname{Tr}(f)(q) =$$ $$60 q - 6 q^{11} + 33 q^{15} + 24 q^{19} - 24 q^{21} - 6 q^{25} + 12 q^{29} - 21 q^{35} - 24 q^{39} - 6 q^{41} - 42 q^{45} + 60 q^{49} + 36 q^{51} - 45 q^{55} + 60 q^{61} - 15 q^{65} - 48 q^{69} + 36 q^{71} - 60 q^{79} - 96 q^{81} + 18 q^{85} - 48 q^{89} - 48 q^{91} - 21 q^{95} - 162 q^{99}+O(q^{100})$$ 60 * q - 6 * q^11 + 33 * q^15 + 24 * q^19 - 24 * q^21 - 6 * q^25 + 12 * q^29 - 21 * q^35 - 24 * q^39 - 6 * q^41 - 42 * q^45 + 60 * q^49 + 36 * q^51 - 45 * q^55 + 60 * q^61 - 15 * q^65 - 48 * q^69 + 36 * q^71 - 60 * q^79 - 96 * q^81 + 18 * q^85 - 48 * q^89 - 48 * q^91 - 21 * q^95 - 162 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 0 −2.21259 2.63686i 0 1.24816 + 1.85529i 0 2.51511 + 1.45210i 0 −1.53655 + 8.71419i 0
9.2 0 −1.23733 1.47460i 0 1.46729 1.68732i 0 −1.29052 0.745084i 0 −0.122496 + 0.694710i 0
9.3 0 −0.907311 1.08129i 0 −1.92890 + 1.13108i 0 −1.09754 0.633667i 0 0.174968 0.992290i 0
9.4 0 −0.817452 0.974201i 0 −2.23594 0.0236276i 0 4.08102 + 2.35618i 0 0.240104 1.36170i 0
9.5 0 −0.587809 0.700524i 0 0.750619 + 2.10632i 0 −3.44494 1.98894i 0 0.375731 2.13087i 0
9.6 0 0.587809 + 0.700524i 0 1.92892 1.13104i 0 3.44494 + 1.98894i 0 0.375731 2.13087i 0
9.7 0 0.817452 + 0.974201i 0 −1.72802 1.41914i 0 −4.08102 2.35618i 0 0.240104 1.36170i 0
9.8 0 0.907311 + 1.08129i 0 −0.750575 2.10633i 0 1.09754 + 0.633667i 0 0.174968 0.992290i 0
9.9 0 1.23733 + 1.47460i 0 0.0394253 + 2.23572i 0 1.29052 + 0.745084i 0 −0.122496 + 0.694710i 0
9.10 0 2.21259 + 2.63686i 0 2.14870 0.618927i 0 −2.51511 1.45210i 0 −1.53655 + 8.71419i 0
149.1 0 −1.00692 2.76648i 0 −1.18175 1.89828i 0 4.39465 2.53725i 0 −4.34139 + 3.64286i 0
149.2 0 −0.920334 2.52860i 0 −0.766078 + 2.10074i 0 −1.72263 + 0.994561i 0 −3.24865 + 2.72594i 0
149.3 0 −0.490777 1.34840i 0 2.12823 + 0.686037i 0 2.27251 1.31204i 0 0.720817 0.604837i 0
149.4 0 −0.423995 1.16492i 0 1.62907 1.53171i 0 −2.04336 + 1.17974i 0 1.12087 0.940523i 0
149.5 0 −0.240889 0.661837i 0 −1.92970 + 1.12972i 0 −0.446593 + 0.257841i 0 1.91813 1.60950i 0
149.6 0 0.240889 + 0.661837i 0 1.42693 + 1.72159i 0 0.446593 0.257841i 0 1.91813 1.60950i 0
149.7 0 0.423995 + 1.16492i 0 −1.00695 1.99651i 0 2.04336 1.17974i 0 1.12087 0.940523i 0
149.8 0 0.490777 + 1.34840i 0 −2.23452 0.0832329i 0 −2.27251 + 1.31204i 0 0.720817 0.604837i 0
149.9 0 0.920334 + 2.52860i 0 0.00138065 + 2.23607i 0 1.72263 0.994561i 0 −3.24865 + 2.72594i 0
149.10 0 1.00692 + 2.76648i 0 1.75973 1.37962i 0 −4.39465 + 2.53725i 0 −4.34139 + 3.64286i 0
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 9.10 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.b even 2 1 inner
19.e even 9 1 inner
95.p even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.bd.a 60
5.b even 2 1 inner 380.2.bd.a 60
19.e even 9 1 inner 380.2.bd.a 60
95.p even 18 1 inner 380.2.bd.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.bd.a 60 1.a even 1 1 trivial
380.2.bd.a 60 5.b even 2 1 inner
380.2.bd.a 60 19.e even 9 1 inner
380.2.bd.a 60 95.p even 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(380, [\chi])$$.