Properties

Label 380.3.bc.a
Level $380$
Weight $3$
Character orbit 380.bc
Analytic conductor $10.354$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [380,3,Mod(29,380)]
 
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, 17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.bc (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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 30 q^{11} - 3 q^{15} + 24 q^{19} - 72 q^{21} + 150 q^{25} + 60 q^{29} + 171 q^{35} + 24 q^{39} - 12 q^{41} - 90 q^{45} + 270 q^{49} - 144 q^{51} + 3 q^{55} + 84 q^{59} + 396 q^{61} - 405 q^{65} + 420 q^{71} + 96 q^{79} - 120 q^{81} + 30 q^{85} + 12 q^{89} - 84 q^{91} + 267 q^{95} + 324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 0 −4.22827 + 3.54794i 0 −3.68991 3.37410i 0 11.4338 + 6.60131i 0 3.72755 21.1400i 0
29.2 0 −4.06848 + 3.41386i 0 4.92418 + 0.867412i 0 −3.35739 1.93839i 0 3.33525 18.9152i 0
29.3 0 −3.70131 + 3.10577i 0 −4.40108 + 2.37286i 0 −5.39353 3.11396i 0 2.49107 14.1276i 0
29.4 0 −2.53827 + 2.12986i 0 1.43585 + 4.78940i 0 7.33686 + 4.23594i 0 0.343673 1.94907i 0
29.5 0 −2.42727 + 2.03672i 0 2.04299 4.56357i 0 −0.860010 0.496527i 0 0.180573 1.02408i 0
29.6 0 −2.15463 + 1.80795i 0 0.381579 + 4.98542i 0 −3.17399 1.83251i 0 −0.189080 + 1.07233i 0
29.7 0 −1.55865 + 1.30786i 0 −3.89091 3.14020i 0 −6.81981 3.93742i 0 −0.843948 + 4.78627i 0
29.8 0 −1.41356 + 1.18612i 0 2.50989 4.32440i 0 6.52791 + 3.76889i 0 −0.971555 + 5.50996i 0
29.9 0 −0.915239 + 0.767977i 0 −4.79298 + 1.42385i 0 4.52632 + 2.61327i 0 −1.31496 + 7.45750i 0
29.10 0 −0.215152 + 0.180534i 0 4.83429 + 1.27657i 0 −9.68534 5.59183i 0 −1.54914 + 8.78559i 0
29.11 0 0.215152 0.180534i 0 4.52385 + 2.12951i 0 9.68534 + 5.59183i 0 −1.54914 + 8.78559i 0
29.12 0 0.915239 0.767977i 0 −2.75641 4.17160i 0 −4.52632 2.61327i 0 −1.31496 + 7.45750i 0
29.13 0 1.41356 1.18612i 0 −0.856980 + 4.92601i 0 −6.52791 3.76889i 0 −0.971555 + 5.50996i 0
29.14 0 1.55865 1.30786i 0 −4.99909 0.0954977i 0 6.81981 + 3.93742i 0 −0.843948 + 4.78627i 0
29.15 0 2.15463 1.80795i 0 3.49687 3.57378i 0 3.17399 + 1.83251i 0 −0.189080 + 1.07233i 0
29.16 0 2.42727 2.03672i 0 −1.36839 + 4.80911i 0 0.860010 + 0.496527i 0 0.180573 1.02408i 0
29.17 0 2.53827 2.12986i 0 4.17849 2.74595i 0 −7.33686 4.23594i 0 0.343673 1.94907i 0
29.18 0 3.70131 3.10577i 0 −1.84618 4.64668i 0 5.39353 + 3.11396i 0 2.49107 14.1276i 0
29.19 0 4.06848 3.41386i 0 4.32971 + 2.50073i 0 3.35739 + 1.93839i 0 3.33525 18.9152i 0
29.20 0 4.22827 3.54794i 0 −4.99547 + 0.212883i 0 −11.4338 6.60131i 0 3.72755 21.1400i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.f odd 18 1 inner
95.o odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.bc.a 120
5.b even 2 1 inner 380.3.bc.a 120
19.f odd 18 1 inner 380.3.bc.a 120
95.o odd 18 1 inner 380.3.bc.a 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.bc.a 120 1.a even 1 1 trivial
380.3.bc.a 120 5.b even 2 1 inner
380.3.bc.a 120 19.f odd 18 1 inner
380.3.bc.a 120 95.o odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).