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Properties

Label 1380.5.k.a
Level $1380$
Weight $5$
Character orbit 1380.k
Analytic conductor $142.651$
Analytic rank $0$
Dimension $96$
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] = [1380,5,Mod(229,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.229");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1380.k (of order \(2\), degree \(1\), 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: \(142.650549056\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2592 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2592 q^{9} - 1248 q^{25} - 2412 q^{29} - 1564 q^{31} + 5148 q^{35} + 2016 q^{39} + 2172 q^{41} + 38660 q^{49} + 8584 q^{55} + 4044 q^{59} - 3168 q^{69} + 13332 q^{71} - 7560 q^{75} + 69984 q^{81} - 38516 q^{85} - 12384 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 0 − 5.19615i 0 −18.3092 + 17.0227i 0 7.95643 0 −27.0000 0
229.2 0 − 5.19615i 0 18.3092 − 17.0227i 0 −7.95643 0 −27.0000 0
229.3 0 5.19615i 0 −18.3092 − 17.0227i 0 7.95643 0 −27.0000 0
229.4 0 5.19615i 0 18.3092 + 17.0227i 0 −7.95643 0 −27.0000 0
229.5 0 − 5.19615i 0 −9.38660 − 23.1709i 0 −93.0466 0 −27.0000 0
229.6 0 − 5.19615i 0 9.38660 + 23.1709i 0 93.0466 0 −27.0000 0
229.7 0 5.19615i 0 −9.38660 + 23.1709i 0 −93.0466 0 −27.0000 0
229.8 0 5.19615i 0 9.38660 − 23.1709i 0 93.0466 0 −27.0000 0
229.9 0 − 5.19615i 0 −11.8318 + 22.0229i 0 −59.8309 0 −27.0000 0
229.10 0 − 5.19615i 0 11.8318 − 22.0229i 0 59.8309 0 −27.0000 0
229.11 0 5.19615i 0 −11.8318 − 22.0229i 0 −59.8309 0 −27.0000 0
229.12 0 5.19615i 0 11.8318 + 22.0229i 0 59.8309 0 −27.0000 0
229.13 0 − 5.19615i 0 −13.6191 − 20.9647i 0 −75.2441 0 −27.0000 0
229.14 0 − 5.19615i 0 13.6191 + 20.9647i 0 75.2441 0 −27.0000 0
229.15 0 5.19615i 0 −13.6191 + 20.9647i 0 −75.2441 0 −27.0000 0
229.16 0 5.19615i 0 13.6191 − 20.9647i 0 75.2441 0 −27.0000 0
229.17 0 − 5.19615i 0 −24.9973 + 0.369248i 0 51.4038 0 −27.0000 0
229.18 0 − 5.19615i 0 24.9973 − 0.369248i 0 −51.4038 0 −27.0000 0
229.19 0 5.19615i 0 −24.9973 − 0.369248i 0 51.4038 0 −27.0000 0
229.20 0 5.19615i 0 24.9973 + 0.369248i 0 −51.4038 0 −27.0000 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.b odd 2 1 inner
115.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.5.k.a ✓ 96
5.b even 2 1 inner 1380.5.k.a ✓ 96
23.b odd 2 1 inner 1380.5.k.a ✓ 96
115.c odd 2 1 inner 1380.5.k.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.5.k.a ✓ 96 1.a even 1 1 trivial
1380.5.k.a ✓ 96 5.b even 2 1 inner
1380.5.k.a ✓ 96 23.b odd 2 1 inner
1380.5.k.a ✓ 96 115.c odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(1380, [\chi])\).