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Properties

Label 1380.5.d.a
Level $1380$
Weight $5$
Character orbit 1380.d
Analytic conductor $142.651$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1380,5,Mod(781,1380)]
 
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, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.781");
 
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.d (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: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 1728 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 1728 q^{9} - 408 q^{13} - 1008 q^{23} - 8000 q^{25} + 732 q^{29} - 2300 q^{31} - 3300 q^{35} + 2016 q^{39} - 1380 q^{41} - 6408 q^{47} - 14788 q^{49} - 4200 q^{55} + 11412 q^{59} + 4176 q^{69} + 10164 q^{71} - 6424 q^{73} + 15480 q^{77} + 46656 q^{81} + 3300 q^{85} + 6696 q^{87} + 8064 q^{93} - 13200 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
781.1 0 −5.19615 0 − 11.1803i 0 − 89.9468i 0 27.0000 0
781.2 0 −5.19615 0 − 11.1803i 0 − 65.7824i 0 27.0000 0
781.3 0 −5.19615 0 − 11.1803i 0 − 65.2196i 0 27.0000 0
781.4 0 −5.19615 0 − 11.1803i 0 − 60.0944i 0 27.0000 0
781.5 0 −5.19615 0 − 11.1803i 0 − 48.7537i 0 27.0000 0
781.6 0 −5.19615 0 − 11.1803i 0 − 42.4865i 0 27.0000 0
781.7 0 −5.19615 0 − 11.1803i 0 − 4.77839i 0 27.0000 0
781.8 0 −5.19615 0 − 11.1803i 0 − 0.190795i 0 27.0000 0
781.9 0 −5.19615 0 − 11.1803i 0 9.05971i 0 27.0000 0
781.10 0 −5.19615 0 − 11.1803i 0 21.8447i 0 27.0000 0
781.11 0 −5.19615 0 − 11.1803i 0 22.0069i 0 27.0000 0
781.12 0 −5.19615 0 − 11.1803i 0 27.3162i 0 27.0000 0
781.13 0 −5.19615 0 − 11.1803i 0 27.9238i 0 27.0000 0
781.14 0 −5.19615 0 − 11.1803i 0 58.7374i 0 27.0000 0
781.15 0 −5.19615 0 − 11.1803i 0 71.9309i 0 27.0000 0
781.16 0 −5.19615 0 − 11.1803i 0 72.3888i 0 27.0000 0
781.17 0 −5.19615 0 11.1803i 0 − 72.3888i 0 27.0000 0
781.18 0 −5.19615 0 11.1803i 0 − 71.9309i 0 27.0000 0
781.19 0 −5.19615 0 11.1803i 0 − 58.7374i 0 27.0000 0
781.20 0 −5.19615 0 11.1803i 0 − 27.9238i 0 27.0000 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 781.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b 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.d.a ✓ 64
23.b odd 2 1 inner 1380.5.d.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.5.d.a ✓ 64 1.a even 1 1 trivial
1380.5.d.a ✓ 64 23.b odd 2 1 inner

Hecke kernels

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