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Properties

Label 1380.2.t.a
Level $1380$
Weight $2$
Character orbit 1380.t
Analytic conductor $11.019$
Analytic rank $0$
Dimension $48$
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] = [1380,2,Mod(1057,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.1057");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.t (of order \(4\), degree \(2\), 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: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{13} - 16 q^{23} - 8 q^{25} + 8 q^{31} + 8 q^{35} - 24 q^{41} + 8 q^{47} - 32 q^{55} - 24 q^{71} + 8 q^{73} + 32 q^{75} + 40 q^{77} - 48 q^{81} + 24 q^{85} - 40 q^{87}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1057.1 0 −0.707107 + 0.707107i 0 −0.811448 − 2.08364i 0 3.03238 − 3.03238i 0 − 1.00000i 0
1057.2 0 −0.707107 + 0.707107i 0 0.811448 + 2.08364i 0 −3.03238 + 3.03238i 0 − 1.00000i 0
1057.3 0 −0.707107 + 0.707107i 0 −1.62167 + 1.53954i 0 2.42328 − 2.42328i 0 − 1.00000i 0
1057.4 0 −0.707107 + 0.707107i 0 1.62167 − 1.53954i 0 −2.42328 + 2.42328i 0 − 1.00000i 0
1057.5 0 −0.707107 + 0.707107i 0 −1.68477 − 1.47022i 0 −2.29608 + 2.29608i 0 − 1.00000i 0
1057.6 0 −0.707107 + 0.707107i 0 1.68477 + 1.47022i 0 2.29608 − 2.29608i 0 − 1.00000i 0
1057.7 0 −0.707107 + 0.707107i 0 −1.35578 + 1.77816i 0 0.243134 − 0.243134i 0 − 1.00000i 0
1057.8 0 −0.707107 + 0.707107i 0 1.35578 − 1.77816i 0 −0.243134 + 0.243134i 0 − 1.00000i 0
1057.9 0 −0.707107 + 0.707107i 0 −2.09989 + 0.768407i 0 −0.0117285 + 0.0117285i 0 − 1.00000i 0
1057.10 0 −0.707107 + 0.707107i 0 2.09989 − 0.768407i 0 0.0117285 − 0.0117285i 0 − 1.00000i 0
1057.11 0 −0.707107 + 0.707107i 0 −0.0653842 − 2.23511i 0 −0.522235 + 0.522235i 0 − 1.00000i 0
1057.12 0 −0.707107 + 0.707107i 0 0.0653842 + 2.23511i 0 0.522235 − 0.522235i 0 − 1.00000i 0
1057.13 0 0.707107 − 0.707107i 0 −0.951596 + 2.02348i 0 3.48236 − 3.48236i 0 − 1.00000i 0
1057.14 0 0.707107 − 0.707107i 0 0.951596 − 2.02348i 0 −3.48236 + 3.48236i 0 − 1.00000i 0
1057.15 0 0.707107 − 0.707107i 0 −2.23126 + 0.146576i 0 2.93317 − 2.93317i 0 − 1.00000i 0
1057.16 0 0.707107 − 0.707107i 0 2.23126 − 0.146576i 0 −2.93317 + 2.93317i 0 − 1.00000i 0
1057.17 0 0.707107 − 0.707107i 0 −1.86909 − 1.22740i 0 0.329148 − 0.329148i 0 − 1.00000i 0
1057.18 0 0.707107 − 0.707107i 0 1.86909 + 1.22740i 0 −0.329148 + 0.329148i 0 − 1.00000i 0
1057.19 0 0.707107 − 0.707107i 0 −0.577284 + 2.16026i 0 −0.575518 + 0.575518i 0 − 1.00000i 0
1057.20 0 0.707107 − 0.707107i 0 0.577284 − 2.16026i 0 0.575518 − 0.575518i 0 − 1.00000i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1057.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.t.a ✓ 48
5.c odd 4 1 inner 1380.2.t.a ✓ 48
23.b odd 2 1 inner 1380.2.t.a ✓ 48
115.e even 4 1 inner 1380.2.t.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.t.a ✓ 48 1.a even 1 1 trivial
1380.2.t.a ✓ 48 5.c odd 4 1 inner
1380.2.t.a ✓ 48 23.b odd 2 1 inner
1380.2.t.a ✓ 48 115.e even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1380, [\chi])\).