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Properties

Label 2016.3.o.a
Level $2016$
Weight $3$
Character orbit 2016.o
Analytic conductor $54.932$
Analytic rank $0$
Dimension $32$
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] = [2016,3,Mod(2015,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.2015");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.o (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: \(54.9320212950\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{11} - 80 q^{23} + 128 q^{25} - 32 q^{35} - 160 q^{37} + 64 q^{49} - 16 q^{71} - 32 q^{85} + 896 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2015.1 0 0 0 −9.13921 0 6.96161 − 0.732092i 0 0 0
2015.2 0 0 0 −9.13921 0 6.96161 + 0.732092i 0 0 0
2015.3 0 0 0 −6.28975 0 0.309983 − 6.99313i 0 0 0
2015.4 0 0 0 −6.28975 0 0.309983 + 6.99313i 0 0 0
2015.5 0 0 0 −5.98281 0 −6.02595 − 3.56202i 0 0 0
2015.6 0 0 0 −5.98281 0 −6.02595 + 3.56202i 0 0 0
2015.7 0 0 0 −5.85510 0 4.04236 + 5.71484i 0 0 0
2015.8 0 0 0 −5.85510 0 4.04236 − 5.71484i 0 0 0
2015.9 0 0 0 −5.41837 0 −3.79876 + 5.87957i 0 0 0
2015.10 0 0 0 −5.41837 0 −3.79876 − 5.87957i 0 0 0
2015.11 0 0 0 −2.94851 0 −6.24598 − 3.16034i 0 0 0
2015.12 0 0 0 −2.94851 0 −6.24598 + 3.16034i 0 0 0
2015.13 0 0 0 −0.840274 0 −6.38482 + 2.86951i 0 0 0
2015.14 0 0 0 −0.840274 0 −6.38482 − 2.86951i 0 0 0
2015.15 0 0 0 −0.281744 0 −2.92889 − 6.35780i 0 0 0
2015.16 0 0 0 −0.281744 0 −2.92889 + 6.35780i 0 0 0
2015.17 0 0 0 0.281744 0 2.92889 + 6.35780i 0 0 0
2015.18 0 0 0 0.281744 0 2.92889 − 6.35780i 0 0 0
2015.19 0 0 0 0.840274 0 6.38482 + 2.86951i 0 0 0
2015.20 0 0 0 0.840274 0 6.38482 − 2.86951i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2015.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
12.b even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.3.o.a ✓ 32
3.b odd 2 1 2016.3.o.b yes 32
4.b odd 2 1 2016.3.o.b yes 32
7.b odd 2 1 inner 2016.3.o.a ✓ 32
12.b even 2 1 inner 2016.3.o.a ✓ 32
21.c even 2 1 2016.3.o.b yes 32
28.d even 2 1 2016.3.o.b yes 32
84.h odd 2 1 inner 2016.3.o.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.3.o.a ✓ 32 1.a even 1 1 trivial
2016.3.o.a ✓ 32 7.b odd 2 1 inner
2016.3.o.a ✓ 32 12.b even 2 1 inner
2016.3.o.a ✓ 32 84.h odd 2 1 inner
2016.3.o.b yes 32 3.b odd 2 1
2016.3.o.b yes 32 4.b odd 2 1
2016.3.o.b yes 32 21.c even 2 1
2016.3.o.b yes 32 28.d even 2 1