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Properties

Label 2016.3.n.a
Level $2016$
Weight $3$
Character orbit 2016.n
Analytic conductor $54.932$
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] = [2016,3,Mod(1457,2016)]
 
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([0, 1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.1457");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.n (of order \(2\), degree \(1\), not 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: \(48\)
Twist minimal: no (minimal twist has level 504)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 240 q^{25} - 256 q^{31} + 336 q^{49} + 512 q^{55} - 640 q^{79}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1457.1 0 0 0 −9.31660 0 2.64575 0 0 0
1457.2 0 0 0 −9.31660 0 2.64575 0 0 0
1457.3 0 0 0 −8.15465 0 −2.64575 0 0 0
1457.4 0 0 0 −8.15465 0 −2.64575 0 0 0
1457.5 0 0 0 −7.03286 0 2.64575 0 0 0
1457.6 0 0 0 −7.03286 0 2.64575 0 0 0
1457.7 0 0 0 −6.52341 0 −2.64575 0 0 0
1457.8 0 0 0 −6.52341 0 −2.64575 0 0 0
1457.9 0 0 0 −6.09763 0 2.64575 0 0 0
1457.10 0 0 0 −6.09763 0 2.64575 0 0 0
1457.11 0 0 0 −5.36369 0 −2.64575 0 0 0
1457.12 0 0 0 −5.36369 0 −2.64575 0 0 0
1457.13 0 0 0 −5.13782 0 −2.64575 0 0 0
1457.14 0 0 0 −5.13782 0 −2.64575 0 0 0
1457.15 0 0 0 −3.92423 0 −2.64575 0 0 0
1457.16 0 0 0 −3.92423 0 −2.64575 0 0 0
1457.17 0 0 0 −1.92658 0 2.64575 0 0 0
1457.18 0 0 0 −1.92658 0 2.64575 0 0 0
1457.19 0 0 0 −1.39175 0 2.64575 0 0 0
1457.20 0 0 0 −1.39175 0 2.64575 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1457.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.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.n.a 48
3.b odd 2 1 inner 2016.3.n.a 48
4.b odd 2 1 504.3.n.a ✓ 48
8.b even 2 1 inner 2016.3.n.a 48
8.d odd 2 1 504.3.n.a ✓ 48
12.b even 2 1 504.3.n.a ✓ 48
24.f even 2 1 504.3.n.a ✓ 48
24.h odd 2 1 inner 2016.3.n.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.3.n.a ✓ 48 4.b odd 2 1
504.3.n.a ✓ 48 8.d odd 2 1
504.3.n.a ✓ 48 12.b even 2 1
504.3.n.a ✓ 48 24.f even 2 1
2016.3.n.a 48 1.a even 1 1 trivial
2016.3.n.a 48 3.b odd 2 1 inner
2016.3.n.a 48 8.b even 2 1 inner
2016.3.n.a 48 24.h odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(2016, [\chi])\).