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Properties

Label 2016.2.cs.d
Level $2016$
Weight $2$
Character orbit 2016.cs
Analytic conductor $16.098$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,2,Mod(703,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.cs (of order \(6\), 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: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{25} + 8 q^{37} - 16 q^{49} - 48 q^{61} + 24 q^{73} - 128 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1 0 0 0 −3.36010 + 1.93995i 0 −1.51299 − 2.17045i 0 0 0
703.2 0 0 0 −3.36010 + 1.93995i 0 1.51299 + 2.17045i 0 0 0
703.3 0 0 0 −2.01791 + 1.16504i 0 −1.90670 + 1.83426i 0 0 0
703.4 0 0 0 −2.01791 + 1.16504i 0 1.90670 − 1.83426i 0 0 0
703.5 0 0 0 −1.01538 + 0.586228i 0 0.488880 + 2.60019i 0 0 0
703.6 0 0 0 −1.01538 + 0.586228i 0 −0.488880 − 2.60019i 0 0 0
703.7 0 0 0 −0.326815 + 0.188687i 0 2.61464 − 0.404519i 0 0 0
703.8 0 0 0 −0.326815 + 0.188687i 0 −2.61464 + 0.404519i 0 0 0
703.9 0 0 0 0.326815 − 0.188687i 0 −2.61464 + 0.404519i 0 0 0
703.10 0 0 0 0.326815 − 0.188687i 0 2.61464 − 0.404519i 0 0 0
703.11 0 0 0 1.01538 − 0.586228i 0 −0.488880 − 2.60019i 0 0 0
703.12 0 0 0 1.01538 − 0.586228i 0 0.488880 + 2.60019i 0 0 0
703.13 0 0 0 2.01791 − 1.16504i 0 1.90670 − 1.83426i 0 0 0
703.14 0 0 0 2.01791 − 1.16504i 0 −1.90670 + 1.83426i 0 0 0
703.15 0 0 0 3.36010 − 1.93995i 0 1.51299 + 2.17045i 0 0 0
703.16 0 0 0 3.36010 − 1.93995i 0 −1.51299 − 2.17045i 0 0 0
1279.1 0 0 0 −3.36010 − 1.93995i 0 −1.51299 + 2.17045i 0 0 0
1279.2 0 0 0 −3.36010 − 1.93995i 0 1.51299 − 2.17045i 0 0 0
1279.3 0 0 0 −2.01791 − 1.16504i 0 −1.90670 − 1.83426i 0 0 0
1279.4 0 0 0 −2.01791 − 1.16504i 0 1.90670 + 1.83426i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.d odd 6 1 inner
12.b even 2 1 inner
21.g even 6 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.cs.d ✓ 32
3.b odd 2 1 inner 2016.2.cs.d ✓ 32
4.b odd 2 1 inner 2016.2.cs.d ✓ 32
7.d odd 6 1 inner 2016.2.cs.d ✓ 32
12.b even 2 1 inner 2016.2.cs.d ✓ 32
21.g even 6 1 inner 2016.2.cs.d ✓ 32
28.f even 6 1 inner 2016.2.cs.d ✓ 32
84.j odd 6 1 inner 2016.2.cs.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.cs.d ✓ 32 1.a even 1 1 trivial
2016.2.cs.d ✓ 32 3.b odd 2 1 inner
2016.2.cs.d ✓ 32 4.b odd 2 1 inner
2016.2.cs.d ✓ 32 7.d odd 6 1 inner
2016.2.cs.d ✓ 32 12.b even 2 1 inner
2016.2.cs.d ✓ 32 21.g even 6 1 inner
2016.2.cs.d ✓ 32 28.f even 6 1 inner
2016.2.cs.d ✓ 32 84.j odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\):

\( T_{5}^{16} - 22T_{5}^{14} + 371T_{5}^{12} - 2230T_{5}^{10} + 9937T_{5}^{8} - 13760T_{5}^{6} + 14576T_{5}^{4} - 2048T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{16} - 46 T_{11}^{14} + 1555 T_{11}^{12} - 23758 T_{11}^{10} + 267361 T_{11}^{8} - 550912 T_{11}^{6} + \cdots + 65536 \) Copy content Toggle raw display