Properties

Label 2016.2.cp.b
Level $2016$
Weight $2$
Character orbit 2016.cp
Analytic conductor $16.098$
Analytic rank $0$
Dimension $56$
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(17,2016)]
 
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([0, 3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.cp (of order \(6\), degree \(2\), 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: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 504)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 20 q^{7} + 8 q^{25} + 36 q^{31} - 28 q^{49} + 72 q^{73} + 12 q^{79}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 0 0 −3.18706 + 1.84005i 0 0.998380 + 2.45015i 0 0 0
17.2 0 0 0 −3.18706 + 1.84005i 0 0.998380 + 2.45015i 0 0 0
17.3 0 0 0 −3.16007 + 1.82447i 0 1.64838 2.06951i 0 0 0
17.4 0 0 0 −3.16007 + 1.82447i 0 1.64838 2.06951i 0 0 0
17.5 0 0 0 −1.87230 + 1.08097i 0 −2.55958 0.669737i 0 0 0
17.6 0 0 0 −1.87230 + 1.08097i 0 −2.55958 0.669737i 0 0 0
17.7 0 0 0 −1.53798 + 0.887954i 0 −0.843933 + 2.50754i 0 0 0
17.8 0 0 0 −1.53798 + 0.887954i 0 −0.843933 + 2.50754i 0 0 0
17.9 0 0 0 −1.00441 + 0.579896i 0 −1.24394 2.33508i 0 0 0
17.10 0 0 0 −1.00441 + 0.579896i 0 −1.24394 2.33508i 0 0 0
17.11 0 0 0 −0.785247 + 0.453362i 0 2.47043 + 0.947077i 0 0 0
17.12 0 0 0 −0.785247 + 0.453362i 0 2.47043 + 0.947077i 0 0 0
17.13 0 0 0 −0.331990 + 0.191675i 0 2.03027 1.69647i 0 0 0
17.14 0 0 0 −0.331990 + 0.191675i 0 2.03027 1.69647i 0 0 0
17.15 0 0 0 0.331990 0.191675i 0 2.03027 1.69647i 0 0 0
17.16 0 0 0 0.331990 0.191675i 0 2.03027 1.69647i 0 0 0
17.17 0 0 0 0.785247 0.453362i 0 2.47043 + 0.947077i 0 0 0
17.18 0 0 0 0.785247 0.453362i 0 2.47043 + 0.947077i 0 0 0
17.19 0 0 0 1.00441 0.579896i 0 −1.24394 2.33508i 0 0 0
17.20 0 0 0 1.00441 0.579896i 0 −1.24394 2.33508i 0 0 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
168.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.cp.b 56
3.b odd 2 1 inner 2016.2.cp.b 56
4.b odd 2 1 504.2.ch.b 56
7.d odd 6 1 inner 2016.2.cp.b 56
8.b even 2 1 inner 2016.2.cp.b 56
8.d odd 2 1 504.2.ch.b 56
12.b even 2 1 504.2.ch.b 56
21.g even 6 1 inner 2016.2.cp.b 56
24.f even 2 1 504.2.ch.b 56
24.h odd 2 1 inner 2016.2.cp.b 56
28.f even 6 1 504.2.ch.b 56
56.j odd 6 1 inner 2016.2.cp.b 56
56.m even 6 1 504.2.ch.b 56
84.j odd 6 1 504.2.ch.b 56
168.ba even 6 1 inner 2016.2.cp.b 56
168.be odd 6 1 504.2.ch.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.ch.b 56 4.b odd 2 1
504.2.ch.b 56 8.d odd 2 1
504.2.ch.b 56 12.b even 2 1
504.2.ch.b 56 24.f even 2 1
504.2.ch.b 56 28.f even 6 1
504.2.ch.b 56 56.m even 6 1
504.2.ch.b 56 84.j odd 6 1
504.2.ch.b 56 168.be odd 6 1
2016.2.cp.b 56 1.a even 1 1 trivial
2016.2.cp.b 56 3.b odd 2 1 inner
2016.2.cp.b 56 7.d odd 6 1 inner
2016.2.cp.b 56 8.b even 2 1 inner
2016.2.cp.b 56 21.g even 6 1 inner
2016.2.cp.b 56 24.h odd 2 1 inner
2016.2.cp.b 56 56.j odd 6 1 inner
2016.2.cp.b 56 168.ba even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} - 37 T_{5}^{26} + 882 T_{5}^{24} - 12429 T_{5}^{22} + 126330 T_{5}^{20} - 820581 T_{5}^{18} + 3879393 T_{5}^{16} - 12338064 T_{5}^{14} + 28587144 T_{5}^{12} - 42674992 T_{5}^{10} + \cdots + 186624 \) acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\). Copy content Toggle raw display