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Properties

Label 2016.3.g.d
Level $2016$
Weight $3$
Character orbit 2016.g
Analytic conductor $54.932$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,3,Mod(1135,2016)]
 
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, 1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.1135");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.g (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: \(24\)
Twist minimal: no (minimal twist has level 168)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 32 q^{11} - 16 q^{17} + 64 q^{19} - 72 q^{25} + 80 q^{41} - 32 q^{43} - 168 q^{49} + 192 q^{65} + 32 q^{67} - 240 q^{73} - 320 q^{83} - 400 q^{89} + 144 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1135.1 0 0 0 − 9.26683i 0 2.64575i 0 0 0
1135.2 0 0 0 − 8.04930i 0 2.64575i 0 0 0
1135.3 0 0 0 − 7.47825i 0 − 2.64575i 0 0 0
1135.4 0 0 0 − 6.16432i 0 2.64575i 0 0 0
1135.5 0 0 0 − 5.94177i 0 − 2.64575i 0 0 0
1135.6 0 0 0 − 4.09875i 0 − 2.64575i 0 0 0
1135.7 0 0 0 − 3.78720i 0 − 2.64575i 0 0 0
1135.8 0 0 0 − 3.19600i 0 − 2.64575i 0 0 0
1135.9 0 0 0 − 2.78060i 0 − 2.64575i 0 0 0
1135.10 0 0 0 − 2.47223i 0 2.64575i 0 0 0
1135.11 0 0 0 − 0.769463i 0 2.64575i 0 0 0
1135.12 0 0 0 − 0.560422i 0 2.64575i 0 0 0
1135.13 0 0 0 0.560422i 0 − 2.64575i 0 0 0
1135.14 0 0 0 0.769463i 0 − 2.64575i 0 0 0
1135.15 0 0 0 2.47223i 0 − 2.64575i 0 0 0
1135.16 0 0 0 2.78060i 0 2.64575i 0 0 0
1135.17 0 0 0 3.19600i 0 2.64575i 0 0 0
1135.18 0 0 0 3.78720i 0 2.64575i 0 0 0
1135.19 0 0 0 4.09875i 0 2.64575i 0 0 0
1135.20 0 0 0 5.94177i 0 2.64575i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1135.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d 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.g.d 24
3.b odd 2 1 672.3.g.a 24
4.b odd 2 1 504.3.g.d 24
8.b even 2 1 504.3.g.d 24
8.d odd 2 1 inner 2016.3.g.d 24
12.b even 2 1 168.3.g.a ✓ 24
24.f even 2 1 672.3.g.a 24
24.h odd 2 1 168.3.g.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.g.a ✓ 24 12.b even 2 1
168.3.g.a ✓ 24 24.h odd 2 1
504.3.g.d 24 4.b odd 2 1
504.3.g.d 24 8.b even 2 1
672.3.g.a 24 3.b odd 2 1
672.3.g.a 24 24.f even 2 1
2016.3.g.d 24 1.a even 1 1 trivial
2016.3.g.d 24 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 336 T_{5}^{22} + 47408 T_{5}^{20} + 3680896 T_{5}^{18} + 173353056 T_{5}^{16} + \cdots + 9028053827584 \) acting on \(S_{3}^{\mathrm{new}}(2016, [\chi])\). Copy content Toggle raw display