Properties

Label 1008.2.bf.i
Level $1008$
Weight $2$
Character orbit 1008.bf
Analytic conductor $8.049$
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] = [1008,2,Mod(31,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bf (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: \(8.04892052375\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{9} - 6 q^{13} - 18 q^{17} - 8 q^{21} - 32 q^{25} - 12 q^{29} + 30 q^{33} + 2 q^{37} + 36 q^{41} + 30 q^{45} + 2 q^{49} - 12 q^{53} - 46 q^{57} + 42 q^{61} + 18 q^{65} - 42 q^{69} - 66 q^{77} - 16 q^{81} - 12 q^{85} - 18 q^{89} + 58 q^{93} - 6 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −1.72863 0.108820i 0 0.435427i 0 2.49096 + 0.891704i 0 2.97632 + 0.376218i 0
31.2 0 −1.72423 0.164453i 0 2.22064i 0 −0.517968 2.59455i 0 2.94591 + 0.567107i 0
31.3 0 −1.33729 + 1.10075i 0 1.09736i 0 1.10288 + 2.40492i 0 0.576701 2.94405i 0
31.4 0 −1.14316 + 1.30123i 0 4.29566i 0 0.334043 2.62458i 0 −0.386381 2.97501i 0
31.5 0 −1.13731 1.30634i 0 0.618621i 0 −1.03639 + 2.43431i 0 −0.413051 + 2.97143i 0
31.6 0 −0.840104 + 1.51467i 0 2.34835i 0 −2.61922 + 0.373730i 0 −1.58845 2.54496i 0
31.7 0 −0.624780 1.61544i 0 2.39618i 0 2.35781 1.20030i 0 −2.21930 + 2.01859i 0
31.8 0 −0.232652 1.71635i 0 3.40340i 0 −2.63727 0.211643i 0 −2.89175 + 0.798628i 0
31.9 0 0.232652 + 1.71635i 0 3.40340i 0 2.63727 + 0.211643i 0 −2.89175 + 0.798628i 0
31.10 0 0.624780 + 1.61544i 0 2.39618i 0 −2.35781 + 1.20030i 0 −2.21930 + 2.01859i 0
31.11 0 0.840104 1.51467i 0 2.34835i 0 2.61922 0.373730i 0 −1.58845 2.54496i 0
31.12 0 1.13731 + 1.30634i 0 0.618621i 0 1.03639 2.43431i 0 −0.413051 + 2.97143i 0
31.13 0 1.14316 1.30123i 0 4.29566i 0 −0.334043 + 2.62458i 0 −0.386381 2.97501i 0
31.14 0 1.33729 1.10075i 0 1.09736i 0 −1.10288 2.40492i 0 0.576701 2.94405i 0
31.15 0 1.72423 + 0.164453i 0 2.22064i 0 0.517968 + 2.59455i 0 2.94591 + 0.567107i 0
31.16 0 1.72863 + 0.108820i 0 0.435427i 0 −2.49096 0.891704i 0 2.97632 + 0.376218i 0
943.1 0 −1.72863 + 0.108820i 0 0.435427i 0 2.49096 0.891704i 0 2.97632 0.376218i 0
943.2 0 −1.72423 + 0.164453i 0 2.22064i 0 −0.517968 + 2.59455i 0 2.94591 0.567107i 0
943.3 0 −1.33729 1.10075i 0 1.09736i 0 1.10288 2.40492i 0 0.576701 + 2.94405i 0
943.4 0 −1.14316 1.30123i 0 4.29566i 0 0.334043 + 2.62458i 0 −0.386381 + 2.97501i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
63.k odd 6 1 inner
252.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.bf.i 32
3.b odd 2 1 3024.2.bf.i 32
4.b odd 2 1 inner 1008.2.bf.i 32
7.d odd 6 1 1008.2.cz.i yes 32
9.c even 3 1 1008.2.cz.i yes 32
9.d odd 6 1 3024.2.cz.i 32
12.b even 2 1 3024.2.bf.i 32
21.g even 6 1 3024.2.cz.i 32
28.f even 6 1 1008.2.cz.i yes 32
36.f odd 6 1 1008.2.cz.i yes 32
36.h even 6 1 3024.2.cz.i 32
63.k odd 6 1 inner 1008.2.bf.i 32
63.s even 6 1 3024.2.bf.i 32
84.j odd 6 1 3024.2.cz.i 32
252.n even 6 1 inner 1008.2.bf.i 32
252.bn odd 6 1 3024.2.bf.i 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.i 32 1.a even 1 1 trivial
1008.2.bf.i 32 4.b odd 2 1 inner
1008.2.bf.i 32 63.k odd 6 1 inner
1008.2.bf.i 32 252.n even 6 1 inner
1008.2.cz.i yes 32 7.d odd 6 1
1008.2.cz.i yes 32 9.c even 3 1
1008.2.cz.i yes 32 28.f even 6 1
1008.2.cz.i yes 32 36.f odd 6 1
3024.2.bf.i 32 3.b odd 2 1
3024.2.bf.i 32 12.b even 2 1
3024.2.bf.i 32 63.s even 6 1
3024.2.bf.i 32 252.bn odd 6 1
3024.2.cz.i 32 9.d odd 6 1
3024.2.cz.i 32 21.g even 6 1
3024.2.cz.i 32 36.h even 6 1
3024.2.cz.i 32 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{16} + 48T_{5}^{14} + 870T_{5}^{12} + 7668T_{5}^{10} + 35001T_{5}^{8} + 79623T_{5}^{6} + 77598T_{5}^{4} + 27459T_{5}^{2} + 2916 \) Copy content Toggle raw display
\( T_{19}^{32} + 146 T_{19}^{30} + 13545 T_{19}^{28} + 760432 T_{19}^{26} + 30956012 T_{19}^{24} + 845979156 T_{19}^{22} + 16893351658 T_{19}^{20} + 222174856073 T_{19}^{18} + \cdots + 156863626279441 \) Copy content Toggle raw display