Properties

Label 416.2.bk.a
Level $416$
Weight $2$
Character orbit 416.bk
Analytic conductor $3.322$
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] = [416,2,Mod(15,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.15");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.bk (of order \(12\), degree \(4\), 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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83} + 12 q^{89} - 104 q^{91} + 4 q^{97} - 168 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 0 −1.49630 2.59167i 0 −1.40823 1.40823i 0 −1.02175 3.81323i 0 −2.97784 + 5.15776i 0
15.2 0 −1.49630 2.59167i 0 1.40823 + 1.40823i 0 1.02175 + 3.81323i 0 −2.97784 + 5.15776i 0
15.3 0 −0.793616 1.37458i 0 −0.203095 0.203095i 0 −0.207385 0.773970i 0 0.240346 0.416291i 0
15.4 0 −0.793616 1.37458i 0 0.203095 + 0.203095i 0 0.207385 + 0.773970i 0 0.240346 0.416291i 0
15.5 0 0.217361 + 0.376480i 0 −1.34711 1.34711i 0 1.10888 + 4.13839i 0 1.40551 2.43441i 0
15.6 0 0.217361 + 0.376480i 0 1.34711 + 1.34711i 0 −1.10888 4.13839i 0 1.40551 2.43441i 0
15.7 0 0.253720 + 0.439456i 0 −2.25762 2.25762i 0 0.0840763 + 0.313777i 0 1.37125 2.37508i 0
15.8 0 0.253720 + 0.439456i 0 2.25762 + 2.25762i 0 −0.0840763 0.313777i 0 1.37125 2.37508i 0
15.9 0 0.958009 + 1.65932i 0 −2.08788 2.08788i 0 −0.981174 3.66179i 0 −0.335561 + 0.581208i 0
15.10 0 0.958009 + 1.65932i 0 2.08788 + 2.08788i 0 0.981174 + 3.66179i 0 −0.335561 + 0.581208i 0
15.11 0 1.36083 + 2.35702i 0 −1.35602 1.35602i 0 0.327458 + 1.22209i 0 −2.20371 + 3.81694i 0
15.12 0 1.36083 + 2.35702i 0 1.35602 + 1.35602i 0 −0.327458 1.22209i 0 −2.20371 + 3.81694i 0
111.1 0 −1.49630 + 2.59167i 0 −1.40823 + 1.40823i 0 −1.02175 + 3.81323i 0 −2.97784 5.15776i 0
111.2 0 −1.49630 + 2.59167i 0 1.40823 1.40823i 0 1.02175 3.81323i 0 −2.97784 5.15776i 0
111.3 0 −0.793616 + 1.37458i 0 −0.203095 + 0.203095i 0 −0.207385 + 0.773970i 0 0.240346 + 0.416291i 0
111.4 0 −0.793616 + 1.37458i 0 0.203095 0.203095i 0 0.207385 0.773970i 0 0.240346 + 0.416291i 0
111.5 0 0.217361 0.376480i 0 −1.34711 + 1.34711i 0 1.10888 4.13839i 0 1.40551 + 2.43441i 0
111.6 0 0.217361 0.376480i 0 1.34711 1.34711i 0 −1.10888 + 4.13839i 0 1.40551 + 2.43441i 0
111.7 0 0.253720 0.439456i 0 −2.25762 + 2.25762i 0 0.0840763 0.313777i 0 1.37125 + 2.37508i 0
111.8 0 0.253720 0.439456i 0 2.25762 2.25762i 0 −0.0840763 + 0.313777i 0 1.37125 + 2.37508i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
13.f odd 12 1 inner
104.u even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.bk.a 48
4.b odd 2 1 104.2.u.a 48
8.b even 2 1 104.2.u.a 48
8.d odd 2 1 inner 416.2.bk.a 48
12.b even 2 1 936.2.ed.d 48
13.f odd 12 1 inner 416.2.bk.a 48
24.h odd 2 1 936.2.ed.d 48
52.l even 12 1 104.2.u.a 48
104.u even 12 1 inner 416.2.bk.a 48
104.x odd 12 1 104.2.u.a 48
156.v odd 12 1 936.2.ed.d 48
312.bo even 12 1 936.2.ed.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.u.a 48 4.b odd 2 1
104.2.u.a 48 8.b even 2 1
104.2.u.a 48 52.l even 12 1
104.2.u.a 48 104.x odd 12 1
416.2.bk.a 48 1.a even 1 1 trivial
416.2.bk.a 48 8.d odd 2 1 inner
416.2.bk.a 48 13.f odd 12 1 inner
416.2.bk.a 48 104.u even 12 1 inner
936.2.ed.d 48 12.b even 2 1
936.2.ed.d 48 24.h odd 2 1
936.2.ed.d 48 156.v odd 12 1
936.2.ed.d 48 312.bo even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(416, [\chi])\).