↑

Properties

Label 4536.2.k.a
Level $4536$
Weight $2$
Character orbit 4536.k
Analytic conductor $36.220$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4536,2,Mod(3401,4536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4536, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4536.3401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4536.k (of order \(2\), degree \(1\), 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: \(36.2201423569\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 504)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{25} - 24 q^{43} - 12 q^{49} + 24 q^{79} - 12 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3401.1 0 0 0 −3.83667 0 −2.13644 + 1.56064i 0 0 0
3401.2 0 0 0 −3.83667 0 −2.13644 − 1.56064i 0 0 0
3401.3 0 0 0 −3.58605 0 0.0701000 + 2.64482i 0 0 0
3401.4 0 0 0 −3.58605 0 0.0701000 − 2.64482i 0 0 0
3401.5 0 0 0 −3.58246 0 0.735140 + 2.54157i 0 0 0
3401.6 0 0 0 −3.58246 0 0.735140 − 2.54157i 0 0 0
3401.7 0 0 0 −3.20581 0 2.51783 + 0.812729i 0 0 0
3401.8 0 0 0 −3.20581 0 2.51783 − 0.812729i 0 0 0
3401.9 0 0 0 −2.53716 0 −2.50698 + 0.845621i 0 0 0
3401.10 0 0 0 −2.53716 0 −2.50698 − 0.845621i 0 0 0
3401.11 0 0 0 −2.32346 0 2.64446 − 0.0825579i 0 0 0
3401.12 0 0 0 −2.32346 0 2.64446 + 0.0825579i 0 0 0
3401.13 0 0 0 −2.22728 0 0.801129 + 2.52155i 0 0 0
3401.14 0 0 0 −2.22728 0 0.801129 − 2.52155i 0 0 0
3401.15 0 0 0 −1.93130 0 −0.600374 − 2.57673i 0 0 0
3401.16 0 0 0 −1.93130 0 −0.600374 + 2.57673i 0 0 0
3401.17 0 0 0 −0.844961 0 −2.43749 + 1.02891i 0 0 0
3401.18 0 0 0 −0.844961 0 −2.43749 − 1.02891i 0 0 0
3401.19 0 0 0 −0.195490 0 2.02407 + 1.70386i 0 0 0
3401.20 0 0 0 −0.195490 0 2.02407 − 1.70386i 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3401.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4536.2.k.a 48
3.b odd 2 1 inner 4536.2.k.a 48
7.b odd 2 1 inner 4536.2.k.a 48
9.c even 3 1 504.2.bu.a ✓ 48
9.c even 3 1 1512.2.bu.a 48
9.d odd 6 1 504.2.bu.a ✓ 48
9.d odd 6 1 1512.2.bu.a 48
21.c even 2 1 inner 4536.2.k.a 48
36.f odd 6 1 1008.2.cc.d 48
36.f odd 6 1 3024.2.cc.d 48
36.h even 6 1 1008.2.cc.d 48
36.h even 6 1 3024.2.cc.d 48
63.l odd 6 1 504.2.bu.a ✓ 48
63.l odd 6 1 1512.2.bu.a 48
63.o even 6 1 504.2.bu.a ✓ 48
63.o even 6 1 1512.2.bu.a 48
252.s odd 6 1 1008.2.cc.d 48
252.s odd 6 1 3024.2.cc.d 48
252.bi even 6 1 1008.2.cc.d 48
252.bi even 6 1 3024.2.cc.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bu.a ✓ 48 9.c even 3 1
504.2.bu.a ✓ 48 9.d odd 6 1
504.2.bu.a ✓ 48 63.l odd 6 1
504.2.bu.a ✓ 48 63.o even 6 1
1008.2.cc.d 48 36.f odd 6 1
1008.2.cc.d 48 36.h even 6 1
1008.2.cc.d 48 252.s odd 6 1
1008.2.cc.d 48 252.bi even 6 1
1512.2.bu.a 48 9.c even 3 1
1512.2.bu.a 48 9.d odd 6 1
1512.2.bu.a 48 63.l odd 6 1
1512.2.bu.a 48 63.o even 6 1
3024.2.cc.d 48 36.f odd 6 1
3024.2.cc.d 48 36.h even 6 1
3024.2.cc.d 48 252.s odd 6 1
3024.2.cc.d 48 252.bi even 6 1
4536.2.k.a 48 1.a even 1 1 trivial
4536.2.k.a 48 3.b odd 2 1 inner
4536.2.k.a 48 7.b odd 2 1 inner
4536.2.k.a 48 21.c even 2 1 inner