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Properties

Label 432.2.l.b
Level $432$
Weight $2$
Character orbit 432.l
Analytic conductor $3.450$
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] = [432,2,Mod(107,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.l (of order \(4\), 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: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{10} - 8 q^{16} - 16 q^{19} + 16 q^{22} + 24 q^{28} + 24 q^{34} - 24 q^{40} - 16 q^{43} + 32 q^{46} + 32 q^{49} + 48 q^{52} - 32 q^{55} + 32 q^{61} - 24 q^{64} - 32 q^{67} - 48 q^{76} - 80 q^{82} + 32 q^{85} - 24 q^{88} - 48 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.41076 − 0.0987658i 0 1.98049 + 0.278670i 0.0308139 − 0.0308139i 0 2.10616 −2.76648 − 0.588741i 0 −0.0465144 + 0.0404277i
107.2 −1.32420 + 0.496490i 0 1.50700 − 1.31490i −1.75903 + 1.75903i 0 −4.05756 −1.34273 + 2.48940i 0 1.45597 − 3.20265i
107.3 −1.19933 + 0.749398i 0 0.876807 − 1.79756i 2.15382 − 2.15382i 0 4.43758 0.295500 + 2.81295i 0 −0.969085 + 4.19723i
107.4 −1.12975 − 0.850687i 0 0.552664 + 1.92212i −1.26575 + 1.26575i 0 1.47880 1.01076 − 2.64166i 0 2.50674 − 0.353222i
107.5 −0.900149 − 1.09075i 0 −0.379462 + 1.96367i 1.29039 − 1.29039i 0 −3.83003 2.48344 − 1.35370i 0 −2.56904 − 0.245948i
107.6 −0.762934 + 1.19077i 0 −0.835863 − 1.81696i −3.06203 + 3.06203i 0 1.75946 2.80128 + 0.390900i 0 −1.31004 − 5.98230i
107.7 −0.379657 + 1.36230i 0 −1.71172 − 1.03441i 0.463925 − 0.463925i 0 −1.85883 2.05905 − 1.93916i 0 0.455873 + 0.808137i
107.8 −0.0710265 + 1.41243i 0 −1.98991 − 0.200640i 1.84591 − 1.84591i 0 −0.0355882 0.424726 − 2.79636i 0 2.47611 + 2.73833i
107.9 0.0710265 − 1.41243i 0 −1.98991 − 0.200640i −1.84591 + 1.84591i 0 −0.0355882 −0.424726 + 2.79636i 0 2.47611 + 2.73833i
107.10 0.379657 − 1.36230i 0 −1.71172 − 1.03441i −0.463925 + 0.463925i 0 −1.85883 −2.05905 + 1.93916i 0 0.455873 + 0.808137i
107.11 0.762934 − 1.19077i 0 −0.835863 − 1.81696i 3.06203 − 3.06203i 0 1.75946 −2.80128 − 0.390900i 0 −1.31004 − 5.98230i
107.12 0.900149 + 1.09075i 0 −0.379462 + 1.96367i −1.29039 + 1.29039i 0 −3.83003 −2.48344 + 1.35370i 0 −2.56904 − 0.245948i
107.13 1.12975 + 0.850687i 0 0.552664 + 1.92212i 1.26575 − 1.26575i 0 1.47880 −1.01076 + 2.64166i 0 2.50674 − 0.353222i
107.14 1.19933 − 0.749398i 0 0.876807 − 1.79756i −2.15382 + 2.15382i 0 4.43758 −0.295500 − 2.81295i 0 −0.969085 + 4.19723i
107.15 1.32420 − 0.496490i 0 1.50700 − 1.31490i 1.75903 − 1.75903i 0 −4.05756 1.34273 − 2.48940i 0 1.45597 − 3.20265i
107.16 1.41076 + 0.0987658i 0 1.98049 + 0.278670i −0.0308139 + 0.0308139i 0 2.10616 2.76648 + 0.588741i 0 −0.0465144 + 0.0404277i
323.1 −1.41076 + 0.0987658i 0 1.98049 − 0.278670i 0.0308139 + 0.0308139i 0 2.10616 −2.76648 + 0.588741i 0 −0.0465144 − 0.0404277i
323.2 −1.32420 − 0.496490i 0 1.50700 + 1.31490i −1.75903 − 1.75903i 0 −4.05756 −1.34273 − 2.48940i 0 1.45597 + 3.20265i
323.3 −1.19933 − 0.749398i 0 0.876807 + 1.79756i 2.15382 + 2.15382i 0 4.43758 0.295500 − 2.81295i 0 −0.969085 − 4.19723i
323.4 −1.12975 + 0.850687i 0 0.552664 − 1.92212i −1.26575 − 1.26575i 0 1.47880 1.01076 + 2.64166i 0 2.50674 + 0.353222i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.l.b ✓ 32
3.b odd 2 1 inner 432.2.l.b ✓ 32
4.b odd 2 1 1728.2.l.b 32
12.b even 2 1 1728.2.l.b 32
16.e even 4 1 1728.2.l.b 32
16.f odd 4 1 inner 432.2.l.b ✓ 32
48.i odd 4 1 1728.2.l.b 32
48.k even 4 1 inner 432.2.l.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.l.b ✓ 32 1.a even 1 1 trivial
432.2.l.b ✓ 32 3.b odd 2 1 inner
432.2.l.b ✓ 32 16.f odd 4 1 inner
432.2.l.b ✓ 32 48.k even 4 1 inner
1728.2.l.b 32 4.b odd 2 1
1728.2.l.b 32 12.b even 2 1
1728.2.l.b 32 16.e even 4 1
1728.2.l.b 32 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 544 T_{5}^{28} + 80512 T_{5}^{24} + 4894464 T_{5}^{20} + 134019200 T_{5}^{16} + 1555154944 T_{5}^{12} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display