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Properties

Label 644.3.l.a
Level $644$
Weight $3$
Character orbit 644.l
Analytic conductor $17.548$
Analytic rank $0$
Dimension $64$
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] = [644,3,Mod(137,644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(644, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("644.137");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 644.l (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: \(17.5477290248\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 112 q^{9} - 23 q^{23} + 222 q^{25} + 60 q^{27} + 12 q^{29} - 6 q^{31} + 132 q^{35} - 104 q^{39} + 244 q^{41} + 98 q^{47} - 240 q^{49} - 456 q^{55} - 140 q^{59} - 146 q^{69} + 184 q^{71} + 160 q^{73} + 92 q^{75} - 158 q^{77} - 360 q^{81} - 152 q^{85} + 102 q^{87} - 28 q^{93} - 102 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 0 −2.90811 + 5.03700i 0 −7.84842 + 4.53128i 0 −2.06919 − 6.68719i 0 −12.4142 − 21.5021i 0
137.2 0 −2.90811 + 5.03700i 0 7.84842 − 4.53128i 0 2.06919 + 6.68719i 0 −12.4142 − 21.5021i 0
137.3 0 −2.46727 + 4.27343i 0 3.99932 − 2.30901i 0 −0.938284 − 6.93683i 0 −7.67483 − 13.2932i 0
137.4 0 −2.46727 + 4.27343i 0 −3.99932 + 2.30901i 0 0.938284 + 6.93683i 0 −7.67483 − 13.2932i 0
137.5 0 −2.13349 + 3.69531i 0 −0.685850 + 0.395976i 0 6.87030 + 1.34124i 0 −4.60352 − 7.97353i 0
137.6 0 −2.13349 + 3.69531i 0 0.685850 − 0.395976i 0 −6.87030 − 1.34124i 0 −4.60352 − 7.97353i 0
137.7 0 −1.58316 + 2.74211i 0 2.93433 − 1.69414i 0 −6.74912 + 1.85726i 0 −0.512791 − 0.888179i 0
137.8 0 −1.58316 + 2.74211i 0 −2.93433 + 1.69414i 0 6.74912 − 1.85726i 0 −0.512791 − 0.888179i 0
137.9 0 −1.40566 + 2.43468i 0 3.68913 − 2.12992i 0 1.00033 − 6.92816i 0 0.548214 + 0.949535i 0
137.10 0 −1.40566 + 2.43468i 0 −3.68913 + 2.12992i 0 −1.00033 + 6.92816i 0 0.548214 + 0.949535i 0
137.11 0 −0.711722 + 1.23274i 0 −5.51457 + 3.18384i 0 −0.907879 − 6.94088i 0 3.48690 + 6.03950i 0
137.12 0 −0.711722 + 1.23274i 0 5.51457 − 3.18384i 0 0.907879 + 6.94088i 0 3.48690 + 6.03950i 0
137.13 0 −0.690959 + 1.19678i 0 −6.75903 + 3.90232i 0 −6.41594 + 2.79924i 0 3.54515 + 6.14038i 0
137.14 0 −0.690959 + 1.19678i 0 6.75903 − 3.90232i 0 6.41594 − 2.79924i 0 3.54515 + 6.14038i 0
137.15 0 −0.618470 + 1.07122i 0 5.81540 − 3.35753i 0 −2.40948 + 6.57224i 0 3.73499 + 6.46919i 0
137.16 0 −0.618470 + 1.07122i 0 −5.81540 + 3.35753i 0 2.40948 − 6.57224i 0 3.73499 + 6.46919i 0
137.17 0 0.324570 − 0.562173i 0 −0.620474 + 0.358231i 0 −5.98238 − 3.63471i 0 4.28931 + 7.42930i 0
137.18 0 0.324570 − 0.562173i 0 0.620474 − 0.358231i 0 5.98238 + 3.63471i 0 4.28931 + 7.42930i 0
137.19 0 0.426440 − 0.738616i 0 −5.42556 + 3.13245i 0 3.96981 + 5.76547i 0 4.13630 + 7.16428i 0
137.20 0 0.426440 − 0.738616i 0 5.42556 − 3.13245i 0 −3.96981 − 5.76547i 0 4.13630 + 7.16428i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.b odd 2 1 inner
161.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 644.3.l.a ✓ 64
7.c even 3 1 inner 644.3.l.a ✓ 64
23.b odd 2 1 inner 644.3.l.a ✓ 64
161.f odd 6 1 inner 644.3.l.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.3.l.a ✓ 64 1.a even 1 1 trivial
644.3.l.a ✓ 64 7.c even 3 1 inner
644.3.l.a ✓ 64 23.b odd 2 1 inner
644.3.l.a ✓ 64 161.f odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(644, [\chi])\).