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Properties

Label 639.2.m.a
Level $639$
Weight $2$
Character orbit 639.m
Analytic conductor $5.102$
Analytic rank $0$
Dimension $96$
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] = [639,2,Mod(17,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.m (of order \(10\), degree \(4\), 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: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 24 q^{4} + 16 q^{10} - 20 q^{16} + 8 q^{19} - 20 q^{22} + 20 q^{28} - 40 q^{37} + 24 q^{40} + 32 q^{43} + 20 q^{46} + 56 q^{49} - 100 q^{52} + 120 q^{55} + 28 q^{58} + 88 q^{64} - 40 q^{67} - 68 q^{73} + 112 q^{76} - 88 q^{79} + 40 q^{82} - 60 q^{85} + 260 q^{88} - 24 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.65366 + 2.27606i 0 −1.82785 − 5.62555i −1.17447 − 0.381609i 0 0.208534 − 0.287022i 10.4754 + 3.40366i 0 2.81074 − 2.04212i
17.2 −1.41825 + 1.95205i 0 −1.18104 − 3.63485i 2.66331 + 0.865361i 0 0.580856 − 0.799479i 4.18087 + 1.35845i 0 −5.46645 + 3.97161i
17.3 −1.26693 + 1.74378i 0 −0.817629 − 2.51640i 3.01804 + 0.980620i 0 −2.63644 + 3.62875i 1.32407 + 0.430215i 0 −5.53364 + 4.02043i
17.4 −1.26240 + 1.73755i 0 −0.807380 − 2.48486i −2.11833 − 0.688287i 0 0.656509 − 0.903607i 1.25158 + 0.406664i 0 3.87012 − 2.81180i
17.5 −1.15082 + 1.58397i 0 −0.566537 − 1.74362i −1.80988 − 0.588067i 0 −0.338736 + 0.466230i −0.310307 − 0.100825i 0 3.01434 − 2.19004i
17.6 −1.08287 + 1.49044i 0 −0.430781 − 1.32581i 0.927667 + 0.301417i 0 1.62345 − 2.23448i −1.06172 − 0.344974i 0 −1.45379 + 1.05624i
17.7 −0.911813 + 1.25500i 0 −0.125595 − 0.386541i 0.936569 + 0.304310i 0 −0.257570 + 0.354514i −2.35106 − 0.763905i 0 −1.23588 + 0.897923i
17.8 −0.753035 + 1.03646i 0 0.110839 + 0.341127i −2.72400 − 0.885082i 0 −2.49756 + 3.43760i −2.87390 − 0.933788i 0 2.96862 − 2.15683i
17.9 −0.552660 + 0.760671i 0 0.344846 + 1.06133i −4.11583 − 1.33732i 0 1.76559 − 2.43013i −2.78635 − 0.905340i 0 3.29191 − 2.39172i
17.10 −0.410816 + 0.565439i 0 0.467082 + 1.43753i 1.72400 + 0.560163i 0 0.305411 − 0.420362i −2.33415 − 0.758411i 0 −1.02499 + 0.744696i
17.11 −0.340048 + 0.468036i 0 0.514609 + 1.58380i −2.00954 − 0.652941i 0 2.35572 − 3.24237i −2.01669 − 0.655262i 0 0.988942 − 0.718508i
17.12 −0.0872184 + 0.120046i 0 0.611230 + 1.88117i 1.82151 + 0.591846i 0 −1.76575 + 2.43035i −0.561382 − 0.182404i 0 −0.229918 + 0.167045i
17.13 0.0872184 − 0.120046i 0 0.611230 + 1.88117i −1.82151 − 0.591846i 0 −1.76575 + 2.43035i 0.561382 + 0.182404i 0 −0.229918 + 0.167045i
17.14 0.340048 − 0.468036i 0 0.514609 + 1.58380i 2.00954 + 0.652941i 0 2.35572 − 3.24237i 2.01669 + 0.655262i 0 0.988942 − 0.718508i
17.15 0.410816 − 0.565439i 0 0.467082 + 1.43753i −1.72400 − 0.560163i 0 0.305411 − 0.420362i 2.33415 + 0.758411i 0 −1.02499 + 0.744696i
17.16 0.552660 − 0.760671i 0 0.344846 + 1.06133i 4.11583 + 1.33732i 0 1.76559 − 2.43013i 2.78635 + 0.905340i 0 3.29191 − 2.39172i
17.17 0.753035 − 1.03646i 0 0.110839 + 0.341127i 2.72400 + 0.885082i 0 −2.49756 + 3.43760i 2.87390 + 0.933788i 0 2.96862 − 2.15683i
17.18 0.911813 − 1.25500i 0 −0.125595 − 0.386541i −0.936569 − 0.304310i 0 −0.257570 + 0.354514i 2.35106 + 0.763905i 0 −1.23588 + 0.897923i
17.19 1.08287 − 1.49044i 0 −0.430781 − 1.32581i −0.927667 − 0.301417i 0 1.62345 − 2.23448i 1.06172 + 0.344974i 0 −1.45379 + 1.05624i
17.20 1.15082 − 1.58397i 0 −0.566537 − 1.74362i 1.80988 + 0.588067i 0 −0.338736 + 0.466230i 0.310307 + 0.100825i 0 3.01434 − 2.19004i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
71.e odd 10 1 inner
213.i even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.m.a ✓ 96
3.b odd 2 1 inner 639.2.m.a ✓ 96
71.e odd 10 1 inner 639.2.m.a ✓ 96
213.i even 10 1 inner 639.2.m.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
639.2.m.a ✓ 96 1.a even 1 1 trivial
639.2.m.a ✓ 96 3.b odd 2 1 inner
639.2.m.a ✓ 96 71.e odd 10 1 inner
639.2.m.a ✓ 96 213.i even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(639, [\chi])\).