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Properties

Label 129.2.h.b
Level $129$
Weight $2$
Character orbit 129.h
Analytic conductor $1.030$
Analytic rank $0$
Dimension $24$
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] = [129,2,Mod(50,129)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(129, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.50");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 129.h (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: \(1.03007018607\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{4} - 3 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{4} - 3 q^{6} - 2 q^{10} - 24 q^{12} + 6 q^{13} + 5 q^{15} - 20 q^{16} - 30 q^{18} + 24 q^{19} - 20 q^{21} + 4 q^{24} - 12 q^{25} - 42 q^{28} + 27 q^{30} + 8 q^{31} - 42 q^{33} - 24 q^{34} - 13 q^{36} + 6 q^{37} - 34 q^{40} + 52 q^{43} + 6 q^{46} - 15 q^{48} + 12 q^{49} - 24 q^{52} + 38 q^{54} + 78 q^{55} - 9 q^{57} + 10 q^{58} + 37 q^{60} + 30 q^{61} + 54 q^{63} - 44 q^{64} + 15 q^{66} - 10 q^{67} - 27 q^{69} + 24 q^{72} - 42 q^{73} + 24 q^{76} + 148 q^{78} + 16 q^{79} - 4 q^{81} + 8 q^{84} + 60 q^{87} + 58 q^{90} - 60 q^{91} - 9 q^{93} + 4 q^{96} - 128 q^{97} + 19 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1 −2.36108 −1.57601 − 0.718473i 3.57471 1.24479 + 2.15604i 3.72108 + 1.69637i −2.70094 − 1.55939i −3.71802 1.96759 + 2.26464i −2.93906 − 5.09060i
50.2 −2.32031 0.205552 + 1.71981i 3.38386 0.346785 + 0.600650i −0.476944 − 3.99050i 0.726168 + 0.419253i −3.21098 −2.91550 + 0.707020i −0.804650 − 1.39370i
50.3 −1.62972 1.71341 + 0.253433i 0.655997 −2.06784 − 3.58161i −2.79238 − 0.413026i −3.43910 − 1.98557i 2.19035 2.87154 + 0.868470i 3.37001 + 5.83703i
50.4 −1.44405 −1.70409 + 0.309989i 0.0852720 −0.901479 − 1.56141i 2.46078 − 0.447639i 3.41367 + 1.97088i 2.76496 2.80781 − 1.05650i 1.30178 + 2.25475i
50.5 −1.11218 1.39814 − 1.02235i −0.763061 1.04069 + 1.80253i −1.55498 + 1.13703i 2.16127 + 1.24781i 3.07301 0.909606 − 2.85878i −1.15743 − 2.00473i
50.6 −0.251446 1.06484 + 1.36606i −1.93677 1.07637 + 1.86432i −0.267750 − 0.343489i −0.161068 − 0.0929928i 0.989886 −0.732219 + 2.90927i −0.270648 − 0.468776i
50.7 0.251446 −0.650618 − 1.60521i −1.93677 −1.07637 − 1.86432i −0.163595 − 0.403623i −0.161068 − 0.0929928i −0.989886 −2.15339 + 2.08876i −0.270648 − 0.468776i
50.8 1.11218 1.58445 − 0.699653i −0.763061 −1.04069 − 1.80253i 1.76219 − 0.778138i 2.16127 + 1.24781i −3.07301 2.02097 − 2.21713i −1.15743 − 2.00473i
50.9 1.44405 −1.12050 + 1.32079i 0.0852720 0.901479 + 1.56141i −1.61806 + 1.90728i 3.41367 + 1.97088i −2.76496 −0.488955 − 2.95989i 1.30178 + 2.25475i
50.10 1.62972 0.637225 − 1.61057i 0.655997 2.06784 + 3.58161i 1.03850 − 2.62479i −3.43910 − 1.98557i −2.19035 −2.18789 − 2.05259i 3.37001 + 5.83703i
50.11 2.32031 −1.38662 − 1.03792i 3.38386 −0.346785 − 0.600650i −3.21740 − 2.40830i 0.726168 + 0.419253i 3.21098 0.845452 + 2.87840i −0.804650 − 1.39370i
50.12 2.36108 −0.165788 + 1.72410i 3.57471 −1.24479 − 2.15604i −0.391439 + 4.07074i −2.70094 − 1.55939i 3.71802 −2.94503 − 0.571669i −2.93906 − 5.09060i
80.1 −2.36108 −1.57601 + 0.718473i 3.57471 1.24479 − 2.15604i 3.72108 − 1.69637i −2.70094 + 1.55939i −3.71802 1.96759 − 2.26464i −2.93906 + 5.09060i
80.2 −2.32031 0.205552 − 1.71981i 3.38386 0.346785 − 0.600650i −0.476944 + 3.99050i 0.726168 − 0.419253i −3.21098 −2.91550 − 0.707020i −0.804650 + 1.39370i
80.3 −1.62972 1.71341 − 0.253433i 0.655997 −2.06784 + 3.58161i −2.79238 + 0.413026i −3.43910 + 1.98557i 2.19035 2.87154 − 0.868470i 3.37001 − 5.83703i
80.4 −1.44405 −1.70409 − 0.309989i 0.0852720 −0.901479 + 1.56141i 2.46078 + 0.447639i 3.41367 − 1.97088i 2.76496 2.80781 + 1.05650i 1.30178 − 2.25475i
80.5 −1.11218 1.39814 + 1.02235i −0.763061 1.04069 − 1.80253i −1.55498 − 1.13703i 2.16127 − 1.24781i 3.07301 0.909606 + 2.85878i −1.15743 + 2.00473i
80.6 −0.251446 1.06484 − 1.36606i −1.93677 1.07637 − 1.86432i −0.267750 + 0.343489i −0.161068 + 0.0929928i 0.989886 −0.732219 − 2.90927i −0.270648 + 0.468776i
80.7 0.251446 −0.650618 + 1.60521i −1.93677 −1.07637 + 1.86432i −0.163595 + 0.403623i −0.161068 + 0.0929928i −0.989886 −2.15339 − 2.08876i −0.270648 + 0.468776i
80.8 1.11218 1.58445 + 0.699653i −0.763061 −1.04069 + 1.80253i 1.76219 + 0.778138i 2.16127 − 1.24781i −3.07301 2.02097 + 2.21713i −1.15743 + 2.00473i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.d odd 6 1 inner
129.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.2.h.b ✓ 24
3.b odd 2 1 inner 129.2.h.b ✓ 24
43.d odd 6 1 inner 129.2.h.b ✓ 24
129.h even 6 1 inner 129.2.h.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.2.h.b ✓ 24 1.a even 1 1 trivial
129.2.h.b ✓ 24 3.b odd 2 1 inner
129.2.h.b ✓ 24 43.d odd 6 1 inner
129.2.h.b ✓ 24 129.h even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 17T_{2}^{10} + 108T_{2}^{8} - 318T_{2}^{6} + 437T_{2}^{4} - 232T_{2}^{2} + 13 \) acting on \(S_{2}^{\mathrm{new}}(129, [\chi])\). Copy content Toggle raw display