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Properties

Label 684.2.bv.a
Level $684$
Weight $2$
Character orbit 684.bv
Analytic conductor $5.462$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,2,Mod(29,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 3, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.bv (of order \(18\), degree \(6\), 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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{3} + 6 q^{9} - 3 q^{13} - 6 q^{15} + 27 q^{17} - 3 q^{19} + 36 q^{23} - 9 q^{27} - 6 q^{33} + 30 q^{39} - 6 q^{43} - 60 q^{49} + 9 q^{51} + 24 q^{57} + 45 q^{59} - 21 q^{61} + 30 q^{63} - 24 q^{67} + 39 q^{73} + 6 q^{79} + 42 q^{81} + 36 q^{83} + 54 q^{87} - 54 q^{89} - 12 q^{91} - 78 q^{93} - 54 q^{95} + 18 q^{97} - 51 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 0 −1.70747 − 0.290745i 0 3.69142 − 0.650898i 0 2.21104 + 3.82964i 0 2.83093 + 0.992879i 0
29.2 0 −1.69319 + 0.364826i 0 −4.10535 + 0.723884i 0 1.48346 + 2.56942i 0 2.73380 − 1.23544i 0
29.3 0 −1.62923 − 0.587891i 0 0.245374 − 0.0432660i 0 −1.11395 − 1.92941i 0 2.30877 + 1.91562i 0
29.4 0 −1.61010 + 0.638425i 0 −0.232288 + 0.0409586i 0 −2.11703 − 3.66680i 0 2.18483 − 2.05585i 0
29.5 0 −1.40766 + 1.00921i 0 0.0943099 − 0.0166294i 0 1.51099 + 2.61711i 0 0.962995 − 2.84124i 0
29.6 0 −1.18109 − 1.26689i 0 −1.58556 + 0.279577i 0 0.746111 + 1.29230i 0 −0.210042 + 2.99264i 0
29.7 0 −1.00889 + 1.40789i 0 1.27952 − 0.225613i 0 −0.527684 − 0.913976i 0 −0.964288 − 2.84080i 0
29.8 0 −0.859122 − 1.50396i 0 4.18868 − 0.738576i 0 −2.24489 − 3.88826i 0 −1.52382 + 2.58418i 0
29.9 0 −0.612135 − 1.62027i 0 −0.801842 + 0.141386i 0 0.313524 + 0.543039i 0 −2.25058 + 1.98365i 0
29.10 0 −0.307575 + 1.70452i 0 3.81083 − 0.671952i 0 0.116179 + 0.201227i 0 −2.81080 − 1.04854i 0
29.11 0 −0.225430 + 1.71732i 0 −3.43631 + 0.605914i 0 −1.29746 − 2.24726i 0 −2.89836 − 0.774269i 0
29.12 0 0.348187 − 1.69669i 0 1.80186 − 0.317717i 0 1.26687 + 2.19428i 0 −2.75753 − 1.18153i 0
29.13 0 0.498579 + 1.65874i 0 −1.34961 + 0.237972i 0 0.466268 + 0.807600i 0 −2.50284 + 1.65403i 0
29.14 0 0.515311 − 1.65362i 0 −2.66107 + 0.469219i 0 −0.510299 − 0.883864i 0 −2.46891 − 1.70425i 0
29.15 0 1.17828 + 1.26951i 0 −1.26826 + 0.223629i 0 2.31728 + 4.01365i 0 −0.223330 + 2.99168i 0
29.16 0 1.31780 − 1.12402i 0 1.43788 − 0.253538i 0 −1.86446 − 3.22935i 0 0.473179 − 2.96245i 0
29.17 0 1.45224 + 0.943925i 0 2.38962 − 0.421354i 0 −0.188096 − 0.325793i 0 1.21801 + 2.74161i 0
29.18 0 1.53236 − 0.807393i 0 1.59676 − 0.281552i 0 2.00848 + 3.47880i 0 1.69623 − 2.47443i 0
29.19 0 1.72945 + 0.0948700i 0 −2.68659 + 0.473719i 0 −0.311143 − 0.538915i 0 2.98200 + 0.328146i 0
29.20 0 1.73000 + 0.0842746i 0 −2.40937 + 0.424838i 0 −1.32552 − 2.29586i 0 2.98580 + 0.291590i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.x even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.bv.a ✓ 120
9.d odd 6 1 684.2.cl.a yes 120
19.f odd 18 1 684.2.cl.a yes 120
171.x even 18 1 inner 684.2.bv.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.bv.a ✓ 120 1.a even 1 1 trivial
684.2.bv.a ✓ 120 171.x even 18 1 inner
684.2.cl.a yes 120 9.d odd 6 1
684.2.cl.a yes 120 19.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).