Properties

Label 592.2.n.a
Level $592$
Weight $2$
Character orbit 592.n
Analytic conductor $4.727$
Analytic rank $0$
Dimension $148$
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] = [592,2,Mod(221,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.n (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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(148\)
Relative dimension: \(74\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 148 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 148 q - 4 q^{3} - 4 q^{10} - 12 q^{11} - 4 q^{12} - 24 q^{16} + 8 q^{21} - 36 q^{26} + 8 q^{27} + 12 q^{28} + 32 q^{30} - 8 q^{33} + 40 q^{34} + 12 q^{36} - 10 q^{37} - 4 q^{38} - 28 q^{40} - 28 q^{44} - 12 q^{46} - 48 q^{47} - 20 q^{48} - 132 q^{49} + 12 q^{53} - 36 q^{58} - 20 q^{62} + 88 q^{63} + 72 q^{64} + 8 q^{65} - 44 q^{67} + 72 q^{70} + 24 q^{74} - 28 q^{75} + 8 q^{77} - 8 q^{78} - 124 q^{81} + 36 q^{83} - 32 q^{84} + 16 q^{85} - 16 q^{86} + 56 q^{90} - 72 q^{95} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1 −1.41366 + 0.0393928i 0.444084 + 0.444084i 1.99690 0.111376i 2.33845 + 2.33845i −0.645280 0.610292i 0.765509i −2.81855 + 0.236112i 2.60558i −3.39791 3.21367i
221.2 −1.40952 + 0.115184i −1.44163 1.44163i 1.97347 0.324708i −0.653168 0.653168i 2.19806 + 1.86595i 0.355889i −2.74423 + 0.684993i 1.15661i 0.995884 + 0.845415i
221.3 −1.40095 + 0.193240i 1.60231 + 1.60231i 1.92532 0.541438i −0.851474 0.851474i −2.55438 1.93512i 3.08751i −2.59264 + 1.13058i 2.13478i 1.35741 + 1.02833i
221.4 −1.39836 0.211131i −2.15533 2.15533i 1.91085 + 0.590476i −0.757230 0.757230i 2.55888 + 3.46899i 4.49827i −2.54739 1.22914i 6.29086i 0.899009 + 1.21876i
221.5 −1.36777 0.359464i 0.510571 + 0.510571i 1.74157 + 0.983327i −0.238361 0.238361i −0.514810 0.881874i 3.62253i −2.02859 1.97099i 2.47863i 0.240339 + 0.411704i
221.6 −1.36596 0.366280i −0.226666 0.226666i 1.73168 + 1.00065i 0.657294 + 0.657294i 0.226593 + 0.392639i 3.10232i −1.99888 2.00112i 2.89724i −0.657082 1.13859i
221.7 −1.34927 0.423634i 2.05501 + 2.05501i 1.64107 + 1.14320i 1.60221 + 1.60221i −1.90220 3.64334i 0.417917i −1.72995 2.23769i 5.44615i −1.48306 2.84056i
221.8 −1.34539 0.435807i −0.798716 0.798716i 1.62015 + 1.17266i −2.82103 2.82103i 0.726498 + 1.42267i 3.41428i −1.66867 2.28375i 1.72411i 2.56596 + 5.02480i
221.9 −1.34339 0.441937i −1.63101 1.63101i 1.60938 + 1.18738i 2.35792 + 2.35792i 1.47028 + 2.91189i 1.63277i −1.63728 2.30636i 2.32041i −2.12555 4.20965i
221.10 −1.31827 + 0.512031i −0.246714 0.246714i 1.47565 1.34998i −2.95295 2.95295i 0.451560 + 0.198910i 1.53917i −1.25406 + 2.53522i 2.87826i 5.40477 + 2.38077i
221.11 −1.27485 + 0.612180i 1.79320 + 1.79320i 1.25047 1.56087i 0.620072 + 0.620072i −3.38382 1.18830i 4.11641i −0.638625 + 2.75539i 3.43115i −1.17009 0.410901i
221.12 −1.26484 + 0.632607i −0.118396 0.118396i 1.19962 1.60029i 1.16800 + 1.16800i 0.224650 + 0.0748536i 2.75252i −0.504968 + 2.78299i 2.97196i −2.21621 0.738444i
221.13 −1.23231 + 0.693846i −0.876614 0.876614i 1.03715 1.71006i −0.196574 0.196574i 1.68849 + 0.472021i 1.91341i −0.0915719 + 2.82694i 1.46310i 0.378631 + 0.105847i
221.14 −1.17685 + 0.784233i −2.12155 2.12155i 0.769957 1.84585i 2.66228 + 2.66228i 4.16055 + 0.832961i 0.00749653i 0.541452 + 2.77612i 6.00199i −5.22095 1.04526i
221.15 −1.16423 0.802848i 0.602306 + 0.602306i 0.710869 + 1.86940i −1.13519 1.13519i −0.217663 1.18478i 2.03778i 0.673230 2.74714i 2.27445i 0.410240 + 2.23302i
221.16 −1.06750 + 0.927602i 1.26290 + 1.26290i 0.279108 1.98043i −1.37978 1.37978i −2.51961 0.176675i 2.37724i 1.53910 + 2.37301i 0.189824i 2.75280 + 0.193026i
221.17 −1.03115 0.967846i −1.40065 1.40065i 0.126548 + 1.99599i 0.708999 + 0.708999i 0.0886691 + 2.79989i 1.67814i 1.80132 2.18065i 0.923627i −0.0448837 1.41729i
221.18 −0.969462 1.02963i −2.41060 2.41060i −0.120287 + 1.99638i −1.05941 1.05941i −0.145048 + 4.81902i 2.94051i 2.17215 1.81156i 8.62199i −0.0637458 + 2.11786i
221.19 −0.926209 + 1.06871i 0.602451 + 0.602451i −0.284273 1.97969i 2.02224 + 2.02224i −1.20184 + 0.0858484i 1.53111i 2.37901 + 1.52981i 2.27411i −4.03421 + 0.288167i
221.20 −0.901343 1.08976i 1.30769 + 1.30769i −0.375163 + 1.96450i −2.45278 2.45278i 0.246394 2.60374i 3.14948i 2.47899 1.36185i 0.420084i −0.462151 + 4.88373i
See next 80 embeddings (of 148 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.74
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
37.b even 2 1 inner
592.n even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.n.a 148
16.e even 4 1 inner 592.2.n.a 148
37.b even 2 1 inner 592.2.n.a 148
592.n even 4 1 inner 592.2.n.a 148
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
592.2.n.a 148 1.a even 1 1 trivial
592.2.n.a 148 16.e even 4 1 inner
592.2.n.a 148 37.b even 2 1 inner
592.2.n.a 148 592.n even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(592, [\chi])\).