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.
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) |
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 \(S_{2}^{\mathrm{new}}(592, [\chi])\).