Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [592,2,Mod(251,592)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(592, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 3, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("592.251");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 592 = 2^{4} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 592.bf (of order \(12\), 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: | \(4.72714379966\) |
Analytic rank: | \(0\) |
Dimension: | \(296\) |
Relative dimension: | \(74\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
251.1 | −1.41416 | + | 0.0124895i | 0.185341 | + | 0.691701i | 1.99969 | − | 0.0353243i | −0.445060 | + | 0.770867i | −0.270740 | − | 0.975861i | −0.429533 | + | 0.743973i | −2.82743 | + | 0.0749293i | 2.15398 | − | 1.24360i | 0.619758 | − | 1.09569i |
251.2 | −1.40808 | + | 0.131531i | 0.751687 | + | 2.80533i | 1.96540 | − | 0.370413i | 1.29725 | − | 2.24691i | −1.42743 | − | 3.85127i | 1.26308 | − | 2.18771i | −2.71873 | + | 0.780083i | −4.70679 | + | 2.71746i | −1.53110 | + | 3.33446i |
251.3 | −1.40096 | − | 0.193131i | 0.114417 | + | 0.427011i | 1.92540 | + | 0.541140i | −1.50227 | + | 2.60201i | −0.0778251 | − | 0.620324i | 2.30014 | − | 3.98397i | −2.59291 | − | 1.12997i | 2.42883 | − | 1.40229i | 2.60715 | − | 3.35518i |
251.4 | −1.39710 | − | 0.219367i | −0.618148 | − | 2.30696i | 1.90376 | + | 0.612955i | −0.349210 | + | 0.604850i | 0.357540 | + | 3.35864i | −0.422689 | + | 0.732118i | −2.52527 | − | 1.27398i | −2.34187 | + | 1.35208i | 0.620564 | − | 0.768428i |
251.5 | −1.37078 | + | 0.347808i | −0.659446 | − | 2.46109i | 1.75806 | − | 0.953534i | −0.530589 | + | 0.919007i | 1.75994 | + | 3.14424i | 0.701035 | − | 1.21423i | −2.07826 | + | 1.91855i | −3.02400 | + | 1.74591i | 0.407681 | − | 1.44430i |
251.6 | −1.37025 | + | 0.349895i | −0.403729 | − | 1.50674i | 1.75515 | − | 0.958884i | −1.73355 | + | 3.00260i | 1.08041 | + | 1.92334i | −2.23939 | + | 3.87874i | −2.06947 | + | 1.92802i | 0.490815 | − | 0.283372i | 1.32480 | − | 4.72086i |
251.7 | −1.36182 | + | 0.381390i | 0.0457446 | + | 0.170721i | 1.70908 | − | 1.03876i | 1.81094 | − | 3.13664i | −0.127407 | − | 0.215044i | 1.16260 | − | 2.01367i | −1.93128 | + | 2.06643i | 2.57102 | − | 1.48438i | −1.26988 | + | 4.96220i |
251.8 | −1.34508 | − | 0.436771i | −0.588609 | − | 2.19672i | 1.61846 | + | 1.17498i | 1.07481 | − | 1.86162i | −0.167738 | + | 3.21184i | 1.22576 | − | 2.12307i | −1.66376 | − | 2.28734i | −1.88103 | + | 1.08601i | −2.25880 | + | 2.03457i |
251.9 | −1.33397 | + | 0.469606i | 0.716356 | + | 2.67348i | 1.55894 | − | 1.25288i | −0.633595 | + | 1.09742i | −2.21108 | − | 3.22993i | −0.406156 | + | 0.703483i | −1.49122 | + | 2.40339i | −4.03624 | + | 2.33033i | 0.329841 | − | 1.76146i |
251.10 | −1.32688 | − | 0.489282i | 0.648209 | + | 2.41915i | 1.52121 | + | 1.29844i | 0.423601 | − | 0.733699i | 0.323553 | − | 3.52707i | −2.29063 | + | 3.96749i | −1.38315 | − | 2.46716i | −2.83403 | + | 1.63623i | −0.921053 | + | 0.766268i |
251.11 | −1.32437 | − | 0.496022i | −0.112260 | − | 0.418959i | 1.50793 | + | 1.31383i | 1.63460 | − | 2.83121i | −0.0591389 | + | 0.610540i | −2.06434 | + | 3.57553i | −1.34536 | − | 2.48797i | 2.43515 | − | 1.40594i | −3.56916 | + | 2.93878i |
251.12 | −1.30534 | + | 0.544131i | −0.729786 | − | 2.72360i | 1.40784 | − | 1.42055i | 1.96845 | − | 3.40945i | 2.43461 | + | 3.15813i | −2.04578 | + | 3.54340i | −1.06475 | + | 2.62036i | −4.28732 | + | 2.47528i | −0.714313 | + | 5.52159i |
251.13 | −1.20413 | + | 0.741666i | −0.257947 | − | 0.962671i | 0.899863 | − | 1.78613i | 0.293162 | − | 0.507772i | 1.02458 | + | 0.967872i | 2.60292 | − | 4.50838i | 0.241156 | + | 2.81813i | 1.73788 | − | 1.00336i | 0.0235913 | + | 0.828852i |
251.14 | −1.19564 | − | 0.755273i | 0.595709 | + | 2.22322i | 0.859125 | + | 1.80607i | −0.155650 | + | 0.269594i | 0.966880 | − | 3.10810i | 0.475920 | − | 0.824318i | 0.336873 | − | 2.80829i | −1.98974 | + | 1.14878i | 0.389719 | − | 0.204780i |
251.15 | −1.18389 | − | 0.773561i | −0.301998 | − | 1.12707i | 0.803206 | + | 1.83163i | −1.40057 | + | 2.42585i | −0.514326 | + | 1.56795i | −0.357830 | + | 0.619780i | 0.465966 | − | 2.78978i | 1.41899 | − | 0.819252i | 3.53467 | − | 1.78853i |
251.16 | −1.14258 | + | 0.833371i | 0.193662 | + | 0.722757i | 0.610987 | − | 1.90439i | 0.285864 | − | 0.495132i | −0.823600 | − | 0.664417i | −1.39274 | + | 2.41230i | 0.888959 | + | 2.68510i | 2.11320 | − | 1.22006i | 0.0860048 | + | 0.803960i |
251.17 | −1.04334 | − | 0.954691i | 0.149173 | + | 0.556722i | 0.177128 | + | 1.99214i | −0.354588 | + | 0.614164i | 0.375859 | − | 0.723267i | 0.432081 | − | 0.748387i | 1.71707 | − | 2.24759i | 2.31039 | − | 1.33390i | 0.956294 | − | 0.302262i |
251.18 | −1.01537 | − | 0.984387i | 0.278090 | + | 1.03785i | 0.0619648 | + | 1.99904i | 2.01095 | − | 3.48306i | 0.739277 | − | 1.32755i | 0.924173 | − | 1.60072i | 1.90491 | − | 2.09077i | 1.59829 | − | 0.922771i | −5.47054 | + | 1.55706i |
251.19 | −0.946318 | + | 1.05094i | 0.0103273 | + | 0.0385418i | −0.208965 | − | 1.98905i | −2.08235 | + | 3.60674i | −0.0502782 | − | 0.0256195i | 0.453175 | − | 0.784923i | 2.28813 | + | 1.66267i | 2.59670 | − | 1.49920i | −1.81991 | − | 5.60156i |
251.20 | −0.892332 | + | 1.09715i | 0.461372 | + | 1.72186i | −0.407487 | − | 1.95805i | 1.92757 | − | 3.33864i | −2.30084 | − | 1.03028i | −0.348836 | + | 0.604201i | 2.51189 | + | 1.30015i | −0.153875 | + | 0.0888400i | 1.94297 | + | 5.09401i |
See next 80 embeddings (of 296 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
592.bf | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 592.2.bf.a | ✓ | 296 |
16.f | odd | 4 | 1 | 592.2.bl.a | yes | 296 | |
37.g | odd | 12 | 1 | 592.2.bl.a | yes | 296 | |
592.bf | even | 12 | 1 | inner | 592.2.bf.a | ✓ | 296 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
592.2.bf.a | ✓ | 296 | 1.a | even | 1 | 1 | trivial |
592.2.bf.a | ✓ | 296 | 592.bf | even | 12 | 1 | inner |
592.2.bl.a | yes | 296 | 16.f | odd | 4 | 1 | |
592.2.bl.a | yes | 296 | 37.g | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(592, [\chi])\).