Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,3,Mod(13,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 7, 11]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.13");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 392.r (of order \(14\), 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: | \(10.6812263629\) |
Analytic rank: | \(0\) |
Dimension: | \(660\) |
Relative dimension: | \(110\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −1.99978 | − | 0.0298008i | −1.84727 | − | 2.31641i | 3.99822 | + | 0.119190i | 4.36230 | + | 5.47015i | 3.62510 | + | 4.68735i | 6.52431 | − | 2.53641i | −7.99201 | − | 0.357504i | 0.0493668 | − | 0.216290i | −8.56062 | − | 11.0691i |
13.2 | −1.99517 | + | 0.138981i | 1.70976 | + | 2.14397i | 3.96137 | − | 0.554578i | −2.50322 | − | 3.13894i | −3.70922 | − | 4.03995i | −6.98044 | + | 0.522965i | −7.82651 | + | 1.65703i | 0.329359 | − | 1.44301i | 5.43060 | + | 5.91481i |
13.3 | −1.99438 | − | 0.149795i | 1.68073 | + | 2.10757i | 3.95512 | + | 0.597496i | −1.71847 | − | 2.15489i | −3.03632 | − | 4.45506i | 6.99470 | − | 0.272369i | −7.79853 | − | 1.78409i | 0.385696 | − | 1.68984i | 3.10449 | + | 4.55509i |
13.4 | −1.99383 | + | 0.156983i | −3.05720 | − | 3.83361i | 3.95071 | − | 0.625994i | 0.514788 | + | 0.645524i | 6.69735 | + | 7.16364i | 0.812904 | + | 6.95264i | −7.77878 | + | 1.86832i | −3.34740 | + | 14.6659i | −1.12774 | − | 1.20625i |
13.5 | −1.96609 | + | 0.366726i | 0.511085 | + | 0.640881i | 3.73102 | − | 1.44203i | 3.48690 | + | 4.37244i | −1.23987 | − | 1.07260i | −6.98216 | − | 0.499436i | −6.80670 | + | 4.20343i | 1.85317 | − | 8.11926i | −8.45905 | − | 7.31787i |
13.6 | −1.96241 | + | 0.385929i | −1.88164 | − | 2.35950i | 3.70212 | − | 1.51470i | −0.816179 | − | 1.02346i | 4.60314 | + | 3.90413i | 1.23528 | − | 6.89014i | −6.68051 | + | 4.40122i | −0.0239853 | + | 0.105086i | 1.99666 | + | 1.69345i |
13.7 | −1.95195 | − | 0.435775i | 2.96624 | + | 3.71955i | 3.62020 | + | 1.70122i | 4.05361 | + | 5.08307i | −4.16906 | − | 8.55298i | 0.462548 | + | 6.98470i | −6.32509 | − | 4.89829i | −3.03377 | + | 13.2918i | −5.69737 | − | 11.6884i |
13.8 | −1.94864 | − | 0.450319i | −1.53176 | − | 1.92077i | 3.59443 | + | 1.75502i | −4.77246 | − | 5.98448i | 2.11990 | + | 4.43268i | −4.17425 | + | 5.61922i | −6.21394 | − | 5.03855i | 0.659631 | − | 2.89003i | 6.60491 | + | 13.8107i |
13.9 | −1.94810 | − | 0.452655i | 2.06701 | + | 2.59195i | 3.59021 | + | 1.76364i | 2.47925 | + | 3.10888i | −2.85349 | − | 5.98503i | −3.31785 | − | 6.16375i | −6.19577 | − | 5.06087i | −0.442986 | + | 1.94085i | −3.42258 | − | 7.17865i |
13.10 | −1.93661 | − | 0.499555i | −2.76411 | − | 3.46609i | 3.50089 | + | 1.93488i | 5.25857 | + | 6.59403i | 3.62150 | + | 8.09328i | −6.83976 | − | 1.48919i | −5.81326 | − | 5.49600i | −2.37076 | + | 10.3870i | −6.88969 | − | 15.3970i |
13.11 | −1.90927 | + | 0.595568i | 0.746108 | + | 0.935590i | 3.29060 | − | 2.27419i | −5.68661 | − | 7.13079i | −1.98173 | − | 1.34193i | 0.995644 | − | 6.92883i | −4.92819 | + | 6.30182i | 1.68404 | − | 7.37825i | 15.1041 | + | 10.2278i |
13.12 | −1.89610 | − | 0.636232i | −0.400805 | − | 0.502593i | 3.19042 | + | 2.41272i | −0.300414 | − | 0.376707i | 0.440202 | + | 1.20797i | 6.96498 | + | 0.699351i | −4.51431 | − | 6.60462i | 1.91073 | − | 8.37147i | 0.329943 | + | 0.905409i |
13.13 | −1.88791 | + | 0.660135i | 3.44719 | + | 4.32263i | 3.12844 | − | 2.49256i | −4.19833 | − | 5.26454i | −9.36152 | − | 5.88516i | 0.385894 | + | 6.98936i | −4.26081 | + | 6.77093i | −4.79939 | + | 21.0275i | 11.4014 | + | 7.16753i |
13.14 | −1.87964 | + | 0.683340i | −0.0855824 | − | 0.107317i | 3.06609 | − | 2.56887i | 2.41719 | + | 3.03106i | 0.234198 | + | 0.143235i | −1.93020 | + | 6.72862i | −4.00774 | + | 6.92373i | 1.99850 | − | 8.75598i | −6.61468 | − | 4.04553i |
13.15 | −1.84922 | − | 0.761826i | −3.59610 | − | 4.50937i | 2.83924 | + | 2.81757i | −4.46752 | − | 5.60209i | 3.21464 | + | 11.0784i | 5.11740 | − | 4.77621i | −3.10389 | − | 7.37332i | −5.39978 | + | 23.6580i | 3.99361 | + | 13.7630i |
13.16 | −1.81490 | + | 0.840317i | −1.58246 | − | 1.98434i | 2.58773 | − | 3.05019i | −3.75053 | − | 4.70302i | 4.53948 | + | 2.27161i | 4.49567 | + | 5.36553i | −2.13336 | + | 7.71030i | 0.569259 | − | 2.49409i | 10.7589 | + | 5.38387i |
13.17 | −1.81351 | − | 0.843320i | 3.58008 | + | 4.48927i | 2.57762 | + | 3.05874i | −3.12084 | − | 3.91341i | −2.70660 | − | 11.1605i | 2.29482 | − | 6.61315i | −2.09504 | − | 7.72080i | −5.33395 | + | 23.3696i | 2.35941 | + | 9.72886i |
13.18 | −1.81340 | + | 0.843560i | 2.52002 | + | 3.16001i | 2.57681 | − | 3.05942i | 3.38083 | + | 4.23943i | −7.23546 | − | 3.60456i | 6.07141 | − | 3.48397i | −2.09198 | + | 7.72163i | −1.63245 | + | 7.15223i | −9.70700 | − | 4.83583i |
13.19 | −1.81153 | − | 0.847560i | −0.878129 | − | 1.10114i | 2.56328 | + | 3.07076i | −1.30470 | − | 1.63604i | 0.657476 | + | 2.73901i | −4.87731 | − | 5.02114i | −2.04081 | − | 7.73531i | 1.56129 | − | 6.84047i | 0.976858 | + | 4.06954i |
13.20 | −1.70471 | + | 1.04593i | −3.23276 | − | 4.05375i | 1.81206 | − | 3.56601i | −0.985719 | − | 1.23605i | 9.75086 | + | 3.52923i | −6.43628 | − | 2.75214i | 0.640762 | + | 7.97430i | −3.97949 | + | 17.4353i | 2.97319 | + | 1.07612i |
See next 80 embeddings (of 660 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
49.f | odd | 14 | 1 | inner |
392.r | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 392.3.r.a | ✓ | 660 |
8.b | even | 2 | 1 | inner | 392.3.r.a | ✓ | 660 |
49.f | odd | 14 | 1 | inner | 392.3.r.a | ✓ | 660 |
392.r | odd | 14 | 1 | inner | 392.3.r.a | ✓ | 660 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
392.3.r.a | ✓ | 660 | 1.a | even | 1 | 1 | trivial |
392.3.r.a | ✓ | 660 | 8.b | even | 2 | 1 | inner |
392.3.r.a | ✓ | 660 | 49.f | odd | 14 | 1 | inner |
392.3.r.a | ✓ | 660 | 392.r | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(392, [\chi])\).