Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [209,2,Mod(8,209)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([9, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("209.8");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 209 = 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 209.t (of order \(30\), degree \(8\), 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: | \(1.66887340224\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8.1 | −0.294203 | − | 2.79916i | −0.462350 | + | 2.17518i | −5.79244 | + | 1.23122i | −0.435401 | + | 0.193853i | 6.22471 | + | 0.654243i | 2.45552 | + | 0.797848i | 3.41103 | + | 10.4981i | −1.77702 | − | 0.791182i | 0.670722 | + | 1.16172i |
8.2 | −0.245338 | − | 2.33423i | −0.0272523 | + | 0.128212i | −3.43217 | + | 0.729530i | 3.51782 | − | 1.56623i | 0.305963 | + | 0.0321580i | −4.02135 | − | 1.30662i | 1.09435 | + | 3.36807i | 2.72494 | + | 1.21322i | −4.51901 | − | 7.82716i |
8.3 | −0.243021 | − | 2.31219i | 0.586000 | − | 2.75691i | −3.33085 | + | 0.707994i | 1.05809 | − | 0.471092i | −6.51691 | − | 0.684955i | 1.61851 | + | 0.525884i | 1.00960 | + | 3.10723i | −4.51654 | − | 2.01089i | −1.34639 | − | 2.33202i |
8.4 | −0.178386 | − | 1.69723i | −0.334130 | + | 1.57196i | −0.892457 | + | 0.189698i | −0.0430015 | + | 0.0191455i | 2.72757 | + | 0.286679i | 1.13464 | + | 0.368667i | −0.573560 | − | 1.76524i | 0.381225 | + | 0.169732i | 0.0401650 | + | 0.0695679i |
8.5 | −0.175935 | − | 1.67391i | −0.355086 | + | 1.67055i | −0.814734 | + | 0.173177i | −3.37205 | + | 1.50133i | 2.85883 | + | 0.300475i | −3.69486 | − | 1.20053i | −0.607010 | − | 1.86818i | 0.0759839 | + | 0.0338302i | 3.10636 | + | 5.38037i |
8.6 | −0.147836 | − | 1.40656i | 0.0962713 | − | 0.452921i | −0.000272568 | 0 | 5.79361e-5i | −0.206887 | + | 0.0921118i | −0.651295 | − | 0.0684538i | 4.81246 | + | 1.56366i | −0.873971 | − | 2.68981i | 2.54477 | + | 1.13300i | 0.160146 | + | 0.277382i |
8.7 | −0.106719 | − | 1.01536i | 0.290180 | − | 1.36519i | 0.936721 | − | 0.199106i | 1.09870 | − | 0.489171i | −1.41713 | − | 0.148947i | −2.19688 | − | 0.713808i | −0.933117 | − | 2.87184i | 0.961096 | + | 0.427908i | −0.613939 | − | 1.06337i |
8.8 | −0.0933513 | − | 0.888178i | 0.578920 | − | 2.72360i | 1.17615 | − | 0.249998i | −3.51240 | + | 1.56382i | −2.47309 | − | 0.259932i | −0.765265 | − | 0.248650i | −0.883786 | − | 2.72001i | −4.34223 | − | 1.93328i | 1.71684 | + | 2.97365i |
8.9 | −0.0555175 | − | 0.528214i | −0.555682 | + | 2.61428i | 1.68037 | − | 0.357173i | 2.67341 | − | 1.19028i | 1.41175 | + | 0.148381i | −0.703891 | − | 0.228708i | −0.610206 | − | 1.87802i | −3.78502 | − | 1.68520i | −0.777142 | − | 1.34605i |
8.10 | 0.0208920 | + | 0.198774i | −0.647125 | + | 3.04448i | 1.91722 | − | 0.407518i | −2.79053 | + | 1.24242i | −0.618682 | − | 0.0650261i | 3.11019 | + | 1.01056i | 0.244584 | + | 0.752752i | −6.10946 | − | 2.72011i | −0.305261 | − | 0.528727i |
8.11 | 0.0303126 | + | 0.288405i | 0.0560461 | − | 0.263676i | 1.87404 | − | 0.398339i | −0.419601 | + | 0.186819i | 0.0777445 | + | 0.00817127i | −0.640936 | − | 0.208253i | 0.350916 | + | 1.08001i | 2.67425 | + | 1.19065i | −0.0665986 | − | 0.115352i |
8.12 | 0.0929138 | + | 0.884016i | 0.0122736 | − | 0.0577427i | 1.18344 | − | 0.251549i | −2.53808 | + | 1.13002i | 0.0521859 | + | 0.00548496i | 0.760554 | + | 0.247119i | 0.881693 | + | 2.71357i | 2.73745 | + | 1.21879i | −1.23478 | − | 2.13870i |
8.13 | 0.105737 | + | 1.00602i | 0.521935 | − | 2.45551i | 0.955395 | − | 0.203075i | 1.56886 | − | 0.698500i | 2.52549 | + | 0.265440i | −2.67054 | − | 0.867712i | 0.930500 | + | 2.86378i | −3.01649 | − | 1.34303i | 0.868593 | + | 1.50445i |
8.14 | 0.164885 | + | 1.56878i | −0.307100 | + | 1.44479i | −0.477587 | + | 0.101514i | 2.63545 | − | 1.17338i | −2.31719 | − | 0.243547i | 0.651771 | + | 0.211773i | 0.736899 | + | 2.26794i | 0.747527 | + | 0.332820i | 2.27531 | + | 3.94096i |
8.15 | 0.180409 | + | 1.71647i | −0.535718 | + | 2.52035i | −0.957438 | + | 0.203510i | −0.0762414 | + | 0.0339449i | −4.42277 | − | 0.464852i | −3.77824 | − | 1.22762i | 0.544633 | + | 1.67621i | −3.32456 | − | 1.48019i | −0.0720200 | − | 0.124742i |
8.16 | 0.212230 | + | 2.01923i | 0.464015 | − | 2.18302i | −2.07595 | + | 0.441257i | −1.43231 | + | 0.637705i | 4.50649 | + | 0.473651i | 4.01889 | + | 1.30582i | −0.0767508 | − | 0.236215i | −1.80962 | − | 0.805694i | −1.59165 | − | 2.75682i |
8.17 | 0.252914 | + | 2.40631i | 0.318632 | − | 1.49905i | −3.77008 | + | 0.801356i | 2.87604 | − | 1.28050i | 3.68776 | + | 0.387599i | 0.551658 | + | 0.179245i | −1.38644 | − | 4.26704i | 0.595024 | + | 0.264922i | 3.80866 | + | 6.59680i |
8.18 | 0.267178 | + | 2.54203i | −0.0599442 | + | 0.282015i | −4.43425 | + | 0.942528i | −1.51541 | + | 0.674705i | −0.732907 | − | 0.0770316i | −2.45125 | − | 0.796461i | −2.00095 | − | 6.15830i | 2.66470 | + | 1.18640i | −2.12001 | − | 3.67196i |
46.1 | −1.80191 | − | 2.00123i | 2.39485 | − | 0.251709i | −0.548964 | + | 5.22304i | 2.42856 | + | 0.516207i | −4.81905 | − | 4.33909i | −2.04537 | + | 2.81521i | 7.08445 | − | 5.14716i | 2.73753 | − | 0.581880i | −3.34301 | − | 5.79027i |
46.2 | −1.69399 | − | 1.88137i | −1.14305 | + | 0.120139i | −0.460883 | + | 4.38501i | 1.39034 | + | 0.295526i | 2.16235 | + | 1.94699i | 2.22157 | − | 3.05773i | 4.93430 | − | 3.58498i | −1.64231 | + | 0.349084i | −1.79924 | − | 3.11637i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
19.d | odd | 6 | 1 | inner |
209.t | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 209.2.t.a | ✓ | 144 |
11.d | odd | 10 | 1 | inner | 209.2.t.a | ✓ | 144 |
19.d | odd | 6 | 1 | inner | 209.2.t.a | ✓ | 144 |
209.t | even | 30 | 1 | inner | 209.2.t.a | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
209.2.t.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
209.2.t.a | ✓ | 144 | 11.d | odd | 10 | 1 | inner |
209.2.t.a | ✓ | 144 | 19.d | odd | 6 | 1 | inner |
209.2.t.a | ✓ | 144 | 209.t | even | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(209, [\chi])\).