Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,2,Mod(3,304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(304, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([18, 27, 26]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("304.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 304.bg (of order \(36\), degree \(12\), 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: | \(2.42745222145\) |
Analytic rank: | \(0\) |
Dimension: | \(456\) |
Relative dimension: | \(38\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −1.41273 | + | 0.0647022i | 0.459790 | + | 0.214404i | 1.99163 | − | 0.182814i | −1.87793 | − | 2.68196i | −0.663433 | − | 0.273146i | 0.233325 | + | 0.404130i | −2.80181 | + | 0.387130i | −1.76292 | − | 2.10097i | 2.82654 | + | 3.66738i |
3.2 | −1.38797 | + | 0.271180i | 2.35855 | + | 1.09981i | 1.85292 | − | 0.752781i | 1.12887 | + | 1.61220i | −3.57184 | − | 0.886910i | 0.814974 | + | 1.41158i | −2.36766 | + | 1.54731i | 2.42481 | + | 2.88978i | −2.00404 | − | 1.93156i |
3.3 | −1.36996 | − | 0.351025i | −1.52863 | − | 0.712812i | 1.75356 | + | 0.961777i | 2.13072 | + | 3.04299i | 1.84394 | + | 1.51311i | 0.591379 | + | 1.02430i | −2.06470 | − | 1.93314i | −0.0997514 | − | 0.118879i | −1.85084 | − | 4.91670i |
3.4 | −1.27934 | − | 0.602732i | −2.12965 | − | 0.993074i | 1.27343 | + | 1.54220i | −0.856388 | − | 1.22305i | 2.12600 | + | 2.55409i | 2.11839 | + | 3.66917i | −0.699613 | − | 2.74054i | 1.62087 | + | 1.93167i | 0.358441 | + | 2.08087i |
3.5 | −1.25803 | + | 0.646026i | −0.566911 | − | 0.264355i | 1.16530 | − | 1.62545i | 0.256415 | + | 0.366198i | 0.883974 | − | 0.0336715i | 1.25795 | + | 2.17884i | −0.415910 | + | 2.79768i | −1.67686 | − | 1.99840i | −0.559152 | − | 0.295039i |
3.6 | −1.23391 | + | 0.690987i | −2.50672 | − | 1.16890i | 1.04507 | − | 1.70523i | 1.02987 | + | 1.47081i | 3.90077 | − | 0.289789i | −0.843507 | − | 1.46100i | −0.111235 | + | 2.82624i | 2.98897 | + | 3.56211i | −2.28708 | − | 1.10322i |
3.7 | −1.20851 | − | 0.734512i | 1.75061 | + | 0.816321i | 0.920986 | + | 1.77533i | 0.635225 | + | 0.907195i | −1.51603 | − | 2.27237i | 0.220626 | + | 0.382136i | 0.190979 | − | 2.82197i | 0.469880 | + | 0.559981i | −0.101329 | − | 1.56293i |
3.8 | −1.15700 | + | 0.813233i | 2.20762 | + | 1.02943i | 0.677304 | − | 1.88182i | −1.33336 | − | 1.90424i | −3.39139 | + | 0.604258i | −2.13323 | − | 3.69486i | 0.746720 | + | 2.72808i | 1.88550 | + | 2.24705i | 3.09129 | + | 1.11887i |
3.9 | −1.09186 | − | 0.898798i | 0.945637 | + | 0.440958i | 0.384323 | + | 1.96273i | −1.51828 | − | 2.16833i | −0.636173 | − | 1.33140i | −0.308642 | − | 0.534584i | 1.34447 | − | 2.48845i | −1.22858 | − | 1.46416i | −0.291138 | + | 3.73214i |
3.10 | −1.01722 | − | 0.982474i | −2.34554 | − | 1.09375i | 0.0694880 | + | 1.99879i | −0.667915 | − | 0.953881i | 1.31137 | + | 3.41702i | −2.50561 | − | 4.33984i | 1.89308 | − | 2.10149i | 2.37694 | + | 2.83272i | −0.257745 | + | 1.62652i |
3.11 | −0.843947 | + | 1.13479i | 0.0677865 | + | 0.0316094i | −0.575508 | − | 1.91541i | 0.591033 | + | 0.844083i | −0.0930783 | + | 0.0502470i | −2.10365 | − | 3.64363i | 2.65929 | + | 0.963421i | −1.92477 | − | 2.29385i | −1.45666 | − | 0.0416610i |
3.12 | −0.799968 | + | 1.16621i | −2.34714 | − | 1.09449i | −0.720104 | − | 1.86586i | −2.31438 | − | 3.30527i | 3.15404 | − | 1.86171i | 0.352854 | + | 0.611162i | 2.75205 | + | 0.652837i | 2.38278 | + | 2.83969i | 5.70607 | − | 0.0549447i |
3.13 | −0.610064 | + | 1.27586i | 0.759059 | + | 0.353955i | −1.25564 | − | 1.55671i | 2.32234 | + | 3.31664i | −0.914672 | + | 0.752519i | 0.950093 | + | 1.64561i | 2.75217 | − | 0.652334i | −1.47748 | − | 1.76079i | −5.64835 | + | 0.939618i |
3.14 | −0.573578 | − | 1.29267i | −1.00251 | − | 0.467479i | −1.34202 | + | 1.48290i | 0.566420 | + | 0.808932i | −0.0292792 | + | 1.56406i | 0.913291 | + | 1.58187i | 2.68666 | + | 0.884231i | −1.14187 | − | 1.36083i | 0.720799 | − | 1.19618i |
3.15 | −0.463438 | − | 1.33612i | 2.94375 | + | 1.37269i | −1.57045 | + | 1.23842i | −1.38143 | − | 1.97288i | 0.469841 | − | 4.56937i | 1.46995 | + | 2.54603i | 2.38249 | + | 1.52438i | 4.85302 | + | 5.78360i | −1.99581 | + | 2.76006i |
3.16 | −0.443631 | + | 1.34283i | 2.49171 | + | 1.16190i | −1.60638 | − | 1.19144i | −0.636818 | − | 0.909471i | −2.66564 | + | 2.83049i | 1.63512 | + | 2.83211i | 2.31255 | − | 1.62854i | 2.93025 | + | 3.49213i | 1.50378 | − | 0.451669i |
3.17 | −0.372821 | − | 1.36419i | 2.26327 | + | 1.05538i | −1.72201 | + | 1.01720i | 2.33412 | + | 3.33347i | 0.595940 | − | 3.48099i | −1.68527 | − | 2.91898i | 2.02965 | + | 1.96991i | 2.08020 | + | 2.47909i | 3.67727 | − | 4.42697i |
3.18 | −0.302259 | + | 1.38154i | 0.00958574 | + | 0.00446990i | −1.81728 | − | 0.835162i | −1.14263 | − | 1.63184i | −0.00907270 | + | 0.0118920i | 0.290298 | + | 0.502811i | 1.70309 | − | 2.25820i | −1.92829 | − | 2.29805i | 2.59982 | − | 1.08534i |
3.19 | −0.256606 | − | 1.39074i | −0.0196774 | − | 0.00917574i | −1.86831 | + | 0.713745i | −0.252165 | − | 0.360129i | −0.00771170 | + | 0.0297207i | −1.27538 | − | 2.20902i | 1.47205 | + | 2.41517i | −1.92806 | − | 2.29777i | −0.436138 | + | 0.443107i |
3.20 | 0.0596919 | − | 1.41295i | −2.91030 | − | 1.35709i | −1.99287 | − | 0.168684i | 1.72144 | + | 2.45847i | −2.09123 | + | 4.03111i | 0.331324 | + | 0.573871i | −0.357301 | + | 2.80577i | 4.69976 | + | 5.60096i | 3.57646 | − | 2.28556i |
See next 80 embeddings (of 456 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.f | odd | 4 | 1 | inner |
19.f | odd | 18 | 1 | inner |
304.bg | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 304.2.bg.a | ✓ | 456 |
16.f | odd | 4 | 1 | inner | 304.2.bg.a | ✓ | 456 |
19.f | odd | 18 | 1 | inner | 304.2.bg.a | ✓ | 456 |
304.bg | even | 36 | 1 | inner | 304.2.bg.a | ✓ | 456 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
304.2.bg.a | ✓ | 456 | 1.a | even | 1 | 1 | trivial |
304.2.bg.a | ✓ | 456 | 16.f | odd | 4 | 1 | inner |
304.2.bg.a | ✓ | 456 | 19.f | odd | 18 | 1 | inner |
304.2.bg.a | ✓ | 456 | 304.bg | even | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(304, [\chi])\).