Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [473,2,Mod(2,473)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(473, base_ring=CyclotomicField(70))
chi = DirichletCharacter(H, H._module([7, 45]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("473.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 473 = 11 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 473.bb (of order \(70\), degree \(24\), 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: | \(3.77692401561\) |
Analytic rank: | \(0\) |
Dimension: | \(1008\) |
Relative dimension: | \(42\) over \(\Q(\zeta_{70})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{70}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −0.372583 | − | 2.75052i | −0.0959763 | − | 0.528873i | −5.49860 | + | 1.51752i | 0.152095 | + | 1.68992i | −1.41892 | + | 0.461033i | 3.09656 | + | 2.24978i | 4.04085 | + | 9.45405i | 2.53821 | − | 0.952606i | 4.59148 | − | 1.04798i |
2.2 | −0.362749 | − | 2.67792i | 0.513347 | + | 2.82877i | −5.11175 | + | 1.41075i | −0.142181 | − | 1.57976i | 7.38901 | − | 2.40084i | −0.688627 | − | 0.500317i | 3.50796 | + | 8.20728i | −4.92973 | + | 1.85016i | −4.17889 | + | 0.953805i |
2.3 | −0.342918 | − | 2.53152i | −0.506523 | − | 2.79117i | −4.36309 | + | 1.20414i | −0.105428 | − | 1.17140i | −6.89222 | + | 2.23942i | −0.320535 | − | 0.232882i | 2.53641 | + | 5.93422i | −4.72536 | + | 1.77346i | −2.92926 | + | 0.668585i |
2.4 | −0.322780 | − | 2.38286i | 0.110576 | + | 0.609324i | −3.64591 | + | 1.00621i | −0.279544 | − | 3.10599i | 1.41624 | − | 0.460165i | −0.654933 | − | 0.475837i | 1.68432 | + | 3.94066i | 2.44966 | − | 0.919371i | −7.31091 | + | 1.66867i |
2.5 | −0.307106 | − | 2.26715i | 0.000236830 | 0.00130504i | −3.11771 | + | 0.860434i | 0.264418 | + | 2.93792i | 0.00288598 | 0.000937712i | −1.62388 | − | 1.17981i | 1.10983 | + | 2.59658i | 2.80870 | − | 1.05412i | 6.57950 | − | 1.50173i | ||
2.6 | −0.295635 | − | 2.18246i | 0.255543 | + | 1.40816i | −2.74783 | + | 0.758352i | 0.228913 | + | 2.54343i | 2.99770 | − | 0.974013i | −2.21775 | − | 1.61129i | 0.736234 | + | 1.72251i | 0.891103 | − | 0.334437i | 5.48328 | − | 1.25152i |
2.7 | −0.273013 | − | 2.01546i | 0.524432 | + | 2.88986i | −2.05962 | + | 0.568418i | 0.168353 | + | 1.87055i | 5.68122 | − | 1.84594i | 3.23413 | + | 2.34973i | 0.109204 | + | 0.255496i | −5.26755 | + | 1.97694i | 3.72406 | − | 0.849993i |
2.8 | −0.269715 | − | 1.99111i | −0.481201 | − | 2.65164i | −1.96386 | + | 0.541990i | 0.239058 | + | 2.65615i | −5.14992 | + | 1.67331i | −1.86622 | − | 1.35589i | 0.0294362 | + | 0.0688694i | −3.99091 | + | 1.49781i | 5.22421 | − | 1.19239i |
2.9 | −0.260060 | − | 1.91984i | −0.361699 | − | 1.99313i | −1.69023 | + | 0.466475i | −0.260769 | − | 2.89738i | −3.73242 | + | 1.21274i | 0.918059 | + | 0.667009i | −0.187755 | − | 0.439274i | −1.03303 | + | 0.387701i | −5.49469 | + | 1.25413i |
2.10 | −0.250506 | − | 1.84931i | −0.126174 | − | 0.695275i | −1.42926 | + | 0.394451i | 0.127317 | + | 1.41460i | −1.25417 | + | 0.407504i | 2.71157 | + | 1.97007i | −0.379425 | − | 0.887710i | 2.34122 | − | 0.878674i | 2.58414 | − | 0.589814i |
2.11 | −0.228008 | − | 1.68322i | 0.417127 | + | 2.29856i | −0.853321 | + | 0.235502i | −0.0721221 | − | 0.801342i | 3.77387 | − | 1.22621i | −2.75063 | − | 1.99845i | −0.744215 | − | 1.74118i | −2.30067 | + | 0.863456i | −1.33239 | + | 0.304110i |
2.12 | −0.182674 | − | 1.34855i | −0.0794355 | − | 0.437725i | 0.142699 | − | 0.0393825i | −0.260312 | − | 2.89230i | −0.575785 | + | 0.187084i | −3.11194 | − | 2.26095i | −1.14889 | − | 2.68796i | 2.62341 | − | 0.984583i | −3.85287 | + | 0.879392i |
2.13 | −0.169153 | − | 1.24874i | 0.347366 | + | 1.91414i | 0.397191 | − | 0.109618i | −0.220300 | − | 2.44773i | 2.33151 | − | 0.757552i | 3.34902 | + | 2.43321i | −1.19461 | − | 2.79492i | −0.734577 | + | 0.275691i | −3.01932 | + | 0.689139i |
2.14 | −0.159508 | − | 1.17754i | −0.219944 | − | 1.21199i | 0.566778 | − | 0.156421i | 0.130059 | + | 1.44508i | −1.39208 | + | 0.452314i | −0.616162 | − | 0.447668i | −1.20865 | − | 2.82778i | 1.38816 | − | 0.520986i | 1.68088 | − | 0.383651i |
2.15 | −0.125050 | − | 0.923153i | 0.226607 | + | 1.24871i | 1.09135 | − | 0.301194i | 0.0615785 | + | 0.684193i | 1.12441 | − | 0.365343i | 0.261946 | + | 0.190315i | −1.14679 | − | 2.68306i | 1.30079 | − | 0.488194i | 0.623915 | − | 0.142404i |
2.16 | −0.104678 | − | 0.772764i | −0.507692 | − | 2.79761i | 1.34172 | − | 0.370291i | 0.246212 | + | 2.73564i | −2.10875 | + | 0.685174i | 4.00390 | + | 2.90901i | −1.03957 | − | 2.43221i | −4.76018 | + | 1.78652i | 2.08823 | − | 0.476626i |
2.17 | −0.0774585 | − | 0.571822i | 0.541596 | + | 2.98444i | 1.60695 | − | 0.443489i | 0.295074 | + | 3.27854i | 1.66462 | − | 0.540867i | −0.530962 | − | 0.385767i | −0.831654 | − | 1.94575i | −5.80485 | + | 2.17860i | 1.85189 | − | 0.422681i |
2.18 | −0.0740138 | − | 0.546392i | −0.0159061 | − | 0.0876497i | 1.63486 | − | 0.451193i | −0.0741801 | − | 0.824208i | −0.0467138 | + | 0.0151782i | 1.61609 | + | 1.17415i | −0.800944 | − | 1.87390i | 2.80128 | − | 1.05134i | −0.444850 | + | 0.101534i |
2.19 | −0.0602079 | − | 0.444473i | −0.557399 | − | 3.07152i | 1.73399 | − | 0.478552i | −0.0730657 | − | 0.811827i | −1.33165 | + | 0.432679i | −2.21122 | − | 1.60654i | −0.669672 | − | 1.56678i | −6.31486 | + | 2.37001i | −0.356435 | + | 0.0813541i |
2.20 | −0.00163409 | − | 0.0120634i | 0.398539 | + | 2.19613i | 1.92778 | − | 0.532034i | −0.324212 | − | 3.60229i | 0.0258415 | − | 0.00839641i | −1.70529 | − | 1.23896i | −0.0191373 | − | 0.0447740i | −1.85546 | + | 0.696365i | −0.0429259 | + | 0.00979757i |
See next 80 embeddings (of 1008 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
43.f | odd | 14 | 1 | inner |
473.bb | even | 70 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 473.2.bb.a | ✓ | 1008 |
11.d | odd | 10 | 1 | inner | 473.2.bb.a | ✓ | 1008 |
43.f | odd | 14 | 1 | inner | 473.2.bb.a | ✓ | 1008 |
473.bb | even | 70 | 1 | inner | 473.2.bb.a | ✓ | 1008 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
473.2.bb.a | ✓ | 1008 | 1.a | even | 1 | 1 | trivial |
473.2.bb.a | ✓ | 1008 | 11.d | odd | 10 | 1 | inner |
473.2.bb.a | ✓ | 1008 | 43.f | odd | 14 | 1 | inner |
473.2.bb.a | ✓ | 1008 | 473.bb | even | 70 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(473, [\chi])\).