Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [588,2,Mod(55,588)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([7, 0, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("588.55");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 588.x (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: | \(4.69520363885\) |
Analytic rank: | \(0\) |
Dimension: | \(168\) |
Relative dimension: | \(28\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 | −1.41421 | + | 0.00366928i | 0.900969 | + | 0.433884i | 1.99997 | − | 0.0103783i | 0.477768 | − | 0.992096i | −1.27575 | − | 0.610296i | 2.27188 | + | 1.35593i | −2.82834 | + | 0.0220155i | 0.623490 | + | 0.781831i | −0.672024 | + | 1.40478i |
55.2 | −1.40198 | + | 0.185596i | 0.900969 | + | 0.433884i | 1.93111 | − | 0.520405i | −1.40445 | + | 2.91638i | −1.34367 | − | 0.441081i | 0.602515 | − | 2.57623i | −2.61079 | + | 1.08801i | 0.623490 | + | 0.781831i | 1.42775 | − | 4.34937i |
55.3 | −1.37146 | + | 0.345091i | 0.900969 | + | 0.433884i | 1.76182 | − | 0.946559i | 1.63442 | − | 3.39390i | −1.38538 | − | 0.284140i | −0.364664 | − | 2.62050i | −2.08963 | + | 1.90616i | 0.623490 | + | 0.781831i | −1.07034 | + | 5.21863i |
55.4 | −1.36844 | − | 0.356880i | 0.900969 | + | 0.433884i | 1.74527 | + | 0.976739i | −0.0842420 | + | 0.174930i | −1.07808 | − | 0.915283i | −2.61856 | − | 0.378316i | −2.03973 | − | 1.95946i | 0.623490 | + | 0.781831i | 0.177709 | − | 0.209318i |
55.5 | −1.25082 | + | 0.659893i | 0.900969 | + | 0.433884i | 1.12908 | − | 1.65081i | −0.617993 | + | 1.28328i | −1.41326 | + | 0.0518343i | −2.62009 | + | 0.367586i | −0.322916 | + | 2.80993i | 0.623490 | + | 0.781831i | −0.0738292 | − | 2.01295i |
55.6 | −1.14486 | − | 0.830241i | 0.900969 | + | 0.433884i | 0.621399 | + | 1.90102i | −0.591080 | + | 1.22739i | −0.671253 | − | 1.24476i | 2.62768 | − | 0.308728i | 0.866889 | − | 2.69230i | 0.623490 | + | 0.781831i | 1.69573 | − | 0.914448i |
55.7 | −1.08805 | − | 0.903405i | 0.900969 | + | 0.433884i | 0.367718 | + | 1.96591i | 1.27172 | − | 2.64076i | −0.588329 | − | 1.28603i | −0.874558 | − | 2.49703i | 1.37591 | − | 2.47121i | 0.623490 | + | 0.781831i | −3.76938 | + | 1.72441i |
55.8 | −0.974763 | + | 1.02462i | 0.900969 | + | 0.433884i | −0.0996732 | − | 1.99751i | 1.10901 | − | 2.30289i | −1.32280 | + | 0.500213i | 0.953739 | + | 2.46787i | 2.14384 | + | 1.84498i | 0.623490 | + | 0.781831i | 1.27855 | + | 3.38109i |
55.9 | −0.778541 | + | 1.18062i | 0.900969 | + | 0.433884i | −0.787748 | − | 1.83833i | −1.73990 | + | 3.61294i | −1.21369 | + | 0.725909i | 2.50782 | + | 0.843102i | 2.78367 | + | 0.501181i | 0.623490 | + | 0.781831i | −2.91094 | − | 4.86699i |
55.10 | −0.611563 | − | 1.27514i | 0.900969 | + | 0.433884i | −1.25198 | + | 1.55966i | −0.776660 | + | 1.61275i | 0.00226428 | − | 1.41421i | −0.448708 | + | 2.60742i | 2.75446 | + | 0.642622i | 0.623490 | + | 0.781831i | 2.53146 | + | 0.00405309i |
55.11 | −0.446111 | + | 1.34201i | 0.900969 | + | 0.433884i | −1.60197 | − | 1.19737i | −0.0202251 | + | 0.0419978i | −0.984208 | + | 1.01555i | −2.50508 | + | 0.851222i | 2.32154 | − | 1.61569i | 0.623490 | + | 0.781831i | −0.0473388 | − | 0.0458779i |
55.12 | −0.315691 | − | 1.37853i | 0.900969 | + | 0.433884i | −1.80068 | + | 0.870378i | 0.236353 | − | 0.490793i | 0.313693 | − | 1.37898i | −2.32580 | + | 1.26121i | 1.76830 | + | 2.20751i | 0.623490 | + | 0.781831i | −0.751186 | − | 0.170881i |
55.13 | −0.302651 | + | 1.38145i | 0.900969 | + | 0.433884i | −1.81681 | − | 0.836193i | 0.0432081 | − | 0.0897226i | −0.872067 | + | 1.11333i | −0.103018 | − | 2.64374i | 1.70502 | − | 2.25675i | 0.623490 | + | 0.781831i | 0.110870 | + | 0.0868444i |
55.14 | −0.225403 | − | 1.39614i | 0.900969 | + | 0.433884i | −1.89839 | + | 0.629386i | 1.60689 | − | 3.33674i | 0.402679 | − | 1.35567i | 2.64502 | + | 0.0624089i | 1.30661 | + | 2.50854i | 0.623490 | + | 0.781831i | −5.02073 | − | 1.49132i |
55.15 | 0.148148 | − | 1.40643i | 0.900969 | + | 0.433884i | −1.95610 | − | 0.416719i | −1.14386 | + | 2.37526i | 0.743705 | − | 1.20287i | 1.90517 | − | 1.83585i | −0.875880 | + | 2.68939i | 0.623490 | + | 0.781831i | 3.17118 | + | 1.96066i |
55.16 | 0.357271 | − | 1.36834i | 0.900969 | + | 0.433884i | −1.74472 | − | 0.977737i | 1.10712 | − | 2.29896i | 0.915591 | − | 1.07782i | −2.45319 | − | 0.990882i | −1.96121 | + | 2.03805i | 0.623490 | + | 0.781831i | −2.75022 | − | 2.33627i |
55.17 | 0.421544 | + | 1.34993i | 0.900969 | + | 0.433884i | −1.64460 | + | 1.13811i | 1.52760 | − | 3.17209i | −0.205913 | + | 1.39914i | 1.99561 | − | 1.73711i | −2.22963 | − | 1.74033i | 0.623490 | + | 0.781831i | 4.92603 | + | 0.724967i |
55.18 | 0.511445 | + | 1.31849i | 0.900969 | + | 0.433884i | −1.47685 | + | 1.34867i | −1.72652 | + | 3.58517i | −0.111277 | + | 1.40983i | −1.74761 | − | 1.98642i | −2.53354 | − | 1.25744i | 0.623490 | + | 0.781831i | −5.61004 | − | 0.442796i |
55.19 | 0.696560 | + | 1.23077i | 0.900969 | + | 0.433884i | −1.02961 | + | 1.71462i | −0.997385 | + | 2.07109i | 0.0935667 | + | 1.41111i | 0.739943 | + | 2.54017i | −2.82749 | − | 0.0728789i | 0.623490 | + | 0.781831i | −3.24378 | + | 0.215085i |
55.20 | 0.761615 | − | 1.19161i | 0.900969 | + | 0.433884i | −0.839884 | − | 1.81510i | −1.00022 | + | 2.07699i | 1.20321 | − | 0.743154i | −0.0377719 | + | 2.64548i | −2.80257 | − | 0.381593i | 0.623490 | + | 0.781831i | 1.71318 | + | 2.77375i |
See next 80 embeddings (of 168 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
196.j | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 588.2.x.b | yes | 168 |
4.b | odd | 2 | 1 | 588.2.x.a | ✓ | 168 | |
49.f | odd | 14 | 1 | 588.2.x.a | ✓ | 168 | |
196.j | even | 14 | 1 | inner | 588.2.x.b | yes | 168 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
588.2.x.a | ✓ | 168 | 4.b | odd | 2 | 1 | |
588.2.x.a | ✓ | 168 | 49.f | odd | 14 | 1 | |
588.2.x.b | yes | 168 | 1.a | even | 1 | 1 | trivial |
588.2.x.b | yes | 168 | 196.j | even | 14 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{168} - 162 T_{11}^{166} + 168 T_{11}^{165} + 15627 T_{11}^{164} - 33768 T_{11}^{163} + \cdots + 89\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\).