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 | −1.40988 | − | 0.110588i | −0.900969 | − | 0.433884i | 1.97554 | + | 0.311833i | 1.52760 | − | 3.17209i | 1.22228 | + | 0.711362i | −1.99561 | + | 1.73711i | −2.75080 | − | 0.658120i | 0.623490 | + | 0.781831i | −2.50453 | + | 4.30333i |
55.2 | −1.39924 | − | 0.205230i | −0.900969 | − | 0.433884i | 1.91576 | + | 0.574332i | −1.72652 | + | 3.58517i | 1.17163 | + | 0.792014i | 1.74761 | + | 1.98642i | −2.56275 | − | 1.19680i | 0.623490 | + | 0.781831i | 3.15161 | − | 4.66218i |
55.3 | −1.35491 | − | 0.405223i | −0.900969 | − | 0.433884i | 1.67159 | + | 1.09809i | −0.997385 | + | 2.07109i | 1.04492 | + | 0.952969i | −0.739943 | − | 2.54017i | −1.81989 | − | 2.16518i | 0.623490 | + | 0.781831i | 2.19063 | − | 2.40199i |
55.4 | −1.27947 | + | 0.602464i | −0.900969 | − | 0.433884i | 1.27407 | − | 1.54167i | 0.0432081 | − | 0.0897226i | 1.41416 | + | 0.0123389i | 0.103018 | + | 2.64374i | −0.701339 | + | 2.74010i | 0.623490 | + | 0.781831i | −0.00122877 | + | 0.140828i |
55.5 | −1.20909 | + | 0.733551i | −0.900969 | − | 0.433884i | 0.923805 | − | 1.77386i | −0.0202251 | + | 0.0419978i | 1.40763 | − | 0.136302i | 2.50508 | − | 0.851222i | 0.184253 | + | 2.82242i | 0.623490 | + | 0.781831i | −0.00635357 | − | 0.0656153i |
55.6 | −1.17933 | − | 0.780496i | −0.900969 | − | 0.433884i | 0.781653 | + | 1.84093i | −0.352634 | + | 0.732252i | 0.723898 | + | 1.21490i | −1.67777 | + | 2.04575i | 0.515009 | − | 2.78114i | 0.623490 | + | 0.781831i | 0.987392 | − | 0.588339i |
55.7 | −1.11034 | − | 0.875865i | −0.900969 | − | 0.433884i | 0.465720 | + | 1.94502i | 0.698680 | − | 1.45082i | 0.620360 | + | 1.27089i | 1.90070 | − | 1.84047i | 1.18647 | − | 2.56755i | 0.623490 | + | 0.781831i | −2.04650 | + | 0.998961i |
55.8 | −0.977782 | + | 1.02174i | −0.900969 | − | 0.433884i | −0.0878851 | − | 1.99807i | −1.73990 | + | 3.61294i | 1.32427 | − | 0.496308i | −2.50782 | − | 0.843102i | 2.12743 | + | 1.86388i | 0.623490 | + | 0.781831i | −1.99022 | − | 5.31038i |
55.9 | −0.782021 | + | 1.17832i | −0.900969 | − | 0.433884i | −0.776887 | − | 1.84295i | 1.10901 | − | 2.30289i | 1.21583 | − | 0.722326i | −0.953739 | − | 2.46787i | 2.77913 | + | 0.525799i | 0.623490 | + | 0.781831i | 1.84628 | + | 3.10769i |
55.10 | −0.712285 | − | 1.22174i | −0.900969 | − | 0.433884i | −0.985301 | + | 1.74045i | 1.07827 | − | 2.23906i | 0.111653 | + | 1.40980i | −2.56980 | − | 0.629382i | 2.82820 | − | 0.0359170i | 0.623490 | + | 0.781831i | −3.50359 | + | 0.277476i |
55.11 | −0.537130 | − | 1.30824i | −0.900969 | − | 0.433884i | −1.42298 | + | 1.40539i | −1.35460 | + | 2.81286i | −0.0836867 | + | 1.41174i | 2.57260 | − | 0.617858i | 2.60291 | + | 1.10673i | 0.623490 | + | 0.781831i | 4.40750 | + | 0.261274i |
55.12 | −0.365015 | + | 1.36630i | −0.900969 | − | 0.433884i | −1.73353 | − | 0.997438i | −0.617993 | + | 1.28328i | 0.921681 | − | 1.07262i | 2.62009 | − | 0.367586i | 1.99556 | − | 2.00443i | 0.623490 | + | 0.781831i | −1.52776 | − | 1.31278i |
55.13 | −0.0562843 | − | 1.41309i | −0.900969 | − | 0.433884i | −1.99366 | + | 0.159070i | 0.302206 | − | 0.627538i | −0.562408 | + | 1.29757i | 1.01653 | + | 2.44268i | 0.336993 | + | 2.80828i | 0.623490 | + | 0.781831i | −0.903779 | − | 0.391725i |
55.14 | −0.0312592 | + | 1.41387i | −0.900969 | − | 0.433884i | −1.99805 | − | 0.0883928i | 1.63442 | − | 3.39390i | 0.641618 | − | 1.26029i | 0.364664 | + | 2.62050i | 0.187433 | − | 2.82221i | 0.623490 | + | 0.781831i | 4.74743 | + | 2.41694i |
55.15 | 0.0809236 | − | 1.41190i | −0.900969 | − | 0.433884i | −1.98690 | − | 0.228511i | −0.881995 | + | 1.83148i | −0.685509 | + | 1.23696i | −1.70818 | − | 2.02043i | −0.483422 | + | 2.78681i | 0.623490 | + | 0.781831i | 2.51449 | + | 1.39349i |
55.16 | 0.131027 | + | 1.40813i | −0.900969 | − | 0.433884i | −1.96566 | + | 0.369007i | −1.40445 | + | 2.91638i | 0.492913 | − | 1.32553i | −0.602515 | + | 2.57623i | −0.777166 | − | 2.71956i | 0.623490 | + | 0.781831i | −4.29066 | − | 1.59553i |
55.17 | 0.311114 | + | 1.37957i | −0.900969 | − | 0.433884i | −1.80642 | + | 0.858405i | 0.477768 | − | 0.992096i | 0.318268 | − | 1.37794i | −2.27188 | − | 1.35593i | −1.74623 | − | 2.22501i | 0.623490 | + | 0.781831i | 1.51730 | + | 0.350459i |
55.18 | 0.505797 | − | 1.32067i | −0.900969 | − | 0.433884i | −1.48834 | − | 1.33598i | 1.55098 | − | 3.22064i | −1.02872 | + | 0.970425i | 1.05984 | − | 2.42420i | −2.51719 | + | 1.28987i | 0.623490 | + | 0.781831i | −3.46892 | − | 3.67732i |
55.19 | 0.652439 | + | 1.25472i | −0.900969 | − | 0.433884i | −1.14865 | + | 1.63726i | −0.0842420 | + | 0.174930i | −0.0434247 | − | 1.41355i | 2.61856 | + | 0.378316i | −2.80372 | − | 0.373018i | 0.623490 | + | 0.781831i | −0.274451 | + | 0.00843125i |
55.20 | 0.745941 | − | 1.20149i | −0.900969 | − | 0.433884i | −0.887143 | − | 1.79248i | 0.0475607 | − | 0.0987608i | −1.19338 | + | 0.758851i | −1.69059 | + | 2.03517i | −2.81540 | − | 0.271193i | 0.623490 | + | 0.781831i | −0.0831823 | − | 0.130813i |
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.a | ✓ | 168 |
4.b | odd | 2 | 1 | 588.2.x.b | yes | 168 | |
49.f | odd | 14 | 1 | 588.2.x.b | yes | 168 | |
196.j | even | 14 | 1 | inner | 588.2.x.a | ✓ | 168 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
588.2.x.a | ✓ | 168 | 1.a | even | 1 | 1 | trivial |
588.2.x.a | ✓ | 168 | 196.j | even | 14 | 1 | inner |
588.2.x.b | yes | 168 | 4.b | odd | 2 | 1 | |
588.2.x.b | yes | 168 | 49.f | odd | 14 | 1 |
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 \(S_{2}^{\mathrm{new}}(588, [\chi])\).