Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [124,3,Mod(7,124)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(124, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 28]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("124.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 124 = 2^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 124.n (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: | \(3.37875527807\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(30\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −1.99926 | + | 0.0544568i | −0.0879632 | − | 0.413834i | 3.99407 | − | 0.217747i | −0.624847 | + | 1.08227i | 0.198397 | + | 0.822572i | −2.70372 | + | 6.07265i | −7.97332 | + | 0.652836i | 8.05839 | − | 3.58783i | 1.19029 | − | 2.19776i |
7.2 | −1.99505 | + | 0.140610i | 0.741912 | + | 3.49042i | 3.96046 | − | 0.561050i | 2.31670 | − | 4.01264i | −1.97094 | − | 6.85925i | 5.02348 | − | 11.2829i | −7.82243 | + | 1.67620i | −3.41071 | + | 1.51855i | −4.05771 | + | 8.33116i |
7.3 | −1.86891 | + | 0.712155i | −1.15158 | − | 5.41775i | 2.98567 | − | 2.66191i | 4.16039 | − | 7.20601i | 6.01048 | + | 9.30520i | 0.685204 | − | 1.53899i | −3.68426 | + | 7.10114i | −19.8040 | + | 8.81730i | −2.64361 | + | 16.4302i |
7.4 | −1.82229 | + | 0.824162i | −0.469244 | − | 2.20762i | 2.64151 | − | 3.00373i | −2.85641 | + | 4.94745i | 2.67454 | + | 3.63620i | 3.33744 | − | 7.49600i | −2.33806 | + | 7.65072i | 3.56852 | − | 1.58881i | 1.12772 | − | 11.3698i |
7.5 | −1.76073 | − | 0.948596i | −1.13434 | − | 5.33667i | 2.20033 | + | 3.34044i | −3.56746 | + | 6.17902i | −3.06507 | + | 10.4725i | −0.864598 | + | 1.94192i | −0.705452 | − | 7.96884i | −18.9714 | + | 8.44661i | 12.1427 | − | 7.49550i |
7.6 | −1.73916 | − | 0.987589i | −0.272138 | − | 1.28031i | 2.04934 | + | 3.43515i | 2.35737 | − | 4.08308i | −0.791127 | + | 2.49542i | −1.03353 | + | 2.32136i | −0.171606 | − | 7.99816i | 6.65678 | − | 2.96379i | −8.13224 | + | 4.77301i |
7.7 | −1.60673 | − | 1.19097i | 0.931035 | + | 4.38017i | 1.16319 | + | 3.82714i | −0.973390 | + | 1.68596i | 3.72072 | − | 8.14661i | −1.13057 | + | 2.53929i | 2.68905 | − | 7.53452i | −10.0972 | + | 4.49556i | 3.57190 | − | 1.54962i |
7.8 | −1.59315 | + | 1.20909i | 0.932435 | + | 4.38676i | 1.07622 | − | 3.85250i | −1.13754 | + | 1.97029i | −6.78947 | − | 5.86135i | −2.59346 | + | 5.82500i | 2.94342 | + | 7.43884i | −10.1523 | + | 4.52010i | −0.569970 | − | 4.51434i |
7.9 | −1.15685 | + | 1.63147i | −0.354741 | − | 1.66892i | −1.32341 | − | 3.77473i | −1.19665 | + | 2.07266i | 3.13318 | + | 1.35194i | −1.19159 | + | 2.67635i | 7.68935 | + | 2.20769i | 5.56244 | − | 2.47656i | −1.99714 | − | 4.35005i |
7.10 | −1.07956 | − | 1.68361i | 0.0275934 | + | 0.129817i | −1.66909 | + | 3.63512i | −2.86810 | + | 4.96769i | 0.188772 | − | 0.186602i | 4.37751 | − | 9.83205i | 7.92202 | − | 1.11424i | 8.20582 | − | 3.65347i | 11.4599 | − | 0.534167i |
7.11 | −0.864102 | + | 1.80370i | 0.286400 | + | 1.34741i | −2.50666 | − | 3.11716i | 3.60353 | − | 6.24150i | −2.67780 | − | 0.647717i | 1.82861 | − | 4.10713i | 7.78842 | − | 1.82771i | 6.48843 | − | 2.88883i | 8.14397 | + | 11.8930i |
7.12 | −0.673616 | − | 1.88315i | −0.736057 | − | 3.46287i | −3.09248 | + | 2.53704i | 3.42633 | − | 5.93457i | −6.02528 | + | 3.71875i | 2.34572 | − | 5.26857i | 6.86076 | + | 4.11461i | −3.22781 | + | 1.43711i | −13.4837 | − | 2.45465i |
7.13 | −0.561919 | − | 1.91944i | 0.736057 | + | 3.46287i | −3.36849 | + | 2.15714i | 3.42633 | − | 5.93457i | 6.23317 | − | 3.35867i | −2.34572 | + | 5.26857i | 6.03332 | + | 5.25348i | −3.22781 | + | 1.43711i | −13.3164 | − | 3.24188i |
7.14 | −0.329666 | + | 1.97264i | −0.974786 | − | 4.58601i | −3.78264 | − | 1.30063i | −0.229111 | + | 0.396833i | 9.36791 | − | 0.411054i | −4.30093 | + | 9.66006i | 3.81268 | − | 7.03303i | −11.8594 | + | 5.28013i | −0.707279 | − | 0.582777i |
7.15 | −0.199060 | + | 1.99007i | 0.547049 | + | 2.57366i | −3.92075 | − | 0.792285i | −4.34583 | + | 7.52719i | −5.23066 | + | 0.576352i | 2.27507 | − | 5.10989i | 2.35717 | − | 7.64485i | 1.89744 | − | 0.844793i | −14.1146 | − | 10.1469i |
7.16 | −0.116218 | − | 1.99662i | −0.0275934 | − | 0.129817i | −3.97299 | + | 0.464087i | −2.86810 | + | 4.96769i | −0.255988 | + | 0.0701807i | −4.37751 | + | 9.83205i | 1.38834 | + | 7.87861i | 8.20582 | − | 3.65347i | 10.2519 | + | 5.14916i |
7.17 | 0.595729 | + | 1.90922i | −0.631742 | − | 2.97211i | −3.29021 | + | 2.27475i | 1.43502 | − | 2.48553i | 5.29806 | − | 2.97670i | 3.93413 | − | 8.83620i | −6.30307 | − | 4.92660i | −0.212439 | + | 0.0945840i | 5.60029 | + | 1.25906i |
7.18 | 0.599843 | − | 1.90793i | −0.931035 | − | 4.38017i | −3.28038 | − | 2.28891i | −0.973390 | + | 1.68596i | −8.91553 | − | 0.851069i | 1.13057 | − | 2.53929i | −6.33479 | + | 4.88573i | −10.0972 | + | 4.49556i | 2.63281 | + | 2.86847i |
7.19 | 0.640254 | + | 1.89475i | 0.631742 | + | 2.97211i | −3.18015 | + | 2.42624i | 1.43502 | − | 2.48553i | −5.22693 | + | 3.09990i | −3.93413 | + | 8.83620i | −6.63323 | − | 4.47217i | −0.212439 | + | 0.0945840i | 5.62823 | + | 1.12763i |
7.20 | 0.826518 | − | 1.82123i | 0.272138 | + | 1.28031i | −2.63374 | − | 3.01055i | 2.35737 | − | 4.08308i | 2.55666 | + | 0.562572i | 1.03353 | − | 2.32136i | −7.65973 | + | 2.30836i | 6.65678 | − | 2.96379i | −5.48781 | − | 7.66804i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
31.g | even | 15 | 1 | inner |
124.n | odd | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 124.3.n.a | ✓ | 240 |
4.b | odd | 2 | 1 | inner | 124.3.n.a | ✓ | 240 |
31.g | even | 15 | 1 | inner | 124.3.n.a | ✓ | 240 |
124.n | odd | 30 | 1 | inner | 124.3.n.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
124.3.n.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
124.3.n.a | ✓ | 240 | 4.b | odd | 2 | 1 | inner |
124.3.n.a | ✓ | 240 | 31.g | even | 15 | 1 | inner |
124.3.n.a | ✓ | 240 | 124.n | odd | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(124, [\chi])\).