Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [385,2,Mod(16,385)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(385, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 10, 12]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("385.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 385 = 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 385.bg (of order \(15\), 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: | \(3.07424047782\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −2.48362 | − | 0.527910i | −0.154959 | + | 1.47433i | 4.06260 | + | 1.80878i | −0.669131 | − | 0.743145i | 1.16317 | − | 3.57988i | −2.63782 | + | 0.204771i | −5.02672 | − | 3.65212i | 0.784795 | + | 0.166813i | 1.26955 | + | 2.19893i |
16.2 | −2.19313 | − | 0.466165i | −0.134536 | + | 1.28002i | 2.76543 | + | 1.23125i | −0.669131 | − | 0.743145i | 0.891757 | − | 2.74454i | 2.11950 | + | 1.58358i | −1.86315 | − | 1.35366i | 1.31408 | + | 0.279317i | 1.12106 | + | 1.94174i |
16.3 | −1.85877 | − | 0.395095i | 0.0857569 | − | 0.815923i | 1.47185 | + | 0.655310i | −0.669131 | − | 0.743145i | −0.481769 | + | 1.48273i | 1.60824 | − | 2.10085i | 0.597821 | + | 0.434342i | 2.27607 | + | 0.483793i | 0.950150 | + | 1.64571i |
16.4 | −1.73586 | − | 0.368968i | 0.324493 | − | 3.08734i | 1.04998 | + | 0.467481i | −0.669131 | − | 0.743145i | −1.70240 | + | 5.23946i | 0.427265 | + | 2.61102i | 1.22130 | + | 0.887324i | −6.49193 | − | 1.37990i | 0.887320 | + | 1.53688i |
16.5 | −1.29673 | − | 0.275629i | 0.0805266 | − | 0.766160i | −0.221545 | − | 0.0986380i | −0.669131 | − | 0.743145i | −0.315598 | + | 0.971309i | −1.82828 | − | 1.91243i | 2.40513 | + | 1.74743i | 2.35393 | + | 0.500343i | 0.662852 | + | 1.14809i |
16.6 | −0.948062 | − | 0.201517i | −0.310072 | + | 2.95014i | −0.968879 | − | 0.431373i | −0.669131 | − | 0.743145i | 0.888470 | − | 2.73443i | −2.08004 | + | 1.63506i | 2.39989 | + | 1.74363i | −5.67274 | − | 1.20578i | 0.484621 | + | 0.839388i |
16.7 | −0.0856524 | − | 0.0182060i | −0.191136 | + | 1.81854i | −1.82009 | − | 0.810355i | −0.669131 | − | 0.743145i | 0.0494794 | − | 0.152282i | 1.21114 | − | 2.35226i | 0.282826 | + | 0.205485i | −0.336098 | − | 0.0714399i | 0.0437829 | + | 0.0758343i |
16.8 | −0.0138669 | − | 0.00294751i | −0.00623667 | + | 0.0593379i | −1.82691 | − | 0.813392i | −0.669131 | − | 0.743145i | 0.000261382 | 0 | 0.000804452i | −1.70824 | + | 2.02037i | 0.0458745 | + | 0.0333298i | 2.93096 | + | 0.622995i | 0.00708836 | + | 0.0122774i |
16.9 | 0.899593 | + | 0.191214i | 0.00620345 | − | 0.0590219i | −1.05439 | − | 0.469443i | −0.669131 | − | 0.743145i | 0.0168664 | − | 0.0519095i | −1.51002 | − | 2.17252i | −2.34684 | − | 1.70508i | 2.93100 | + | 0.623003i | −0.459845 | − | 0.796475i |
16.10 | 1.00398 | + | 0.213403i | 0.279619 | − | 2.66040i | −0.864648 | − | 0.384966i | −0.669131 | − | 0.743145i | 0.848472 | − | 2.61133i | 2.64455 | + | 0.0798230i | −2.44671 | − | 1.77764i | −4.06510 | − | 0.864063i | −0.513207 | − | 0.888900i |
16.11 | 1.27591 | + | 0.271203i | 0.276533 | − | 2.63103i | −0.272693 | − | 0.121411i | −0.669131 | − | 0.743145i | 1.06638 | − | 3.28197i | −2.31419 | + | 1.28239i | −2.42559 | − | 1.76230i | −3.91143 | − | 0.831400i | −0.652208 | − | 1.12966i |
16.12 | 1.52639 | + | 0.324444i | −0.319852 | + | 3.04319i | 0.397510 | + | 0.176983i | −0.669131 | − | 0.743145i | −1.47557 | + | 4.54132i | 1.59067 | + | 2.11418i | −1.97559 | − | 1.43535i | −6.22427 | − | 1.32301i | −0.780245 | − | 1.35142i |
16.13 | 1.91336 | + | 0.406697i | −0.00809976 | + | 0.0770640i | 1.66844 | + | 0.742838i | −0.669131 | − | 0.743145i | −0.0468394 | + | 0.144157i | 2.64565 | − | 0.0229757i | −0.274827 | − | 0.199673i | 2.92857 | + | 0.622487i | −0.978051 | − | 1.69403i |
16.14 | 2.41002 | + | 0.512266i | −0.00445547 | + | 0.0423909i | 3.71869 | + | 1.65567i | −0.669131 | − | 0.743145i | −0.0324532 | + | 0.0998806i | 0.537679 | + | 2.59054i | 4.12736 | + | 2.99871i | 2.93267 | + | 0.623357i | −1.23193 | − | 2.13377i |
16.15 | 2.47196 | + | 0.525431i | 0.228008 | − | 2.16935i | 4.00741 | + | 1.78421i | −0.669131 | − | 0.743145i | 1.70347 | − | 5.24275i | −1.75901 | − | 1.97633i | 4.87959 | + | 3.54523i | −1.71967 | − | 0.365527i | −1.26359 | − | 2.18860i |
16.16 | 2.65346 | + | 0.564011i | −0.320924 | + | 3.05338i | 4.89566 | + | 2.17969i | −0.669131 | − | 0.743145i | −2.57370 | + | 7.92104i | −0.256108 | − | 2.63333i | 7.37178 | + | 5.35591i | −6.28572 | − | 1.33607i | −1.35637 | − | 2.34930i |
81.1 | −0.267970 | + | 2.54956i | −0.289753 | − | 0.321804i | −4.47216 | − | 0.950588i | −0.913545 | − | 0.406737i | 0.898104 | − | 0.652510i | −1.48138 | − | 2.19215i | 2.03759 | − | 6.27107i | 0.293985 | − | 2.79708i | 1.28180 | − | 2.22015i |
81.2 | −0.248098 | + | 2.36050i | −0.446775 | − | 0.496193i | −3.55411 | − | 0.755449i | −0.913545 | − | 0.406737i | 1.28211 | − | 0.931506i | −0.591520 | + | 2.57878i | 1.19810 | − | 3.68738i | 0.266985 | − | 2.54019i | 1.18675 | − | 2.05551i |
81.3 | −0.211129 | + | 2.00876i | 1.98285 | + | 2.20217i | −2.03426 | − | 0.432395i | −0.913545 | − | 0.406737i | −4.84228 | + | 3.51812i | −2.64106 | + | 0.157528i | 0.0497481 | − | 0.153109i | −0.604305 | + | 5.74958i | 1.00991 | − | 1.74922i |
81.4 | −0.200627 | + | 1.90883i | −1.92283 | − | 2.13552i | −1.64710 | − | 0.350102i | −0.913545 | − | 0.406737i | 4.46213 | − | 3.24193i | 2.30574 | − | 1.29752i | −0.187483 | + | 0.577013i | −0.549585 | + | 5.22896i | 0.959674 | − | 1.66220i |
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
11.c | even | 5 | 1 | inner |
77.m | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 385.2.bg.a | ✓ | 128 |
7.c | even | 3 | 1 | inner | 385.2.bg.a | ✓ | 128 |
11.c | even | 5 | 1 | inner | 385.2.bg.a | ✓ | 128 |
77.m | even | 15 | 1 | inner | 385.2.bg.a | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
385.2.bg.a | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
385.2.bg.a | ✓ | 128 | 7.c | even | 3 | 1 | inner |
385.2.bg.a | ✓ | 128 | 11.c | even | 5 | 1 | inner |
385.2.bg.a | ✓ | 128 | 77.m | even | 15 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} - 26 T_{2}^{126} - 20 T_{2}^{125} + 278 T_{2}^{124} + 416 T_{2}^{123} - 710 T_{2}^{122} + \cdots + 923521 \) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\).