Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [760,2,Mod(267,760)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(760, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("760.267");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 760 = 2^{3} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 760.w (of order \(4\), degree \(2\), 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: | \(6.06863055362\) |
Analytic rank: | \(0\) |
Dimension: | \(108\) |
Relative dimension: | \(54\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
267.1 | −1.40857 | + | 0.126238i | −1.36883 | + | 1.36883i | 1.96813 | − | 0.355630i | 0.375211 | − | 2.20436i | 1.75529 | − | 2.10089i | 2.05093 | − | 2.05093i | −2.72735 | + | 0.749382i | − | 0.747394i | −0.250235 | + | 3.15236i | |
267.2 | −1.39849 | + | 0.210323i | 1.23627 | − | 1.23627i | 1.91153 | − | 0.588268i | −1.98563 | − | 1.02824i | −1.46890 | + | 1.98893i | −2.63738 | + | 2.63738i | −2.54952 | + | 1.22472i | − | 0.0567502i | 2.99314 | + | 1.02036i | |
267.3 | −1.39309 | − | 0.243545i | −2.03277 | + | 2.03277i | 1.88137 | + | 0.678557i | −2.21833 | − | 0.281128i | 3.32689 | − | 2.33675i | −2.44756 | + | 2.44756i | −2.45565 | − | 1.40349i | − | 5.26427i | 3.02185 | + | 0.931897i | |
267.4 | −1.39205 | + | 0.249374i | −0.527859 | + | 0.527859i | 1.87562 | − | 0.694285i | −1.43303 | + | 1.71651i | 0.603173 | − | 0.866442i | 1.24713 | − | 1.24713i | −2.43783 | + | 1.43421i | 2.44273i | 1.56680 | − | 2.74684i | ||
267.5 | −1.38724 | − | 0.274874i | −1.68946 | + | 1.68946i | 1.84889 | + | 0.762634i | 1.63822 | + | 1.52192i | 2.80809 | − | 1.87931i | −1.48938 | + | 1.48938i | −2.35523 | − | 1.56617i | − | 2.70857i | −1.85427 | − | 2.56158i | |
267.6 | −1.38164 | + | 0.301766i | 2.05043 | − | 2.05043i | 1.81787 | − | 0.833866i | 1.90605 | − | 1.16918i | −2.21421 | + | 3.45171i | 1.97140 | − | 1.97140i | −2.26002 | + | 1.70068i | − | 5.40854i | −2.28066 | + | 2.19057i | |
267.7 | −1.37681 | + | 0.323095i | −0.336310 | + | 0.336310i | 1.79122 | − | 0.889683i | 1.81791 | + | 1.30200i | 0.354375 | − | 0.571695i | −0.881910 | + | 0.881910i | −2.17872 | + | 1.80366i | 2.77379i | −2.92359 | − | 1.20525i | ||
267.8 | −1.31439 | − | 0.521896i | 0.746757 | − | 0.746757i | 1.45525 | + | 1.37195i | −2.20913 | − | 0.346067i | −1.37126 | + | 0.591801i | 2.83517 | − | 2.83517i | −1.19675 | − | 2.56277i | 1.88471i | 2.72305 | + | 1.60780i | ||
267.9 | −1.25201 | − | 0.657626i | 2.08229 | − | 2.08229i | 1.13506 | + | 1.64671i | 0.127264 | − | 2.23244i | −3.97642 | + | 1.23768i | 0.638542 | − | 0.638542i | −0.338182 | − | 2.80814i | − | 5.67187i | −1.62745 | + | 2.71135i | |
267.10 | −1.13965 | − | 0.837382i | 0.654350 | − | 0.654350i | 0.597584 | + | 1.90864i | 0.582206 | + | 2.15894i | −1.29367 | + | 0.197786i | 0.531937 | − | 0.531937i | 0.917224 | − | 2.67557i | 2.14365i | 1.14435 | − | 2.94796i | ||
267.11 | −1.09829 | + | 0.890935i | 1.15450 | − | 1.15450i | 0.412468 | − | 1.95701i | −1.79430 | + | 1.33434i | −0.239388 | + | 2.29656i | −0.577109 | + | 0.577109i | 1.29056 | + | 2.51684i | 0.334254i | 0.781847 | − | 3.06410i | ||
267.12 | −1.07153 | + | 0.922945i | 0.0817665 | − | 0.0817665i | 0.296346 | − | 1.97792i | 1.80920 | − | 1.31408i | −0.0121492 | + | 0.163081i | 1.71397 | − | 1.71397i | 1.50797 | + | 2.39291i | 2.98663i | −0.725784 | + | 3.07786i | ||
267.13 | −1.03664 | + | 0.961964i | −1.93572 | + | 1.93572i | 0.149250 | − | 1.99442i | −2.23450 | + | 0.0837651i | 0.144554 | − | 3.86873i | 2.96042 | − | 2.96042i | 1.76384 | + | 2.21108i | − | 4.49399i | 2.23579 | − | 2.23634i | |
267.14 | −1.02073 | − | 0.978826i | −0.723718 | + | 0.723718i | 0.0837985 | + | 1.99824i | −2.17752 | + | 0.508321i | 1.44712 | − | 0.0303299i | 1.00583 | − | 1.00583i | 1.87040 | − | 2.12170i | 1.95246i | 2.72023 | + | 1.61256i | ||
267.15 | −0.978826 | − | 1.02073i | −0.723718 | + | 0.723718i | −0.0837985 | + | 1.99824i | 2.17752 | − | 0.508321i | 1.44712 | + | 0.0303299i | −1.00583 | + | 1.00583i | 2.12170 | − | 1.87040i | 1.95246i | −2.65028 | − | 1.72512i | ||
267.16 | −0.848677 | + | 1.13126i | −1.23252 | + | 1.23252i | −0.559496 | − | 1.92015i | −0.104679 | + | 2.23362i | −0.348290 | − | 2.44032i | −3.04449 | + | 3.04449i | 2.64702 | + | 0.996648i | − | 0.0382290i | −2.43796 | − | 2.01404i | |
267.17 | −0.837382 | − | 1.13965i | 0.654350 | − | 0.654350i | −0.597584 | + | 1.90864i | −0.582206 | − | 2.15894i | −1.29367 | − | 0.197786i | −0.531937 | + | 0.531937i | 2.67557 | − | 0.917224i | 2.14365i | −1.97290 | + | 2.47137i | ||
267.18 | −0.679797 | + | 1.24011i | 2.16703 | − | 2.16703i | −1.07575 | − | 1.68605i | 0.704730 | + | 2.12211i | 1.21422 | + | 4.16050i | 3.03658 | − | 3.03658i | 2.82218 | − | 0.187884i | − | 6.39203i | −3.11073 | − | 0.568660i | |
267.19 | −0.657626 | − | 1.25201i | 2.08229 | − | 2.08229i | −1.13506 | + | 1.64671i | −0.127264 | + | 2.23244i | −3.97642 | − | 1.23768i | −0.638542 | + | 0.638542i | 2.80814 | + | 0.338182i | − | 5.67187i | 2.87873 | − | 1.30878i | |
267.20 | −0.634707 | + | 1.26378i | −2.43641 | + | 2.43641i | −1.19429 | − | 1.60426i | −0.120344 | − | 2.23283i | −1.53269 | − | 4.62549i | −2.17928 | + | 2.17928i | 2.78547 | − | 0.491093i | − | 8.87215i | 2.89819 | + | 1.26510i | |
See next 80 embeddings (of 108 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
8.d | odd | 2 | 1 | inner |
40.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 760.2.w.d | ✓ | 108 |
5.c | odd | 4 | 1 | inner | 760.2.w.d | ✓ | 108 |
8.d | odd | 2 | 1 | inner | 760.2.w.d | ✓ | 108 |
40.k | even | 4 | 1 | inner | 760.2.w.d | ✓ | 108 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
760.2.w.d | ✓ | 108 | 1.a | even | 1 | 1 | trivial |
760.2.w.d | ✓ | 108 | 5.c | odd | 4 | 1 | inner |
760.2.w.d | ✓ | 108 | 8.d | odd | 2 | 1 | inner |
760.2.w.d | ✓ | 108 | 40.k | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\):
\( T_{3}^{54} + 376 T_{3}^{50} - 20 T_{3}^{49} + 320 T_{3}^{47} + 56862 T_{3}^{46} - 3220 T_{3}^{45} + \cdots + 131072 \) |
\( T_{7}^{108} + 2860 T_{7}^{104} + 3729602 T_{7}^{100} + 2938272260 T_{7}^{96} + 1562341888351 T_{7}^{92} + \cdots + 10\!\cdots\!24 \) |