Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [460,2,Mod(47,460)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([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("460.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 460.j (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: | \(3.67311849298\) |
Analytic rank: | \(0\) |
Dimension: | \(132\) |
Relative dimension: | \(66\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −1.41322 | + | 0.0530604i | 1.60974 | + | 1.60974i | 1.99437 | − | 0.149972i | −1.92782 | − | 1.13292i | −2.36032 | − | 2.18950i | 1.60901 | − | 1.60901i | −2.81052 | + | 0.317765i | 2.18251i | 2.78454 | + | 1.49877i | ||
47.2 | −1.41199 | + | 0.0791888i | 0.613496 | + | 0.613496i | 1.98746 | − | 0.223628i | 0.184594 | + | 2.22844i | −0.914836 | − | 0.817672i | 1.46274 | − | 1.46274i | −2.78857 | + | 0.473147i | − | 2.24724i | −0.437114 | − | 3.13192i | |
47.3 | −1.40975 | + | 0.112316i | −1.50989 | − | 1.50989i | 1.97477 | − | 0.316673i | −2.17152 | + | 0.533381i | 2.29815 | + | 1.95898i | 1.06422 | − | 1.06422i | −2.74836 | + | 0.668226i | 1.55955i | 3.00139 | − | 0.995827i | ||
47.4 | −1.39809 | − | 0.212930i | −0.766366 | − | 0.766366i | 1.90932 | + | 0.595390i | 2.12262 | − | 0.703182i | 0.908268 | + | 1.23463i | −1.45641 | + | 1.45641i | −2.54263 | − | 1.23896i | − | 1.82537i | −3.11735 | + | 0.531143i | |
47.5 | −1.38544 | + | 0.283802i | 2.27633 | + | 2.27633i | 1.83891 | − | 0.786383i | 2.17204 | + | 0.531258i | −3.79975 | − | 2.50770i | 1.09581 | − | 1.09581i | −2.32454 | + | 1.61138i | 7.36333i | −3.16002 | − | 0.119599i | ||
47.6 | −1.37758 | − | 0.319819i | 0.818006 | + | 0.818006i | 1.79543 | + | 0.881151i | −0.0316596 | + | 2.23584i | −0.865251 | − | 1.38848i | −1.96436 | + | 1.96436i | −2.19153 | − | 1.78807i | − | 1.66173i | 0.758679 | − | 3.06992i | |
47.7 | −1.35036 | + | 0.420152i | −2.31541 | − | 2.31541i | 1.64695 | − | 1.13471i | 0.145684 | − | 2.23132i | 4.09946 | + | 2.15381i | −1.71433 | + | 1.71433i | −1.74722 | + | 2.22424i | 7.72225i | 0.740766 | + | 3.07429i | ||
47.8 | −1.34829 | + | 0.426761i | 0.785861 | + | 0.785861i | 1.63575 | − | 1.15079i | −0.890562 | − | 2.05107i | −1.39494 | − | 0.724190i | −2.71252 | + | 2.71252i | −1.71435 | + | 2.24967i | − | 1.76485i | 2.07605 | + | 2.38538i | |
47.9 | −1.33173 | − | 0.475911i | −1.70820 | − | 1.70820i | 1.54702 | + | 1.26757i | 1.45963 | + | 1.69395i | 1.46191 | + | 3.08781i | 3.00205 | − | 3.00205i | −1.45696 | − | 2.42431i | 2.83587i | −1.13766 | − | 2.95055i | ||
47.10 | −1.30050 | − | 0.555604i | −1.11072 | − | 1.11072i | 1.38261 | + | 1.44513i | −1.45914 | − | 1.69437i | 0.827374 | + | 2.06162i | 1.00241 | − | 1.00241i | −0.995163 | − | 2.64757i | − | 0.532591i | 0.956214 | + | 3.01424i | |
47.11 | −1.29378 | + | 0.571084i | −0.0565025 | − | 0.0565025i | 1.34773 | − | 1.47771i | 2.13204 | − | 0.674090i | 0.105369 | + | 0.0408341i | 2.63515 | − | 2.63515i | −0.899763 | + | 2.68150i | − | 2.99361i | −2.37343 | + | 2.08970i | |
47.12 | −1.22458 | − | 0.707387i | 1.89438 | + | 1.89438i | 0.999207 | + | 1.73251i | 1.67434 | − | 1.48208i | −0.979766 | − | 3.65988i | −2.51212 | + | 2.51212i | 0.00194187 | − | 2.82843i | 4.17735i | −3.09878 | + | 0.630526i | ||
47.13 | −1.19633 | + | 0.754187i | −0.274826 | − | 0.274826i | 0.862404 | − | 1.80451i | −2.10134 | + | 0.764452i | 0.536053 | + | 0.121512i | −0.802532 | + | 0.802532i | 0.329221 | + | 2.80920i | − | 2.84894i | 1.93735 | − | 2.49934i | |
47.14 | −1.19284 | − | 0.759687i | 0.971542 | + | 0.971542i | 0.845751 | + | 1.81238i | 0.361417 | − | 2.20667i | −0.420830 | − | 1.89697i | 2.60162 | − | 2.60162i | 0.367989 | − | 2.80439i | − | 1.11221i | −2.10749 | + | 2.35764i | |
47.15 | −1.15450 | − | 0.816781i | −2.22730 | − | 2.22730i | 0.665738 | + | 1.88595i | −0.469590 | + | 2.18620i | 0.752200 | + | 4.39063i | −3.32421 | + | 3.32421i | 0.771810 | − | 2.72109i | 6.92171i | 2.32779 | − | 2.14042i | ||
47.16 | −1.12735 | + | 0.853866i | −1.12763 | − | 1.12763i | 0.541827 | − | 1.92521i | 1.64258 | + | 1.51721i | 2.23408 | + | 0.308387i | −1.95321 | + | 1.95321i | 1.03304 | + | 2.63303i | − | 0.456898i | −3.14725 | − | 0.307880i | |
47.17 | −0.956422 | + | 1.04176i | 1.18769 | + | 1.18769i | −0.170514 | − | 1.99272i | 1.54331 | − | 1.61809i | −2.37321 | + | 0.101351i | −0.438743 | + | 0.438743i | 2.23901 | + | 1.72824i | − | 0.178792i | 0.209602 | + | 3.15532i | |
47.18 | −0.865562 | + | 1.11839i | −2.09555 | − | 2.09555i | −0.501606 | − | 1.93608i | −0.928013 | + | 2.03440i | 4.15748 | − | 0.529822i | 2.24593 | − | 2.24593i | 2.59947 | + | 1.11480i | 5.78270i | −1.47201 | − | 2.79878i | ||
47.19 | −0.816781 | − | 1.15450i | 2.22730 | + | 2.22730i | −0.665738 | + | 1.88595i | −0.469590 | + | 2.18620i | 0.752200 | − | 4.39063i | 3.32421 | − | 3.32421i | 2.72109 | − | 0.771810i | 6.92171i | 2.90752 | − | 1.24351i | ||
47.20 | −0.759687 | − | 1.19284i | −0.971542 | − | 0.971542i | −0.845751 | + | 1.81238i | 0.361417 | − | 2.20667i | −0.420830 | + | 1.89697i | −2.60162 | + | 2.60162i | 2.80439 | − | 0.367989i | − | 1.11221i | −2.90677 | + | 1.24526i | |
See next 80 embeddings (of 132 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
20.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 460.2.j.a | ✓ | 132 |
4.b | odd | 2 | 1 | inner | 460.2.j.a | ✓ | 132 |
5.c | odd | 4 | 1 | inner | 460.2.j.a | ✓ | 132 |
20.e | even | 4 | 1 | inner | 460.2.j.a | ✓ | 132 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
460.2.j.a | ✓ | 132 | 1.a | even | 1 | 1 | trivial |
460.2.j.a | ✓ | 132 | 4.b | odd | 2 | 1 | inner |
460.2.j.a | ✓ | 132 | 5.c | odd | 4 | 1 | inner |
460.2.j.a | ✓ | 132 | 20.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(460, [\chi])\).