Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,2,Mod(19,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 25, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 462.ba (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: | \(3.68908857338\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(8\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −0.994522 | + | 0.104528i | −0.743145 | − | 0.669131i | 0.978148 | − | 0.207912i | −1.44711 | − | 3.25025i | 0.809017 | + | 0.587785i | 1.04897 | + | 2.42892i | −0.951057 | + | 0.309017i | 0.104528 | + | 0.994522i | 1.77892 | + | 3.08119i |
19.2 | −0.994522 | + | 0.104528i | −0.743145 | − | 0.669131i | 0.978148 | − | 0.207912i | −1.26939 | − | 2.85109i | 0.809017 | + | 0.587785i | −0.613482 | − | 2.57364i | −0.951057 | + | 0.309017i | 0.104528 | + | 0.994522i | 1.56045 | + | 2.70278i |
19.3 | −0.994522 | + | 0.104528i | −0.743145 | − | 0.669131i | 0.978148 | − | 0.207912i | 0.0572849 | + | 0.128664i | 0.809017 | + | 0.587785i | −2.64421 | + | 0.0903816i | −0.951057 | + | 0.309017i | 0.104528 | + | 0.994522i | −0.0704201 | − | 0.121971i |
19.4 | −0.994522 | + | 0.104528i | −0.743145 | − | 0.669131i | 0.978148 | − | 0.207912i | 0.949965 | + | 2.13366i | 0.809017 | + | 0.587785i | 2.43658 | + | 1.03106i | −0.951057 | + | 0.309017i | 0.104528 | + | 0.994522i | −1.16779 | − | 2.02267i |
19.5 | 0.994522 | − | 0.104528i | 0.743145 | + | 0.669131i | 0.978148 | − | 0.207912i | −1.01671 | − | 2.28357i | 0.809017 | + | 0.587785i | −0.151373 | − | 2.64142i | 0.951057 | − | 0.309017i | 0.104528 | + | 0.994522i | −1.24984 | − | 2.16479i |
19.6 | 0.994522 | − | 0.104528i | 0.743145 | + | 0.669131i | 0.978148 | − | 0.207912i | −0.748568 | − | 1.68131i | 0.809017 | + | 0.587785i | 2.42705 | + | 1.05329i | 0.951057 | − | 0.309017i | 0.104528 | + | 0.994522i | −0.920212 | − | 1.59385i |
19.7 | 0.994522 | − | 0.104528i | 0.743145 | + | 0.669131i | 0.978148 | − | 0.207912i | 0.316797 | + | 0.711538i | 0.809017 | + | 0.587785i | −0.370234 | + | 2.61972i | 0.951057 | − | 0.309017i | 0.104528 | + | 0.994522i | 0.389438 | + | 0.674526i |
19.8 | 0.994522 | − | 0.104528i | 0.743145 | + | 0.669131i | 0.978148 | − | 0.207912i | 1.33064 | + | 2.98866i | 0.809017 | + | 0.587785i | 1.90285 | − | 1.83825i | 0.951057 | − | 0.309017i | 0.104528 | + | 0.994522i | 1.63575 | + | 2.83319i |
61.1 | −0.207912 | + | 0.978148i | 0.994522 | + | 0.104528i | −0.913545 | − | 0.406737i | −3.24579 | + | 2.92252i | −0.309017 | + | 0.951057i | −0.190680 | − | 2.63887i | 0.587785 | − | 0.809017i | 0.978148 | + | 0.207912i | −2.18382 | − | 3.78249i |
61.2 | −0.207912 | + | 0.978148i | 0.994522 | + | 0.104528i | −0.913545 | − | 0.406737i | −0.468951 | + | 0.422245i | −0.309017 | + | 0.951057i | −0.177812 | + | 2.63977i | 0.587785 | − | 0.809017i | 0.978148 | + | 0.207912i | −0.315518 | − | 0.546493i |
61.3 | −0.207912 | + | 0.978148i | 0.994522 | + | 0.104528i | −0.913545 | − | 0.406737i | 0.131864 | − | 0.118731i | −0.309017 | + | 0.951057i | −1.17519 | − | 2.37043i | 0.587785 | − | 0.809017i | 0.978148 | + | 0.207912i | 0.0887206 | + | 0.153669i |
61.4 | −0.207912 | + | 0.978148i | 0.994522 | + | 0.104528i | −0.913545 | − | 0.406737i | 1.55595 | − | 1.40099i | −0.309017 | + | 0.951057i | 2.62525 | − | 0.328695i | 0.587785 | − | 0.809017i | 0.978148 | + | 0.207912i | 1.04687 | + | 1.81324i |
61.5 | 0.207912 | − | 0.978148i | −0.994522 | − | 0.104528i | −0.913545 | − | 0.406737i | −1.40387 | + | 1.26405i | −0.309017 | + | 0.951057i | 1.39244 | − | 2.24969i | −0.587785 | + | 0.809017i | 0.978148 | + | 0.207912i | 0.944546 | + | 1.63600i |
61.6 | 0.207912 | − | 0.978148i | −0.994522 | − | 0.104528i | −0.913545 | − | 0.406737i | −0.716223 | + | 0.644890i | −0.309017 | + | 0.951057i | 2.17783 | + | 1.50235i | −0.587785 | + | 0.809017i | 0.978148 | + | 0.207912i | 0.481886 | + | 0.834652i |
61.7 | 0.207912 | − | 0.978148i | −0.994522 | − | 0.104528i | −0.913545 | − | 0.406737i | 0.192170 | − | 0.173031i | −0.309017 | + | 0.951057i | −1.35536 | + | 2.27222i | −0.587785 | + | 0.809017i | 0.978148 | + | 0.207912i | −0.129295 | − | 0.223946i |
61.8 | 0.207912 | − | 0.978148i | −0.994522 | − | 0.104528i | −0.913545 | − | 0.406737i | 2.61658 | − | 2.35598i | −0.309017 | + | 0.951057i | 1.99807 | − | 1.73428i | −0.587785 | + | 0.809017i | 0.978148 | + | 0.207912i | −1.76048 | − | 3.04924i |
73.1 | −0.994522 | − | 0.104528i | −0.743145 | + | 0.669131i | 0.978148 | + | 0.207912i | −1.44711 | + | 3.25025i | 0.809017 | − | 0.587785i | 1.04897 | − | 2.42892i | −0.951057 | − | 0.309017i | 0.104528 | − | 0.994522i | 1.77892 | − | 3.08119i |
73.2 | −0.994522 | − | 0.104528i | −0.743145 | + | 0.669131i | 0.978148 | + | 0.207912i | −1.26939 | + | 2.85109i | 0.809017 | − | 0.587785i | −0.613482 | + | 2.57364i | −0.951057 | − | 0.309017i | 0.104528 | − | 0.994522i | 1.56045 | − | 2.70278i |
73.3 | −0.994522 | − | 0.104528i | −0.743145 | + | 0.669131i | 0.978148 | + | 0.207912i | 0.0572849 | − | 0.128664i | 0.809017 | − | 0.587785i | −2.64421 | − | 0.0903816i | −0.951057 | − | 0.309017i | 0.104528 | − | 0.994522i | −0.0704201 | + | 0.121971i |
73.4 | −0.994522 | − | 0.104528i | −0.743145 | + | 0.669131i | 0.978148 | + | 0.207912i | 0.949965 | − | 2.13366i | 0.809017 | − | 0.587785i | 2.43658 | − | 1.03106i | −0.951057 | − | 0.309017i | 0.104528 | − | 0.994522i | −1.16779 | + | 2.02267i |
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
77.n | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 462.2.ba.b | yes | 64 |
7.d | odd | 6 | 1 | 462.2.ba.a | ✓ | 64 | |
11.d | odd | 10 | 1 | 462.2.ba.a | ✓ | 64 | |
77.n | even | 30 | 1 | inner | 462.2.ba.b | yes | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
462.2.ba.a | ✓ | 64 | 7.d | odd | 6 | 1 | |
462.2.ba.a | ✓ | 64 | 11.d | odd | 10 | 1 | |
462.2.ba.b | yes | 64 | 1.a | even | 1 | 1 | trivial |
462.2.ba.b | yes | 64 | 77.n | even | 30 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} + 2 T_{5}^{63} + 22 T_{5}^{62} + 8 T_{5}^{61} + 52 T_{5}^{60} - 1232 T_{5}^{59} + \cdots + 1291129965841 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).