Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,2,Mod(81,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.81");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 605.g (of order \(5\), degree \(4\), not 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: | \(4.83094932229\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
81.1 | −2.12272 | − | 1.54225i | 0.414045 | − | 1.27430i | 1.50939 | + | 4.64542i | −0.809017 | + | 0.587785i | −2.84419 | + | 2.06642i | −0.810808 | − | 2.49541i | 2.33876 | − | 7.19797i | 0.974647 | + | 0.708122i | 2.62383 | ||
81.2 | −1.11052 | − | 0.806841i | 1.00795 | − | 3.10216i | −0.0357685 | − | 0.110084i | −0.809017 | + | 0.587785i | −3.62230 | + | 2.63176i | −0.424181 | − | 1.30550i | −0.897462 | + | 2.76210i | −6.18037 | − | 4.49030i | 1.37268 | ||
81.3 | −0.389057 | − | 0.282666i | −0.494946 | + | 1.52329i | −0.546569 | − | 1.68217i | −0.809017 | + | 0.587785i | 0.623145 | − | 0.452741i | −0.148607 | − | 0.457364i | −0.560059 | + | 1.72368i | 0.351618 | + | 0.255466i | 0.480901 | ||
81.4 | 0.389057 | + | 0.282666i | −0.494946 | + | 1.52329i | −0.546569 | − | 1.68217i | −0.809017 | + | 0.587785i | −0.623145 | + | 0.452741i | 0.148607 | + | 0.457364i | 0.560059 | − | 1.72368i | 0.351618 | + | 0.255466i | −0.480901 | ||
81.5 | 1.11052 | + | 0.806841i | 1.00795 | − | 3.10216i | −0.0357685 | − | 0.110084i | −0.809017 | + | 0.587785i | 3.62230 | − | 2.63176i | 0.424181 | + | 1.30550i | 0.897462 | − | 2.76210i | −6.18037 | − | 4.49030i | −1.37268 | ||
81.6 | 2.12272 | + | 1.54225i | 0.414045 | − | 1.27430i | 1.50939 | + | 4.64542i | −0.809017 | + | 0.587785i | 2.84419 | − | 2.06642i | 0.810808 | + | 2.49541i | −2.33876 | + | 7.19797i | 0.974647 | + | 0.708122i | −2.62383 | ||
251.1 | −0.810808 | + | 2.49541i | −1.08398 | + | 0.787560i | −3.95163 | − | 2.87103i | 0.309017 | + | 0.951057i | −1.08638 | − | 3.34354i | −2.12272 | − | 1.54225i | 6.12296 | − | 4.44859i | −0.372282 | + | 1.14577i | −2.62383 | ||
251.2 | −0.424181 | + | 1.30550i | −2.63885 | + | 1.91724i | 0.0936432 | + | 0.0680358i | 0.309017 | + | 0.951057i | −1.38360 | − | 4.25827i | −1.11052 | − | 0.806841i | −2.34959 | + | 1.70707i | 2.36069 | − | 7.26546i | −1.37268 | ||
251.3 | −0.148607 | + | 0.457364i | 1.29579 | − | 0.941443i | 1.43094 | + | 1.03964i | 0.309017 | + | 0.951057i | 0.238020 | + | 0.732550i | −0.389057 | − | 0.282666i | −1.46625 | + | 1.06529i | −0.134306 | + | 0.413352i | −0.480901 | ||
251.4 | 0.148607 | − | 0.457364i | 1.29579 | − | 0.941443i | 1.43094 | + | 1.03964i | 0.309017 | + | 0.951057i | −0.238020 | − | 0.732550i | 0.389057 | + | 0.282666i | 1.46625 | − | 1.06529i | −0.134306 | + | 0.413352i | 0.480901 | ||
251.5 | 0.424181 | − | 1.30550i | −2.63885 | + | 1.91724i | 0.0936432 | + | 0.0680358i | 0.309017 | + | 0.951057i | 1.38360 | + | 4.25827i | 1.11052 | + | 0.806841i | 2.34959 | − | 1.70707i | 2.36069 | − | 7.26546i | 1.37268 | ||
251.6 | 0.810808 | − | 2.49541i | −1.08398 | + | 0.787560i | −3.95163 | − | 2.87103i | 0.309017 | + | 0.951057i | 1.08638 | + | 3.34354i | 2.12272 | + | 1.54225i | −6.12296 | + | 4.44859i | −0.372282 | + | 1.14577i | 2.62383 | ||
366.1 | −2.12272 | + | 1.54225i | 0.414045 | + | 1.27430i | 1.50939 | − | 4.64542i | −0.809017 | − | 0.587785i | −2.84419 | − | 2.06642i | −0.810808 | + | 2.49541i | 2.33876 | + | 7.19797i | 0.974647 | − | 0.708122i | 2.62383 | ||
366.2 | −1.11052 | + | 0.806841i | 1.00795 | + | 3.10216i | −0.0357685 | + | 0.110084i | −0.809017 | − | 0.587785i | −3.62230 | − | 2.63176i | −0.424181 | + | 1.30550i | −0.897462 | − | 2.76210i | −6.18037 | + | 4.49030i | 1.37268 | ||
366.3 | −0.389057 | + | 0.282666i | −0.494946 | − | 1.52329i | −0.546569 | + | 1.68217i | −0.809017 | − | 0.587785i | 0.623145 | + | 0.452741i | −0.148607 | + | 0.457364i | −0.560059 | − | 1.72368i | 0.351618 | − | 0.255466i | 0.480901 | ||
366.4 | 0.389057 | − | 0.282666i | −0.494946 | − | 1.52329i | −0.546569 | + | 1.68217i | −0.809017 | − | 0.587785i | −0.623145 | − | 0.452741i | 0.148607 | − | 0.457364i | 0.560059 | + | 1.72368i | 0.351618 | − | 0.255466i | −0.480901 | ||
366.5 | 1.11052 | − | 0.806841i | 1.00795 | + | 3.10216i | −0.0357685 | + | 0.110084i | −0.809017 | − | 0.587785i | 3.62230 | + | 2.63176i | 0.424181 | − | 1.30550i | 0.897462 | + | 2.76210i | −6.18037 | + | 4.49030i | −1.37268 | ||
366.6 | 2.12272 | − | 1.54225i | 0.414045 | + | 1.27430i | 1.50939 | − | 4.64542i | −0.809017 | − | 0.587785i | 2.84419 | + | 2.06642i | 0.810808 | − | 2.49541i | −2.33876 | − | 7.19797i | 0.974647 | − | 0.708122i | −2.62383 | ||
511.1 | −0.810808 | − | 2.49541i | −1.08398 | − | 0.787560i | −3.95163 | + | 2.87103i | 0.309017 | − | 0.951057i | −1.08638 | + | 3.34354i | −2.12272 | + | 1.54225i | 6.12296 | + | 4.44859i | −0.372282 | − | 1.14577i | −2.62383 | ||
511.2 | −0.424181 | − | 1.30550i | −2.63885 | − | 1.91724i | 0.0936432 | − | 0.0680358i | 0.309017 | − | 0.951057i | −1.38360 | + | 4.25827i | −1.11052 | + | 0.806841i | −2.34959 | − | 1.70707i | 2.36069 | + | 7.26546i | −1.37268 | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
11.c | even | 5 | 3 | inner |
11.d | odd | 10 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 605.2.g.q | 24 | |
11.b | odd | 2 | 1 | inner | 605.2.g.q | 24 | |
11.c | even | 5 | 1 | 605.2.a.m | ✓ | 6 | |
11.c | even | 5 | 3 | inner | 605.2.g.q | 24 | |
11.d | odd | 10 | 1 | 605.2.a.m | ✓ | 6 | |
11.d | odd | 10 | 3 | inner | 605.2.g.q | 24 | |
33.f | even | 10 | 1 | 5445.2.a.bx | 6 | ||
33.h | odd | 10 | 1 | 5445.2.a.bx | 6 | ||
44.g | even | 10 | 1 | 9680.2.a.cw | 6 | ||
44.h | odd | 10 | 1 | 9680.2.a.cw | 6 | ||
55.h | odd | 10 | 1 | 3025.2.a.bg | 6 | ||
55.j | even | 10 | 1 | 3025.2.a.bg | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
605.2.a.m | ✓ | 6 | 11.c | even | 5 | 1 | |
605.2.a.m | ✓ | 6 | 11.d | odd | 10 | 1 | |
605.2.g.q | 24 | 1.a | even | 1 | 1 | trivial | |
605.2.g.q | 24 | 11.b | odd | 2 | 1 | inner | |
605.2.g.q | 24 | 11.c | even | 5 | 3 | inner | |
605.2.g.q | 24 | 11.d | odd | 10 | 3 | inner | |
3025.2.a.bg | 6 | 55.h | odd | 10 | 1 | ||
3025.2.a.bg | 6 | 55.j | even | 10 | 1 | ||
5445.2.a.bx | 6 | 33.f | even | 10 | 1 | ||
5445.2.a.bx | 6 | 33.h | odd | 10 | 1 | ||
9680.2.a.cw | 6 | 44.g | even | 10 | 1 | ||
9680.2.a.cw | 6 | 44.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):
\( T_{2}^{24} + 9 T_{2}^{22} + 66 T_{2}^{20} + 462 T_{2}^{18} + 3195 T_{2}^{16} + 6534 T_{2}^{14} + \cdots + 81 \) |
\( T_{3}^{12} + 3 T_{3}^{11} + 12 T_{3}^{10} + 38 T_{3}^{9} + 129 T_{3}^{8} + 54 T_{3}^{7} + 283 T_{3}^{6} + \cdots + 2401 \) |