Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(251,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.251");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.q (of order \(6\), 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: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
251.1 | 0.500000 | − | 0.866025i | −1.68382 | − | 0.405895i | −0.500000 | − | 0.866025i | − | 0.465645i | −1.19343 | + | 1.25528i | 2.62938 | + | 0.293915i | −1.00000 | 2.67050 | + | 1.36691i | −0.403261 | − | 0.232823i | |||
251.2 | 0.500000 | − | 0.866025i | −1.63915 | + | 0.559644i | −0.500000 | − | 0.866025i | − | 2.94869i | −0.334907 | + | 1.69936i | −1.08009 | − | 2.41524i | −1.00000 | 2.37360 | − | 1.83468i | −2.55364 | − | 1.47434i | |||
251.3 | 0.500000 | − | 0.866025i | −1.48647 | + | 0.889052i | −0.500000 | − | 0.866025i | − | 0.655349i | 0.0267080 | + | 1.73184i | 0.402891 | + | 2.61490i | −1.00000 | 1.41917 | − | 2.64309i | −0.567549 | − | 0.327675i | |||
251.4 | 0.500000 | − | 0.866025i | −1.17531 | + | 1.27226i | −0.500000 | − | 0.866025i | 3.58412i | 0.514158 | + | 1.65398i | −2.54363 | + | 0.727985i | −1.00000 | −0.237306 | − | 2.99060i | 3.10394 | + | 1.79206i | ||||
251.5 | 0.500000 | − | 0.866025i | −0.702948 | − | 1.58299i | −0.500000 | − | 0.866025i | 3.28289i | −1.72239 | − | 0.182725i | −0.203837 | + | 2.63789i | −1.00000 | −2.01173 | + | 2.22552i | 2.84307 | + | 1.64144i | ||||
251.6 | 0.500000 | − | 0.866025i | −0.377999 | − | 1.69030i | −0.500000 | − | 0.866025i | 0.188901i | −1.65284 | − | 0.517794i | −1.93422 | + | 1.80522i | −1.00000 | −2.71423 | + | 1.27786i | 0.163593 | + | 0.0944507i | ||||
251.7 | 0.500000 | − | 0.866025i | 0.377999 | + | 1.69030i | −0.500000 | − | 0.866025i | − | 0.188901i | 1.65284 | + | 0.517794i | −2.53047 | + | 0.772472i | −1.00000 | −2.71423 | + | 1.27786i | −0.163593 | − | 0.0944507i | |||
251.8 | 0.500000 | − | 0.866025i | 0.702948 | + | 1.58299i | −0.500000 | − | 0.866025i | − | 3.28289i | 1.72239 | + | 0.182725i | −2.38640 | − | 1.14242i | −1.00000 | −2.01173 | + | 2.22552i | −2.84307 | − | 1.64144i | |||
251.9 | 0.500000 | − | 0.866025i | 1.17531 | − | 1.27226i | −0.500000 | − | 0.866025i | − | 3.58412i | −0.514158 | − | 1.65398i | −1.90227 | + | 1.83885i | −1.00000 | −0.237306 | − | 2.99060i | −3.10394 | − | 1.79206i | |||
251.10 | 0.500000 | − | 0.866025i | 1.48647 | − | 0.889052i | −0.500000 | − | 0.866025i | 0.655349i | −0.0267080 | − | 1.73184i | −2.06312 | − | 1.65636i | −1.00000 | 1.41917 | − | 2.64309i | 0.567549 | + | 0.327675i | ||||
251.11 | 0.500000 | − | 0.866025i | 1.63915 | − | 0.559644i | −0.500000 | − | 0.866025i | 2.94869i | 0.334907 | − | 1.69936i | 1.55161 | + | 2.14301i | −1.00000 | 2.37360 | − | 1.83468i | 2.55364 | + | 1.47434i | ||||
251.12 | 0.500000 | − | 0.866025i | 1.68382 | + | 0.405895i | −0.500000 | − | 0.866025i | 0.465645i | 1.19343 | − | 1.25528i | 1.06015 | − | 2.42406i | −1.00000 | 2.67050 | + | 1.36691i | 0.403261 | + | 0.232823i | ||||
335.1 | 0.500000 | + | 0.866025i | −1.68382 | + | 0.405895i | −0.500000 | + | 0.866025i | 0.465645i | −1.19343 | − | 1.25528i | 2.62938 | − | 0.293915i | −1.00000 | 2.67050 | − | 1.36691i | −0.403261 | + | 0.232823i | ||||
335.2 | 0.500000 | + | 0.866025i | −1.63915 | − | 0.559644i | −0.500000 | + | 0.866025i | 2.94869i | −0.334907 | − | 1.69936i | −1.08009 | + | 2.41524i | −1.00000 | 2.37360 | + | 1.83468i | −2.55364 | + | 1.47434i | ||||
335.3 | 0.500000 | + | 0.866025i | −1.48647 | − | 0.889052i | −0.500000 | + | 0.866025i | 0.655349i | 0.0267080 | − | 1.73184i | 0.402891 | − | 2.61490i | −1.00000 | 1.41917 | + | 2.64309i | −0.567549 | + | 0.327675i | ||||
335.4 | 0.500000 | + | 0.866025i | −1.17531 | − | 1.27226i | −0.500000 | + | 0.866025i | − | 3.58412i | 0.514158 | − | 1.65398i | −2.54363 | − | 0.727985i | −1.00000 | −0.237306 | + | 2.99060i | 3.10394 | − | 1.79206i | |||
335.5 | 0.500000 | + | 0.866025i | −0.702948 | + | 1.58299i | −0.500000 | + | 0.866025i | − | 3.28289i | −1.72239 | + | 0.182725i | −0.203837 | − | 2.63789i | −1.00000 | −2.01173 | − | 2.22552i | 2.84307 | − | 1.64144i | |||
335.6 | 0.500000 | + | 0.866025i | −0.377999 | + | 1.69030i | −0.500000 | + | 0.866025i | − | 0.188901i | −1.65284 | + | 0.517794i | −1.93422 | − | 1.80522i | −1.00000 | −2.71423 | − | 1.27786i | 0.163593 | − | 0.0944507i | |||
335.7 | 0.500000 | + | 0.866025i | 0.377999 | − | 1.69030i | −0.500000 | + | 0.866025i | 0.188901i | 1.65284 | − | 0.517794i | −2.53047 | − | 0.772472i | −1.00000 | −2.71423 | − | 1.27786i | −0.163593 | + | 0.0944507i | ||||
335.8 | 0.500000 | + | 0.866025i | 0.702948 | − | 1.58299i | −0.500000 | + | 0.866025i | 3.28289i | 1.72239 | − | 0.182725i | −2.38640 | + | 1.14242i | −1.00000 | −2.01173 | − | 2.22552i | −2.84307 | + | 1.64144i | ||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
39.h | odd | 6 | 1 | inner |
273.u | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.q.j | yes | 24 |
3.b | odd | 2 | 1 | 546.2.q.i | ✓ | 24 | |
7.b | odd | 2 | 1 | inner | 546.2.q.j | yes | 24 |
13.e | even | 6 | 1 | 546.2.q.i | ✓ | 24 | |
21.c | even | 2 | 1 | 546.2.q.i | ✓ | 24 | |
39.h | odd | 6 | 1 | inner | 546.2.q.j | yes | 24 |
91.t | odd | 6 | 1 | 546.2.q.i | ✓ | 24 | |
273.u | even | 6 | 1 | inner | 546.2.q.j | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.q.i | ✓ | 24 | 3.b | odd | 2 | 1 | |
546.2.q.i | ✓ | 24 | 13.e | even | 6 | 1 | |
546.2.q.i | ✓ | 24 | 21.c | even | 2 | 1 | |
546.2.q.i | ✓ | 24 | 91.t | odd | 6 | 1 | |
546.2.q.j | yes | 24 | 1.a | even | 1 | 1 | trivial |
546.2.q.j | yes | 24 | 7.b | odd | 2 | 1 | inner |
546.2.q.j | yes | 24 | 39.h | odd | 6 | 1 | inner |
546.2.q.j | yes | 24 | 273.u | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\):
\( T_{5}^{12} + 33T_{5}^{10} + 366T_{5}^{8} + 1442T_{5}^{6} + 861T_{5}^{4} + 141T_{5}^{2} + 4 \) |
\( T_{11}^{12} + 36 T_{11}^{10} - 96 T_{11}^{9} + 1080 T_{11}^{8} - 2160 T_{11}^{7} + 10464 T_{11}^{6} - 20736 T_{11}^{5} + 74304 T_{11}^{4} - 111744 T_{11}^{3} + 145152 T_{11}^{2} - 82944 T_{11} + 36864 \) |
\( T_{17}^{24} + 111 T_{17}^{22} + 8355 T_{17}^{20} + 322054 T_{17}^{18} + 8816949 T_{17}^{16} + 156438945 T_{17}^{14} + 2020732761 T_{17}^{12} + 16041014910 T_{17}^{10} + 87372493479 T_{17}^{8} + \cdots + 102627966736 \) |