Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [186,2,Mod(11,186)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(186, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 23]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("186.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 186 = 2 \cdot 3 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 186.p (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: | \(1.48521747760\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(10\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −0.951057 | + | 0.309017i | −1.55075 | − | 0.771484i | 0.809017 | − | 0.587785i | −1.31329 | + | 0.758228i | 1.71325 | + | 0.254518i | 0.0465366 | + | 0.442766i | −0.587785 | + | 0.809017i | 1.80963 | + | 2.39275i | 1.01471 | − | 1.12695i |
11.2 | −0.951057 | + | 0.309017i | −0.878235 | + | 1.49288i | 0.809017 | − | 0.587785i | −2.08872 | + | 1.20592i | 0.373925 | − | 1.69121i | 0.0316894 | + | 0.301504i | −0.587785 | + | 0.809017i | −1.45741 | − | 2.62221i | 1.61384 | − | 1.79235i |
11.3 | −0.951057 | + | 0.309017i | −0.127863 | − | 1.72732i | 0.809017 | − | 0.587785i | 1.72504 | − | 0.995952i | 0.655377 | + | 1.60327i | −0.0669056 | − | 0.636564i | −0.587785 | + | 0.809017i | −2.96730 | + | 0.441720i | −1.33284 | + | 1.48027i |
11.4 | −0.951057 | + | 0.309017i | 1.70155 | + | 0.323596i | 0.809017 | − | 0.587785i | 1.52111 | − | 0.878214i | −1.71827 | + | 0.218051i | −0.525722 | − | 5.00191i | −0.587785 | + | 0.809017i | 2.79057 | + | 1.10123i | −1.17528 | + | 1.30528i |
11.5 | −0.951057 | + | 0.309017i | 1.72891 | − | 0.104282i | 0.809017 | − | 0.587785i | −1.24540 | + | 0.719033i | −1.61207 | + | 0.633440i | 0.425612 | + | 4.04943i | −0.587785 | + | 0.809017i | 2.97825 | − | 0.360587i | 0.962254 | − | 1.06869i |
11.6 | 0.951057 | − | 0.309017i | −1.57651 | + | 0.717375i | 0.809017 | − | 0.587785i | 2.08872 | − | 1.20592i | −1.27767 | + | 1.16943i | 0.0316894 | + | 0.301504i | 0.587785 | − | 0.809017i | 1.97075 | − | 2.26189i | 1.61384 | − | 1.79235i |
11.7 | 0.951057 | − | 0.309017i | −0.143963 | − | 1.72606i | 0.809017 | − | 0.587785i | −1.52111 | + | 0.878214i | −0.670298 | − | 1.59709i | −0.525722 | − | 5.00191i | 0.587785 | − | 0.809017i | −2.95855 | + | 0.496976i | −1.17528 | + | 1.30528i |
11.8 | 0.951057 | − | 0.309017i | 0.284431 | − | 1.70854i | 0.809017 | − | 0.587785i | 1.24540 | − | 0.719033i | −0.257457 | − | 1.71281i | 0.425612 | + | 4.04943i | 0.587785 | − | 0.809017i | −2.83820 | − | 0.971921i | 0.962254 | − | 1.06869i |
11.9 | 0.951057 | − | 0.309017i | 0.605161 | + | 1.62289i | 0.809017 | − | 0.587785i | 1.31329 | − | 0.758228i | 1.07704 | + | 1.35646i | 0.0465366 | + | 0.442766i | 0.587785 | − | 0.809017i | −2.26756 | + | 1.96422i | 1.01471 | − | 1.12695i |
11.10 | 0.951057 | − | 0.309017i | 1.70450 | + | 0.307717i | 0.809017 | − | 0.587785i | −1.72504 | + | 0.995952i | 1.71616 | − | 0.234063i | −0.0669056 | − | 0.636564i | 0.587785 | − | 0.809017i | 2.81062 | + | 1.04900i | −1.33284 | + | 1.48027i |
17.1 | −0.951057 | − | 0.309017i | −1.55075 | + | 0.771484i | 0.809017 | + | 0.587785i | −1.31329 | − | 0.758228i | 1.71325 | − | 0.254518i | 0.0465366 | − | 0.442766i | −0.587785 | − | 0.809017i | 1.80963 | − | 2.39275i | 1.01471 | + | 1.12695i |
17.2 | −0.951057 | − | 0.309017i | −0.878235 | − | 1.49288i | 0.809017 | + | 0.587785i | −2.08872 | − | 1.20592i | 0.373925 | + | 1.69121i | 0.0316894 | − | 0.301504i | −0.587785 | − | 0.809017i | −1.45741 | + | 2.62221i | 1.61384 | + | 1.79235i |
17.3 | −0.951057 | − | 0.309017i | −0.127863 | + | 1.72732i | 0.809017 | + | 0.587785i | 1.72504 | + | 0.995952i | 0.655377 | − | 1.60327i | −0.0669056 | + | 0.636564i | −0.587785 | − | 0.809017i | −2.96730 | − | 0.441720i | −1.33284 | − | 1.48027i |
17.4 | −0.951057 | − | 0.309017i | 1.70155 | − | 0.323596i | 0.809017 | + | 0.587785i | 1.52111 | + | 0.878214i | −1.71827 | − | 0.218051i | −0.525722 | + | 5.00191i | −0.587785 | − | 0.809017i | 2.79057 | − | 1.10123i | −1.17528 | − | 1.30528i |
17.5 | −0.951057 | − | 0.309017i | 1.72891 | + | 0.104282i | 0.809017 | + | 0.587785i | −1.24540 | − | 0.719033i | −1.61207 | − | 0.633440i | 0.425612 | − | 4.04943i | −0.587785 | − | 0.809017i | 2.97825 | + | 0.360587i | 0.962254 | + | 1.06869i |
17.6 | 0.951057 | + | 0.309017i | −1.57651 | − | 0.717375i | 0.809017 | + | 0.587785i | 2.08872 | + | 1.20592i | −1.27767 | − | 1.16943i | 0.0316894 | − | 0.301504i | 0.587785 | + | 0.809017i | 1.97075 | + | 2.26189i | 1.61384 | + | 1.79235i |
17.7 | 0.951057 | + | 0.309017i | −0.143963 | + | 1.72606i | 0.809017 | + | 0.587785i | −1.52111 | − | 0.878214i | −0.670298 | + | 1.59709i | −0.525722 | + | 5.00191i | 0.587785 | + | 0.809017i | −2.95855 | − | 0.496976i | −1.17528 | − | 1.30528i |
17.8 | 0.951057 | + | 0.309017i | 0.284431 | + | 1.70854i | 0.809017 | + | 0.587785i | 1.24540 | + | 0.719033i | −0.257457 | + | 1.71281i | 0.425612 | − | 4.04943i | 0.587785 | + | 0.809017i | −2.83820 | + | 0.971921i | 0.962254 | + | 1.06869i |
17.9 | 0.951057 | + | 0.309017i | 0.605161 | − | 1.62289i | 0.809017 | + | 0.587785i | 1.31329 | + | 0.758228i | 1.07704 | − | 1.35646i | 0.0465366 | − | 0.442766i | 0.587785 | + | 0.809017i | −2.26756 | − | 1.96422i | 1.01471 | + | 1.12695i |
17.10 | 0.951057 | + | 0.309017i | 1.70450 | − | 0.307717i | 0.809017 | + | 0.587785i | −1.72504 | − | 0.995952i | 1.71616 | + | 0.234063i | −0.0669056 | + | 0.636564i | 0.587785 | + | 0.809017i | 2.81062 | − | 1.04900i | −1.33284 | − | 1.48027i |
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
31.h | odd | 30 | 1 | inner |
93.p | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 186.2.p.a | ✓ | 80 |
3.b | odd | 2 | 1 | inner | 186.2.p.a | ✓ | 80 |
31.h | odd | 30 | 1 | inner | 186.2.p.a | ✓ | 80 |
93.p | even | 30 | 1 | inner | 186.2.p.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
186.2.p.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
186.2.p.a | ✓ | 80 | 3.b | odd | 2 | 1 | inner |
186.2.p.a | ✓ | 80 | 31.h | odd | 30 | 1 | inner |
186.2.p.a | ✓ | 80 | 93.p | even | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(186, [\chi])\).