Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,7,Mod(3,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([13]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.3");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 38.f (of order \(18\), degree \(6\), 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: | \(8.74205517755\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −3.63616 | + | 4.33340i | −14.5675 | − | 40.0239i | −5.55674 | − | 31.5138i | 14.4780 | − | 82.1091i | 226.409 | + | 82.4062i | −61.1665 | − | 105.943i | 156.767 | + | 90.5097i | −831.251 | + | 697.502i | 303.167 | + | 361.301i |
3.2 | −3.63616 | + | 4.33340i | −7.01359 | − | 19.2697i | −5.55674 | − | 31.5138i | −32.9273 | + | 186.740i | 109.006 | + | 39.6749i | −111.397 | − | 192.946i | 156.767 | + | 90.5097i | 236.316 | − | 198.293i | −689.491 | − | 821.703i |
3.3 | −3.63616 | + | 4.33340i | 0.501194 | + | 1.37702i | −5.55674 | − | 31.5138i | 6.94044 | − | 39.3612i | −7.78959 | − | 2.83518i | 159.899 | + | 276.953i | 156.767 | + | 90.5097i | 556.801 | − | 467.212i | 145.331 | + | 173.199i |
3.4 | −3.63616 | + | 4.33340i | 10.0117 | + | 27.5069i | −5.55674 | − | 31.5138i | 26.4694 | − | 150.115i | −155.603 | − | 56.6347i | −232.836 | − | 403.285i | 156.767 | + | 90.5097i | −97.9501 | + | 82.1899i | 554.263 | + | 660.545i |
3.5 | −3.63616 | + | 4.33340i | 12.5750 | + | 34.5495i | −5.55674 | − | 31.5138i | −24.6848 | + | 139.995i | −195.442 | − | 71.1350i | 136.699 | + | 236.770i | 156.767 | + | 90.5097i | −477.093 | + | 400.329i | −516.895 | − | 616.012i |
3.6 | 3.63616 | − | 4.33340i | −14.7348 | − | 40.4836i | −5.55674 | − | 31.5138i | 40.1148 | − | 227.502i | −229.010 | − | 83.3528i | 276.187 | + | 478.370i | −156.767 | − | 90.5097i | −863.360 | + | 724.445i | −839.995 | − | 1001.07i |
3.7 | 3.63616 | − | 4.33340i | −10.8314 | − | 29.7590i | −5.55674 | − | 31.5138i | −11.6174 | + | 65.8855i | −168.342 | − | 61.2716i | −205.116 | − | 355.271i | −156.767 | − | 90.5097i | −209.833 | + | 176.071i | 243.266 | + | 289.913i |
3.8 | 3.63616 | − | 4.33340i | −1.10346 | − | 3.03174i | −5.55674 | − | 31.5138i | −30.9885 | + | 175.744i | −17.1501 | − | 6.24213i | 294.726 | + | 510.481i | −156.767 | − | 90.5097i | 550.473 | − | 461.901i | 648.892 | + | 773.320i |
3.9 | 3.63616 | − | 4.33340i | 4.25575 | + | 11.6926i | −5.55674 | − | 31.5138i | 13.0241 | − | 73.8634i | 66.1433 | + | 24.0742i | −97.4265 | − | 168.748i | −156.767 | − | 90.5097i | 439.841 | − | 369.071i | −272.722 | − | 325.017i |
3.10 | 3.63616 | − | 4.33340i | 16.5102 | + | 45.3614i | −5.55674 | − | 31.5138i | −20.2573 | + | 114.885i | 256.603 | + | 93.3957i | −72.0498 | − | 124.794i | −156.767 | − | 90.5097i | −1226.62 | + | 1029.26i | 424.183 | + | 505.522i |
13.1 | −3.63616 | − | 4.33340i | −14.5675 | + | 40.0239i | −5.55674 | + | 31.5138i | 14.4780 | + | 82.1091i | 226.409 | − | 82.4062i | −61.1665 | + | 105.943i | 156.767 | − | 90.5097i | −831.251 | − | 697.502i | 303.167 | − | 361.301i |
13.2 | −3.63616 | − | 4.33340i | −7.01359 | + | 19.2697i | −5.55674 | + | 31.5138i | −32.9273 | − | 186.740i | 109.006 | − | 39.6749i | −111.397 | + | 192.946i | 156.767 | − | 90.5097i | 236.316 | + | 198.293i | −689.491 | + | 821.703i |
13.3 | −3.63616 | − | 4.33340i | 0.501194 | − | 1.37702i | −5.55674 | + | 31.5138i | 6.94044 | + | 39.3612i | −7.78959 | + | 2.83518i | 159.899 | − | 276.953i | 156.767 | − | 90.5097i | 556.801 | + | 467.212i | 145.331 | − | 173.199i |
13.4 | −3.63616 | − | 4.33340i | 10.0117 | − | 27.5069i | −5.55674 | + | 31.5138i | 26.4694 | + | 150.115i | −155.603 | + | 56.6347i | −232.836 | + | 403.285i | 156.767 | − | 90.5097i | −97.9501 | − | 82.1899i | 554.263 | − | 660.545i |
13.5 | −3.63616 | − | 4.33340i | 12.5750 | − | 34.5495i | −5.55674 | + | 31.5138i | −24.6848 | − | 139.995i | −195.442 | + | 71.1350i | 136.699 | − | 236.770i | 156.767 | − | 90.5097i | −477.093 | − | 400.329i | −516.895 | + | 616.012i |
13.6 | 3.63616 | + | 4.33340i | −14.7348 | + | 40.4836i | −5.55674 | + | 31.5138i | 40.1148 | + | 227.502i | −229.010 | + | 83.3528i | 276.187 | − | 478.370i | −156.767 | + | 90.5097i | −863.360 | − | 724.445i | −839.995 | + | 1001.07i |
13.7 | 3.63616 | + | 4.33340i | −10.8314 | + | 29.7590i | −5.55674 | + | 31.5138i | −11.6174 | − | 65.8855i | −168.342 | + | 61.2716i | −205.116 | + | 355.271i | −156.767 | + | 90.5097i | −209.833 | − | 176.071i | 243.266 | − | 289.913i |
13.8 | 3.63616 | + | 4.33340i | −1.10346 | + | 3.03174i | −5.55674 | + | 31.5138i | −30.9885 | − | 175.744i | −17.1501 | + | 6.24213i | 294.726 | − | 510.481i | −156.767 | + | 90.5097i | 550.473 | + | 461.901i | 648.892 | − | 773.320i |
13.9 | 3.63616 | + | 4.33340i | 4.25575 | − | 11.6926i | −5.55674 | + | 31.5138i | 13.0241 | + | 73.8634i | 66.1433 | − | 24.0742i | −97.4265 | + | 168.748i | −156.767 | + | 90.5097i | 439.841 | + | 369.071i | −272.722 | + | 325.017i |
13.10 | 3.63616 | + | 4.33340i | 16.5102 | − | 45.3614i | −5.55674 | + | 31.5138i | −20.2573 | − | 114.885i | 256.603 | − | 93.3957i | −72.0498 | + | 124.794i | −156.767 | + | 90.5097i | −1226.62 | − | 1029.26i | 424.183 | − | 505.522i |
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.f | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 38.7.f.a | ✓ | 60 |
19.f | odd | 18 | 1 | inner | 38.7.f.a | ✓ | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.7.f.a | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
38.7.f.a | ✓ | 60 | 19.f | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(38, [\chi])\).