Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [148,3,Mod(147,148)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(148, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("148.147");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 148 = 2^{2} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 148.b (of order \(2\), degree \(1\), 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: | \(4.03270791253\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
147.1 | −1.98085 | − | 0.276105i | − | 5.42142i | 3.84753 | + | 1.09385i | 6.41467i | −1.49688 | + | 10.7390i | 11.2188i | −7.31936 | − | 3.22907i | −20.3918 | 1.77112 | − | 12.7065i | |||||||
147.2 | −1.98085 | + | 0.276105i | 5.42142i | 3.84753 | − | 1.09385i | − | 6.41467i | −1.49688 | − | 10.7390i | − | 11.2188i | −7.31936 | + | 3.22907i | −20.3918 | 1.77112 | + | 12.7065i | ||||||
147.3 | −1.80028 | − | 0.871193i | 4.32601i | 2.48205 | + | 3.13679i | 4.81171i | 3.76879 | − | 7.78804i | 7.95088i | −1.73564 | − | 7.80945i | −9.71433 | 4.19193 | − | 8.66244i | ||||||||
147.4 | −1.80028 | + | 0.871193i | − | 4.32601i | 2.48205 | − | 3.13679i | − | 4.81171i | 3.76879 | + | 7.78804i | − | 7.95088i | −1.73564 | + | 7.80945i | −9.71433 | 4.19193 | + | 8.66244i | |||||
147.5 | −1.74750 | − | 0.972749i | − | 2.88771i | 2.10752 | + | 3.39976i | − | 8.99975i | −2.80901 | + | 5.04627i | 1.82117i | −0.375785 | − | 7.99117i | 0.661152 | −8.75450 | + | 15.7271i | ||||||
147.6 | −1.74750 | + | 0.972749i | 2.88771i | 2.10752 | − | 3.39976i | 8.99975i | −2.80901 | − | 5.04627i | − | 1.82117i | −0.375785 | + | 7.99117i | 0.661152 | −8.75450 | − | 15.7271i | |||||||
147.7 | −1.60738 | − | 1.19010i | − | 2.23022i | 1.16732 | + | 3.82588i | 5.95375i | −2.65419 | + | 3.58480i | − | 10.9654i | 2.67687 | − | 7.53886i | 4.02613 | 7.08557 | − | 9.56992i | ||||||
147.8 | −1.60738 | + | 1.19010i | 2.23022i | 1.16732 | − | 3.82588i | − | 5.95375i | −2.65419 | − | 3.58480i | 10.9654i | 2.67687 | + | 7.53886i | 4.02613 | 7.08557 | + | 9.56992i | |||||||
147.9 | −1.47472 | − | 1.35100i | 2.95699i | 0.349594 | + | 3.98469i | − | 1.64530i | 3.99490 | − | 4.36073i | − | 5.87725i | 4.86777 | − | 6.34861i | 0.256206 | −2.22279 | + | 2.42635i | ||||||
147.10 | −1.47472 | + | 1.35100i | − | 2.95699i | 0.349594 | − | 3.98469i | 1.64530i | 3.99490 | + | 4.36073i | 5.87725i | 4.86777 | + | 6.34861i | 0.256206 | −2.22279 | − | 2.42635i | |||||||
147.11 | −0.933862 | − | 1.76859i | − | 1.32495i | −2.25580 | + | 3.30323i | 1.38998i | −2.34329 | + | 1.23732i | 11.9333i | 7.94867 | + | 0.904826i | 7.24450 | 2.45830 | − | 1.29805i | |||||||
147.12 | −0.933862 | + | 1.76859i | 1.32495i | −2.25580 | − | 3.30323i | − | 1.38998i | −2.34329 | − | 1.23732i | − | 11.9333i | 7.94867 | − | 0.904826i | 7.24450 | 2.45830 | + | 1.29805i | ||||||
147.13 | −0.655926 | − | 1.88938i | 2.57244i | −3.13952 | + | 2.47859i | − | 5.32643i | 4.86032 | − | 1.68733i | − | 0.691466i | 6.74229 | + | 4.30599i | 2.38255 | −10.0637 | + | 3.49374i | ||||||
147.14 | −0.655926 | + | 1.88938i | − | 2.57244i | −3.13952 | − | 2.47859i | 5.32643i | 4.86032 | + | 1.68733i | 0.691466i | 6.74229 | − | 4.30599i | 2.38255 | −10.0637 | − | 3.49374i | |||||||
147.15 | −0.469745 | − | 1.94405i | − | 5.24065i | −3.55868 | + | 1.82642i | − | 0.496890i | −10.1881 | + | 2.46177i | − | 4.23582i | 5.22232 | + | 6.06031i | −18.4644 | −0.965980 | + | 0.233411i | |||||
147.16 | −0.469745 | + | 1.94405i | 5.24065i | −3.55868 | − | 1.82642i | 0.496890i | −10.1881 | − | 2.46177i | 4.23582i | 5.22232 | − | 6.06031i | −18.4644 | −0.965980 | − | 0.233411i | ||||||||
147.17 | 0.469745 | − | 1.94405i | 5.24065i | −3.55868 | − | 1.82642i | − | 0.496890i | 10.1881 | + | 2.46177i | 4.23582i | −5.22232 | + | 6.06031i | −18.4644 | −0.965980 | − | 0.233411i | |||||||
147.18 | 0.469745 | + | 1.94405i | − | 5.24065i | −3.55868 | + | 1.82642i | 0.496890i | 10.1881 | − | 2.46177i | − | 4.23582i | −5.22232 | − | 6.06031i | −18.4644 | −0.965980 | + | 0.233411i | ||||||
147.19 | 0.655926 | − | 1.88938i | − | 2.57244i | −3.13952 | − | 2.47859i | − | 5.32643i | −4.86032 | − | 1.68733i | 0.691466i | −6.74229 | + | 4.30599i | 2.38255 | −10.0637 | − | 3.49374i | ||||||
147.20 | 0.655926 | + | 1.88938i | 2.57244i | −3.13952 | + | 2.47859i | 5.32643i | −4.86032 | + | 1.68733i | − | 0.691466i | −6.74229 | − | 4.30599i | 2.38255 | −10.0637 | + | 3.49374i | |||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
37.b | even | 2 | 1 | inner |
148.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 148.3.b.d | ✓ | 32 |
4.b | odd | 2 | 1 | inner | 148.3.b.d | ✓ | 32 |
37.b | even | 2 | 1 | inner | 148.3.b.d | ✓ | 32 |
148.b | odd | 2 | 1 | inner | 148.3.b.d | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
148.3.b.d | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
148.3.b.d | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
148.3.b.d | ✓ | 32 | 37.b | even | 2 | 1 | inner |
148.3.b.d | ✓ | 32 | 148.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(148, [\chi])\):
\( T_{3}^{16} + 106 T_{3}^{14} + 4525 T_{3}^{12} + 100754 T_{3}^{10} + 1273980 T_{3}^{8} + 9354466 T_{3}^{6} + 38751853 T_{3}^{4} + 81462714 T_{3}^{2} + 63644625 \) |
\( T_{5}^{16} + 214 T_{5}^{14} + 17467 T_{5}^{12} + 698102 T_{5}^{10} + 14237025 T_{5}^{8} + 135883168 T_{5}^{6} + 450840064 T_{5}^{4} + 509091840 T_{5}^{2} + 100204544 \) |
\( T_{19}^{16} - 1912 T_{19}^{14} + 1490544 T_{19}^{12} - 608337792 T_{19}^{10} + 138711081216 T_{19}^{8} - 17413582249984 T_{19}^{6} + \cdots + 23\!\cdots\!60 \) |