Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [345,3,Mod(229,345)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(345, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("345.229");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 345 = 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 345.d (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: | \(9.40056912043\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
229.1 | − | 3.87170i | 1.73205i | −10.9901 | −3.20378 | + | 3.83872i | 6.70598 | 6.81780 | 27.0634i | −3.00000 | 14.8624 | + | 12.4041i | |||||||||||||
229.2 | − | 3.87170i | 1.73205i | −10.9901 | 3.20378 | − | 3.83872i | 6.70598 | −6.81780 | 27.0634i | −3.00000 | −14.8624 | − | 12.4041i | |||||||||||||
229.3 | − | 3.64193i | − | 1.73205i | −9.26363 | −0.940387 | − | 4.91077i | −6.30800 | −2.95540 | 19.1697i | −3.00000 | −17.8847 | + | 3.42482i | ||||||||||||
229.4 | − | 3.64193i | − | 1.73205i | −9.26363 | 0.940387 | + | 4.91077i | −6.30800 | 2.95540 | 19.1697i | −3.00000 | 17.8847 | − | 3.42482i | ||||||||||||
229.5 | − | 3.09614i | − | 1.73205i | −5.58609 | −4.94474 | + | 0.741284i | −5.36267 | −4.84190 | 4.91075i | −3.00000 | 2.29512 | + | 15.3096i | ||||||||||||
229.6 | − | 3.09614i | − | 1.73205i | −5.58609 | 4.94474 | − | 0.741284i | −5.36267 | 4.84190 | 4.91075i | −3.00000 | −2.29512 | − | 15.3096i | ||||||||||||
229.7 | − | 2.96442i | 1.73205i | −4.78781 | −3.79693 | − | 3.25320i | 5.13453 | 2.22266 | 2.33540i | −3.00000 | −9.64386 | + | 11.2557i | |||||||||||||
229.8 | − | 2.96442i | 1.73205i | −4.78781 | 3.79693 | + | 3.25320i | 5.13453 | −2.22266 | 2.33540i | −3.00000 | 9.64386 | − | 11.2557i | |||||||||||||
229.9 | − | 2.73585i | 1.73205i | −3.48487 | −0.698940 | + | 4.95091i | 4.73863 | −5.09771 | − | 1.40933i | −3.00000 | 13.5449 | + | 1.91219i | ||||||||||||
229.10 | − | 2.73585i | 1.73205i | −3.48487 | 0.698940 | − | 4.95091i | 4.73863 | 5.09771 | − | 1.40933i | −3.00000 | −13.5449 | − | 1.91219i | ||||||||||||
229.11 | − | 2.58060i | − | 1.73205i | −2.65950 | −2.43175 | + | 4.36882i | −4.46973 | 10.6156 | − | 3.45929i | −3.00000 | 11.2742 | + | 6.27538i | |||||||||||
229.12 | − | 2.58060i | − | 1.73205i | −2.65950 | 2.43175 | − | 4.36882i | −4.46973 | −10.6156 | − | 3.45929i | −3.00000 | −11.2742 | − | 6.27538i | |||||||||||
229.13 | − | 2.02853i | 1.73205i | −0.114950 | −4.74723 | + | 1.56966i | 3.51352 | 11.5567 | − | 7.88096i | −3.00000 | 3.18411 | + | 9.62991i | ||||||||||||
229.14 | − | 2.02853i | 1.73205i | −0.114950 | 4.74723 | − | 1.56966i | 3.51352 | −11.5567 | − | 7.88096i | −3.00000 | −3.18411 | − | 9.62991i | ||||||||||||
229.15 | − | 1.48820i | − | 1.73205i | 1.78525 | −0.794027 | + | 4.93655i | −2.57765 | −12.7168 | − | 8.60963i | −3.00000 | 7.34659 | + | 1.18167i | |||||||||||
229.16 | − | 1.48820i | − | 1.73205i | 1.78525 | 0.794027 | − | 4.93655i | −2.57765 | 12.7168 | − | 8.60963i | −3.00000 | −7.34659 | − | 1.18167i | |||||||||||
229.17 | − | 1.17949i | − | 1.73205i | 2.60880 | −3.95152 | − | 3.06357i | −2.04294 | 0.208857 | − | 7.79502i | −3.00000 | −3.61345 | + | 4.66079i | |||||||||||
229.18 | − | 1.17949i | − | 1.73205i | 2.60880 | 3.95152 | + | 3.06357i | −2.04294 | −0.208857 | − | 7.79502i | −3.00000 | 3.61345 | − | 4.66079i | |||||||||||
229.19 | − | 1.12284i | 1.73205i | 2.73924 | −3.66308 | − | 3.40321i | 1.94481 | −10.5997 | − | 7.56706i | −3.00000 | −3.82125 | + | 4.11303i | ||||||||||||
229.20 | − | 1.12284i | 1.73205i | 2.73924 | 3.66308 | + | 3.40321i | 1.94481 | 10.5997 | − | 7.56706i | −3.00000 | 3.82125 | − | 4.11303i | ||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
115.c | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 345.3.d.a | ✓ | 48 |
5.b | even | 2 | 1 | inner | 345.3.d.a | ✓ | 48 |
23.b | odd | 2 | 1 | inner | 345.3.d.a | ✓ | 48 |
115.c | odd | 2 | 1 | inner | 345.3.d.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
345.3.d.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
345.3.d.a | ✓ | 48 | 5.b | even | 2 | 1 | inner |
345.3.d.a | ✓ | 48 | 23.b | odd | 2 | 1 | inner |
345.3.d.a | ✓ | 48 | 115.c | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(345, [\chi])\).