Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [380,3,Mod(29,380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(380, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 9, 17]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("380.29");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 380.bc (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: | \(10.3542500457\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | 0 | −4.22827 | + | 3.54794i | 0 | −3.68991 | − | 3.37410i | 0 | 11.4338 | + | 6.60131i | 0 | 3.72755 | − | 21.1400i | 0 | ||||||||||
29.2 | 0 | −4.06848 | + | 3.41386i | 0 | 4.92418 | + | 0.867412i | 0 | −3.35739 | − | 1.93839i | 0 | 3.33525 | − | 18.9152i | 0 | ||||||||||
29.3 | 0 | −3.70131 | + | 3.10577i | 0 | −4.40108 | + | 2.37286i | 0 | −5.39353 | − | 3.11396i | 0 | 2.49107 | − | 14.1276i | 0 | ||||||||||
29.4 | 0 | −2.53827 | + | 2.12986i | 0 | 1.43585 | + | 4.78940i | 0 | 7.33686 | + | 4.23594i | 0 | 0.343673 | − | 1.94907i | 0 | ||||||||||
29.5 | 0 | −2.42727 | + | 2.03672i | 0 | 2.04299 | − | 4.56357i | 0 | −0.860010 | − | 0.496527i | 0 | 0.180573 | − | 1.02408i | 0 | ||||||||||
29.6 | 0 | −2.15463 | + | 1.80795i | 0 | 0.381579 | + | 4.98542i | 0 | −3.17399 | − | 1.83251i | 0 | −0.189080 | + | 1.07233i | 0 | ||||||||||
29.7 | 0 | −1.55865 | + | 1.30786i | 0 | −3.89091 | − | 3.14020i | 0 | −6.81981 | − | 3.93742i | 0 | −0.843948 | + | 4.78627i | 0 | ||||||||||
29.8 | 0 | −1.41356 | + | 1.18612i | 0 | 2.50989 | − | 4.32440i | 0 | 6.52791 | + | 3.76889i | 0 | −0.971555 | + | 5.50996i | 0 | ||||||||||
29.9 | 0 | −0.915239 | + | 0.767977i | 0 | −4.79298 | + | 1.42385i | 0 | 4.52632 | + | 2.61327i | 0 | −1.31496 | + | 7.45750i | 0 | ||||||||||
29.10 | 0 | −0.215152 | + | 0.180534i | 0 | 4.83429 | + | 1.27657i | 0 | −9.68534 | − | 5.59183i | 0 | −1.54914 | + | 8.78559i | 0 | ||||||||||
29.11 | 0 | 0.215152 | − | 0.180534i | 0 | 4.52385 | + | 2.12951i | 0 | 9.68534 | + | 5.59183i | 0 | −1.54914 | + | 8.78559i | 0 | ||||||||||
29.12 | 0 | 0.915239 | − | 0.767977i | 0 | −2.75641 | − | 4.17160i | 0 | −4.52632 | − | 2.61327i | 0 | −1.31496 | + | 7.45750i | 0 | ||||||||||
29.13 | 0 | 1.41356 | − | 1.18612i | 0 | −0.856980 | + | 4.92601i | 0 | −6.52791 | − | 3.76889i | 0 | −0.971555 | + | 5.50996i | 0 | ||||||||||
29.14 | 0 | 1.55865 | − | 1.30786i | 0 | −4.99909 | − | 0.0954977i | 0 | 6.81981 | + | 3.93742i | 0 | −0.843948 | + | 4.78627i | 0 | ||||||||||
29.15 | 0 | 2.15463 | − | 1.80795i | 0 | 3.49687 | − | 3.57378i | 0 | 3.17399 | + | 1.83251i | 0 | −0.189080 | + | 1.07233i | 0 | ||||||||||
29.16 | 0 | 2.42727 | − | 2.03672i | 0 | −1.36839 | + | 4.80911i | 0 | 0.860010 | + | 0.496527i | 0 | 0.180573 | − | 1.02408i | 0 | ||||||||||
29.17 | 0 | 2.53827 | − | 2.12986i | 0 | 4.17849 | − | 2.74595i | 0 | −7.33686 | − | 4.23594i | 0 | 0.343673 | − | 1.94907i | 0 | ||||||||||
29.18 | 0 | 3.70131 | − | 3.10577i | 0 | −1.84618 | − | 4.64668i | 0 | 5.39353 | + | 3.11396i | 0 | 2.49107 | − | 14.1276i | 0 | ||||||||||
29.19 | 0 | 4.06848 | − | 3.41386i | 0 | 4.32971 | + | 2.50073i | 0 | 3.35739 | + | 1.93839i | 0 | 3.33525 | − | 18.9152i | 0 | ||||||||||
29.20 | 0 | 4.22827 | − | 3.54794i | 0 | −4.99547 | + | 0.212883i | 0 | −11.4338 | − | 6.60131i | 0 | 3.72755 | − | 21.1400i | 0 | ||||||||||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.f | odd | 18 | 1 | inner |
95.o | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 380.3.bc.a | ✓ | 120 |
5.b | even | 2 | 1 | inner | 380.3.bc.a | ✓ | 120 |
19.f | odd | 18 | 1 | inner | 380.3.bc.a | ✓ | 120 |
95.o | odd | 18 | 1 | inner | 380.3.bc.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.3.bc.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
380.3.bc.a | ✓ | 120 | 5.b | even | 2 | 1 | inner |
380.3.bc.a | ✓ | 120 | 19.f | odd | 18 | 1 | inner |
380.3.bc.a | ✓ | 120 | 95.o | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).