Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,2,Mod(9,230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 230.j (of order \(22\), degree \(10\), 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: | \(1.83655924649\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | −0.281733 | − | 0.959493i | −1.95972 | + | 1.69811i | −0.841254 | + | 0.540641i | −0.0903172 | − | 2.23424i | 2.18144 | + | 1.40193i | 0.748675 | + | 0.341908i | 0.755750 | + | 0.654861i | 0.529991 | − | 3.68617i | −2.11830 | + | 0.716118i |
9.2 | −0.281733 | − | 0.959493i | −1.62154 | + | 1.40507i | −0.841254 | + | 0.540641i | 1.91000 | + | 1.16271i | 1.80499 | + | 1.16000i | −4.06165 | − | 1.85489i | 0.755750 | + | 0.654861i | 0.228216 | − | 1.58728i | 0.577502 | − | 2.16021i |
9.3 | −0.281733 | − | 0.959493i | −0.0697998 | + | 0.0604819i | −0.841254 | + | 0.540641i | −0.948149 | + | 2.02510i | 0.0776969 | + | 0.0499327i | 1.04108 | + | 0.475445i | 0.755750 | + | 0.654861i | −0.425731 | + | 2.96102i | 2.21019 | + | 0.339207i |
9.4 | −0.281733 | − | 0.959493i | 0.951117 | − | 0.824147i | −0.841254 | + | 0.540641i | −1.90749 | − | 1.16682i | −1.05872 | − | 0.680401i | −4.71597 | − | 2.15371i | 0.755750 | + | 0.654861i | −0.201540 | + | 1.40174i | −0.582157 | + | 2.15896i |
9.5 | −0.281733 | − | 0.959493i | 1.78070 | − | 1.54299i | −0.841254 | + | 0.540641i | −0.352110 | − | 2.20817i | −1.98217 | − | 1.27386i | 3.91611 | + | 1.78843i | 0.755750 | + | 0.654861i | 0.363147 | − | 2.52574i | −2.01952 | + | 0.959961i |
9.6 | −0.281733 | − | 0.959493i | 2.36951 | − | 2.05319i | −0.841254 | + | 0.540641i | 1.30136 | + | 1.81837i | −2.63759 | − | 1.69508i | −1.55660 | − | 0.710876i | 0.755750 | + | 0.654861i | 0.972037 | − | 6.76066i | 1.37808 | − | 1.76094i |
9.7 | 0.281733 | + | 0.959493i | −2.36951 | + | 2.05319i | −0.841254 | + | 0.540641i | −0.736350 | + | 2.11135i | −2.63759 | − | 1.69508i | 1.55660 | + | 0.710876i | −0.755750 | − | 0.654861i | 0.972037 | − | 6.76066i | −2.23328 | − | 0.111687i |
9.8 | 0.281733 | + | 0.959493i | −1.78070 | + | 1.54299i | −0.841254 | + | 0.540641i | −0.284267 | − | 2.21793i | −1.98217 | − | 1.27386i | −3.91611 | − | 1.78843i | −0.755750 | − | 0.654861i | 0.363147 | − | 2.52574i | 2.04800 | − | 0.897613i |
9.9 | 0.281733 | + | 0.959493i | −0.951117 | + | 0.824147i | −0.841254 | + | 0.540641i | 1.50149 | − | 1.65696i | −1.05872 | − | 0.680401i | 4.71597 | + | 2.15371i | −0.755750 | − | 0.654861i | −0.201540 | + | 1.40174i | 2.01286 | + | 0.973851i |
9.10 | 0.281733 | + | 0.959493i | 0.0697998 | − | 0.0604819i | −0.841254 | + | 0.540641i | 1.48028 | + | 1.67594i | 0.0776969 | + | 0.0499327i | −1.04108 | − | 0.475445i | −0.755750 | − | 0.654861i | −0.425731 | + | 2.96102i | −1.19101 | + | 1.89248i |
9.11 | 0.281733 | + | 0.959493i | 1.62154 | − | 1.40507i | −0.841254 | + | 0.540641i | −1.50506 | + | 1.65372i | 1.80499 | + | 1.16000i | 4.06165 | + | 1.85489i | −0.755750 | − | 0.654861i | 0.228216 | − | 1.58728i | −2.01076 | − | 0.978188i |
9.12 | 0.281733 | + | 0.959493i | 1.95972 | − | 1.69811i | −0.841254 | + | 0.540641i | −0.542800 | − | 2.16919i | 2.18144 | + | 1.40193i | −0.748675 | − | 0.341908i | −0.755750 | − | 0.654861i | 0.529991 | − | 3.68617i | 1.92839 | − | 1.13194i |
29.1 | −0.909632 | − | 0.415415i | −0.768680 | + | 2.61788i | 0.654861 | + | 0.755750i | 1.33202 | + | 1.79603i | 1.78672 | − | 2.06199i | 1.27046 | + | 0.182665i | −0.281733 | − | 0.959493i | −3.73869 | − | 2.40271i | −0.465548 | − | 2.18707i |
29.2 | −0.909632 | − | 0.415415i | −0.642787 | + | 2.18913i | 0.654861 | + | 0.755750i | −1.99872 | − | 1.00255i | 1.49410 | − | 1.72428i | −2.70319 | − | 0.388660i | −0.281733 | − | 0.959493i | −1.85536 | − | 1.19237i | 1.40163 | + | 1.74225i |
29.3 | −0.909632 | − | 0.415415i | 0.0359844 | − | 0.122552i | 0.654861 | + | 0.755750i | −1.96836 | + | 1.06092i | −0.0836423 | + | 0.0965284i | −0.277941 | − | 0.0399619i | −0.281733 | − | 0.959493i | 2.51004 | + | 1.61310i | 2.23121 | − | 0.147361i |
29.4 | −0.909632 | − | 0.415415i | 0.192005 | − | 0.653908i | 0.654861 | + | 0.755750i | 2.23607 | + | 0.00163718i | −0.446297 | + | 0.515054i | −0.838442 | − | 0.120550i | −0.281733 | − | 0.959493i | 2.13303 | + | 1.37082i | −2.03332 | − | 0.930385i |
29.5 | −0.909632 | − | 0.415415i | 0.537451 | − | 1.83039i | 0.654861 | + | 0.755750i | −0.575877 | − | 2.16064i | −1.24925 | + | 1.44172i | 2.93634 | + | 0.422181i | −0.281733 | − | 0.959493i | −0.537711 | − | 0.345566i | −0.373726 | + | 2.20462i |
29.6 | −0.909632 | − | 0.415415i | 0.880099 | − | 2.99734i | 0.654861 | + | 0.755750i | −1.33842 | + | 1.79127i | −2.04571 | + | 2.36087i | −3.80587 | − | 0.547202i | −0.281733 | − | 0.959493i | −5.68573 | − | 3.65400i | 1.96159 | − | 1.07339i |
29.7 | 0.909632 | + | 0.415415i | −0.880099 | + | 2.99734i | 0.654861 | + | 0.755750i | −2.18539 | + | 0.473349i | −2.04571 | + | 2.36087i | 3.80587 | + | 0.547202i | 0.281733 | + | 0.959493i | −5.68573 | − | 3.65400i | −2.18454 | − | 0.477271i |
29.8 | 0.909632 | + | 0.415415i | −0.537451 | + | 1.83039i | 0.654861 | + | 0.755750i | 1.72616 | + | 1.42140i | −1.24925 | + | 1.44172i | −2.93634 | − | 0.422181i | 0.281733 | + | 0.959493i | −0.537711 | − | 0.345566i | 0.979700 | + | 2.01002i |
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 |
23.c | even | 11 | 1 | inner |
115.j | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 230.2.j.a | ✓ | 120 |
5.b | even | 2 | 1 | inner | 230.2.j.a | ✓ | 120 |
23.c | even | 11 | 1 | inner | 230.2.j.a | ✓ | 120 |
115.j | even | 22 | 1 | inner | 230.2.j.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.2.j.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
230.2.j.a | ✓ | 120 | 5.b | even | 2 | 1 | inner |
230.2.j.a | ✓ | 120 | 23.c | even | 11 | 1 | inner |
230.2.j.a | ✓ | 120 | 115.j | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(230, [\chi])\).