Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,4,Mod(31,230)]
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([0, 6]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.31");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 230.g (of order \(11\), 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: | \(13.5704393013\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 0.284630 | − | 1.97964i | −3.91607 | − | 8.57500i | −3.83797 | − | 1.12693i | −3.27430 | − | 3.77875i | −18.0901 | + | 5.31172i | 13.7433 | + | 8.83226i | −3.32332 | + | 7.27706i | −40.5137 | + | 46.7553i | −8.41254 | + | 5.40641i |
31.2 | 0.284630 | − | 1.97964i | −1.76914 | − | 3.87387i | −3.83797 | − | 1.12693i | −3.27430 | − | 3.77875i | −8.17242 | + | 2.39964i | −26.5942 | − | 17.0910i | −3.32332 | + | 7.27706i | 5.80424 | − | 6.69844i | −8.41254 | + | 5.40641i |
31.3 | 0.284630 | − | 1.97964i | −0.871814 | − | 1.90901i | −3.83797 | − | 1.12693i | −3.27430 | − | 3.77875i | −4.02729 | + | 1.18252i | 25.9987 | + | 16.7083i | −3.32332 | + | 7.27706i | 14.7970 | − | 17.0766i | −8.41254 | + | 5.40641i |
31.4 | 0.284630 | − | 1.97964i | −0.654798 | − | 1.43381i | −3.83797 | − | 1.12693i | −3.27430 | − | 3.77875i | −3.02480 | + | 0.888162i | 7.64693 | + | 4.91439i | −3.32332 | + | 7.27706i | 16.0542 | − | 18.5275i | −8.41254 | + | 5.40641i |
31.5 | 0.284630 | − | 1.97964i | 2.61500 | + | 5.72604i | −3.83797 | − | 1.12693i | −3.27430 | − | 3.77875i | 12.0798 | − | 3.54696i | −5.58110 | − | 3.58676i | −3.32332 | + | 7.27706i | −8.26814 | + | 9.54194i | −8.41254 | + | 5.40641i |
31.6 | 0.284630 | − | 1.97964i | 2.99586 | + | 6.56001i | −3.83797 | − | 1.12693i | −3.27430 | − | 3.77875i | 13.8392 | − | 4.06355i | 2.05971 | + | 1.32369i | −3.32332 | + | 7.27706i | −16.3774 | + | 18.9005i | −8.41254 | + | 5.40641i |
41.1 | 1.91899 | + | 0.563465i | −4.61458 | + | 5.32551i | 3.36501 | + | 2.16256i | −0.711574 | + | 4.94911i | −11.8561 | + | 7.61942i | 10.2472 | + | 22.4383i | 5.23889 | + | 6.04600i | −3.22420 | − | 22.4248i | −4.15415 | + | 9.09632i |
41.2 | 1.91899 | + | 0.563465i | −2.52693 | + | 2.91624i | 3.36501 | + | 2.16256i | −0.711574 | + | 4.94911i | −6.49235 | + | 4.17238i | −13.4966 | − | 29.5535i | 5.23889 | + | 6.04600i | 1.72346 | + | 11.9869i | −4.15415 | + | 9.09632i |
41.3 | 1.91899 | + | 0.563465i | 0.460964 | − | 0.531981i | 3.36501 | + | 2.16256i | −0.711574 | + | 4.94911i | 1.18434 | − | 0.761126i | −1.15601 | − | 2.53131i | 5.23889 | + | 6.04600i | 3.77198 | + | 26.2347i | −4.15415 | + | 9.09632i |
41.4 | 1.91899 | + | 0.563465i | 1.94747 | − | 2.24751i | 3.36501 | + | 2.16256i | −0.711574 | + | 4.94911i | 5.00357 | − | 3.21560i | 8.06153 | + | 17.6523i | 5.23889 | + | 6.04600i | 2.58388 | + | 17.9713i | −4.15415 | + | 9.09632i |
41.5 | 1.91899 | + | 0.563465i | 3.92901 | − | 4.53431i | 3.36501 | + | 2.16256i | −0.711574 | + | 4.94911i | 10.0946 | − | 6.48743i | −9.52105 | − | 20.8482i | 5.23889 | + | 6.04600i | −1.28042 | − | 8.90551i | −4.15415 | + | 9.09632i |
41.6 | 1.91899 | + | 0.563465i | 6.53866 | − | 7.54601i | 3.36501 | + | 2.16256i | −0.711574 | + | 4.94911i | 16.7995 | − | 10.7964i | 14.9302 | + | 32.6926i | 5.23889 | + | 6.04600i | −10.3458 | − | 71.9564i | −4.15415 | + | 9.09632i |
71.1 | −1.68251 | − | 1.08128i | −1.10005 | − | 7.65101i | 1.66166 | + | 3.63853i | −4.79746 | − | 1.40866i | −6.42205 | + | 14.0623i | −11.9276 | + | 13.7651i | 1.13852 | − | 7.91857i | −31.4215 | + | 9.22619i | 6.54861 | + | 7.55750i |
71.2 | −1.68251 | − | 1.08128i | −0.719442 | − | 5.00383i | 1.66166 | + | 3.63853i | −4.79746 | − | 1.40866i | −4.20008 | + | 9.19689i | 9.79868 | − | 11.3083i | 1.13852 | − | 7.91857i | 1.38563 | − | 0.406859i | 6.54861 | + | 7.55750i |
71.3 | −1.68251 | − | 1.08128i | −0.259790 | − | 1.80688i | 1.66166 | + | 3.63853i | −4.79746 | − | 1.40866i | −1.51665 | + | 3.32099i | −3.50130 | + | 4.04072i | 1.13852 | − | 7.91857i | 22.7090 | − | 6.66796i | 6.54861 | + | 7.55750i |
71.4 | −1.68251 | − | 1.08128i | −0.161442 | − | 1.12285i | 1.66166 | + | 3.63853i | −4.79746 | − | 1.40866i | −0.942491 | + | 2.06377i | 20.3265 | − | 23.4580i | 1.13852 | − | 7.91857i | 24.6716 | − | 7.24423i | 6.54861 | + | 7.55750i |
71.5 | −1.68251 | − | 1.08128i | 0.839469 | + | 5.83864i | 1.66166 | + | 3.63853i | −4.79746 | − | 1.40866i | 4.90080 | − | 10.7312i | 5.10412 | − | 5.89047i | 1.13852 | − | 7.91857i | −7.47864 | + | 2.19593i | 6.54861 | + | 7.55750i |
71.6 | −1.68251 | − | 1.08128i | 1.10986 | + | 7.71925i | 1.66166 | + | 3.63853i | −4.79746 | − | 1.40866i | 6.47933 | − | 14.1878i | −5.92245 | + | 6.83487i | 1.13852 | − | 7.91857i | −32.4487 | + | 9.52780i | 6.54861 | + | 7.55750i |
81.1 | −1.68251 | + | 1.08128i | −1.10005 | + | 7.65101i | 1.66166 | − | 3.63853i | −4.79746 | + | 1.40866i | −6.42205 | − | 14.0623i | −11.9276 | − | 13.7651i | 1.13852 | + | 7.91857i | −31.4215 | − | 9.22619i | 6.54861 | − | 7.55750i |
81.2 | −1.68251 | + | 1.08128i | −0.719442 | + | 5.00383i | 1.66166 | − | 3.63853i | −4.79746 | + | 1.40866i | −4.20008 | − | 9.19689i | 9.79868 | + | 11.3083i | 1.13852 | + | 7.91857i | 1.38563 | + | 0.406859i | 6.54861 | − | 7.55750i |
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 230.4.g.b | ✓ | 60 |
23.c | even | 11 | 1 | inner | 230.4.g.b | ✓ | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.4.g.b | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
230.4.g.b | ✓ | 60 | 23.c | even | 11 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{60} + 3 T_{3}^{59} + 51 T_{3}^{58} + 469 T_{3}^{57} + 6348 T_{3}^{56} + 46774 T_{3}^{55} + \cdots + 52\!\cdots\!61 \)
acting on \(S_{4}^{\mathrm{new}}(230, [\chi])\).