Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [369,2,Mod(124,369)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(369, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("369.124");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 369 = 3^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 369.e (of order \(3\), degree \(2\), 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: | \(2.94647983459\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
124.1 | −1.26536 | + | 2.19167i | 1.69326 | − | 0.364528i | −2.20228 | − | 3.81446i | 0.592169 | + | 1.02567i | −1.34366 | + | 4.17232i | 1.45439 | − | 2.51909i | 6.08526 | 2.73424 | − | 1.23448i | −2.99723 | ||||
124.2 | −0.987647 | + | 1.71066i | −1.61969 | + | 0.613673i | −0.950894 | − | 1.64700i | −0.800059 | − | 1.38574i | 0.549903 | − | 3.37683i | 0.367319 | − | 0.636215i | −0.193998 | 2.24681 | − | 1.98792i | 3.16071 | ||||
124.3 | −0.966816 | + | 1.67457i | 1.03213 | − | 1.39094i | −0.869467 | − | 1.50596i | 1.86893 | + | 3.23707i | 1.33135 | + | 3.07316i | −0.233357 | + | 0.404186i | −0.504807 | −0.869426 | − | 2.87125i | −7.22763 | ||||
124.4 | −0.954563 | + | 1.65335i | −0.947667 | − | 1.44980i | −0.822382 | − | 1.42441i | 0.112697 | + | 0.195198i | 3.30164 | − | 0.182900i | 0.630622 | − | 1.09227i | −0.678190 | −1.20385 | + | 2.74786i | −0.430307 | ||||
124.5 | −0.484849 | + | 0.839783i | −0.272051 | + | 1.71055i | 0.529843 | + | 0.917715i | 0.894340 | + | 1.54904i | −1.30459 | − | 1.05782i | −0.911059 | + | 1.57800i | −2.96697 | −2.85198 | − | 0.930716i | −1.73448 | ||||
124.6 | −0.395185 | + | 0.684480i | −1.70940 | − | 0.279205i | 0.687658 | + | 1.19106i | −0.0693987 | − | 0.120202i | 0.866638 | − | 1.05971i | 2.32472 | − | 4.02654i | −2.66775 | 2.84409 | + | 0.954544i | 0.109701 | ||||
124.7 | −0.373128 | + | 0.646277i | −0.160252 | + | 1.72462i | 0.721551 | + | 1.24976i | −1.18163 | − | 2.04664i | −1.05479 | − | 0.747072i | 0.193114 | − | 0.334484i | −2.56944 | −2.94864 | − | 0.552748i | 1.76359 | ||||
124.8 | −0.111854 | + | 0.193736i | 1.67652 | − | 0.435046i | 0.974977 | + | 1.68871i | 0.0946835 | + | 0.163997i | −0.103241 | + | 0.373465i | −1.37723 | + | 2.38544i | −0.883634 | 2.62147 | − | 1.45873i | −0.0423628 | ||||
124.9 | 0.0766551 | − | 0.132771i | 1.63859 | + | 0.561255i | 0.988248 | + | 1.71170i | −1.89658 | − | 3.28497i | 0.200125 | − | 0.174534i | 1.48279 | − | 2.56827i | 0.609637 | 2.36999 | + | 1.83934i | −0.581529 | ||||
124.10 | 0.178971 | − | 0.309987i | −1.39349 | + | 1.02868i | 0.935939 | + | 1.62109i | 1.28139 | + | 2.21943i | 0.0694843 | + | 0.616068i | 0.644358 | − | 1.11606i | 1.38591 | 0.883621 | − | 2.86692i | 0.917326 | ||||
124.11 | 0.474444 | − | 0.821762i | −0.261726 | − | 1.71216i | 0.549805 | + | 0.952290i | −1.41059 | − | 2.44321i | −1.53116 | − | 0.597249i | −0.217294 | + | 0.376365i | 2.94119 | −2.86300 | + | 0.896235i | −2.67698 | ||||
124.12 | 0.779106 | − | 1.34945i | −1.72925 | − | 0.0985029i | −0.214012 | − | 0.370680i | 0.276060 | + | 0.478149i | −1.48019 | + | 2.25679i | −1.91437 | + | 3.31579i | 2.44947 | 2.98059 | + | 0.340672i | 0.860319 | ||||
124.13 | 0.820512 | − | 1.42117i | 1.55494 | + | 0.763004i | −0.346479 | − | 0.600120i | 0.171853 | + | 0.297657i | 2.36020 | − | 1.58377i | 0.00181771 | − | 0.00314836i | 2.14489 | 1.83565 | + | 2.37285i | 0.564028 | ||||
124.14 | 0.881207 | − | 1.52630i | −0.938111 | − | 1.45600i | −0.553053 | − | 0.957916i | 2.03169 | + | 3.51900i | −3.04896 | + | 0.148793i | 2.50362 | − | 4.33640i | 1.57541 | −1.23990 | + | 2.73179i | 7.16138 | ||||
124.15 | 1.12092 | − | 1.94148i | 0.323299 | − | 1.70161i | −1.51291 | − | 2.62043i | −0.671092 | − | 1.16236i | −2.94126 | − | 2.53504i | −0.114415 | + | 0.198173i | −2.29971 | −2.79096 | − | 1.10026i | −3.00895 | ||||
124.16 | 1.20759 | − | 2.09161i | 0.612897 | + | 1.61999i | −1.91655 | − | 3.31956i | −0.794470 | − | 1.37606i | 4.12851 | + | 0.674340i | 1.66497 | − | 2.88381i | −4.42726 | −2.24871 | + | 1.98577i | −3.83758 | ||||
247.1 | −1.26536 | − | 2.19167i | 1.69326 | + | 0.364528i | −2.20228 | + | 3.81446i | 0.592169 | − | 1.02567i | −1.34366 | − | 4.17232i | 1.45439 | + | 2.51909i | 6.08526 | 2.73424 | + | 1.23448i | −2.99723 | ||||
247.2 | −0.987647 | − | 1.71066i | −1.61969 | − | 0.613673i | −0.950894 | + | 1.64700i | −0.800059 | + | 1.38574i | 0.549903 | + | 3.37683i | 0.367319 | + | 0.636215i | −0.193998 | 2.24681 | + | 1.98792i | 3.16071 | ||||
247.3 | −0.966816 | − | 1.67457i | 1.03213 | + | 1.39094i | −0.869467 | + | 1.50596i | 1.86893 | − | 3.23707i | 1.33135 | − | 3.07316i | −0.233357 | − | 0.404186i | −0.504807 | −0.869426 | + | 2.87125i | −7.22763 | ||||
247.4 | −0.954563 | − | 1.65335i | −0.947667 | + | 1.44980i | −0.822382 | + | 1.42441i | 0.112697 | − | 0.195198i | 3.30164 | + | 0.182900i | 0.630622 | + | 1.09227i | −0.678190 | −1.20385 | − | 2.74786i | −0.430307 | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 369.2.e.a | ✓ | 32 |
3.b | odd | 2 | 1 | 1107.2.e.a | 32 | ||
9.c | even | 3 | 1 | inner | 369.2.e.a | ✓ | 32 |
9.c | even | 3 | 1 | 3321.2.a.i | 16 | ||
9.d | odd | 6 | 1 | 1107.2.e.a | 32 | ||
9.d | odd | 6 | 1 | 3321.2.a.j | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
369.2.e.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
369.2.e.a | ✓ | 32 | 9.c | even | 3 | 1 | inner |
1107.2.e.a | 32 | 3.b | odd | 2 | 1 | ||
1107.2.e.a | 32 | 9.d | odd | 6 | 1 | ||
3321.2.a.i | 16 | 9.c | even | 3 | 1 | ||
3321.2.a.j | 16 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 20 T_{2}^{30} + 242 T_{2}^{28} + 2 T_{2}^{27} + 1902 T_{2}^{26} + 55 T_{2}^{25} + 11055 T_{2}^{24} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(369, [\chi])\).