Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,2,Mod(43,378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(378, base_ring=CyclotomicField(18))
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("378.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.u (of order \(9\), 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: | \(3.01834519640\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 | −0.766044 | − | 0.642788i | −1.73154 | − | 0.0419083i | 0.173648 | + | 0.984808i | −0.354259 | − | 0.128940i | 1.29950 | + | 1.14512i | 0.173648 | − | 0.984808i | 0.500000 | − | 0.866025i | 2.99649 | + | 0.145132i | 0.188497 | + | 0.326487i |
43.2 | −0.766044 | − | 0.642788i | −1.15570 | − | 1.29010i | 0.173648 | + | 0.984808i | −2.06967 | − | 0.753297i | 0.0560567 | + | 1.73114i | 0.173648 | − | 0.984808i | 0.500000 | − | 0.866025i | −0.328718 | + | 2.98194i | 1.10125 | + | 1.90741i |
43.3 | −0.766044 | − | 0.642788i | −0.910669 | + | 1.47332i | 0.173648 | + | 0.984808i | 1.95320 | + | 0.710905i | 1.64465 | − | 0.543265i | 0.173648 | − | 0.984808i | 0.500000 | − | 0.866025i | −1.34137 | − | 2.68342i | −1.03927 | − | 1.80007i |
43.4 | −0.766044 | − | 0.642788i | 0.391603 | − | 1.68720i | 0.173648 | + | 0.984808i | 2.08466 | + | 0.758756i | −1.38450 | + | 1.04075i | 0.173648 | − | 0.984808i | 0.500000 | − | 0.866025i | −2.69329 | − | 1.32143i | −1.10923 | − | 1.92124i |
43.5 | −0.766044 | − | 0.642788i | 1.44903 | + | 0.948848i | 0.173648 | + | 0.984808i | −3.88962 | − | 1.41571i | −0.500114 | − | 1.65828i | 0.173648 | − | 0.984808i | 0.500000 | − | 0.866025i | 1.19938 | + | 2.74982i | 2.06962 | + | 3.58470i |
43.6 | −0.766044 | − | 0.642788i | 1.69123 | + | 0.373800i | 0.173648 | + | 0.984808i | 3.54173 | + | 1.28909i | −1.05529 | − | 1.37345i | 0.173648 | − | 0.984808i | 0.500000 | − | 0.866025i | 2.72055 | + | 1.26437i | −1.88452 | − | 3.26408i |
85.1 | 0.939693 | + | 0.342020i | −1.62983 | − | 0.586228i | 0.766044 | + | 0.642788i | 0.184482 | − | 1.04625i | −1.33103 | − | 1.10831i | 0.766044 | − | 0.642788i | 0.500000 | + | 0.866025i | 2.31267 | + | 1.91090i | 0.531195 | − | 0.920056i |
85.2 | 0.939693 | + | 0.342020i | −1.33644 | + | 1.10178i | 0.766044 | + | 0.642788i | −0.619299 | + | 3.51222i | −1.63268 | + | 0.578248i | 0.766044 | − | 0.642788i | 0.500000 | + | 0.866025i | 0.572147 | − | 2.94494i | −1.78320 | + | 3.08859i |
85.3 | 0.939693 | + | 0.342020i | 0.873169 | − | 1.49585i | 0.766044 | + | 0.642788i | −0.605889 | + | 3.43617i | 1.33212 | − | 1.10700i | 0.766044 | − | 0.642788i | 0.500000 | + | 0.866025i | −1.47515 | − | 2.61227i | −1.74459 | + | 3.02172i |
85.4 | 0.939693 | + | 0.342020i | 0.927705 | + | 1.46266i | 0.766044 | + | 0.642788i | 0.587615 | − | 3.33253i | 0.371500 | + | 1.69174i | 0.766044 | − | 0.642788i | 0.500000 | + | 0.866025i | −1.27873 | + | 2.71383i | 1.69197 | − | 2.93058i |
85.5 | 0.939693 | + | 0.342020i | 1.28430 | + | 1.16214i | 0.766044 | + | 0.642788i | −0.345781 | + | 1.96102i | 0.809368 | + | 1.53131i | 0.766044 | − | 0.642788i | 0.500000 | + | 0.866025i | 0.298838 | + | 2.98508i | −0.995637 | + | 1.72449i |
85.6 | 0.939693 | + | 0.342020i | 1.32079 | − | 1.12050i | 0.766044 | + | 0.642788i | 0.359180 | − | 2.03701i | 1.62437 | − | 0.601188i | 0.766044 | − | 0.642788i | 0.500000 | + | 0.866025i | 0.488967 | − | 2.95988i | 1.03422 | − | 1.79132i |
169.1 | 0.939693 | − | 0.342020i | −1.62983 | + | 0.586228i | 0.766044 | − | 0.642788i | 0.184482 | + | 1.04625i | −1.33103 | + | 1.10831i | 0.766044 | + | 0.642788i | 0.500000 | − | 0.866025i | 2.31267 | − | 1.91090i | 0.531195 | + | 0.920056i |
169.2 | 0.939693 | − | 0.342020i | −1.33644 | − | 1.10178i | 0.766044 | − | 0.642788i | −0.619299 | − | 3.51222i | −1.63268 | − | 0.578248i | 0.766044 | + | 0.642788i | 0.500000 | − | 0.866025i | 0.572147 | + | 2.94494i | −1.78320 | − | 3.08859i |
169.3 | 0.939693 | − | 0.342020i | 0.873169 | + | 1.49585i | 0.766044 | − | 0.642788i | −0.605889 | − | 3.43617i | 1.33212 | + | 1.10700i | 0.766044 | + | 0.642788i | 0.500000 | − | 0.866025i | −1.47515 | + | 2.61227i | −1.74459 | − | 3.02172i |
169.4 | 0.939693 | − | 0.342020i | 0.927705 | − | 1.46266i | 0.766044 | − | 0.642788i | 0.587615 | + | 3.33253i | 0.371500 | − | 1.69174i | 0.766044 | + | 0.642788i | 0.500000 | − | 0.866025i | −1.27873 | − | 2.71383i | 1.69197 | + | 2.93058i |
169.5 | 0.939693 | − | 0.342020i | 1.28430 | − | 1.16214i | 0.766044 | − | 0.642788i | −0.345781 | − | 1.96102i | 0.809368 | − | 1.53131i | 0.766044 | + | 0.642788i | 0.500000 | − | 0.866025i | 0.298838 | − | 2.98508i | −0.995637 | − | 1.72449i |
169.6 | 0.939693 | − | 0.342020i | 1.32079 | + | 1.12050i | 0.766044 | − | 0.642788i | 0.359180 | + | 2.03701i | 1.62437 | + | 0.601188i | 0.766044 | + | 0.642788i | 0.500000 | − | 0.866025i | 0.488967 | + | 2.95988i | 1.03422 | + | 1.79132i |
211.1 | −0.766044 | + | 0.642788i | −1.73154 | + | 0.0419083i | 0.173648 | − | 0.984808i | −0.354259 | + | 0.128940i | 1.29950 | − | 1.14512i | 0.173648 | + | 0.984808i | 0.500000 | + | 0.866025i | 2.99649 | − | 0.145132i | 0.188497 | − | 0.326487i |
211.2 | −0.766044 | + | 0.642788i | −1.15570 | + | 1.29010i | 0.173648 | − | 0.984808i | −2.06967 | + | 0.753297i | 0.0560567 | − | 1.73114i | 0.173648 | + | 0.984808i | 0.500000 | + | 0.866025i | −0.328718 | − | 2.98194i | 1.10125 | − | 1.90741i |
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.2.u.e | ✓ | 36 |
27.e | even | 9 | 1 | inner | 378.2.u.e | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.2.u.e | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
378.2.u.e | ✓ | 36 | 27.e | even | 9 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{36} - 3 T_{5}^{35} + 6 T_{5}^{34} - 15 T_{5}^{33} - 9 T_{5}^{32} + 297 T_{5}^{31} + 2244 T_{5}^{30} - 11889 T_{5}^{29} + 44739 T_{5}^{28} - 143496 T_{5}^{27} + 493074 T_{5}^{26} - 1076814 T_{5}^{25} + \cdots + 7766544384 \)
acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).