Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [370,2,Mod(71,370)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(370, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("370.71");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 370 = 2 \cdot 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 370.o (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: | \(2.95446487479\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
71.1 | −0.766044 | − | 0.642788i | −2.54754 | + | 2.13764i | 0.173648 | + | 0.984808i | −0.939693 | − | 0.342020i | 3.32558 | −3.31719 | − | 1.20736i | 0.500000 | − | 0.866025i | 1.39951 | − | 7.93703i | 0.500000 | + | 0.866025i | ||
71.2 | −0.766044 | − | 0.642788i | 0.204920 | − | 0.171948i | 0.173648 | + | 0.984808i | −0.939693 | − | 0.342020i | −0.267503 | −3.03750 | − | 1.10556i | 0.500000 | − | 0.866025i | −0.508519 | + | 2.88395i | 0.500000 | + | 0.866025i | ||
71.3 | −0.766044 | − | 0.642788i | 0.727578 | − | 0.610510i | 0.173648 | + | 0.984808i | −0.939693 | − | 0.342020i | −0.949785 | 4.16326 | + | 1.51530i | 0.500000 | − | 0.866025i | −0.364298 | + | 2.06604i | 0.500000 | + | 0.866025i | ||
71.4 | −0.766044 | − | 0.642788i | 2.38109 | − | 1.99797i | 0.173648 | + | 0.984808i | −0.939693 | − | 0.342020i | −3.10829 | −0.187965 | − | 0.0684136i | 0.500000 | − | 0.866025i | 1.15675 | − | 6.56026i | 0.500000 | + | 0.866025i | ||
81.1 | 0.939693 | − | 0.342020i | −2.67866 | − | 0.974954i | 0.766044 | − | 0.642788i | 0.173648 | + | 0.984808i | −2.85057 | −0.751904 | − | 4.26426i | 0.500000 | − | 0.866025i | 3.92657 | + | 3.29478i | 0.500000 | + | 0.866025i | ||
81.2 | 0.939693 | − | 0.342020i | −2.36335 | − | 0.860188i | 0.766044 | − | 0.642788i | 0.173648 | + | 0.984808i | −2.51502 | 0.832620 | + | 4.72202i | 0.500000 | − | 0.866025i | 2.54735 | + | 2.13748i | 0.500000 | + | 0.866025i | ||
81.3 | 0.939693 | − | 0.342020i | 1.85873 | + | 0.676523i | 0.766044 | − | 0.642788i | 0.173648 | + | 0.984808i | 1.97802 | 0.241589 | + | 1.37012i | 0.500000 | − | 0.866025i | 0.699071 | + | 0.586590i | 0.500000 | + | 0.866025i | ||
81.4 | 0.939693 | − | 0.342020i | 2.24358 | + | 0.816598i | 0.766044 | − | 0.642788i | 0.173648 | + | 0.984808i | 2.38757 | −0.475008 | − | 2.69391i | 0.500000 | − | 0.866025i | 2.06871 | + | 1.73585i | 0.500000 | + | 0.866025i | ||
181.1 | −0.173648 | + | 0.984808i | −0.420953 | − | 2.38734i | −0.939693 | − | 0.342020i | 0.766044 | − | 0.642788i | 2.42417 | 0.480222 | − | 0.402954i | 0.500000 | − | 0.866025i | −2.70313 | + | 0.983860i | 0.500000 | + | 0.866025i | ||
181.2 | −0.173648 | + | 0.984808i | −0.116582 | − | 0.661169i | −0.939693 | − | 0.342020i | 0.766044 | − | 0.642788i | 0.671368 | −1.79162 | + | 1.50334i | 0.500000 | − | 0.866025i | 2.39553 | − | 0.871900i | 0.500000 | + | 0.866025i | ||
181.3 | −0.173648 | + | 0.984808i | 0.160396 | + | 0.909649i | −0.939693 | − | 0.342020i | 0.766044 | − | 0.642788i | −0.923681 | 1.63147 | − | 1.36896i | 0.500000 | − | 0.866025i | 2.01734 | − | 0.734253i | 0.500000 | + | 0.866025i | ||
181.4 | −0.173648 | + | 0.984808i | 0.550788 | + | 3.12367i | −0.939693 | − | 0.342020i | 0.766044 | − | 0.642788i | −3.17186 | 0.712016 | − | 0.597452i | 0.500000 | − | 0.866025i | −6.63488 | + | 2.41490i | 0.500000 | + | 0.866025i | ||
201.1 | 0.939693 | + | 0.342020i | −2.67866 | + | 0.974954i | 0.766044 | + | 0.642788i | 0.173648 | − | 0.984808i | −2.85057 | −0.751904 | + | 4.26426i | 0.500000 | + | 0.866025i | 3.92657 | − | 3.29478i | 0.500000 | − | 0.866025i | ||
201.2 | 0.939693 | + | 0.342020i | −2.36335 | + | 0.860188i | 0.766044 | + | 0.642788i | 0.173648 | − | 0.984808i | −2.51502 | 0.832620 | − | 4.72202i | 0.500000 | + | 0.866025i | 2.54735 | − | 2.13748i | 0.500000 | − | 0.866025i | ||
201.3 | 0.939693 | + | 0.342020i | 1.85873 | − | 0.676523i | 0.766044 | + | 0.642788i | 0.173648 | − | 0.984808i | 1.97802 | 0.241589 | − | 1.37012i | 0.500000 | + | 0.866025i | 0.699071 | − | 0.586590i | 0.500000 | − | 0.866025i | ||
201.4 | 0.939693 | + | 0.342020i | 2.24358 | − | 0.816598i | 0.766044 | + | 0.642788i | 0.173648 | − | 0.984808i | 2.38757 | −0.475008 | + | 2.69391i | 0.500000 | + | 0.866025i | 2.06871 | − | 1.73585i | 0.500000 | − | 0.866025i | ||
231.1 | −0.173648 | − | 0.984808i | −0.420953 | + | 2.38734i | −0.939693 | + | 0.342020i | 0.766044 | + | 0.642788i | 2.42417 | 0.480222 | + | 0.402954i | 0.500000 | + | 0.866025i | −2.70313 | − | 0.983860i | 0.500000 | − | 0.866025i | ||
231.2 | −0.173648 | − | 0.984808i | −0.116582 | + | 0.661169i | −0.939693 | + | 0.342020i | 0.766044 | + | 0.642788i | 0.671368 | −1.79162 | − | 1.50334i | 0.500000 | + | 0.866025i | 2.39553 | + | 0.871900i | 0.500000 | − | 0.866025i | ||
231.3 | −0.173648 | − | 0.984808i | 0.160396 | − | 0.909649i | −0.939693 | + | 0.342020i | 0.766044 | + | 0.642788i | −0.923681 | 1.63147 | + | 1.36896i | 0.500000 | + | 0.866025i | 2.01734 | + | 0.734253i | 0.500000 | − | 0.866025i | ||
231.4 | −0.173648 | − | 0.984808i | 0.550788 | − | 3.12367i | −0.939693 | + | 0.342020i | 0.766044 | + | 0.642788i | −3.17186 | 0.712016 | + | 0.597452i | 0.500000 | + | 0.866025i | −6.63488 | − | 2.41490i | 0.500000 | − | 0.866025i | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.f | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 370.2.o.c | ✓ | 24 |
37.f | even | 9 | 1 | inner | 370.2.o.c | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
370.2.o.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
370.2.o.c | ✓ | 24 | 37.f | even | 9 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 6 T_{3}^{22} + 10 T_{3}^{21} + 54 T_{3}^{20} - 189 T_{3}^{19} + 550 T_{3}^{18} + \cdots + 179776 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).