Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [390,2,Mod(67,390)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(390, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("390.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 390.bn (of order \(12\), degree \(4\), 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: | \(3.11416567883\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −0.500000 | − | 0.866025i | −0.965926 | − | 0.258819i | −0.500000 | + | 0.866025i | −2.18531 | + | 0.473708i | 0.258819 | + | 0.965926i | 2.35599 | + | 1.36023i | 1.00000 | 0.866025 | + | 0.500000i | 1.50290 | + | 1.65568i | ||
67.2 | −0.500000 | − | 0.866025i | −0.965926 | − | 0.258819i | −0.500000 | + | 0.866025i | 0.0142140 | − | 2.23602i | 0.258819 | + | 0.965926i | −3.50522 | − | 2.02374i | 1.00000 | 0.866025 | + | 0.500000i | −1.94356 | + | 1.10570i | ||
67.3 | −0.500000 | − | 0.866025i | −0.965926 | − | 0.258819i | −0.500000 | + | 0.866025i | 0.933504 | − | 2.03189i | 0.258819 | + | 0.965926i | 3.73901 | + | 2.15872i | 1.00000 | 0.866025 | + | 0.500000i | −2.22642 | + | 0.207506i | ||
67.4 | −0.500000 | − | 0.866025i | −0.965926 | − | 0.258819i | −0.500000 | + | 0.866025i | 1.23760 | + | 1.86235i | 0.258819 | + | 0.965926i | −3.19321 | − | 1.84360i | 1.00000 | 0.866025 | + | 0.500000i | 0.994046 | − | 2.00297i | ||
67.5 | −0.500000 | − | 0.866025i | 0.965926 | + | 0.258819i | −0.500000 | + | 0.866025i | −2.23570 | + | 0.0407428i | −0.258819 | − | 0.965926i | 2.13987 | + | 1.23545i | 1.00000 | 0.866025 | + | 0.500000i | 1.15313 | + | 1.91580i | ||
67.6 | −0.500000 | − | 0.866025i | 0.965926 | + | 0.258819i | −0.500000 | + | 0.866025i | −1.64894 | + | 1.51030i | −0.258819 | − | 0.965926i | −3.98101 | − | 2.29844i | 1.00000 | 0.866025 | + | 0.500000i | 2.13243 | + | 0.672876i | ||
67.7 | −0.500000 | − | 0.866025i | 0.965926 | + | 0.258819i | −0.500000 | + | 0.866025i | 1.83584 | + | 1.27660i | −0.258819 | − | 0.965926i | 0.838691 | + | 0.484219i | 1.00000 | 0.866025 | + | 0.500000i | 0.187644 | − | 2.22818i | ||
67.8 | −0.500000 | − | 0.866025i | 0.965926 | + | 0.258819i | −0.500000 | + | 0.866025i | 2.04880 | − | 0.895783i | −0.258819 | − | 0.965926i | −1.39412 | − | 0.804897i | 1.00000 | 0.866025 | + | 0.500000i | −1.80017 | − | 1.32642i | ||
97.1 | −0.500000 | + | 0.866025i | −0.258819 | − | 0.965926i | −0.500000 | − | 0.866025i | −2.04652 | + | 0.900968i | 0.965926 | + | 0.258819i | −0.0517653 | + | 0.0298867i | 1.00000 | −0.866025 | + | 0.500000i | 0.243000 | − | 2.22283i | ||
97.2 | −0.500000 | + | 0.866025i | −0.258819 | − | 0.965926i | −0.500000 | − | 0.866025i | −0.849511 | − | 2.06841i | 0.965926 | + | 0.258819i | −1.91935 | + | 1.10814i | 1.00000 | −0.866025 | + | 0.500000i | 2.21605 | + | 0.298508i | ||
97.3 | −0.500000 | + | 0.866025i | −0.258819 | − | 0.965926i | −0.500000 | − | 0.866025i | 1.08182 | + | 1.95695i | 0.965926 | + | 0.258819i | −4.11737 | + | 2.37716i | 1.00000 | −0.866025 | + | 0.500000i | −2.23568 | − | 0.0415914i | ||
97.4 | −0.500000 | + | 0.866025i | −0.258819 | − | 0.965926i | −0.500000 | − | 0.866025i | 1.81421 | − | 1.30715i | 0.965926 | + | 0.258819i | 1.24242 | − | 0.717312i | 1.00000 | −0.866025 | + | 0.500000i | 0.224915 | + | 2.22473i | ||
97.5 | −0.500000 | + | 0.866025i | 0.258819 | + | 0.965926i | −0.500000 | − | 0.866025i | −1.65293 | − | 1.50593i | −0.965926 | − | 0.258819i | −1.08688 | + | 0.627513i | 1.00000 | −0.866025 | + | 0.500000i | 2.13064 | − | 0.678517i | ||
97.6 | −0.500000 | + | 0.866025i | 0.258819 | + | 0.965926i | −0.500000 | − | 0.866025i | −0.743195 | + | 2.10895i | −0.965926 | − | 0.258819i | −0.0594459 | + | 0.0343211i | 1.00000 | −0.866025 | + | 0.500000i | −1.45481 | − | 1.69810i | ||
97.7 | −0.500000 | + | 0.866025i | 0.258819 | + | 0.965926i | −0.500000 | − | 0.866025i | 1.13085 | − | 1.92904i | −0.965926 | − | 0.258819i | −0.245433 | + | 0.141701i | 1.00000 | −0.866025 | + | 0.500000i | 1.10517 | + | 1.94386i | ||
97.8 | −0.500000 | + | 0.866025i | 0.258819 | + | 0.965926i | −0.500000 | − | 0.866025i | 1.26528 | + | 1.84366i | −0.965926 | − | 0.258819i | 3.23783 | − | 1.86936i | 1.00000 | −0.866025 | + | 0.500000i | −2.22929 | + | 0.173937i | ||
163.1 | −0.500000 | + | 0.866025i | −0.965926 | + | 0.258819i | −0.500000 | − | 0.866025i | −2.18531 | − | 0.473708i | 0.258819 | − | 0.965926i | 2.35599 | − | 1.36023i | 1.00000 | 0.866025 | − | 0.500000i | 1.50290 | − | 1.65568i | ||
163.2 | −0.500000 | + | 0.866025i | −0.965926 | + | 0.258819i | −0.500000 | − | 0.866025i | 0.0142140 | + | 2.23602i | 0.258819 | − | 0.965926i | −3.50522 | + | 2.02374i | 1.00000 | 0.866025 | − | 0.500000i | −1.94356 | − | 1.10570i | ||
163.3 | −0.500000 | + | 0.866025i | −0.965926 | + | 0.258819i | −0.500000 | − | 0.866025i | 0.933504 | + | 2.03189i | 0.258819 | − | 0.965926i | 3.73901 | − | 2.15872i | 1.00000 | 0.866025 | − | 0.500000i | −2.22642 | − | 0.207506i | ||
163.4 | −0.500000 | + | 0.866025i | −0.965926 | + | 0.258819i | −0.500000 | − | 0.866025i | 1.23760 | − | 1.86235i | 0.258819 | − | 0.965926i | −3.19321 | + | 1.84360i | 1.00000 | 0.866025 | − | 0.500000i | 0.994046 | + | 2.00297i | ||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.o | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 390.2.bn.c | yes | 32 |
5.c | odd | 4 | 1 | 390.2.bd.c | ✓ | 32 | |
13.f | odd | 12 | 1 | 390.2.bd.c | ✓ | 32 | |
65.o | even | 12 | 1 | inner | 390.2.bn.c | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
390.2.bd.c | ✓ | 32 | 5.c | odd | 4 | 1 | |
390.2.bd.c | ✓ | 32 | 13.f | odd | 12 | 1 | |
390.2.bn.c | yes | 32 | 1.a | even | 1 | 1 | trivial |
390.2.bn.c | yes | 32 | 65.o | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{32} + 12 T_{7}^{31} + 6 T_{7}^{30} - 504 T_{7}^{29} - 1189 T_{7}^{28} + 14124 T_{7}^{27} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).