Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(439,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.439");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.s (of order \(6\), 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: | \(7.42608738798\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
439.1 | − | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | −2.16404 | − | 0.562979i | 0.500000 | − | 0.866025i | 0.759514 | + | 0.438506i | 1.00000i | 0.500000 | + | 0.866025i | −0.562979 | + | 2.16404i | |||||
439.2 | − | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | −1.82738 | + | 1.28868i | 0.500000 | − | 0.866025i | −2.60552 | − | 1.50429i | 1.00000i | 0.500000 | + | 0.866025i | 1.28868 | + | 1.82738i | |||||
439.3 | − | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | −0.670801 | − | 2.13308i | 0.500000 | − | 0.866025i | −1.77376 | − | 1.02408i | 1.00000i | 0.500000 | + | 0.866025i | −2.13308 | + | 0.670801i | |||||
439.4 | − | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | −0.347530 | + | 2.20890i | 0.500000 | − | 0.866025i | 3.10811 | + | 1.79447i | 1.00000i | 0.500000 | + | 0.866025i | 2.20890 | + | 0.347530i | |||||
439.5 | − | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | 0.836743 | − | 2.07361i | 0.500000 | − | 0.866025i | 1.24405 | + | 0.718253i | 1.00000i | 0.500000 | + | 0.866025i | −2.07361 | − | 0.836743i | |||||
439.6 | − | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | 1.02320 | + | 1.98823i | 0.500000 | − | 0.866025i | −3.96572 | − | 2.28961i | 1.00000i | 0.500000 | + | 0.866025i | 1.98823 | − | 1.02320i | |||||
439.7 | − | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | 1.91775 | + | 1.14989i | 0.500000 | − | 0.866025i | −0.230788 | − | 0.133246i | 1.00000i | 0.500000 | + | 0.866025i | 1.14989 | − | 1.91775i | |||||
439.8 | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | −2.21417 | − | 0.312164i | 0.500000 | − | 0.866025i | −1.24405 | − | 0.718253i | − | 1.00000i | 0.500000 | + | 0.866025i | 0.312164 | − | 2.21417i | |||||
439.9 | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | −1.51190 | − | 1.64747i | 0.500000 | − | 0.866025i | 1.77376 | + | 1.02408i | − | 1.00000i | 0.500000 | + | 0.866025i | 1.64747 | − | 1.51190i | |||||
439.10 | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | 0.0369558 | + | 2.23576i | 0.500000 | − | 0.866025i | 0.230788 | + | 0.133246i | − | 1.00000i | 0.500000 | + | 0.866025i | −2.23576 | + | 0.0369558i | |||||
439.11 | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | 0.594464 | − | 2.15560i | 0.500000 | − | 0.866025i | −0.759514 | − | 0.438506i | − | 1.00000i | 0.500000 | + | 0.866025i | 2.15560 | + | 0.594464i | |||||
439.12 | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | 1.21026 | + | 1.88023i | 0.500000 | − | 0.866025i | 3.96572 | + | 2.28961i | − | 1.00000i | 0.500000 | + | 0.866025i | −1.88023 | + | 1.21026i | |||||
439.13 | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | 2.02972 | − | 0.938214i | 0.500000 | − | 0.866025i | 2.60552 | + | 1.50429i | − | 1.00000i | 0.500000 | + | 0.866025i | 0.938214 | + | 2.02972i | |||||
439.14 | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | 2.08673 | + | 0.803479i | 0.500000 | − | 0.866025i | −3.10811 | − | 1.79447i | − | 1.00000i | 0.500000 | + | 0.866025i | −0.803479 | + | 2.08673i | |||||
769.1 | − | 1.00000i | −0.866025 | + | 0.500000i | −1.00000 | −2.21417 | + | 0.312164i | 0.500000 | + | 0.866025i | −1.24405 | + | 0.718253i | 1.00000i | 0.500000 | − | 0.866025i | 0.312164 | + | 2.21417i | |||||
769.2 | − | 1.00000i | −0.866025 | + | 0.500000i | −1.00000 | −1.51190 | + | 1.64747i | 0.500000 | + | 0.866025i | 1.77376 | − | 1.02408i | 1.00000i | 0.500000 | − | 0.866025i | 1.64747 | + | 1.51190i | |||||
769.3 | − | 1.00000i | −0.866025 | + | 0.500000i | −1.00000 | 0.0369558 | − | 2.23576i | 0.500000 | + | 0.866025i | 0.230788 | − | 0.133246i | 1.00000i | 0.500000 | − | 0.866025i | −2.23576 | − | 0.0369558i | |||||
769.4 | − | 1.00000i | −0.866025 | + | 0.500000i | −1.00000 | 0.594464 | + | 2.15560i | 0.500000 | + | 0.866025i | −0.759514 | + | 0.438506i | 1.00000i | 0.500000 | − | 0.866025i | 2.15560 | − | 0.594464i | |||||
769.5 | − | 1.00000i | −0.866025 | + | 0.500000i | −1.00000 | 1.21026 | − | 1.88023i | 0.500000 | + | 0.866025i | 3.96572 | − | 2.28961i | 1.00000i | 0.500000 | − | 0.866025i | −1.88023 | − | 1.21026i | |||||
769.6 | − | 1.00000i | −0.866025 | + | 0.500000i | −1.00000 | 2.02972 | + | 0.938214i | 0.500000 | + | 0.866025i | 2.60552 | − | 1.50429i | 1.00000i | 0.500000 | − | 0.866025i | 0.938214 | − | 2.02972i | |||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
31.c | even | 3 | 1 | inner |
155.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 930.2.s.d | ✓ | 28 |
5.b | even | 2 | 1 | inner | 930.2.s.d | ✓ | 28 |
31.c | even | 3 | 1 | inner | 930.2.s.d | ✓ | 28 |
155.j | even | 6 | 1 | inner | 930.2.s.d | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.s.d | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
930.2.s.d | ✓ | 28 | 5.b | even | 2 | 1 | inner |
930.2.s.d | ✓ | 28 | 31.c | even | 3 | 1 | inner |
930.2.s.d | ✓ | 28 | 155.j | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{28} - 50 T_{7}^{26} + 1605 T_{7}^{24} - 30462 T_{7}^{22} + 418090 T_{7}^{20} - 3858942 T_{7}^{18} + 26113589 T_{7}^{16} - 115770466 T_{7}^{14} + 370013242 T_{7}^{12} - 723541658 T_{7}^{10} + \cdots + 1336336 \)
acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).