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: | \(24\) |
Relative dimension: | \(12\) 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 | −1.89131 | + | 1.19288i | −0.500000 | + | 0.866025i | 2.10840 | + | 1.21729i | 1.00000i | 0.500000 | + | 0.866025i | 1.19288 | + | 1.89131i | |||||
439.2 | − | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | −0.968769 | + | 2.01531i | −0.500000 | + | 0.866025i | −3.61895 | − | 2.08940i | 1.00000i | 0.500000 | + | 0.866025i | 2.01531 | + | 0.968769i | |||||
439.3 | − | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | −0.291362 | − | 2.21700i | −0.500000 | + | 0.866025i | 3.11266 | + | 1.79709i | 1.00000i | 0.500000 | + | 0.866025i | −2.21700 | + | 0.291362i | |||||
439.4 | − | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | 0.875319 | − | 2.05762i | −0.500000 | + | 0.866025i | 0.765960 | + | 0.442227i | 1.00000i | 0.500000 | + | 0.866025i | −2.05762 | − | 0.875319i | |||||
439.5 | − | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | 1.24897 | + | 1.85475i | −0.500000 | + | 0.866025i | −0.669647 | − | 0.386621i | 1.00000i | 0.500000 | + | 0.866025i | 1.85475 | − | 1.24897i | |||||
439.6 | − | 1.00000i | −0.866025 | − | 0.500000i | −1.00000 | 2.02715 | + | 0.943742i | −0.500000 | + | 0.866025i | 3.49773 | + | 2.01941i | 1.00000i | 0.500000 | + | 0.866025i | 0.943742 | − | 2.02715i | |||||
439.7 | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | −2.21961 | − | 0.270763i | −0.500000 | + | 0.866025i | −0.765960 | − | 0.442227i | − | 1.00000i | 0.500000 | + | 0.866025i | 0.270763 | − | 2.21961i | |||||
439.8 | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | −1.77430 | − | 1.36083i | −0.500000 | + | 0.866025i | −3.11266 | − | 1.79709i | − | 1.00000i | 0.500000 | + | 0.866025i | 1.36083 | − | 1.77430i | |||||
439.9 | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | −0.196273 | + | 2.22744i | −0.500000 | + | 0.866025i | −3.49773 | − | 2.01941i | − | 1.00000i | 0.500000 | + | 0.866025i | −2.22744 | − | 0.196273i | |||||
439.10 | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | 0.981775 | + | 2.00901i | −0.500000 | + | 0.866025i | 0.669647 | + | 0.386621i | − | 1.00000i | 0.500000 | + | 0.866025i | −2.00901 | + | 0.981775i | |||||
439.11 | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | 1.97872 | − | 1.04148i | −0.500000 | + | 0.866025i | −2.10840 | − | 1.21729i | − | 1.00000i | 0.500000 | + | 0.866025i | 1.04148 | + | 1.97872i | |||||
439.12 | 1.00000i | 0.866025 | + | 0.500000i | −1.00000 | 2.22970 | + | 0.168678i | −0.500000 | + | 0.866025i | 3.61895 | + | 2.08940i | − | 1.00000i | 0.500000 | + | 0.866025i | −0.168678 | + | 2.22970i | |||||
769.1 | − | 1.00000i | 0.866025 | − | 0.500000i | −1.00000 | −2.21961 | + | 0.270763i | −0.500000 | − | 0.866025i | −0.765960 | + | 0.442227i | 1.00000i | 0.500000 | − | 0.866025i | 0.270763 | + | 2.21961i | |||||
769.2 | − | 1.00000i | 0.866025 | − | 0.500000i | −1.00000 | −1.77430 | + | 1.36083i | −0.500000 | − | 0.866025i | −3.11266 | + | 1.79709i | 1.00000i | 0.500000 | − | 0.866025i | 1.36083 | + | 1.77430i | |||||
769.3 | − | 1.00000i | 0.866025 | − | 0.500000i | −1.00000 | −0.196273 | − | 2.22744i | −0.500000 | − | 0.866025i | −3.49773 | + | 2.01941i | 1.00000i | 0.500000 | − | 0.866025i | −2.22744 | + | 0.196273i | |||||
769.4 | − | 1.00000i | 0.866025 | − | 0.500000i | −1.00000 | 0.981775 | − | 2.00901i | −0.500000 | − | 0.866025i | 0.669647 | − | 0.386621i | 1.00000i | 0.500000 | − | 0.866025i | −2.00901 | − | 0.981775i | |||||
769.5 | − | 1.00000i | 0.866025 | − | 0.500000i | −1.00000 | 1.97872 | + | 1.04148i | −0.500000 | − | 0.866025i | −2.10840 | + | 1.21729i | 1.00000i | 0.500000 | − | 0.866025i | 1.04148 | − | 1.97872i | |||||
769.6 | − | 1.00000i | 0.866025 | − | 0.500000i | −1.00000 | 2.22970 | − | 0.168678i | −0.500000 | − | 0.866025i | 3.61895 | − | 2.08940i | 1.00000i | 0.500000 | − | 0.866025i | −0.168678 | − | 2.22970i | |||||
769.7 | 1.00000i | −0.866025 | + | 0.500000i | −1.00000 | −1.89131 | − | 1.19288i | −0.500000 | − | 0.866025i | 2.10840 | − | 1.21729i | − | 1.00000i | 0.500000 | − | 0.866025i | 1.19288 | − | 1.89131i | |||||
769.8 | 1.00000i | −0.866025 | + | 0.500000i | −1.00000 | −0.968769 | − | 2.01531i | −0.500000 | − | 0.866025i | −3.61895 | + | 2.08940i | − | 1.00000i | 0.500000 | − | 0.866025i | 2.01531 | − | 0.968769i | |||||
See all 24 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.c | ✓ | 24 |
5.b | even | 2 | 1 | inner | 930.2.s.c | ✓ | 24 |
31.c | even | 3 | 1 | inner | 930.2.s.c | ✓ | 24 |
155.j | even | 6 | 1 | inner | 930.2.s.c | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.s.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
930.2.s.c | ✓ | 24 | 5.b | even | 2 | 1 | inner |
930.2.s.c | ✓ | 24 | 31.c | even | 3 | 1 | inner |
930.2.s.c | ✓ | 24 | 155.j | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} - 54 T_{7}^{22} + 1845 T_{7}^{20} - 39122 T_{7}^{18} + 608562 T_{7}^{16} - 6462558 T_{7}^{14} + \cdots + 104060401 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).