Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(41,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.41");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 630.bl (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: | \(5.03057532734\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 | −0.866025 | + | 0.500000i | −1.54760 | − | 0.777773i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 1.72915 | − | 0.100230i | −1.61204 | − | 2.09793i | 1.00000i | 1.79014 | + | 2.40736i | 1.00000i | ||||
41.2 | −0.866025 | + | 0.500000i | −1.23327 | − | 1.21616i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 1.67612 | + | 0.436589i | 2.26973 | + | 1.35953i | 1.00000i | 0.0419137 | + | 2.99971i | 1.00000i | ||||
41.3 | −0.866025 | + | 0.500000i | −1.02449 | + | 1.39657i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 0.188953 | − | 1.72171i | −1.87969 | − | 1.86193i | 1.00000i | −0.900821 | − | 2.86156i | 1.00000i | ||||
41.4 | −0.866025 | + | 0.500000i | 0.258130 | + | 1.71271i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −1.07990 | − | 1.35418i | 2.39121 | − | 1.13230i | 1.00000i | −2.86674 | + | 0.884201i | 1.00000i | ||||
41.5 | −0.866025 | + | 0.500000i | 0.325592 | − | 1.70117i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 0.568616 | + | 1.63606i | −1.21776 | + | 2.34884i | 1.00000i | −2.78798 | − | 1.10778i | 1.00000i | ||||
41.6 | −0.866025 | + | 0.500000i | 0.628600 | + | 1.61396i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −1.35136 | − | 1.08343i | −2.64350 | − | 0.109131i | 1.00000i | −2.20972 | + | 2.02907i | 1.00000i | ||||
41.7 | −0.866025 | + | 0.500000i | 0.998657 | − | 1.41516i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −0.157281 | + | 1.72489i | 2.33995 | − | 1.23477i | 1.00000i | −1.00537 | − | 2.82652i | 1.00000i | ||||
41.8 | −0.866025 | + | 0.500000i | 1.72836 | − | 0.112970i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −1.44032 | + | 0.962017i | −1.87995 | + | 1.86166i | 1.00000i | 2.97448 | − | 0.390508i | 1.00000i | ||||
41.9 | 0.866025 | − | 0.500000i | −1.62160 | − | 0.608614i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −1.70865 | + | 0.283725i | 2.49147 | + | 0.890259i | − | 1.00000i | 2.25918 | + | 1.97386i | − | 1.00000i | ||
41.10 | 0.866025 | − | 0.500000i | −1.49242 | + | 0.879021i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −0.852964 | + | 1.50747i | −2.63873 | − | 0.192579i | − | 1.00000i | 1.45464 | − | 2.62374i | − | 1.00000i | ||
41.11 | 0.866025 | − | 0.500000i | −0.600052 | + | 1.62479i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 0.292734 | + | 1.70713i | −0.0777835 | + | 2.64461i | − | 1.00000i | −2.27988 | − | 1.94991i | − | 1.00000i | ||
41.12 | 0.866025 | − | 0.500000i | 0.150193 | − | 1.72553i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −0.732692 | − | 1.56945i | 1.07735 | − | 2.41647i | − | 1.00000i | −2.95488 | − | 0.518324i | − | 1.00000i | ||
41.13 | 0.866025 | − | 0.500000i | 0.954824 | + | 1.44510i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 1.54945 | + | 0.774080i | −1.08530 | − | 2.41291i | − | 1.00000i | −1.17662 | + | 2.75963i | − | 1.00000i | ||
41.14 | 0.866025 | − | 0.500000i | 1.17115 | − | 1.27609i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 0.376205 | − | 1.69070i | −2.62158 | − | 0.356853i | − | 1.00000i | −0.256800 | − | 2.98899i | − | 1.00000i | ||
41.15 | 0.866025 | − | 0.500000i | 1.62688 | + | 0.594351i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 1.70610 | − | 0.298718i | 2.41754 | − | 1.07493i | − | 1.00000i | 2.29349 | + | 1.93388i | − | 1.00000i | ||
41.16 | 0.866025 | − | 0.500000i | 1.67705 | − | 0.433031i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 1.23585 | − | 1.21354i | 1.66908 | + | 2.05285i | − | 1.00000i | 2.62497 | − | 1.45243i | − | 1.00000i | ||
461.1 | −0.866025 | − | 0.500000i | −1.54760 | + | 0.777773i | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 1.72915 | + | 0.100230i | −1.61204 | + | 2.09793i | − | 1.00000i | 1.79014 | − | 2.40736i | − | 1.00000i | ||
461.2 | −0.866025 | − | 0.500000i | −1.23327 | + | 1.21616i | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 1.67612 | − | 0.436589i | 2.26973 | − | 1.35953i | − | 1.00000i | 0.0419137 | − | 2.99971i | − | 1.00000i | ||
461.3 | −0.866025 | − | 0.500000i | −1.02449 | − | 1.39657i | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 0.188953 | + | 1.72171i | −1.87969 | + | 1.86193i | − | 1.00000i | −0.900821 | + | 2.86156i | − | 1.00000i | ||
461.4 | −0.866025 | − | 0.500000i | 0.258130 | − | 1.71271i | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | −1.07990 | + | 1.35418i | 2.39121 | + | 1.13230i | − | 1.00000i | −2.86674 | − | 0.884201i | − | 1.00000i | ||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.o | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 630.2.bl.b | yes | 32 |
3.b | odd | 2 | 1 | 1890.2.bl.a | 32 | ||
7.b | odd | 2 | 1 | 630.2.bl.a | ✓ | 32 | |
9.c | even | 3 | 1 | 1890.2.bl.b | 32 | ||
9.d | odd | 6 | 1 | 630.2.bl.a | ✓ | 32 | |
21.c | even | 2 | 1 | 1890.2.bl.b | 32 | ||
63.l | odd | 6 | 1 | 1890.2.bl.a | 32 | ||
63.o | even | 6 | 1 | inner | 630.2.bl.b | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.2.bl.a | ✓ | 32 | 7.b | odd | 2 | 1 | |
630.2.bl.a | ✓ | 32 | 9.d | odd | 6 | 1 | |
630.2.bl.b | yes | 32 | 1.a | even | 1 | 1 | trivial |
630.2.bl.b | yes | 32 | 63.o | even | 6 | 1 | inner |
1890.2.bl.a | 32 | 3.b | odd | 2 | 1 | ||
1890.2.bl.a | 32 | 63.l | odd | 6 | 1 | ||
1890.2.bl.b | 32 | 9.c | even | 3 | 1 | ||
1890.2.bl.b | 32 | 21.c | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{32} - 6 T_{13}^{31} - 105 T_{13}^{30} + 702 T_{13}^{29} + 7017 T_{13}^{28} + \cdots + 22\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\).