Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1008,2,Mod(223,1008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1008.223");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1008.cx (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: | \(8.04892052375\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
223.1 | 0 | −1.71721 | − | 0.226267i | 0 | 1.72020 | − | 0.993161i | 0 | 1.81723 | + | 1.92293i | 0 | 2.89761 | + | 0.777096i | 0 | ||||||||||
223.2 | 0 | −1.69365 | + | 0.362704i | 0 | −2.10269 | + | 1.21399i | 0 | 1.15057 | − | 2.38247i | 0 | 2.73689 | − | 1.22858i | 0 | ||||||||||
223.3 | 0 | −1.44146 | + | 0.960311i | 0 | 0.675942 | − | 0.390255i | 0 | −1.89912 | + | 1.84210i | 0 | 1.15560 | − | 2.76850i | 0 | ||||||||||
223.4 | 0 | −1.21473 | − | 1.23468i | 0 | −3.37919 | + | 1.95098i | 0 | 0.667355 | + | 2.56020i | 0 | −0.0488500 | + | 2.99960i | 0 | ||||||||||
223.5 | 0 | −1.11570 | − | 1.32484i | 0 | −0.263424 | + | 0.152088i | 0 | −2.20588 | − | 1.46085i | 0 | −0.510411 | + | 2.95626i | 0 | ||||||||||
223.6 | 0 | −0.940521 | + | 1.45445i | 0 | 3.48918 | − | 2.01448i | 0 | −1.90020 | − | 1.84099i | 0 | −1.23084 | − | 2.73588i | 0 | ||||||||||
223.7 | 0 | 0.940521 | − | 1.45445i | 0 | −3.48918 | + | 2.01448i | 0 | −2.54444 | − | 0.725126i | 0 | −1.23084 | − | 2.73588i | 0 | ||||||||||
223.8 | 0 | 1.11570 | + | 1.32484i | 0 | 0.263424 | − | 0.152088i | 0 | −2.36807 | − | 1.17993i | 0 | −0.510411 | + | 2.95626i | 0 | ||||||||||
223.9 | 0 | 1.21473 | + | 1.23468i | 0 | 3.37919 | − | 1.95098i | 0 | 2.55088 | − | 0.702155i | 0 | −0.0488500 | + | 2.99960i | 0 | ||||||||||
223.10 | 0 | 1.44146 | − | 0.960311i | 0 | −0.675942 | + | 0.390255i | 0 | 0.645750 | − | 2.56574i | 0 | 1.15560 | − | 2.76850i | 0 | ||||||||||
223.11 | 0 | 1.69365 | − | 0.362704i | 0 | 2.10269 | − | 1.21399i | 0 | −1.48800 | + | 2.18766i | 0 | 2.73689 | − | 1.22858i | 0 | ||||||||||
223.12 | 0 | 1.71721 | + | 0.226267i | 0 | −1.72020 | + | 0.993161i | 0 | 2.57392 | + | 0.612306i | 0 | 2.89761 | + | 0.777096i | 0 | ||||||||||
895.1 | 0 | −1.71721 | + | 0.226267i | 0 | 1.72020 | + | 0.993161i | 0 | 1.81723 | − | 1.92293i | 0 | 2.89761 | − | 0.777096i | 0 | ||||||||||
895.2 | 0 | −1.69365 | − | 0.362704i | 0 | −2.10269 | − | 1.21399i | 0 | 1.15057 | + | 2.38247i | 0 | 2.73689 | + | 1.22858i | 0 | ||||||||||
895.3 | 0 | −1.44146 | − | 0.960311i | 0 | 0.675942 | + | 0.390255i | 0 | −1.89912 | − | 1.84210i | 0 | 1.15560 | + | 2.76850i | 0 | ||||||||||
895.4 | 0 | −1.21473 | + | 1.23468i | 0 | −3.37919 | − | 1.95098i | 0 | 0.667355 | − | 2.56020i | 0 | −0.0488500 | − | 2.99960i | 0 | ||||||||||
895.5 | 0 | −1.11570 | + | 1.32484i | 0 | −0.263424 | − | 0.152088i | 0 | −2.20588 | + | 1.46085i | 0 | −0.510411 | − | 2.95626i | 0 | ||||||||||
895.6 | 0 | −0.940521 | − | 1.45445i | 0 | 3.48918 | + | 2.01448i | 0 | −1.90020 | + | 1.84099i | 0 | −1.23084 | + | 2.73588i | 0 | ||||||||||
895.7 | 0 | 0.940521 | + | 1.45445i | 0 | −3.48918 | − | 2.01448i | 0 | −2.54444 | + | 0.725126i | 0 | −1.23084 | + | 2.73588i | 0 | ||||||||||
895.8 | 0 | 1.11570 | − | 1.32484i | 0 | 0.263424 | + | 0.152088i | 0 | −2.36807 | + | 1.17993i | 0 | −0.510411 | − | 2.95626i | 0 | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
36.f | odd | 6 | 1 | inner |
252.bi | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1008.2.cx.i | ✓ | 24 |
3.b | odd | 2 | 1 | 3024.2.cx.i | 24 | ||
4.b | odd | 2 | 1 | 1008.2.cx.j | yes | 24 | |
7.b | odd | 2 | 1 | inner | 1008.2.cx.i | ✓ | 24 |
9.c | even | 3 | 1 | 1008.2.cx.j | yes | 24 | |
9.d | odd | 6 | 1 | 3024.2.cx.j | 24 | ||
12.b | even | 2 | 1 | 3024.2.cx.j | 24 | ||
21.c | even | 2 | 1 | 3024.2.cx.i | 24 | ||
28.d | even | 2 | 1 | 1008.2.cx.j | yes | 24 | |
36.f | odd | 6 | 1 | inner | 1008.2.cx.i | ✓ | 24 |
36.h | even | 6 | 1 | 3024.2.cx.i | 24 | ||
63.l | odd | 6 | 1 | 1008.2.cx.j | yes | 24 | |
63.o | even | 6 | 1 | 3024.2.cx.j | 24 | ||
84.h | odd | 2 | 1 | 3024.2.cx.j | 24 | ||
252.s | odd | 6 | 1 | 3024.2.cx.i | 24 | ||
252.bi | even | 6 | 1 | inner | 1008.2.cx.i | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1008.2.cx.i | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
1008.2.cx.i | ✓ | 24 | 7.b | odd | 2 | 1 | inner |
1008.2.cx.i | ✓ | 24 | 36.f | odd | 6 | 1 | inner |
1008.2.cx.i | ✓ | 24 | 252.bi | even | 6 | 1 | inner |
1008.2.cx.j | yes | 24 | 4.b | odd | 2 | 1 | |
1008.2.cx.j | yes | 24 | 9.c | even | 3 | 1 | |
1008.2.cx.j | yes | 24 | 28.d | even | 2 | 1 | |
1008.2.cx.j | yes | 24 | 63.l | odd | 6 | 1 | |
3024.2.cx.i | 24 | 3.b | odd | 2 | 1 | ||
3024.2.cx.i | 24 | 21.c | even | 2 | 1 | ||
3024.2.cx.i | 24 | 36.h | even | 6 | 1 | ||
3024.2.cx.i | 24 | 252.s | odd | 6 | 1 | ||
3024.2.cx.j | 24 | 9.d | odd | 6 | 1 | ||
3024.2.cx.j | 24 | 12.b | even | 2 | 1 | ||
3024.2.cx.j | 24 | 63.o | even | 6 | 1 | ||
3024.2.cx.j | 24 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):
\( T_{5}^{24} - 42 T_{5}^{22} + 1155 T_{5}^{20} - 18432 T_{5}^{18} + 212814 T_{5}^{16} - 1508085 T_{5}^{14} + \cdots + 104976 \) |
\( T_{11}^{12} - 42 T_{11}^{10} + 1455 T_{11}^{8} - 81 T_{11}^{7} - 12627 T_{11}^{6} + 2781 T_{11}^{5} + \cdots + 26244 \) |