Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1008,2,Mod(31,1008)]
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, 2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1008.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1008.bf (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: | \(32\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 0 | −1.72863 | − | 0.108820i | 0 | − | 0.435427i | 0 | 2.49096 | + | 0.891704i | 0 | 2.97632 | + | 0.376218i | 0 | |||||||||||
31.2 | 0 | −1.72423 | − | 0.164453i | 0 | − | 2.22064i | 0 | −0.517968 | − | 2.59455i | 0 | 2.94591 | + | 0.567107i | 0 | |||||||||||
31.3 | 0 | −1.33729 | + | 1.10075i | 0 | 1.09736i | 0 | 1.10288 | + | 2.40492i | 0 | 0.576701 | − | 2.94405i | 0 | ||||||||||||
31.4 | 0 | −1.14316 | + | 1.30123i | 0 | 4.29566i | 0 | 0.334043 | − | 2.62458i | 0 | −0.386381 | − | 2.97501i | 0 | ||||||||||||
31.5 | 0 | −1.13731 | − | 1.30634i | 0 | 0.618621i | 0 | −1.03639 | + | 2.43431i | 0 | −0.413051 | + | 2.97143i | 0 | ||||||||||||
31.6 | 0 | −0.840104 | + | 1.51467i | 0 | − | 2.34835i | 0 | −2.61922 | + | 0.373730i | 0 | −1.58845 | − | 2.54496i | 0 | |||||||||||
31.7 | 0 | −0.624780 | − | 1.61544i | 0 | 2.39618i | 0 | 2.35781 | − | 1.20030i | 0 | −2.21930 | + | 2.01859i | 0 | ||||||||||||
31.8 | 0 | −0.232652 | − | 1.71635i | 0 | − | 3.40340i | 0 | −2.63727 | − | 0.211643i | 0 | −2.89175 | + | 0.798628i | 0 | |||||||||||
31.9 | 0 | 0.232652 | + | 1.71635i | 0 | − | 3.40340i | 0 | 2.63727 | + | 0.211643i | 0 | −2.89175 | + | 0.798628i | 0 | |||||||||||
31.10 | 0 | 0.624780 | + | 1.61544i | 0 | 2.39618i | 0 | −2.35781 | + | 1.20030i | 0 | −2.21930 | + | 2.01859i | 0 | ||||||||||||
31.11 | 0 | 0.840104 | − | 1.51467i | 0 | − | 2.34835i | 0 | 2.61922 | − | 0.373730i | 0 | −1.58845 | − | 2.54496i | 0 | |||||||||||
31.12 | 0 | 1.13731 | + | 1.30634i | 0 | 0.618621i | 0 | 1.03639 | − | 2.43431i | 0 | −0.413051 | + | 2.97143i | 0 | ||||||||||||
31.13 | 0 | 1.14316 | − | 1.30123i | 0 | 4.29566i | 0 | −0.334043 | + | 2.62458i | 0 | −0.386381 | − | 2.97501i | 0 | ||||||||||||
31.14 | 0 | 1.33729 | − | 1.10075i | 0 | 1.09736i | 0 | −1.10288 | − | 2.40492i | 0 | 0.576701 | − | 2.94405i | 0 | ||||||||||||
31.15 | 0 | 1.72423 | + | 0.164453i | 0 | − | 2.22064i | 0 | 0.517968 | + | 2.59455i | 0 | 2.94591 | + | 0.567107i | 0 | |||||||||||
31.16 | 0 | 1.72863 | + | 0.108820i | 0 | − | 0.435427i | 0 | −2.49096 | − | 0.891704i | 0 | 2.97632 | + | 0.376218i | 0 | |||||||||||
943.1 | 0 | −1.72863 | + | 0.108820i | 0 | 0.435427i | 0 | 2.49096 | − | 0.891704i | 0 | 2.97632 | − | 0.376218i | 0 | ||||||||||||
943.2 | 0 | −1.72423 | + | 0.164453i | 0 | 2.22064i | 0 | −0.517968 | + | 2.59455i | 0 | 2.94591 | − | 0.567107i | 0 | ||||||||||||
943.3 | 0 | −1.33729 | − | 1.10075i | 0 | − | 1.09736i | 0 | 1.10288 | − | 2.40492i | 0 | 0.576701 | + | 2.94405i | 0 | |||||||||||
943.4 | 0 | −1.14316 | − | 1.30123i | 0 | − | 4.29566i | 0 | 0.334043 | + | 2.62458i | 0 | −0.386381 | + | 2.97501i | 0 | |||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
63.k | odd | 6 | 1 | inner |
252.n | 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.bf.i | ✓ | 32 |
3.b | odd | 2 | 1 | 3024.2.bf.i | 32 | ||
4.b | odd | 2 | 1 | inner | 1008.2.bf.i | ✓ | 32 |
7.d | odd | 6 | 1 | 1008.2.cz.i | yes | 32 | |
9.c | even | 3 | 1 | 1008.2.cz.i | yes | 32 | |
9.d | odd | 6 | 1 | 3024.2.cz.i | 32 | ||
12.b | even | 2 | 1 | 3024.2.bf.i | 32 | ||
21.g | even | 6 | 1 | 3024.2.cz.i | 32 | ||
28.f | even | 6 | 1 | 1008.2.cz.i | yes | 32 | |
36.f | odd | 6 | 1 | 1008.2.cz.i | yes | 32 | |
36.h | even | 6 | 1 | 3024.2.cz.i | 32 | ||
63.k | odd | 6 | 1 | inner | 1008.2.bf.i | ✓ | 32 |
63.s | even | 6 | 1 | 3024.2.bf.i | 32 | ||
84.j | odd | 6 | 1 | 3024.2.cz.i | 32 | ||
252.n | even | 6 | 1 | inner | 1008.2.bf.i | ✓ | 32 |
252.bn | odd | 6 | 1 | 3024.2.bf.i | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1008.2.bf.i | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
1008.2.bf.i | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
1008.2.bf.i | ✓ | 32 | 63.k | odd | 6 | 1 | inner |
1008.2.bf.i | ✓ | 32 | 252.n | even | 6 | 1 | inner |
1008.2.cz.i | yes | 32 | 7.d | odd | 6 | 1 | |
1008.2.cz.i | yes | 32 | 9.c | even | 3 | 1 | |
1008.2.cz.i | yes | 32 | 28.f | even | 6 | 1 | |
1008.2.cz.i | yes | 32 | 36.f | odd | 6 | 1 | |
3024.2.bf.i | 32 | 3.b | odd | 2 | 1 | ||
3024.2.bf.i | 32 | 12.b | even | 2 | 1 | ||
3024.2.bf.i | 32 | 63.s | even | 6 | 1 | ||
3024.2.bf.i | 32 | 252.bn | odd | 6 | 1 | ||
3024.2.cz.i | 32 | 9.d | odd | 6 | 1 | ||
3024.2.cz.i | 32 | 21.g | even | 6 | 1 | ||
3024.2.cz.i | 32 | 36.h | even | 6 | 1 | ||
3024.2.cz.i | 32 | 84.j | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):
\( T_{5}^{16} + 48T_{5}^{14} + 870T_{5}^{12} + 7668T_{5}^{10} + 35001T_{5}^{8} + 79623T_{5}^{6} + 77598T_{5}^{4} + 27459T_{5}^{2} + 2916 \) |
\( T_{19}^{32} + 146 T_{19}^{30} + 13545 T_{19}^{28} + 760432 T_{19}^{26} + 30956012 T_{19}^{24} + 845979156 T_{19}^{22} + 16893351658 T_{19}^{20} + 222174856073 T_{19}^{18} + \cdots + 156863626279441 \) |