Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [228,3,Mod(125,228)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(228, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("228.125");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 228 = 2^{2} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 228.o (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: | \(6.21255002741\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
125.1 | 0 | −2.99994 | − | 0.0182875i | 0 | 1.08094 | + | 0.624082i | 0 | −4.29950 | 0 | 8.99933 | + | 0.109723i | 0 | ||||||||||||
125.2 | 0 | −2.60151 | − | 1.49404i | 0 | −0.973208 | − | 0.561882i | 0 | 6.36523 | 0 | 4.53572 | + | 7.77350i | 0 | ||||||||||||
125.3 | 0 | −2.35298 | + | 1.86104i | 0 | 2.07590 | + | 1.19852i | 0 | −6.50893 | 0 | 2.07305 | − | 8.75799i | 0 | ||||||||||||
125.4 | 0 | −2.27725 | + | 1.95298i | 0 | −7.75465 | − | 4.47715i | 0 | 10.5834 | 0 | 1.37176 | − | 8.89485i | 0 | ||||||||||||
125.5 | 0 | −1.73353 | − | 2.44845i | 0 | 8.34937 | + | 4.82051i | 0 | −1.90607 | 0 | −2.98977 | + | 8.48889i | 0 | ||||||||||||
125.6 | 0 | −1.11376 | − | 2.78560i | 0 | −5.28947 | − | 3.05387i | 0 | −2.23411 | 0 | −6.51909 | + | 6.20495i | 0 | ||||||||||||
125.7 | 0 | −0.552701 | + | 2.94865i | 0 | 7.75465 | + | 4.47715i | 0 | 10.5834 | 0 | −8.38904 | − | 3.25944i | 0 | ||||||||||||
125.8 | 0 | −0.435218 | + | 2.96826i | 0 | −2.07590 | − | 1.19852i | 0 | −6.50893 | 0 | −8.62117 | − | 2.58368i | 0 | ||||||||||||
125.9 | 0 | 1.51581 | + | 2.58888i | 0 | −1.08094 | − | 0.624082i | 0 | −4.29950 | 0 | −4.40464 | + | 7.84851i | 0 | ||||||||||||
125.10 | 0 | 2.59463 | + | 1.50596i | 0 | 0.973208 | + | 0.561882i | 0 | 6.36523 | 0 | 4.46419 | + | 7.81480i | 0 | ||||||||||||
125.11 | 0 | 2.96928 | − | 0.428257i | 0 | 5.28947 | + | 3.05387i | 0 | −2.23411 | 0 | 8.63319 | − | 2.54322i | 0 | ||||||||||||
125.12 | 0 | 2.98718 | + | 0.277055i | 0 | −8.34937 | − | 4.82051i | 0 | −1.90607 | 0 | 8.84648 | + | 1.65522i | 0 | ||||||||||||
197.1 | 0 | −2.99994 | + | 0.0182875i | 0 | 1.08094 | − | 0.624082i | 0 | −4.29950 | 0 | 8.99933 | − | 0.109723i | 0 | ||||||||||||
197.2 | 0 | −2.60151 | + | 1.49404i | 0 | −0.973208 | + | 0.561882i | 0 | 6.36523 | 0 | 4.53572 | − | 7.77350i | 0 | ||||||||||||
197.3 | 0 | −2.35298 | − | 1.86104i | 0 | 2.07590 | − | 1.19852i | 0 | −6.50893 | 0 | 2.07305 | + | 8.75799i | 0 | ||||||||||||
197.4 | 0 | −2.27725 | − | 1.95298i | 0 | −7.75465 | + | 4.47715i | 0 | 10.5834 | 0 | 1.37176 | + | 8.89485i | 0 | ||||||||||||
197.5 | 0 | −1.73353 | + | 2.44845i | 0 | 8.34937 | − | 4.82051i | 0 | −1.90607 | 0 | −2.98977 | − | 8.48889i | 0 | ||||||||||||
197.6 | 0 | −1.11376 | + | 2.78560i | 0 | −5.28947 | + | 3.05387i | 0 | −2.23411 | 0 | −6.51909 | − | 6.20495i | 0 | ||||||||||||
197.7 | 0 | −0.552701 | − | 2.94865i | 0 | 7.75465 | − | 4.47715i | 0 | 10.5834 | 0 | −8.38904 | + | 3.25944i | 0 | ||||||||||||
197.8 | 0 | −0.435218 | − | 2.96826i | 0 | −2.07590 | + | 1.19852i | 0 | −6.50893 | 0 | −8.62117 | + | 2.58368i | 0 | ||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
57.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 228.3.o.c | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 228.3.o.c | ✓ | 24 |
19.c | even | 3 | 1 | inner | 228.3.o.c | ✓ | 24 |
57.h | odd | 6 | 1 | inner | 228.3.o.c | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
228.3.o.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
228.3.o.c | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
228.3.o.c | ✓ | 24 | 19.c | even | 3 | 1 | inner |
228.3.o.c | ✓ | 24 | 57.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(228, [\chi])\):
\( T_{5}^{24} - 219 T_{5}^{22} + 32229 T_{5}^{20} - 2643260 T_{5}^{18} + 157034700 T_{5}^{16} - 5159171592 T_{5}^{14} + 118202291248 T_{5}^{12} - 894201641520 T_{5}^{10} + \cdots + 9877191840000 \) |
\( T_{7}^{6} - 2T_{7}^{5} - 109T_{7}^{4} - 136T_{7}^{3} + 2562T_{7}^{2} + 8886T_{7} + 8028 \) |