Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2268,4,Mod(1133,2268)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2268, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2268.1133");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 2268.f (of order \(2\), degree \(1\), 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: | \(133.816331893\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | no (minimal twist has level 252) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1133.1 | 0 | 0 | 0 | −21.1141 | 0 | −13.5800 | + | 12.5930i | 0 | 0 | 0 | ||||||||||||||||
1133.2 | 0 | 0 | 0 | −21.1141 | 0 | −13.5800 | − | 12.5930i | 0 | 0 | 0 | ||||||||||||||||
1133.3 | 0 | 0 | 0 | −18.2402 | 0 | 18.5185 | − | 0.256813i | 0 | 0 | 0 | ||||||||||||||||
1133.4 | 0 | 0 | 0 | −18.2402 | 0 | 18.5185 | + | 0.256813i | 0 | 0 | 0 | ||||||||||||||||
1133.5 | 0 | 0 | 0 | −16.5975 | 0 | −11.9446 | − | 14.1537i | 0 | 0 | 0 | ||||||||||||||||
1133.6 | 0 | 0 | 0 | −16.5975 | 0 | −11.9446 | + | 14.1537i | 0 | 0 | 0 | ||||||||||||||||
1133.7 | 0 | 0 | 0 | −15.6490 | 0 | −3.87814 | − | 18.1097i | 0 | 0 | 0 | ||||||||||||||||
1133.8 | 0 | 0 | 0 | −15.6490 | 0 | −3.87814 | + | 18.1097i | 0 | 0 | 0 | ||||||||||||||||
1133.9 | 0 | 0 | 0 | −12.0714 | 0 | 14.8834 | + | 11.0221i | 0 | 0 | 0 | ||||||||||||||||
1133.10 | 0 | 0 | 0 | −12.0714 | 0 | 14.8834 | − | 11.0221i | 0 | 0 | 0 | ||||||||||||||||
1133.11 | 0 | 0 | 0 | −10.9938 | 0 | 11.4708 | − | 14.5403i | 0 | 0 | 0 | ||||||||||||||||
1133.12 | 0 | 0 | 0 | −10.9938 | 0 | 11.4708 | + | 14.5403i | 0 | 0 | 0 | ||||||||||||||||
1133.13 | 0 | 0 | 0 | −10.3297 | 0 | 2.73026 | + | 18.3179i | 0 | 0 | 0 | ||||||||||||||||
1133.14 | 0 | 0 | 0 | −10.3297 | 0 | 2.73026 | − | 18.3179i | 0 | 0 | 0 | ||||||||||||||||
1133.15 | 0 | 0 | 0 | −7.06893 | 0 | −18.4258 | − | 1.86777i | 0 | 0 | 0 | ||||||||||||||||
1133.16 | 0 | 0 | 0 | −7.06893 | 0 | −18.4258 | + | 1.86777i | 0 | 0 | 0 | ||||||||||||||||
1133.17 | 0 | 0 | 0 | −5.99994 | 0 | −15.8480 | − | 9.58340i | 0 | 0 | 0 | ||||||||||||||||
1133.18 | 0 | 0 | 0 | −5.99994 | 0 | −15.8480 | + | 9.58340i | 0 | 0 | 0 | ||||||||||||||||
1133.19 | 0 | 0 | 0 | −4.68539 | 0 | −7.47522 | − | 16.9446i | 0 | 0 | 0 | ||||||||||||||||
1133.20 | 0 | 0 | 0 | −4.68539 | 0 | −7.47522 | + | 16.9446i | 0 | 0 | 0 | ||||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2268.4.f.a | 48 | |
3.b | odd | 2 | 1 | inner | 2268.4.f.a | 48 | |
7.b | odd | 2 | 1 | inner | 2268.4.f.a | 48 | |
9.c | even | 3 | 1 | 252.4.x.a | ✓ | 48 | |
9.c | even | 3 | 1 | 756.4.x.a | 48 | ||
9.d | odd | 6 | 1 | 252.4.x.a | ✓ | 48 | |
9.d | odd | 6 | 1 | 756.4.x.a | 48 | ||
21.c | even | 2 | 1 | inner | 2268.4.f.a | 48 | |
63.l | odd | 6 | 1 | 252.4.x.a | ✓ | 48 | |
63.l | odd | 6 | 1 | 756.4.x.a | 48 | ||
63.o | even | 6 | 1 | 252.4.x.a | ✓ | 48 | |
63.o | even | 6 | 1 | 756.4.x.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.4.x.a | ✓ | 48 | 9.c | even | 3 | 1 | |
252.4.x.a | ✓ | 48 | 9.d | odd | 6 | 1 | |
252.4.x.a | ✓ | 48 | 63.l | odd | 6 | 1 | |
252.4.x.a | ✓ | 48 | 63.o | even | 6 | 1 | |
756.4.x.a | 48 | 9.c | even | 3 | 1 | ||
756.4.x.a | 48 | 9.d | odd | 6 | 1 | ||
756.4.x.a | 48 | 63.l | odd | 6 | 1 | ||
756.4.x.a | 48 | 63.o | even | 6 | 1 | ||
2268.4.f.a | 48 | 1.a | even | 1 | 1 | trivial | |
2268.4.f.a | 48 | 3.b | odd | 2 | 1 | inner | |
2268.4.f.a | 48 | 7.b | odd | 2 | 1 | inner | |
2268.4.f.a | 48 | 21.c | even | 2 | 1 | inner |