Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2736,2,Mod(1151,2736)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2736.1151");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2736.cg (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: | \(21.8470699930\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1151.1 | 0 | 0 | 0 | −3.43403 | − | 1.98264i | 0 | 2.01359i | 0 | 0 | 0 | ||||||||||||||||
1151.2 | 0 | 0 | 0 | −2.27729 | − | 1.31480i | 0 | 3.01718i | 0 | 0 | 0 | ||||||||||||||||
1151.3 | 0 | 0 | 0 | −2.15944 | − | 1.24676i | 0 | − | 0.872468i | 0 | 0 | 0 | |||||||||||||||
1151.4 | 0 | 0 | 0 | −1.62081 | − | 0.935772i | 0 | − | 2.07786i | 0 | 0 | 0 | |||||||||||||||
1151.5 | 0 | 0 | 0 | −1.02690 | − | 0.592883i | 0 | − | 4.00663i | 0 | 0 | 0 | |||||||||||||||
1151.6 | 0 | 0 | 0 | −0.420258 | − | 0.242636i | 0 | 1.92619i | 0 | 0 | 0 | ||||||||||||||||
1151.7 | 0 | 0 | 0 | 0.420258 | + | 0.242636i | 0 | 1.92619i | 0 | 0 | 0 | ||||||||||||||||
1151.8 | 0 | 0 | 0 | 1.02690 | + | 0.592883i | 0 | − | 4.00663i | 0 | 0 | 0 | |||||||||||||||
1151.9 | 0 | 0 | 0 | 1.62081 | + | 0.935772i | 0 | − | 2.07786i | 0 | 0 | 0 | |||||||||||||||
1151.10 | 0 | 0 | 0 | 2.15944 | + | 1.24676i | 0 | − | 0.872468i | 0 | 0 | 0 | |||||||||||||||
1151.11 | 0 | 0 | 0 | 2.27729 | + | 1.31480i | 0 | 3.01718i | 0 | 0 | 0 | ||||||||||||||||
1151.12 | 0 | 0 | 0 | 3.43403 | + | 1.98264i | 0 | 2.01359i | 0 | 0 | 0 | ||||||||||||||||
2591.1 | 0 | 0 | 0 | −3.43403 | + | 1.98264i | 0 | − | 2.01359i | 0 | 0 | 0 | |||||||||||||||
2591.2 | 0 | 0 | 0 | −2.27729 | + | 1.31480i | 0 | − | 3.01718i | 0 | 0 | 0 | |||||||||||||||
2591.3 | 0 | 0 | 0 | −2.15944 | + | 1.24676i | 0 | 0.872468i | 0 | 0 | 0 | ||||||||||||||||
2591.4 | 0 | 0 | 0 | −1.62081 | + | 0.935772i | 0 | 2.07786i | 0 | 0 | 0 | ||||||||||||||||
2591.5 | 0 | 0 | 0 | −1.02690 | + | 0.592883i | 0 | 4.00663i | 0 | 0 | 0 | ||||||||||||||||
2591.6 | 0 | 0 | 0 | −0.420258 | + | 0.242636i | 0 | − | 1.92619i | 0 | 0 | 0 | |||||||||||||||
2591.7 | 0 | 0 | 0 | 0.420258 | − | 0.242636i | 0 | − | 1.92619i | 0 | 0 | 0 | |||||||||||||||
2591.8 | 0 | 0 | 0 | 1.02690 | − | 0.592883i | 0 | 4.00663i | 0 | 0 | 0 | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
76.g | odd | 6 | 1 | inner |
228.m | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2736.2.cg.a | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 2736.2.cg.a | ✓ | 24 |
4.b | odd | 2 | 1 | 2736.2.cg.b | yes | 24 | |
12.b | even | 2 | 1 | 2736.2.cg.b | yes | 24 | |
19.c | even | 3 | 1 | 2736.2.cg.b | yes | 24 | |
57.h | odd | 6 | 1 | 2736.2.cg.b | yes | 24 | |
76.g | odd | 6 | 1 | inner | 2736.2.cg.a | ✓ | 24 |
228.m | even | 6 | 1 | inner | 2736.2.cg.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2736.2.cg.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
2736.2.cg.a | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
2736.2.cg.a | ✓ | 24 | 76.g | odd | 6 | 1 | inner |
2736.2.cg.a | ✓ | 24 | 228.m | even | 6 | 1 | inner |
2736.2.cg.b | yes | 24 | 4.b | odd | 2 | 1 | |
2736.2.cg.b | yes | 24 | 12.b | even | 2 | 1 | |
2736.2.cg.b | yes | 24 | 19.c | even | 3 | 1 | |
2736.2.cg.b | yes | 24 | 57.h | odd | 6 | 1 |