Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,4,Mod(307,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, 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("462.307");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 462.e (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: | \(27.2588824227\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
307.1 | − | 2.00000i | 3.00000i | −4.00000 | − | 21.6833i | 6.00000 | 14.4011 | + | 11.6451i | 8.00000i | −9.00000 | −43.3665 | ||||||||||||||
307.2 | − | 2.00000i | 3.00000i | −4.00000 | − | 18.5526i | 6.00000 | −11.3241 | + | 14.6549i | 8.00000i | −9.00000 | −37.1052 | ||||||||||||||
307.3 | − | 2.00000i | 3.00000i | −4.00000 | − | 12.2530i | 6.00000 | 8.77074 | − | 16.3118i | 8.00000i | −9.00000 | −24.5060 | ||||||||||||||
307.4 | − | 2.00000i | 3.00000i | −4.00000 | − | 8.14463i | 6.00000 | −16.3752 | − | 8.65177i | 8.00000i | −9.00000 | −16.2893 | ||||||||||||||
307.5 | − | 2.00000i | 3.00000i | −4.00000 | − | 6.71871i | 6.00000 | −17.1062 | + | 7.09771i | 8.00000i | −9.00000 | −13.4374 | ||||||||||||||
307.6 | − | 2.00000i | 3.00000i | −4.00000 | − | 6.45926i | 6.00000 | 17.9060 | − | 4.73009i | 8.00000i | −9.00000 | −12.9185 | ||||||||||||||
307.7 | − | 2.00000i | 3.00000i | −4.00000 | − | 3.90709i | 6.00000 | −9.50728 | − | 15.8938i | 8.00000i | −9.00000 | −7.81419 | ||||||||||||||
307.8 | − | 2.00000i | 3.00000i | −4.00000 | 7.44234i | 6.00000 | 5.85787 | + | 17.5694i | 8.00000i | −9.00000 | 14.8847 | |||||||||||||||
307.9 | − | 2.00000i | 3.00000i | −4.00000 | 7.94213i | 6.00000 | −16.9601 | + | 7.44012i | 8.00000i | −9.00000 | 15.8843 | |||||||||||||||
307.10 | − | 2.00000i | 3.00000i | −4.00000 | 12.6816i | 6.00000 | 2.63864 | − | 18.3313i | 8.00000i | −9.00000 | 25.3632 | |||||||||||||||
307.11 | − | 2.00000i | 3.00000i | −4.00000 | 17.5107i | 6.00000 | 18.3856 | + | 2.22927i | 8.00000i | −9.00000 | 35.0213 | |||||||||||||||
307.12 | − | 2.00000i | 3.00000i | −4.00000 | 18.1418i | 6.00000 | −14.6872 | + | 11.2822i | 8.00000i | −9.00000 | 36.2836 | |||||||||||||||
307.13 | 2.00000i | − | 3.00000i | −4.00000 | − | 18.1418i | 6.00000 | −14.6872 | − | 11.2822i | − | 8.00000i | −9.00000 | 36.2836 | |||||||||||||
307.14 | 2.00000i | − | 3.00000i | −4.00000 | − | 17.5107i | 6.00000 | 18.3856 | − | 2.22927i | − | 8.00000i | −9.00000 | 35.0213 | |||||||||||||
307.15 | 2.00000i | − | 3.00000i | −4.00000 | − | 12.6816i | 6.00000 | 2.63864 | + | 18.3313i | − | 8.00000i | −9.00000 | 25.3632 | |||||||||||||
307.16 | 2.00000i | − | 3.00000i | −4.00000 | − | 7.94213i | 6.00000 | −16.9601 | − | 7.44012i | − | 8.00000i | −9.00000 | 15.8843 | |||||||||||||
307.17 | 2.00000i | − | 3.00000i | −4.00000 | − | 7.44234i | 6.00000 | 5.85787 | − | 17.5694i | − | 8.00000i | −9.00000 | 14.8847 | |||||||||||||
307.18 | 2.00000i | − | 3.00000i | −4.00000 | 3.90709i | 6.00000 | −9.50728 | + | 15.8938i | − | 8.00000i | −9.00000 | −7.81419 | ||||||||||||||
307.19 | 2.00000i | − | 3.00000i | −4.00000 | 6.45926i | 6.00000 | 17.9060 | + | 4.73009i | − | 8.00000i | −9.00000 | −12.9185 | ||||||||||||||
307.20 | 2.00000i | − | 3.00000i | −4.00000 | 6.71871i | 6.00000 | −17.1062 | − | 7.09771i | − | 8.00000i | −9.00000 | −13.4374 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
77.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 462.4.e.b | yes | 24 |
7.b | odd | 2 | 1 | 462.4.e.a | ✓ | 24 | |
11.b | odd | 2 | 1 | 462.4.e.a | ✓ | 24 | |
77.b | even | 2 | 1 | inner | 462.4.e.b | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
462.4.e.a | ✓ | 24 | 7.b | odd | 2 | 1 | |
462.4.e.a | ✓ | 24 | 11.b | odd | 2 | 1 | |
462.4.e.b | yes | 24 | 1.a | even | 1 | 1 | trivial |
462.4.e.b | yes | 24 | 77.b | even | 2 | 1 | inner |