Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [360,4,Mod(251,360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("360.251");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 360.b (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: | \(21.2406876021\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
251.1 | −2.74252 | − | 0.691792i | 0 | 7.04285 | + | 3.79451i | −5.00000 | 0 | 25.7913i | −16.6901 | − | 15.2787i | 0 | 13.7126 | + | 3.45896i | ||||||||||
251.2 | −2.74252 | + | 0.691792i | 0 | 7.04285 | − | 3.79451i | −5.00000 | 0 | − | 25.7913i | −16.6901 | + | 15.2787i | 0 | 13.7126 | − | 3.45896i | |||||||||
251.3 | −2.61333 | − | 1.08191i | 0 | 5.65895 | + | 5.65476i | −5.00000 | 0 | − | 24.2359i | −8.67075 | − | 20.9002i | 0 | 13.0666 | + | 5.40954i | |||||||||
251.4 | −2.61333 | + | 1.08191i | 0 | 5.65895 | − | 5.65476i | −5.00000 | 0 | 24.2359i | −8.67075 | + | 20.9002i | 0 | 13.0666 | − | 5.40954i | ||||||||||
251.5 | −2.38691 | − | 1.51745i | 0 | 3.39471 | + | 7.24403i | −5.00000 | 0 | 27.9651i | 2.88957 | − | 22.4422i | 0 | 11.9346 | + | 7.58724i | ||||||||||
251.6 | −2.38691 | + | 1.51745i | 0 | 3.39471 | − | 7.24403i | −5.00000 | 0 | − | 27.9651i | 2.88957 | + | 22.4422i | 0 | 11.9346 | − | 7.58724i | |||||||||
251.7 | −2.18228 | − | 1.79934i | 0 | 1.52472 | + | 7.85336i | −5.00000 | 0 | − | 12.2479i | 10.8035 | − | 19.8817i | 0 | 10.9114 | + | 8.99672i | |||||||||
251.8 | −2.18228 | + | 1.79934i | 0 | 1.52472 | − | 7.85336i | −5.00000 | 0 | 12.2479i | 10.8035 | + | 19.8817i | 0 | 10.9114 | − | 8.99672i | ||||||||||
251.9 | −1.70898 | − | 2.25375i | 0 | −2.15877 | + | 7.70323i | −5.00000 | 0 | − | 15.4755i | 21.0504 | − | 8.29935i | 0 | 8.54491 | + | 11.2687i | |||||||||
251.10 | −1.70898 | + | 2.25375i | 0 | −2.15877 | − | 7.70323i | −5.00000 | 0 | 15.4755i | 21.0504 | + | 8.29935i | 0 | 8.54491 | − | 11.2687i | ||||||||||
251.11 | −0.671247 | − | 2.74762i | 0 | −7.09886 | + | 3.68866i | −5.00000 | 0 | − | 2.99301i | 14.9001 | + | 17.0290i | 0 | 3.35623 | + | 13.7381i | |||||||||
251.12 | −0.671247 | + | 2.74762i | 0 | −7.09886 | − | 3.68866i | −5.00000 | 0 | 2.99301i | 14.9001 | − | 17.0290i | 0 | 3.35623 | − | 13.7381i | ||||||||||
251.13 | −0.281738 | − | 2.81436i | 0 | −7.84125 | + | 1.58583i | −5.00000 | 0 | − | 11.3970i | 6.67226 | + | 21.6213i | 0 | 1.40869 | + | 14.0718i | |||||||||
251.14 | −0.281738 | + | 2.81436i | 0 | −7.84125 | − | 1.58583i | −5.00000 | 0 | 11.3970i | 6.67226 | − | 21.6213i | 0 | 1.40869 | − | 14.0718i | ||||||||||
251.15 | 0.666534 | − | 2.74877i | 0 | −7.11147 | − | 3.66429i | −5.00000 | 0 | 22.7533i | −14.8123 | + | 17.1054i | 0 | −3.33267 | + | 13.7438i | ||||||||||
251.16 | 0.666534 | + | 2.74877i | 0 | −7.11147 | + | 3.66429i | −5.00000 | 0 | − | 22.7533i | −14.8123 | − | 17.1054i | 0 | −3.33267 | − | 13.7438i | |||||||||
251.17 | 1.51037 | − | 2.39140i | 0 | −3.43758 | − | 7.22378i | −5.00000 | 0 | − | 35.1660i | −22.4670 | − | 2.68996i | 0 | −7.55184 | + | 11.9570i | |||||||||
251.18 | 1.51037 | + | 2.39140i | 0 | −3.43758 | + | 7.22378i | −5.00000 | 0 | 35.1660i | −22.4670 | + | 2.68996i | 0 | −7.55184 | − | 11.9570i | ||||||||||
251.19 | 2.19727 | − | 1.78101i | 0 | 1.65600 | − | 7.82673i | −5.00000 | 0 | − | 4.37103i | −10.3008 | − | 20.1468i | 0 | −10.9864 | + | 8.90506i | |||||||||
251.20 | 2.19727 | + | 1.78101i | 0 | 1.65600 | + | 7.82673i | −5.00000 | 0 | 4.37103i | −10.3008 | + | 20.1468i | 0 | −10.9864 | − | 8.90506i | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 360.4.b.a | ✓ | 24 |
3.b | odd | 2 | 1 | 360.4.b.b | yes | 24 | |
4.b | odd | 2 | 1 | 1440.4.b.a | 24 | ||
8.b | even | 2 | 1 | 1440.4.b.b | 24 | ||
8.d | odd | 2 | 1 | 360.4.b.b | yes | 24 | |
12.b | even | 2 | 1 | 1440.4.b.b | 24 | ||
24.f | even | 2 | 1 | inner | 360.4.b.a | ✓ | 24 |
24.h | odd | 2 | 1 | 1440.4.b.a | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
360.4.b.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
360.4.b.a | ✓ | 24 | 24.f | even | 2 | 1 | inner |
360.4.b.b | yes | 24 | 3.b | odd | 2 | 1 | |
360.4.b.b | yes | 24 | 8.d | odd | 2 | 1 | |
1440.4.b.a | 24 | 4.b | odd | 2 | 1 | ||
1440.4.b.a | 24 | 24.h | odd | 2 | 1 | ||
1440.4.b.b | 24 | 8.b | even | 2 | 1 | ||
1440.4.b.b | 24 | 12.b | even | 2 | 1 |