Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,6,Mod(139,230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.139");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 230.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: | \(36.8882785570\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
139.1 | − | 4.00000i | − | 28.7267i | −16.0000 | −53.1448 | − | 17.3388i | −114.907 | 115.551i | 64.0000i | −582.223 | −69.3553 | + | 212.579i | ||||||||||||
139.2 | − | 4.00000i | − | 24.7103i | −16.0000 | 44.5324 | + | 33.7915i | −98.8414 | − | 141.158i | 64.0000i | −367.601 | 135.166 | − | 178.130i | |||||||||||
139.3 | − | 4.00000i | − | 18.9029i | −16.0000 | −54.8757 | + | 10.6611i | −75.6115 | − | 198.078i | 64.0000i | −114.319 | 42.6446 | + | 219.503i | |||||||||||
139.4 | − | 4.00000i | − | 16.5762i | −16.0000 | 35.3277 | − | 43.3238i | −66.3049 | 169.564i | 64.0000i | −31.7715 | −173.295 | − | 141.311i | ||||||||||||
139.5 | − | 4.00000i | − | 10.9586i | −16.0000 | −39.6919 | − | 39.3644i | −43.8345 | − | 12.6560i | 64.0000i | 122.909 | −157.458 | + | 158.767i | |||||||||||
139.6 | − | 4.00000i | − | 4.21065i | −16.0000 | 53.1232 | − | 17.4048i | −16.8426 | − | 167.782i | 64.0000i | 225.270 | −69.6193 | − | 212.493i | |||||||||||
139.7 | − | 4.00000i | − | 2.12202i | −16.0000 | 19.1932 | + | 52.5036i | −8.48807 | 64.0472i | 64.0000i | 238.497 | 210.014 | − | 76.7726i | ||||||||||||
139.8 | − | 4.00000i | 0.0569538i | −16.0000 | −30.6335 | − | 46.7609i | 0.227815 | 149.136i | 64.0000i | 242.997 | −187.044 | + | 122.534i | |||||||||||||
139.9 | − | 4.00000i | 8.39668i | −16.0000 | −49.8703 | + | 25.2578i | 33.5867 | − | 95.8251i | 64.0000i | 172.496 | 101.031 | + | 199.481i | ||||||||||||
139.10 | − | 4.00000i | 10.0068i | −16.0000 | 7.97591 | + | 55.3298i | 40.0274 | 231.737i | 64.0000i | 142.863 | 221.319 | − | 31.9036i | |||||||||||||
139.11 | − | 4.00000i | 14.1177i | −16.0000 | 13.1286 | − | 54.3382i | 56.4709 | − | 69.8012i | 64.0000i | 43.6900 | −217.353 | − | 52.5144i | ||||||||||||
139.12 | − | 4.00000i | 19.5360i | −16.0000 | −18.7601 | + | 52.6598i | 78.1441 | − | 153.006i | 64.0000i | −138.657 | 210.639 | + | 75.0405i | ||||||||||||
139.13 | − | 4.00000i | 26.2766i | −16.0000 | 54.8977 | + | 10.5471i | 105.107 | − | 233.425i | 64.0000i | −447.461 | 42.1883 | − | 219.591i | ||||||||||||
139.14 | − | 4.00000i | 27.0471i | −16.0000 | 54.9312 | + | 10.3712i | 108.188 | 238.280i | 64.0000i | −488.545 | 41.4850 | − | 219.725i | |||||||||||||
139.15 | − | 4.00000i | 27.7695i | −16.0000 | −51.1336 | − | 22.5909i | 111.078 | 2.41627i | 64.0000i | −528.145 | −90.3638 | + | 204.535i | |||||||||||||
139.16 | 4.00000i | − | 27.7695i | −16.0000 | −51.1336 | + | 22.5909i | 111.078 | − | 2.41627i | − | 64.0000i | −528.145 | −90.3638 | − | 204.535i | |||||||||||
139.17 | 4.00000i | − | 27.0471i | −16.0000 | 54.9312 | − | 10.3712i | 108.188 | − | 238.280i | − | 64.0000i | −488.545 | 41.4850 | + | 219.725i | |||||||||||
139.18 | 4.00000i | − | 26.2766i | −16.0000 | 54.8977 | − | 10.5471i | 105.107 | 233.425i | − | 64.0000i | −447.461 | 42.1883 | + | 219.591i | ||||||||||||
139.19 | 4.00000i | − | 19.5360i | −16.0000 | −18.7601 | − | 52.6598i | 78.1441 | 153.006i | − | 64.0000i | −138.657 | 210.639 | − | 75.0405i | ||||||||||||
139.20 | 4.00000i | − | 14.1177i | −16.0000 | 13.1286 | + | 54.3382i | 56.4709 | 69.8012i | − | 64.0000i | 43.6900 | −217.353 | + | 52.5144i | ||||||||||||
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 230.6.b.b | ✓ | 30 |
5.b | even | 2 | 1 | inner | 230.6.b.b | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.6.b.b | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
230.6.b.b | ✓ | 30 | 5.b | even | 2 | 1 | inner |