Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,5,Mod(229,230)]
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, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.229");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 230.c (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: | \(23.7750915093\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
229.1 | − | 2.82843i | − | 9.09812i | −8.00000 | −21.5641 | + | 12.6486i | −25.7334 | 94.0651 | 22.6274i | −1.77586 | 35.7757 | + | 60.9926i | ||||||||||||
229.2 | 2.82843i | 9.09812i | −8.00000 | −21.5641 | − | 12.6486i | −25.7334 | 94.0651 | − | 22.6274i | −1.77586 | 35.7757 | − | 60.9926i | |||||||||||||
229.3 | − | 2.82843i | − | 15.3728i | −8.00000 | −24.9424 | + | 1.69556i | −43.4808 | −85.1277 | 22.6274i | −155.323 | 4.79578 | + | 70.5479i | ||||||||||||
229.4 | 2.82843i | 15.3728i | −8.00000 | −24.9424 | − | 1.69556i | −43.4808 | −85.1277 | − | 22.6274i | −155.323 | 4.79578 | − | 70.5479i | |||||||||||||
229.5 | − | 2.82843i | 13.4757i | −8.00000 | 20.1787 | − | 14.7587i | 38.1149 | 78.6574 | 22.6274i | −100.593 | −41.7439 | − | 57.0741i | |||||||||||||
229.6 | 2.82843i | − | 13.4757i | −8.00000 | 20.1787 | + | 14.7587i | 38.1149 | 78.6574 | − | 22.6274i | −100.593 | −41.7439 | + | 57.0741i | ||||||||||||
229.7 | − | 2.82843i | − | 0.832711i | −8.00000 | 19.4576 | − | 15.6972i | −2.35526 | 66.8572 | 22.6274i | 80.3066 | −44.3985 | − | 55.0343i | ||||||||||||
229.8 | 2.82843i | 0.832711i | −8.00000 | 19.4576 | + | 15.6972i | −2.35526 | 66.8572 | − | 22.6274i | 80.3066 | −44.3985 | + | 55.0343i | |||||||||||||
229.9 | − | 2.82843i | 6.81504i | −8.00000 | 6.15914 | + | 24.2294i | 19.2758 | 68.2555 | 22.6274i | 34.5552 | 68.5312 | − | 17.4207i | |||||||||||||
229.10 | 2.82843i | − | 6.81504i | −8.00000 | 6.15914 | − | 24.2294i | 19.2758 | 68.2555 | − | 22.6274i | 34.5552 | 68.5312 | + | 17.4207i | ||||||||||||
229.11 | − | 2.82843i | 7.41746i | −8.00000 | 23.4452 | + | 8.67897i | 20.9797 | −42.4567 | 22.6274i | 25.9814 | 24.5478 | − | 66.3129i | |||||||||||||
229.12 | 2.82843i | − | 7.41746i | −8.00000 | 23.4452 | − | 8.67897i | 20.9797 | −42.4567 | − | 22.6274i | 25.9814 | 24.5478 | + | 66.3129i | ||||||||||||
229.13 | − | 2.82843i | 10.7328i | −8.00000 | 12.7779 | − | 21.4878i | 30.3568 | −32.1609 | 22.6274i | −34.1923 | −60.7766 | − | 36.1415i | |||||||||||||
229.14 | 2.82843i | − | 10.7328i | −8.00000 | 12.7779 | + | 21.4878i | 30.3568 | −32.1609 | − | 22.6274i | −34.1923 | −60.7766 | + | 36.1415i | ||||||||||||
229.15 | − | 2.82843i | 15.5407i | −8.00000 | 22.6151 | + | 10.6563i | 43.9558 | −25.0660 | 22.6274i | −160.514 | 30.1404 | − | 63.9653i | |||||||||||||
229.16 | 2.82843i | − | 15.5407i | −8.00000 | 22.6151 | − | 10.6563i | 43.9558 | −25.0660 | − | 22.6274i | −160.514 | 30.1404 | + | 63.9653i | ||||||||||||
229.17 | − | 2.82843i | − | 15.7740i | −8.00000 | −12.1414 | − | 21.8538i | −44.6155 | 29.1236 | 22.6274i | −167.818 | −61.8118 | + | 34.3409i | ||||||||||||
229.18 | 2.82843i | 15.7740i | −8.00000 | −12.1414 | + | 21.8538i | −44.6155 | 29.1236 | − | 22.6274i | −167.818 | −61.8118 | − | 34.3409i | |||||||||||||
229.19 | − | 2.82843i | − | 1.19309i | −8.00000 | −1.65984 | − | 24.9448i | −3.37456 | 28.6899 | 22.6274i | 79.5765 | −70.5547 | + | 4.69474i | ||||||||||||
229.20 | 2.82843i | 1.19309i | −8.00000 | −1.65984 | + | 24.9448i | −3.37456 | 28.6899 | − | 22.6274i | 79.5765 | −70.5547 | − | 4.69474i | |||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
115.c | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 230.5.c.a | ✓ | 48 |
5.b | even | 2 | 1 | inner | 230.5.c.a | ✓ | 48 |
23.b | odd | 2 | 1 | inner | 230.5.c.a | ✓ | 48 |
115.c | odd | 2 | 1 | inner | 230.5.c.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.5.c.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
230.5.c.a | ✓ | 48 | 5.b | even | 2 | 1 | inner |
230.5.c.a | ✓ | 48 | 23.b | odd | 2 | 1 | inner |
230.5.c.a | ✓ | 48 | 115.c | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(230, [\chi])\).