Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,5,Mod(47,230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.47");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 230.f (of order \(4\), 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: | \(23.7750915093\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | 2.00000 | + | 2.00000i | −11.2624 | + | 11.2624i | 8.00000i | −23.2954 | − | 9.07330i | −45.0495 | 2.86636 | + | 2.86636i | −16.0000 | + | 16.0000i | − | 172.682i | −28.4442 | − | 64.7374i | |||||
47.2 | 2.00000 | + | 2.00000i | −11.0455 | + | 11.0455i | 8.00000i | 3.15467 | + | 24.8002i | −44.1818 | −2.46420 | − | 2.46420i | −16.0000 | + | 16.0000i | − | 163.004i | −43.2910 | + | 55.9097i | |||||
47.3 | 2.00000 | + | 2.00000i | −9.95475 | + | 9.95475i | 8.00000i | 24.5881 | + | 4.51957i | −39.8190 | −57.5399 | − | 57.5399i | −16.0000 | + | 16.0000i | − | 117.194i | 40.1370 | + | 58.2153i | |||||
47.4 | 2.00000 | + | 2.00000i | −9.27707 | + | 9.27707i | 8.00000i | 23.6357 | − | 8.14565i | −37.1083 | 15.7230 | + | 15.7230i | −16.0000 | + | 16.0000i | − | 91.1281i | 63.5628 | + | 30.9802i | |||||
47.5 | 2.00000 | + | 2.00000i | −8.96878 | + | 8.96878i | 8.00000i | 1.96751 | − | 24.9225i | −35.8751 | 53.1255 | + | 53.1255i | −16.0000 | + | 16.0000i | − | 79.8782i | 53.7799 | − | 45.9099i | |||||
47.6 | 2.00000 | + | 2.00000i | −5.57799 | + | 5.57799i | 8.00000i | −24.4867 | − | 5.04022i | −22.3120 | −57.8690 | − | 57.8690i | −16.0000 | + | 16.0000i | 18.7720i | −38.8929 | − | 59.0537i | ||||||
47.7 | 2.00000 | + | 2.00000i | −4.66373 | + | 4.66373i | 8.00000i | −12.4209 | + | 21.6961i | −18.6549 | −24.9944 | − | 24.9944i | −16.0000 | + | 16.0000i | 37.4993i | −68.2341 | + | 18.5503i | ||||||
47.8 | 2.00000 | + | 2.00000i | −4.44879 | + | 4.44879i | 8.00000i | 18.7807 | + | 16.5011i | −17.7952 | 10.7782 | + | 10.7782i | −16.0000 | + | 16.0000i | 41.4165i | 4.55931 | + | 70.5635i | ||||||
47.9 | 2.00000 | + | 2.00000i | −4.05030 | + | 4.05030i | 8.00000i | −24.3623 | + | 5.61047i | −16.2012 | 52.2645 | + | 52.2645i | −16.0000 | + | 16.0000i | 48.1901i | −59.9456 | − | 37.5037i | ||||||
47.10 | 2.00000 | + | 2.00000i | −3.68907 | + | 3.68907i | 8.00000i | 6.02861 | − | 24.2622i | −14.7563 | −4.81567 | − | 4.81567i | −16.0000 | + | 16.0000i | 53.7815i | 60.5817 | − | 36.4673i | ||||||
47.11 | 2.00000 | + | 2.00000i | −0.310392 | + | 0.310392i | 8.00000i | 6.21241 | + | 24.2158i | −1.24157 | 40.7961 | + | 40.7961i | −16.0000 | + | 16.0000i | 80.8073i | −36.0068 | + | 60.8565i | ||||||
47.12 | 2.00000 | + | 2.00000i | 1.43653 | − | 1.43653i | 8.00000i | 23.6990 | − | 7.95979i | 5.74612 | −17.1725 | − | 17.1725i | −16.0000 | + | 16.0000i | 76.8728i | 63.3175 | + | 31.4784i | ||||||
47.13 | 2.00000 | + | 2.00000i | 2.64936 | − | 2.64936i | 8.00000i | 22.4224 | + | 11.0560i | 10.5974 | −62.5314 | − | 62.5314i | −16.0000 | + | 16.0000i | 66.9618i | 22.7329 | + | 66.9568i | ||||||
47.14 | 2.00000 | + | 2.00000i | 3.59966 | − | 3.59966i | 8.00000i | −10.4188 | − | 22.7255i | 14.3986 | 7.45573 | + | 7.45573i | −16.0000 | + | 16.0000i | 55.0849i | 24.6134 | − | 66.2886i | ||||||
47.15 | 2.00000 | + | 2.00000i | 4.87410 | − | 4.87410i | 8.00000i | 23.2684 | − | 9.14227i | 19.4964 | 49.4466 | + | 49.4466i | −16.0000 | + | 16.0000i | 33.4862i | 64.8214 | + | 28.2523i | ||||||
47.16 | 2.00000 | + | 2.00000i | 5.56222 | − | 5.56222i | 8.00000i | −17.4021 | + | 17.9490i | 22.2489 | −33.3744 | − | 33.3744i | −16.0000 | + | 16.0000i | 19.1234i | −70.7022 | + | 1.09393i | ||||||
47.17 | 2.00000 | + | 2.00000i | 7.12415 | − | 7.12415i | 8.00000i | −12.4603 | + | 21.6735i | 28.4966 | −34.8776 | − | 34.8776i | −16.0000 | + | 16.0000i | − | 20.5070i | −68.2676 | + | 18.4264i | |||||
47.18 | 2.00000 | + | 2.00000i | 7.50341 | − | 7.50341i | 8.00000i | −23.5955 | − | 8.26153i | 30.0136 | 38.1189 | + | 38.1189i | −16.0000 | + | 16.0000i | − | 31.6022i | −30.6679 | − | 63.7140i | |||||
47.19 | 2.00000 | + | 2.00000i | 7.61108 | − | 7.61108i | 8.00000i | −1.36901 | + | 24.9625i | 30.4443 | 45.0835 | + | 45.0835i | −16.0000 | + | 16.0000i | − | 34.8570i | −52.6630 | + | 47.1870i | |||||
47.20 | 2.00000 | + | 2.00000i | 9.67771 | − | 9.67771i | 8.00000i | 14.1269 | − | 20.6259i | 38.7108 | −43.7928 | − | 43.7928i | −16.0000 | + | 16.0000i | − | 106.316i | 69.5058 | − | 12.9980i | |||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 230.5.f.b | ✓ | 44 |
5.c | odd | 4 | 1 | inner | 230.5.f.b | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.5.f.b | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
230.5.f.b | ✓ | 44 | 5.c | odd | 4 | 1 | inner |