Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [35,6,Mod(13,35)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("35.13");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 35 = 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 35.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: | \(5.61343369345\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −7.19923 | + | 7.19923i | −13.8307 | + | 13.8307i | − | 71.6578i | 8.68156 | − | 55.2235i | − | 199.140i | −83.2487 | + | 99.3813i | 285.505 | + | 285.505i | − | 139.575i | 335.066 | + | 460.067i | |||
13.2 | −7.19923 | + | 7.19923i | 13.8307 | − | 13.8307i | − | 71.6578i | −8.68156 | + | 55.2235i | 199.140i | −99.3813 | + | 83.2487i | 285.505 | + | 285.505i | − | 139.575i | −335.066 | − | 460.067i | ||||
13.3 | −5.71951 | + | 5.71951i | −2.93091 | + | 2.93091i | − | 33.4255i | −55.3132 | + | 8.08990i | − | 33.5267i | 33.7470 | − | 125.172i | 8.15308 | + | 8.15308i | 225.820i | 270.094 | − | 362.635i | ||||
13.4 | −5.71951 | + | 5.71951i | 2.93091 | − | 2.93091i | − | 33.4255i | 55.3132 | − | 8.08990i | 33.5267i | 125.172 | − | 33.7470i | 8.15308 | + | 8.15308i | 225.820i | −270.094 | + | 362.635i | |||||
13.5 | −3.69988 | + | 3.69988i | −18.2186 | + | 18.2186i | 4.62173i | −9.39510 | + | 55.1066i | − | 134.813i | 44.6849 | + | 121.697i | −135.496 | − | 135.496i | − | 420.835i | −169.127 | − | 238.649i | ||||
13.6 | −3.69988 | + | 3.69988i | 18.2186 | − | 18.2186i | 4.62173i | 9.39510 | − | 55.1066i | 134.813i | −121.697 | − | 44.6849i | −135.496 | − | 135.496i | − | 420.835i | 169.127 | + | 238.649i | |||||
13.7 | −2.47595 | + | 2.47595i | −6.55592 | + | 6.55592i | 19.7394i | 52.0184 | + | 20.4716i | − | 32.4643i | −114.937 | − | 59.9710i | −128.104 | − | 128.104i | 157.040i | −179.482 | + | 78.1083i | |||||
13.8 | −2.47595 | + | 2.47595i | 6.55592 | − | 6.55592i | 19.7394i | −52.0184 | − | 20.4716i | 32.4643i | 59.9710 | + | 114.937i | −128.104 | − | 128.104i | 157.040i | 179.482 | − | 78.1083i | ||||||
13.9 | −0.295977 | + | 0.295977i | −17.5765 | + | 17.5765i | 31.8248i | −16.1154 | − | 53.5284i | − | 10.4045i | 32.8881 | − | 125.401i | −18.8906 | − | 18.8906i | − | 374.868i | 20.6129 | + | 11.0734i | ||||
13.10 | −0.295977 | + | 0.295977i | 17.5765 | − | 17.5765i | 31.8248i | 16.1154 | + | 53.5284i | 10.4045i | 125.401 | − | 32.8881i | −18.8906 | − | 18.8906i | − | 374.868i | −20.6129 | − | 11.0734i | |||||
13.11 | 1.35053 | − | 1.35053i | −1.76991 | + | 1.76991i | 28.3521i | 33.8544 | − | 44.4846i | 4.78065i | 33.8584 | + | 125.142i | 81.5075 | + | 81.5075i | 236.735i | −14.3563 | − | 105.799i | ||||||
13.12 | 1.35053 | − | 1.35053i | 1.76991 | − | 1.76991i | 28.3521i | −33.8544 | + | 44.4846i | − | 4.78065i | −125.142 | − | 33.8584i | 81.5075 | + | 81.5075i | 236.735i | 14.3563 | + | 105.799i | |||||
13.13 | 4.47623 | − | 4.47623i | −10.4953 | + | 10.4953i | − | 8.07328i | 34.0119 | + | 44.3643i | 93.9592i | 122.053 | − | 43.7039i | 107.102 | + | 107.102i | 22.6955i | 350.830 | + | 46.3396i | |||||
13.14 | 4.47623 | − | 4.47623i | 10.4953 | − | 10.4953i | − | 8.07328i | −34.0119 | − | 44.3643i | − | 93.9592i | 43.7039 | − | 122.053i | 107.102 | + | 107.102i | 22.6955i | −350.830 | − | 46.3396i | ||||
13.15 | 5.10578 | − | 5.10578i | −14.8441 | + | 14.8441i | − | 20.1381i | −55.8880 | − | 1.23744i | 151.582i | −74.7452 | + | 105.925i | 60.5645 | + | 60.5645i | − | 197.695i | −291.670 | + | 279.034i | ||||
13.16 | 5.10578 | − | 5.10578i | 14.8441 | − | 14.8441i | − | 20.1381i | 55.8880 | + | 1.23744i | − | 151.582i | −105.925 | + | 74.7452i | 60.5645 | + | 60.5645i | − | 197.695i | 291.670 | − | 279.034i | |||
13.17 | 7.45800 | − | 7.45800i | −9.59993 | + | 9.59993i | − | 79.2434i | 39.6521 | − | 39.4045i | 143.193i | −88.8292 | − | 94.4265i | −352.341 | − | 352.341i | 58.6826i | 1.84648 | − | 589.603i | |||||
13.18 | 7.45800 | − | 7.45800i | 9.59993 | − | 9.59993i | − | 79.2434i | −39.6521 | + | 39.4045i | − | 143.193i | 94.4265 | + | 88.8292i | −352.341 | − | 352.341i | 58.6826i | −1.84648 | + | 589.603i | ||||
27.1 | −7.19923 | − | 7.19923i | −13.8307 | − | 13.8307i | 71.6578i | 8.68156 | + | 55.2235i | 199.140i | −83.2487 | − | 99.3813i | 285.505 | − | 285.505i | 139.575i | 335.066 | − | 460.067i | ||||||
27.2 | −7.19923 | − | 7.19923i | 13.8307 | + | 13.8307i | 71.6578i | −8.68156 | − | 55.2235i | − | 199.140i | −99.3813 | − | 83.2487i | 285.505 | − | 285.505i | 139.575i | −335.066 | + | 460.067i | |||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.b | odd | 2 | 1 | inner |
35.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 35.6.f.a | ✓ | 36 |
5.c | odd | 4 | 1 | inner | 35.6.f.a | ✓ | 36 |
7.b | odd | 2 | 1 | inner | 35.6.f.a | ✓ | 36 |
35.f | even | 4 | 1 | inner | 35.6.f.a | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
35.6.f.a | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
35.6.f.a | ✓ | 36 | 5.c | odd | 4 | 1 | inner |
35.6.f.a | ✓ | 36 | 7.b | odd | 2 | 1 | inner |
35.6.f.a | ✓ | 36 | 35.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(35, [\chi])\).