Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [100,6,Mod(21,100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("100.21");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 100.g (of order \(5\), degree \(4\), 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: | \(16.0383819813\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Relative dimension: | \(13\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
21.1 | 0 | −23.4577 | − | 17.0430i | 0 | 52.3337 | − | 19.6516i | 0 | 137.238 | 0 | 184.708 | + | 568.472i | 0 | ||||||||||||
21.2 | 0 | −23.1010 | − | 16.7838i | 0 | −35.4147 | + | 43.2528i | 0 | −103.774 | 0 | 176.867 | + | 544.339i | 0 | ||||||||||||
21.3 | 0 | −12.3717 | − | 8.98853i | 0 | −17.8429 | − | 52.9776i | 0 | −136.040 | 0 | −2.82702 | − | 8.70068i | 0 | ||||||||||||
21.4 | 0 | −11.3077 | − | 8.21554i | 0 | −48.4390 | − | 27.9045i | 0 | 250.128 | 0 | −14.7217 | − | 45.3088i | 0 | ||||||||||||
21.5 | 0 | −9.09673 | − | 6.60916i | 0 | 38.0613 | + | 40.9431i | 0 | −35.2774 | 0 | −36.0216 | − | 110.863i | 0 | ||||||||||||
21.6 | 0 | −6.56136 | − | 4.76711i | 0 | −37.0820 | + | 41.8321i | 0 | −4.12841 | 0 | −54.7650 | − | 168.549i | 0 | ||||||||||||
21.7 | 0 | 1.62583 | + | 1.18123i | 0 | 43.6006 | − | 34.9856i | 0 | 115.151 | 0 | −73.8431 | − | 227.266i | 0 | ||||||||||||
21.8 | 0 | 5.48004 | + | 3.98148i | 0 | 53.5629 | − | 16.0003i | 0 | −205.169 | 0 | −60.9125 | − | 187.469i | 0 | ||||||||||||
21.9 | 0 | 10.7159 | + | 7.78556i | 0 | −54.0256 | + | 14.3608i | 0 | −52.1911 | 0 | −20.8755 | − | 64.2481i | 0 | ||||||||||||
21.10 | 0 | 11.4456 | + | 8.31568i | 0 | 9.13427 | + | 55.1504i | 0 | 206.581 | 0 | −13.2410 | − | 40.7517i | 0 | ||||||||||||
21.11 | 0 | 14.0415 | + | 10.2018i | 0 | −37.5181 | − | 41.4415i | 0 | −48.5199 | 0 | 17.9974 | + | 55.3901i | 0 | ||||||||||||
21.12 | 0 | 20.8083 | + | 15.1181i | 0 | 18.7145 | − | 52.6761i | 0 | 120.829 | 0 | 129.338 | + | 398.061i | 0 | ||||||||||||
21.13 | 0 | 22.2789 | + | 16.1866i | 0 | 31.6166 | + | 46.1020i | 0 | −146.877 | 0 | 159.254 | + | 490.133i | 0 | ||||||||||||
41.1 | 0 | −8.70966 | − | 26.8056i | 0 | 43.6109 | + | 34.9726i | 0 | −69.2087 | 0 | −446.089 | + | 324.103i | 0 | ||||||||||||
41.2 | 0 | −7.57963 | − | 23.3277i | 0 | −36.9556 | − | 41.9438i | 0 | −216.676 | 0 | −290.140 | + | 210.799i | 0 | ||||||||||||
41.3 | 0 | −5.66638 | − | 17.4393i | 0 | −28.2721 | + | 48.2254i | 0 | 144.604 | 0 | −75.4310 | + | 54.8038i | 0 | ||||||||||||
41.4 | 0 | −5.45584 | − | 16.7914i | 0 | 38.9708 | − | 40.0784i | 0 | 174.835 | 0 | −55.5923 | + | 40.3902i | 0 | ||||||||||||
41.5 | 0 | −2.99812 | − | 9.22728i | 0 | −51.6337 | − | 21.4233i | 0 | 70.0668 | 0 | 120.437 | − | 87.5028i | 0 | ||||||||||||
41.6 | 0 | −1.45295 | − | 4.47171i | 0 | 45.8662 | − | 31.9577i | 0 | −98.5151 | 0 | 178.706 | − | 129.838i | 0 | ||||||||||||
41.7 | 0 | 0.438384 | + | 1.34921i | 0 | 37.4300 | + | 41.5210i | 0 | −49.5774 | 0 | 194.963 | − | 141.649i | 0 | ||||||||||||
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 100.6.g.a | ✓ | 52 |
25.d | even | 5 | 1 | inner | 100.6.g.a | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
100.6.g.a | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
100.6.g.a | ✓ | 52 | 25.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(100, [\chi])\).