Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,8,Mod(17,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.17");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 63.p (of order \(6\), 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: | \(19.6802566055\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −18.8163 | + | 10.8636i | 0 | 172.035 | − | 297.973i | 38.2128 | + | 66.1865i | 0 | 884.855 | + | 201.432i | 4694.58i | 0 | −1438.04 | − | 830.255i | ||||||||
17.2 | −16.6651 | + | 9.62158i | 0 | 121.150 | − | 209.837i | 115.111 | + | 199.378i | 0 | −904.118 | + | 78.1943i | 2199.48i | 0 | −3836.67 | − | 2215.10i | ||||||||
17.3 | −13.9069 | + | 8.02916i | 0 | 64.9347 | − | 112.470i | −95.0205 | − | 164.580i | 0 | 874.061 | + | 244.050i | 30.0190i | 0 | 2642.88 | + | 1525.87i | ||||||||
17.4 | −13.0229 | + | 7.51875i | 0 | 49.0633 | − | 84.9800i | −217.264 | − | 376.313i | 0 | −593.116 | − | 686.845i | − | 449.223i | 0 | 5658.81 | + | 3267.11i | |||||||
17.5 | −11.8878 | + | 6.86345i | 0 | 30.2140 | − | 52.3321i | 99.9401 | + | 173.101i | 0 | −671.322 | + | 610.631i | − | 927.556i | 0 | −2376.15 | − | 1371.87i | |||||||
17.6 | −6.95719 | + | 4.01673i | 0 | −31.7317 | + | 54.9609i | 79.8320 | + | 138.273i | 0 | 726.865 | − | 543.333i | − | 1538.12i | 0 | −1110.81 | − | 641.328i | |||||||
17.7 | −6.63630 | + | 3.83147i | 0 | −34.6397 | + | 59.9977i | 101.591 | + | 175.960i | 0 | −81.3062 | − | 903.843i | − | 1511.74i | 0 | −1348.37 | − | 778.484i | |||||||
17.8 | −4.38065 | + | 2.52917i | 0 | −51.2066 | + | 88.6924i | −258.225 | − | 447.259i | 0 | −900.609 | − | 111.567i | − | 1165.51i | 0 | 2262.39 | + | 1306.19i | |||||||
17.9 | −0.522037 | + | 0.301398i | 0 | −63.8183 | + | 110.537i | 136.040 | + | 235.628i | 0 | 146.190 | + | 895.640i | − | 154.097i | 0 | −142.035 | − | 82.0042i | |||||||
17.10 | 0.522037 | − | 0.301398i | 0 | −63.8183 | + | 110.537i | −136.040 | − | 235.628i | 0 | 146.190 | + | 895.640i | 154.097i | 0 | −142.035 | − | 82.0042i | ||||||||
17.11 | 4.38065 | − | 2.52917i | 0 | −51.2066 | + | 88.6924i | 258.225 | + | 447.259i | 0 | −900.609 | − | 111.567i | 1165.51i | 0 | 2262.39 | + | 1306.19i | ||||||||
17.12 | 6.63630 | − | 3.83147i | 0 | −34.6397 | + | 59.9977i | −101.591 | − | 175.960i | 0 | −81.3062 | − | 903.843i | 1511.74i | 0 | −1348.37 | − | 778.484i | ||||||||
17.13 | 6.95719 | − | 4.01673i | 0 | −31.7317 | + | 54.9609i | −79.8320 | − | 138.273i | 0 | 726.865 | − | 543.333i | 1538.12i | 0 | −1110.81 | − | 641.328i | ||||||||
17.14 | 11.8878 | − | 6.86345i | 0 | 30.2140 | − | 52.3321i | −99.9401 | − | 173.101i | 0 | −671.322 | + | 610.631i | 927.556i | 0 | −2376.15 | − | 1371.87i | ||||||||
17.15 | 13.0229 | − | 7.51875i | 0 | 49.0633 | − | 84.9800i | 217.264 | + | 376.313i | 0 | −593.116 | − | 686.845i | 449.223i | 0 | 5658.81 | + | 3267.11i | ||||||||
17.16 | 13.9069 | − | 8.02916i | 0 | 64.9347 | − | 112.470i | 95.0205 | + | 164.580i | 0 | 874.061 | + | 244.050i | − | 30.0190i | 0 | 2642.88 | + | 1525.87i | |||||||
17.17 | 16.6651 | − | 9.62158i | 0 | 121.150 | − | 209.837i | −115.111 | − | 199.378i | 0 | −904.118 | + | 78.1943i | − | 2199.48i | 0 | −3836.67 | − | 2215.10i | |||||||
17.18 | 18.8163 | − | 10.8636i | 0 | 172.035 | − | 297.973i | −38.2128 | − | 66.1865i | 0 | 884.855 | + | 201.432i | − | 4694.58i | 0 | −1438.04 | − | 830.255i | |||||||
26.1 | −18.8163 | − | 10.8636i | 0 | 172.035 | + | 297.973i | 38.2128 | − | 66.1865i | 0 | 884.855 | − | 201.432i | − | 4694.58i | 0 | −1438.04 | + | 830.255i | |||||||
26.2 | −16.6651 | − | 9.62158i | 0 | 121.150 | + | 209.837i | 115.111 | − | 199.378i | 0 | −904.118 | − | 78.1943i | − | 2199.48i | 0 | −3836.67 | + | 2215.10i | |||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 63.8.p.a | ✓ | 36 |
3.b | odd | 2 | 1 | inner | 63.8.p.a | ✓ | 36 |
7.c | even | 3 | 1 | 441.8.c.a | 36 | ||
7.d | odd | 6 | 1 | inner | 63.8.p.a | ✓ | 36 |
7.d | odd | 6 | 1 | 441.8.c.a | 36 | ||
21.g | even | 6 | 1 | inner | 63.8.p.a | ✓ | 36 |
21.g | even | 6 | 1 | 441.8.c.a | 36 | ||
21.h | odd | 6 | 1 | 441.8.c.a | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.8.p.a | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
63.8.p.a | ✓ | 36 | 3.b | odd | 2 | 1 | inner |
63.8.p.a | ✓ | 36 | 7.d | odd | 6 | 1 | inner |
63.8.p.a | ✓ | 36 | 21.g | even | 6 | 1 | inner |
441.8.c.a | 36 | 7.c | even | 3 | 1 | ||
441.8.c.a | 36 | 7.d | odd | 6 | 1 | ||
441.8.c.a | 36 | 21.g | even | 6 | 1 | ||
441.8.c.a | 36 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(63, [\chi])\).