Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [195,2,Mod(38,195)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(195, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("195.38");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 195.s (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: | \(1.55708283941\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
38.1 | −1.57280 | + | 1.57280i | −1.73164 | + | 0.0375613i | − | 2.94741i | 2.12590 | + | 0.693215i | 2.66446 | − | 2.78261i | 3.19872 | + | 3.19872i | 1.49009 | + | 1.49009i | 2.99718 | − | 0.130086i | −4.43391 | + | 2.25333i | |
38.2 | −1.57280 | + | 1.57280i | 0.0375613 | − | 1.73164i | − | 2.94741i | 2.12590 | + | 0.693215i | 2.66446 | + | 2.78261i | −3.19872 | − | 3.19872i | 1.49009 | + | 1.49009i | −2.99718 | − | 0.130086i | −4.43391 | + | 2.25333i | |
38.3 | −1.25684 | + | 1.25684i | −1.06336 | + | 1.36721i | − | 1.15927i | −0.261006 | − | 2.22078i | −0.381882 | − | 3.05483i | −1.60133 | − | 1.60133i | −1.05665 | − | 1.05665i | −0.738515 | − | 2.90768i | 3.11920 | + | 2.46312i | |
38.4 | −1.25684 | + | 1.25684i | 1.36721 | − | 1.06336i | − | 1.15927i | −0.261006 | − | 2.22078i | −0.381882 | + | 3.05483i | 1.60133 | + | 1.60133i | −1.05665 | − | 1.05665i | 0.738515 | − | 2.90768i | 3.11920 | + | 2.46312i | |
38.5 | −0.597551 | + | 0.597551i | −1.67038 | + | 0.458083i | 1.28586i | −0.289977 | + | 2.21719i | 0.724408 | − | 1.27186i | −2.39819 | − | 2.39819i | −1.96347 | − | 1.96347i | 2.58032 | − | 1.53034i | −1.15161 | − | 1.49816i | ||
38.6 | −0.597551 | + | 0.597551i | 0.458083 | − | 1.67038i | 1.28586i | −0.289977 | + | 2.21719i | 0.724408 | + | 1.27186i | 2.39819 | + | 2.39819i | −1.96347 | − | 1.96347i | −2.58032 | − | 1.53034i | −1.15161 | − | 1.49816i | ||
38.7 | −0.299314 | + | 0.299314i | −0.125002 | + | 1.72753i | 1.82082i | 2.19735 | − | 0.414323i | −0.479660 | − | 0.554490i | 0.976029 | + | 0.976029i | −1.14363 | − | 1.14363i | −2.96875 | − | 0.431892i | −0.533684 | + | 0.781709i | ||
38.8 | −0.299314 | + | 0.299314i | 1.72753 | − | 0.125002i | 1.82082i | 2.19735 | − | 0.414323i | −0.479660 | + | 0.554490i | −0.976029 | − | 0.976029i | −1.14363 | − | 1.14363i | 2.96875 | − | 0.431892i | −0.533684 | + | 0.781709i | ||
38.9 | 0.299314 | − | 0.299314i | −0.125002 | + | 1.72753i | 1.82082i | −2.19735 | + | 0.414323i | 0.479660 | + | 0.554490i | −0.976029 | − | 0.976029i | 1.14363 | + | 1.14363i | −2.96875 | − | 0.431892i | −0.533684 | + | 0.781709i | ||
38.10 | 0.299314 | − | 0.299314i | 1.72753 | − | 0.125002i | 1.82082i | −2.19735 | + | 0.414323i | 0.479660 | − | 0.554490i | 0.976029 | + | 0.976029i | 1.14363 | + | 1.14363i | 2.96875 | − | 0.431892i | −0.533684 | + | 0.781709i | ||
38.11 | 0.597551 | − | 0.597551i | −1.67038 | + | 0.458083i | 1.28586i | 0.289977 | − | 2.21719i | −0.724408 | + | 1.27186i | 2.39819 | + | 2.39819i | 1.96347 | + | 1.96347i | 2.58032 | − | 1.53034i | −1.15161 | − | 1.49816i | ||
38.12 | 0.597551 | − | 0.597551i | 0.458083 | − | 1.67038i | 1.28586i | 0.289977 | − | 2.21719i | −0.724408 | − | 1.27186i | −2.39819 | − | 2.39819i | 1.96347 | + | 1.96347i | −2.58032 | − | 1.53034i | −1.15161 | − | 1.49816i | ||
38.13 | 1.25684 | − | 1.25684i | −1.06336 | + | 1.36721i | − | 1.15927i | 0.261006 | + | 2.22078i | 0.381882 | + | 3.05483i | 1.60133 | + | 1.60133i | 1.05665 | + | 1.05665i | −0.738515 | − | 2.90768i | 3.11920 | + | 2.46312i | |
38.14 | 1.25684 | − | 1.25684i | 1.36721 | − | 1.06336i | − | 1.15927i | 0.261006 | + | 2.22078i | 0.381882 | − | 3.05483i | −1.60133 | − | 1.60133i | 1.05665 | + | 1.05665i | 0.738515 | − | 2.90768i | 3.11920 | + | 2.46312i | |
38.15 | 1.57280 | − | 1.57280i | −1.73164 | + | 0.0375613i | − | 2.94741i | −2.12590 | − | 0.693215i | −2.66446 | + | 2.78261i | −3.19872 | − | 3.19872i | −1.49009 | − | 1.49009i | 2.99718 | − | 0.130086i | −4.43391 | + | 2.25333i | |
38.16 | 1.57280 | − | 1.57280i | 0.0375613 | − | 1.73164i | − | 2.94741i | −2.12590 | − | 0.693215i | −2.66446 | − | 2.78261i | 3.19872 | + | 3.19872i | −1.49009 | − | 1.49009i | −2.99718 | − | 0.130086i | −4.43391 | + | 2.25333i | |
77.1 | −1.57280 | − | 1.57280i | −1.73164 | − | 0.0375613i | 2.94741i | 2.12590 | − | 0.693215i | 2.66446 | + | 2.78261i | 3.19872 | − | 3.19872i | 1.49009 | − | 1.49009i | 2.99718 | + | 0.130086i | −4.43391 | − | 2.25333i | ||
77.2 | −1.57280 | − | 1.57280i | 0.0375613 | + | 1.73164i | 2.94741i | 2.12590 | − | 0.693215i | 2.66446 | − | 2.78261i | −3.19872 | + | 3.19872i | 1.49009 | − | 1.49009i | −2.99718 | + | 0.130086i | −4.43391 | − | 2.25333i | ||
77.3 | −1.25684 | − | 1.25684i | −1.06336 | − | 1.36721i | 1.15927i | −0.261006 | + | 2.22078i | −0.381882 | + | 3.05483i | −1.60133 | + | 1.60133i | −1.05665 | + | 1.05665i | −0.738515 | + | 2.90768i | 3.11920 | − | 2.46312i | ||
77.4 | −1.25684 | − | 1.25684i | 1.36721 | + | 1.06336i | 1.15927i | −0.261006 | + | 2.22078i | −0.381882 | − | 3.05483i | 1.60133 | − | 1.60133i | −1.05665 | + | 1.05665i | 0.738515 | + | 2.90768i | 3.11920 | − | 2.46312i | ||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
13.b | even | 2 | 1 | inner |
15.e | even | 4 | 1 | inner |
39.d | odd | 2 | 1 | inner |
65.h | odd | 4 | 1 | inner |
195.s | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 195.2.s.b | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 195.2.s.b | ✓ | 32 |
5.b | even | 2 | 1 | 975.2.s.e | 32 | ||
5.c | odd | 4 | 1 | inner | 195.2.s.b | ✓ | 32 |
5.c | odd | 4 | 1 | 975.2.s.e | 32 | ||
13.b | even | 2 | 1 | inner | 195.2.s.b | ✓ | 32 |
15.d | odd | 2 | 1 | 975.2.s.e | 32 | ||
15.e | even | 4 | 1 | inner | 195.2.s.b | ✓ | 32 |
15.e | even | 4 | 1 | 975.2.s.e | 32 | ||
39.d | odd | 2 | 1 | inner | 195.2.s.b | ✓ | 32 |
65.d | even | 2 | 1 | 975.2.s.e | 32 | ||
65.h | odd | 4 | 1 | inner | 195.2.s.b | ✓ | 32 |
65.h | odd | 4 | 1 | 975.2.s.e | 32 | ||
195.e | odd | 2 | 1 | 975.2.s.e | 32 | ||
195.s | even | 4 | 1 | inner | 195.2.s.b | ✓ | 32 |
195.s | even | 4 | 1 | 975.2.s.e | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
195.2.s.b | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
195.2.s.b | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
195.2.s.b | ✓ | 32 | 5.c | odd | 4 | 1 | inner |
195.2.s.b | ✓ | 32 | 13.b | even | 2 | 1 | inner |
195.2.s.b | ✓ | 32 | 15.e | even | 4 | 1 | inner |
195.2.s.b | ✓ | 32 | 39.d | odd | 2 | 1 | inner |
195.2.s.b | ✓ | 32 | 65.h | odd | 4 | 1 | inner |
195.2.s.b | ✓ | 32 | 195.s | even | 4 | 1 | inner |
975.2.s.e | 32 | 5.b | even | 2 | 1 | ||
975.2.s.e | 32 | 5.c | odd | 4 | 1 | ||
975.2.s.e | 32 | 15.d | odd | 2 | 1 | ||
975.2.s.e | 32 | 15.e | even | 4 | 1 | ||
975.2.s.e | 32 | 65.d | even | 2 | 1 | ||
975.2.s.e | 32 | 65.h | odd | 4 | 1 | ||
975.2.s.e | 32 | 195.e | odd | 2 | 1 | ||
975.2.s.e | 32 | 195.s | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 35T_{2}^{12} + 263T_{2}^{8} + 133T_{2}^{4} + 4 \) acting on \(S_{2}^{\mathrm{new}}(195, [\chi])\).