Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,3,Mod(26,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.26");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.l (of order \(10\), 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: | \(2.69755461717\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 | −1.91284 | + | 2.63279i | 0 | −2.03659 | − | 6.26797i | −0.936271 | − | 1.28867i | 0 | −3.36204 | − | 10.3473i | 8.01778 | + | 2.60514i | 0 | 5.18373 | ||||||||
26.2 | −1.75181 | + | 2.41117i | 0 | −1.50880 | − | 4.64360i | 4.61811 | + | 6.35629i | 0 | 0.867699 | + | 2.67050i | 2.50164 | + | 0.812833i | 0 | −23.4161 | ||||||||
26.3 | −1.15480 | + | 1.58945i | 0 | 0.0432885 | + | 0.133228i | −5.65603 | − | 7.78486i | 0 | −1.61633 | − | 4.97456i | −7.73579 | − | 2.51351i | 0 | 18.9052 | ||||||||
26.4 | −0.480038 | + | 0.660715i | 0 | 1.02996 | + | 3.16989i | −0.615389 | − | 0.847011i | 0 | 2.11067 | + | 6.49597i | −5.69568 | − | 1.85064i | 0 | 0.855043 | ||||||||
26.5 | 0.480038 | − | 0.660715i | 0 | 1.02996 | + | 3.16989i | 0.615389 | + | 0.847011i | 0 | 2.11067 | + | 6.49597i | 5.69568 | + | 1.85064i | 0 | 0.855043 | ||||||||
26.6 | 1.15480 | − | 1.58945i | 0 | 0.0432885 | + | 0.133228i | 5.65603 | + | 7.78486i | 0 | −1.61633 | − | 4.97456i | 7.73579 | + | 2.51351i | 0 | 18.9052 | ||||||||
26.7 | 1.75181 | − | 2.41117i | 0 | −1.50880 | − | 4.64360i | −4.61811 | − | 6.35629i | 0 | 0.867699 | + | 2.67050i | −2.50164 | − | 0.812833i | 0 | −23.4161 | ||||||||
26.8 | 1.91284 | − | 2.63279i | 0 | −2.03659 | − | 6.26797i | 0.936271 | + | 1.28867i | 0 | −3.36204 | − | 10.3473i | −8.01778 | − | 2.60514i | 0 | 5.18373 | ||||||||
53.1 | −3.44320 | − | 1.11876i | 0 | 7.36792 | + | 5.35311i | −0.157113 | + | 0.0510491i | 0 | −8.33195 | − | 6.05352i | −10.8683 | − | 14.9589i | 0 | 0.598083 | ||||||||
53.2 | −2.84833 | − | 0.925479i | 0 | 4.02041 | + | 2.92100i | −1.53284 | + | 0.498051i | 0 | 2.69388 | + | 1.95722i | −1.70668 | − | 2.34904i | 0 | 4.82698 | ||||||||
53.3 | −1.28527 | − | 0.417608i | 0 | −1.75856 | − | 1.27767i | −4.85485 | + | 1.57744i | 0 | 10.6531 | + | 7.73991i | 4.90400 | + | 6.74978i | 0 | 6.89852 | ||||||||
53.4 | −0.296118 | − | 0.0962144i | 0 | −3.15764 | − | 2.29416i | 5.65537 | − | 1.83754i | 0 | −7.01499 | − | 5.09669i | 1.44634 | + | 1.99072i | 0 | −1.85145 | ||||||||
53.5 | 0.296118 | + | 0.0962144i | 0 | −3.15764 | − | 2.29416i | −5.65537 | + | 1.83754i | 0 | −7.01499 | − | 5.09669i | −1.44634 | − | 1.99072i | 0 | −1.85145 | ||||||||
53.6 | 1.28527 | + | 0.417608i | 0 | −1.75856 | − | 1.27767i | 4.85485 | − | 1.57744i | 0 | 10.6531 | + | 7.73991i | −4.90400 | − | 6.74978i | 0 | 6.89852 | ||||||||
53.7 | 2.84833 | + | 0.925479i | 0 | 4.02041 | + | 2.92100i | 1.53284 | − | 0.498051i | 0 | 2.69388 | + | 1.95722i | 1.70668 | + | 2.34904i | 0 | 4.82698 | ||||||||
53.8 | 3.44320 | + | 1.11876i | 0 | 7.36792 | + | 5.35311i | 0.157113 | − | 0.0510491i | 0 | −8.33195 | − | 6.05352i | 10.8683 | + | 14.9589i | 0 | 0.598083 | ||||||||
71.1 | −3.44320 | + | 1.11876i | 0 | 7.36792 | − | 5.35311i | −0.157113 | − | 0.0510491i | 0 | −8.33195 | + | 6.05352i | −10.8683 | + | 14.9589i | 0 | 0.598083 | ||||||||
71.2 | −2.84833 | + | 0.925479i | 0 | 4.02041 | − | 2.92100i | −1.53284 | − | 0.498051i | 0 | 2.69388 | − | 1.95722i | −1.70668 | + | 2.34904i | 0 | 4.82698 | ||||||||
71.3 | −1.28527 | + | 0.417608i | 0 | −1.75856 | + | 1.27767i | −4.85485 | − | 1.57744i | 0 | 10.6531 | − | 7.73991i | 4.90400 | − | 6.74978i | 0 | 6.89852 | ||||||||
71.4 | −0.296118 | + | 0.0962144i | 0 | −3.15764 | + | 2.29416i | 5.65537 | + | 1.83754i | 0 | −7.01499 | + | 5.09669i | 1.44634 | − | 1.99072i | 0 | −1.85145 | ||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.c | even | 5 | 1 | inner |
33.h | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.3.l.a | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 99.3.l.a | ✓ | 32 |
11.c | even | 5 | 1 | inner | 99.3.l.a | ✓ | 32 |
11.c | even | 5 | 1 | 1089.3.b.i | 16 | ||
11.d | odd | 10 | 1 | 1089.3.b.j | 16 | ||
33.f | even | 10 | 1 | 1089.3.b.j | 16 | ||
33.h | odd | 10 | 1 | inner | 99.3.l.a | ✓ | 32 |
33.h | odd | 10 | 1 | 1089.3.b.i | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.3.l.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
99.3.l.a | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
99.3.l.a | ✓ | 32 | 11.c | even | 5 | 1 | inner |
99.3.l.a | ✓ | 32 | 33.h | odd | 10 | 1 | inner |
1089.3.b.i | 16 | 11.c | even | 5 | 1 | ||
1089.3.b.i | 16 | 33.h | odd | 10 | 1 | ||
1089.3.b.j | 16 | 11.d | odd | 10 | 1 | ||
1089.3.b.j | 16 | 33.f | even | 10 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(99, [\chi])\).