Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,3,Mod(128,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.128");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.n (of order \(6\), degree \(2\), not 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: | \(12.0163796583\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 63) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
128.1 | −2.98832 | + | 1.72531i | 1.91247 | − | 2.31137i | 3.95337 | − | 6.84744i | − | 0.987479i | −1.72725 | + | 10.2067i | 0 | 13.4807i | −1.68490 | − | 8.84088i | 1.70370 | + | 2.95090i | |||||
128.2 | −2.65531 | + | 1.53305i | −2.90200 | − | 0.760509i | 2.70046 | − | 4.67733i | − | 0.260810i | 8.87162 | − | 2.42952i | 0 | 4.29534i | 7.84325 | + | 4.41400i | 0.399833 | + | 0.692532i | |||||
128.3 | −2.27188 | + | 1.31167i | 2.33893 | + | 1.87867i | 1.44095 | − | 2.49580i | 8.11665i | −7.77795 | − | 1.20021i | 0 | − | 2.93316i | 1.94118 | + | 8.78816i | −10.6464 | − | 18.4400i | |||||
128.4 | −1.64693 | + | 0.950855i | −1.46820 | + | 2.61618i | −0.191750 | + | 0.332121i | − | 1.64023i | −0.0695945 | − | 5.70470i | 0 | − | 8.33614i | −4.68880 | − | 7.68213i | 1.55962 | + | 2.70135i | ||||
128.5 | −0.744550 | + | 0.429866i | −0.107768 | − | 2.99806i | −1.63043 | + | 2.82399i | − | 6.44598i | 1.36900 | + | 2.18588i | 0 | − | 6.24239i | −8.97677 | + | 0.646193i | 2.77091 | + | 4.79936i | ||||
128.6 | −0.526549 | + | 0.304003i | 2.98425 | − | 0.307033i | −1.81516 | + | 3.14396i | − | 1.05593i | −1.47801 | + | 1.06889i | 0 | − | 4.63929i | 8.81146 | − | 1.83252i | 0.321007 | + | 0.556001i | ||||
128.7 | −0.296130 | + | 0.170971i | −1.73716 | − | 2.44587i | −1.94154 | + | 3.36284i | 8.90671i | 0.932598 | + | 0.427294i | 0 | − | 2.69555i | −2.96458 | + | 8.49772i | −1.52279 | − | 2.63755i | |||||
128.8 | 0.649615 | − | 0.375055i | 0.894137 | + | 2.86365i | −1.71867 | + | 2.97682i | − | 3.09558i | 1.65487 | + | 1.52492i | 0 | 5.57882i | −7.40104 | + | 5.12100i | −1.16101 | − | 2.01093i | |||||
128.9 | 1.73625 | − | 1.00242i | 2.54076 | − | 1.59516i | 0.00971148 | − | 0.0168208i | 4.93918i | 2.81237 | − | 5.31652i | 0 | 7.98046i | 3.91093 | − | 8.10584i | 4.95116 | + | 8.57566i | ||||||
128.10 | 2.50746 | − | 1.44768i | −0.711128 | + | 2.91450i | 2.19156 | − | 3.79590i | 7.88080i | 2.43614 | + | 8.33747i | 0 | − | 1.10929i | −7.98859 | − | 4.14516i | 11.4089 | + | 19.7608i | |||||
128.11 | 2.95047 | − | 1.70346i | 2.84264 | + | 0.958863i | 3.80352 | − | 6.58789i | − | 7.90772i | 10.0205 | − | 2.01321i | 0 | − | 12.2888i | 7.16116 | + | 5.45140i | −13.4704 | − | 23.3315i | ||||
128.12 | 3.28587 | − | 1.89710i | −1.58693 | − | 2.54591i | 5.19798 | − | 9.00316i | 1.94269i | −10.0443 | − | 5.35497i | 0 | − | 24.2675i | −3.96330 | + | 8.08036i | 3.68547 | + | 6.38343i | |||||
410.1 | −2.98832 | − | 1.72531i | 1.91247 | + | 2.31137i | 3.95337 | + | 6.84744i | 0.987479i | −1.72725 | − | 10.2067i | 0 | − | 13.4807i | −1.68490 | + | 8.84088i | 1.70370 | − | 2.95090i | |||||
410.2 | −2.65531 | − | 1.53305i | −2.90200 | + | 0.760509i | 2.70046 | + | 4.67733i | 0.260810i | 8.87162 | + | 2.42952i | 0 | − | 4.29534i | 7.84325 | − | 4.41400i | 0.399833 | − | 0.692532i | |||||
410.3 | −2.27188 | − | 1.31167i | 2.33893 | − | 1.87867i | 1.44095 | + | 2.49580i | − | 8.11665i | −7.77795 | + | 1.20021i | 0 | 2.93316i | 1.94118 | − | 8.78816i | −10.6464 | + | 18.4400i | |||||
410.4 | −1.64693 | − | 0.950855i | −1.46820 | − | 2.61618i | −0.191750 | − | 0.332121i | 1.64023i | −0.0695945 | + | 5.70470i | 0 | 8.33614i | −4.68880 | + | 7.68213i | 1.55962 | − | 2.70135i | ||||||
410.5 | −0.744550 | − | 0.429866i | −0.107768 | + | 2.99806i | −1.63043 | − | 2.82399i | 6.44598i | 1.36900 | − | 2.18588i | 0 | 6.24239i | −8.97677 | − | 0.646193i | 2.77091 | − | 4.79936i | ||||||
410.6 | −0.526549 | − | 0.304003i | 2.98425 | + | 0.307033i | −1.81516 | − | 3.14396i | 1.05593i | −1.47801 | − | 1.06889i | 0 | 4.63929i | 8.81146 | + | 1.83252i | 0.321007 | − | 0.556001i | ||||||
410.7 | −0.296130 | − | 0.170971i | −1.73716 | + | 2.44587i | −1.94154 | − | 3.36284i | − | 8.90671i | 0.932598 | − | 0.427294i | 0 | 2.69555i | −2.96458 | − | 8.49772i | −1.52279 | + | 2.63755i | |||||
410.8 | 0.649615 | + | 0.375055i | 0.894137 | − | 2.86365i | −1.71867 | − | 2.97682i | 3.09558i | 1.65487 | − | 1.52492i | 0 | − | 5.57882i | −7.40104 | − | 5.12100i | −1.16101 | + | 2.01093i | |||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.n | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.3.n.h | 24 | |
7.b | odd | 2 | 1 | 441.3.n.g | 24 | ||
7.c | even | 3 | 1 | 441.3.j.g | 24 | ||
7.c | even | 3 | 1 | 441.3.r.h | 24 | ||
7.d | odd | 6 | 1 | 63.3.r.a | ✓ | 24 | |
7.d | odd | 6 | 1 | 441.3.j.h | 24 | ||
9.d | odd | 6 | 1 | 441.3.j.g | 24 | ||
21.g | even | 6 | 1 | 189.3.r.a | 24 | ||
63.i | even | 6 | 1 | 63.3.r.a | ✓ | 24 | |
63.j | odd | 6 | 1 | 441.3.r.h | 24 | ||
63.k | odd | 6 | 1 | 567.3.b.a | 24 | ||
63.n | odd | 6 | 1 | inner | 441.3.n.h | 24 | |
63.o | even | 6 | 1 | 441.3.j.h | 24 | ||
63.s | even | 6 | 1 | 441.3.n.g | 24 | ||
63.s | even | 6 | 1 | 567.3.b.a | 24 | ||
63.t | odd | 6 | 1 | 189.3.r.a | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.3.r.a | ✓ | 24 | 7.d | odd | 6 | 1 | |
63.3.r.a | ✓ | 24 | 63.i | even | 6 | 1 | |
189.3.r.a | 24 | 21.g | even | 6 | 1 | ||
189.3.r.a | 24 | 63.t | odd | 6 | 1 | ||
441.3.j.g | 24 | 7.c | even | 3 | 1 | ||
441.3.j.g | 24 | 9.d | odd | 6 | 1 | ||
441.3.j.h | 24 | 7.d | odd | 6 | 1 | ||
441.3.j.h | 24 | 63.o | even | 6 | 1 | ||
441.3.n.g | 24 | 7.b | odd | 2 | 1 | ||
441.3.n.g | 24 | 63.s | even | 6 | 1 | ||
441.3.n.h | 24 | 1.a | even | 1 | 1 | trivial | |
441.3.n.h | 24 | 63.n | odd | 6 | 1 | inner | |
441.3.r.h | 24 | 7.c | even | 3 | 1 | ||
441.3.r.h | 24 | 63.j | odd | 6 | 1 | ||
567.3.b.a | 24 | 63.k | odd | 6 | 1 | ||
567.3.b.a | 24 | 63.s | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\):
\( T_{2}^{24} - 36 T_{2}^{22} + 828 T_{2}^{20} + 198 T_{2}^{19} - 11558 T_{2}^{18} - 4860 T_{2}^{17} + \cdots + 281961 \) |
\( T_{13}^{24} + 1149 T_{13}^{22} + 4320 T_{13}^{21} + 839334 T_{13}^{20} + 4046076 T_{13}^{19} + \cdots + 15\!\cdots\!36 \) |