Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,2,Mod(362,441)]
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, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.362");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.s (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: | \(3.52140272914\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
362.1 | −2.23278 | + | 1.28910i | −0.625134 | − | 1.61530i | 2.32354 | − | 4.02449i | 2.33189 | 3.47807 | + | 2.80076i | 0 | 6.82470i | −2.21841 | + | 2.01956i | −5.20660 | + | 3.00603i | ||||||
362.2 | −2.23278 | + | 1.28910i | 0.625134 | + | 1.61530i | 2.32354 | − | 4.02449i | −2.33189 | −3.47807 | − | 2.80076i | 0 | 6.82470i | −2.21841 | + | 2.01956i | 5.20660 | − | 3.00603i | ||||||
362.3 | −1.80506 | + | 1.04215i | −1.73189 | − | 0.0239080i | 1.17216 | − | 2.03024i | 3.30465 | 3.15107 | − | 1.76173i | 0 | 0.717672i | 2.99886 | + | 0.0828118i | −5.96509 | + | 3.44395i | ||||||
362.4 | −1.80506 | + | 1.04215i | 1.73189 | + | 0.0239080i | 1.17216 | − | 2.03024i | −3.30465 | −3.15107 | + | 1.76173i | 0 | 0.717672i | 2.99886 | + | 0.0828118i | 5.96509 | − | 3.44395i | ||||||
362.5 | −1.61855 | + | 0.934468i | −0.710525 | + | 1.57961i | 0.746462 | − | 1.29291i | 2.50573 | −0.326074 | − | 3.22063i | 0 | − | 0.947692i | −1.99031 | − | 2.24470i | −4.05565 | + | 2.34153i | |||||
362.6 | −1.61855 | + | 0.934468i | 0.710525 | − | 1.57961i | 0.746462 | − | 1.29291i | −2.50573 | 0.326074 | + | 3.22063i | 0 | − | 0.947692i | −1.99031 | − | 2.24470i | 4.05565 | − | 2.34153i | |||||
362.7 | −1.58658 | + | 0.916012i | −0.108803 | + | 1.72863i | 0.678156 | − | 1.17460i | −0.645568 | −1.41082 | − | 2.84227i | 0 | − | 1.17925i | −2.97632 | − | 0.376160i | 1.02425 | − | 0.591348i | |||||
362.8 | −1.58658 | + | 0.916012i | 0.108803 | − | 1.72863i | 0.678156 | − | 1.17460i | 0.645568 | 1.41082 | + | 2.84227i | 0 | − | 1.17925i | −2.97632 | − | 0.376160i | −1.02425 | + | 0.591348i | |||||
362.9 | −0.575298 | + | 0.332148i | −0.537154 | − | 1.64665i | −0.779355 | + | 1.34988i | 0.0283039 | 0.855956 | + | 0.768901i | 0 | − | 2.36404i | −2.42293 | + | 1.76901i | −0.0162832 | + | 0.00940110i | |||||
362.10 | −0.575298 | + | 0.332148i | 0.537154 | + | 1.64665i | −0.779355 | + | 1.34988i | −0.0283039 | −0.855956 | − | 0.768901i | 0 | − | 2.36404i | −2.42293 | + | 1.76901i | 0.0162832 | − | 0.00940110i | |||||
362.11 | −0.105953 | + | 0.0611722i | −1.73002 | + | 0.0838860i | −0.992516 | + | 1.71909i | 0.529430 | 0.178170 | − | 0.114717i | 0 | − | 0.487547i | 2.98593 | − | 0.290249i | −0.0560949 | + | 0.0323864i | |||||
362.12 | −0.105953 | + | 0.0611722i | 1.73002 | − | 0.0838860i | −0.992516 | + | 1.71909i | −0.529430 | −0.178170 | + | 0.114717i | 0 | − | 0.487547i | 2.98593 | − | 0.290249i | 0.0560949 | − | 0.0323864i | |||||
362.13 | 0.367369 | − | 0.212101i | −1.71145 | − | 0.266368i | −0.910027 | + | 1.57621i | −3.60763 | −0.685229 | + | 0.265143i | 0 | 1.62047i | 2.85810 | + | 0.911750i | −1.32533 | + | 0.765180i | ||||||
362.14 | 0.367369 | − | 0.212101i | 1.71145 | + | 0.266368i | −0.910027 | + | 1.57621i | 3.60763 | 0.685229 | − | 0.265143i | 0 | 1.62047i | 2.85810 | + | 0.911750i | 1.32533 | − | 0.765180i | ||||||
362.15 | 0.850109 | − | 0.490811i | −0.831570 | + | 1.51937i | −0.518210 | + | 0.897565i | −1.88120 | 0.0387990 | + | 1.69978i | 0 | 2.98061i | −1.61698 | − | 2.52693i | −1.59922 | + | 0.923312i | ||||||
362.16 | 0.850109 | − | 0.490811i | 0.831570 | − | 1.51937i | −0.518210 | + | 0.897565i | 1.88120 | −0.0387990 | − | 1.69978i | 0 | 2.98061i | −1.61698 | − | 2.52693i | 1.59922 | − | 0.923312i | ||||||
362.17 | 1.02035 | − | 0.589100i | −1.34152 | − | 1.09560i | −0.305921 | + | 0.529871i | 4.33202 | −2.01424 | − | 0.327604i | 0 | 3.07728i | 0.599340 | + | 2.93952i | 4.42019 | − | 2.55200i | ||||||
362.18 | 1.02035 | − | 0.589100i | 1.34152 | + | 1.09560i | −0.305921 | + | 0.529871i | −4.33202 | 2.01424 | + | 0.327604i | 0 | 3.07728i | 0.599340 | + | 2.93952i | −4.42019 | + | 2.55200i | ||||||
362.19 | 1.28562 | − | 0.742253i | −1.72526 | + | 0.153190i | 0.101880 | − | 0.176462i | −0.308431 | −2.10433 | + | 1.47753i | 0 | 2.66653i | 2.95307 | − | 0.528586i | −0.396525 | + | 0.228934i | ||||||
362.20 | 1.28562 | − | 0.742253i | 1.72526 | − | 0.153190i | 0.101880 | − | 0.176462i | 0.308431 | 2.10433 | − | 1.47753i | 0 | 2.66653i | 2.95307 | − | 0.528586i | 0.396525 | − | 0.228934i | ||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
63.n | odd | 6 | 1 | inner |
63.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.2.s.d | 48 | |
3.b | odd | 2 | 1 | 1323.2.s.d | 48 | ||
7.b | odd | 2 | 1 | inner | 441.2.s.d | 48 | |
7.c | even | 3 | 1 | 441.2.i.d | 48 | ||
7.c | even | 3 | 1 | 441.2.o.e | ✓ | 48 | |
7.d | odd | 6 | 1 | 441.2.i.d | 48 | ||
7.d | odd | 6 | 1 | 441.2.o.e | ✓ | 48 | |
9.c | even | 3 | 1 | 1323.2.i.d | 48 | ||
9.d | odd | 6 | 1 | 441.2.i.d | 48 | ||
21.c | even | 2 | 1 | 1323.2.s.d | 48 | ||
21.g | even | 6 | 1 | 1323.2.i.d | 48 | ||
21.g | even | 6 | 1 | 1323.2.o.e | 48 | ||
21.h | odd | 6 | 1 | 1323.2.i.d | 48 | ||
21.h | odd | 6 | 1 | 1323.2.o.e | 48 | ||
63.g | even | 3 | 1 | 1323.2.s.d | 48 | ||
63.h | even | 3 | 1 | 1323.2.o.e | 48 | ||
63.i | even | 6 | 1 | 441.2.o.e | ✓ | 48 | |
63.j | odd | 6 | 1 | 441.2.o.e | ✓ | 48 | |
63.k | odd | 6 | 1 | 1323.2.s.d | 48 | ||
63.l | odd | 6 | 1 | 1323.2.i.d | 48 | ||
63.n | odd | 6 | 1 | inner | 441.2.s.d | 48 | |
63.o | even | 6 | 1 | 441.2.i.d | 48 | ||
63.s | even | 6 | 1 | inner | 441.2.s.d | 48 | |
63.t | odd | 6 | 1 | 1323.2.o.e | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.2.i.d | 48 | 7.c | even | 3 | 1 | ||
441.2.i.d | 48 | 7.d | odd | 6 | 1 | ||
441.2.i.d | 48 | 9.d | odd | 6 | 1 | ||
441.2.i.d | 48 | 63.o | even | 6 | 1 | ||
441.2.o.e | ✓ | 48 | 7.c | even | 3 | 1 | |
441.2.o.e | ✓ | 48 | 7.d | odd | 6 | 1 | |
441.2.o.e | ✓ | 48 | 63.i | even | 6 | 1 | |
441.2.o.e | ✓ | 48 | 63.j | odd | 6 | 1 | |
441.2.s.d | 48 | 1.a | even | 1 | 1 | trivial | |
441.2.s.d | 48 | 7.b | odd | 2 | 1 | inner | |
441.2.s.d | 48 | 63.n | odd | 6 | 1 | inner | |
441.2.s.d | 48 | 63.s | even | 6 | 1 | inner | |
1323.2.i.d | 48 | 9.c | even | 3 | 1 | ||
1323.2.i.d | 48 | 21.g | even | 6 | 1 | ||
1323.2.i.d | 48 | 21.h | odd | 6 | 1 | ||
1323.2.i.d | 48 | 63.l | odd | 6 | 1 | ||
1323.2.o.e | 48 | 21.g | even | 6 | 1 | ||
1323.2.o.e | 48 | 21.h | odd | 6 | 1 | ||
1323.2.o.e | 48 | 63.h | even | 3 | 1 | ||
1323.2.o.e | 48 | 63.t | odd | 6 | 1 | ||
1323.2.s.d | 48 | 3.b | odd | 2 | 1 | ||
1323.2.s.d | 48 | 21.c | even | 2 | 1 | ||
1323.2.s.d | 48 | 63.g | even | 3 | 1 | ||
1323.2.s.d | 48 | 63.k | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 18 T_{2}^{22} + 207 T_{2}^{20} + 12 T_{2}^{19} - 1434 T_{2}^{18} - 108 T_{2}^{17} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).