Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1441,2,Mod(1,1441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1441, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1441.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1441 = 11 \cdot 131 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1441.a (trivial) |
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: | yes |
Analytic conductor: | \(11.5064429313\) |
Analytic rank: | \(1\) |
Dimension: | \(23\) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.66894 | 0.662508 | 5.12325 | −2.84363 | −1.76820 | −0.180196 | −8.33576 | −2.56108 | 7.58947 | ||||||||||||||||||
1.2 | −2.52155 | −0.648204 | 4.35820 | 2.39834 | 1.63448 | −0.583352 | −5.94631 | −2.57983 | −6.04753 | ||||||||||||||||||
1.3 | −2.47542 | 2.04682 | 4.12769 | −2.23639 | −5.06674 | 4.08338 | −5.26693 | 1.18947 | 5.53600 | ||||||||||||||||||
1.4 | −2.16219 | −2.96733 | 2.67508 | 2.79492 | 6.41593 | 0.0561777 | −1.45964 | 5.80503 | −6.04316 | ||||||||||||||||||
1.5 | −2.10794 | 1.73442 | 2.44341 | 0.240614 | −3.65606 | 0.0249073 | −0.934681 | 0.00822254 | −0.507199 | ||||||||||||||||||
1.6 | −2.06686 | −1.75987 | 2.27189 | −3.28939 | 3.63739 | 1.71043 | −0.561957 | 0.0971338 | 6.79870 | ||||||||||||||||||
1.7 | −1.61515 | −2.31019 | 0.608714 | −1.59855 | 3.73131 | −4.36629 | 2.24714 | 2.33698 | 2.58190 | ||||||||||||||||||
1.8 | −1.47853 | 1.20449 | 0.186052 | 1.76198 | −1.78087 | −2.63188 | 2.68198 | −1.54921 | −2.60515 | ||||||||||||||||||
1.9 | −1.26006 | 0.450963 | −0.412243 | 1.25945 | −0.568242 | 3.91553 | 3.03958 | −2.79663 | −1.58699 | ||||||||||||||||||
1.10 | −0.850138 | −0.411295 | −1.27727 | −3.81435 | 0.349658 | 2.53746 | 2.78613 | −2.83084 | 3.24272 | ||||||||||||||||||
1.11 | −0.793163 | 2.98559 | −1.37089 | −2.40573 | −2.36806 | −3.11236 | 2.67367 | 5.91372 | 1.90814 | ||||||||||||||||||
1.12 | −0.265643 | −2.11945 | −1.92943 | 2.15570 | 0.563016 | −4.44888 | 1.04383 | 1.49206 | −0.572646 | ||||||||||||||||||
1.13 | −0.248714 | −2.31541 | −1.93814 | 3.78233 | 0.575874 | 0.228936 | 0.979470 | 2.36112 | −0.940718 | ||||||||||||||||||
1.14 | −0.200388 | 2.28550 | −1.95984 | −0.516505 | −0.457986 | −2.27895 | 0.793506 | 2.22349 | 0.103502 | ||||||||||||||||||
1.15 | 0.465389 | 1.44715 | −1.78341 | 0.0108050 | 0.673486 | 0.934440 | −1.76076 | −0.905766 | 0.00502854 | ||||||||||||||||||
1.16 | 0.947276 | −2.80681 | −1.10267 | −2.22661 | −2.65882 | 3.55035 | −2.93908 | 4.87818 | −2.10922 | ||||||||||||||||||
1.17 | 1.04685 | 0.462154 | −0.904108 | 2.89071 | 0.483806 | −3.03594 | −3.04016 | −2.78641 | 3.02614 | ||||||||||||||||||
1.18 | 1.39370 | 1.35746 | −0.0576006 | −3.24463 | 1.89189 | 2.34571 | −2.86768 | −1.15730 | −4.52204 | ||||||||||||||||||
1.19 | 1.54502 | 2.73368 | 0.387096 | −3.25406 | 4.22360 | −3.29481 | −2.49197 | 4.47303 | −5.02759 | ||||||||||||||||||
1.20 | 1.83773 | −0.0182258 | 1.37726 | 1.49121 | −0.0334942 | −3.57754 | −1.14443 | −2.99967 | 2.74044 | ||||||||||||||||||
See all 23 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(11\) | \(1\) |
\(131\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1441.2.a.d | ✓ | 23 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1441.2.a.d | ✓ | 23 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{23} + 7 T_{2}^{22} - 8 T_{2}^{21} - 156 T_{2}^{20} - 142 T_{2}^{19} + 1401 T_{2}^{18} + 2517 T_{2}^{17} + \cdots - 76 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1441))\).