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: | \(0\) |
Dimension: | \(31\) |
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 | −2.67025 | −2.19223 | 5.13021 | 1.01111 | 5.85378 | −2.59765 | −8.35844 | 1.80585 | −2.69992 | ||||||||||||||||||
1.2 | −2.51757 | −0.0469654 | 4.33818 | 2.78146 | 0.118239 | 3.17303 | −5.88653 | −2.99779 | −7.00252 | ||||||||||||||||||
1.3 | −2.29335 | 2.59523 | 3.25943 | 0.281199 | −5.95177 | −4.71795 | −2.88831 | 3.73524 | −0.644885 | ||||||||||||||||||
1.4 | −2.27484 | 0.433632 | 3.17492 | −2.29703 | −0.986446 | −4.86805 | −2.67276 | −2.81196 | 5.22539 | ||||||||||||||||||
1.5 | −2.10874 | −1.73148 | 2.44680 | 1.13214 | 3.65125 | 3.68428 | −0.942192 | −0.00197058 | −2.38740 | ||||||||||||||||||
1.6 | −1.89426 | 2.98322 | 1.58823 | 3.71742 | −5.65101 | 2.24937 | 0.780003 | 5.89963 | −7.04177 | ||||||||||||||||||
1.7 | −1.61641 | −1.01358 | 0.612772 | 4.09374 | 1.63835 | −3.10004 | 2.24232 | −1.97266 | −6.61715 | ||||||||||||||||||
1.8 | −1.51851 | 2.67622 | 0.305884 | −2.61923 | −4.06387 | 2.29369 | 2.57254 | 4.16214 | 3.97734 | ||||||||||||||||||
1.9 | −1.31847 | −2.88980 | −0.261634 | −1.34376 | 3.81011 | 0.384750 | 2.98190 | 5.35092 | 1.77171 | ||||||||||||||||||
1.10 | −1.09944 | −1.67977 | −0.791230 | −0.843613 | 1.84680 | 0.723048 | 3.06879 | −0.178383 | 0.927502 | ||||||||||||||||||
1.11 | −0.999239 | 0.560764 | −1.00152 | 0.738008 | −0.560337 | −0.683383 | 2.99924 | −2.68554 | −0.737446 | ||||||||||||||||||
1.12 | −0.749638 | 0.436865 | −1.43804 | −3.66899 | −0.327491 | −2.83791 | 2.57729 | −2.80915 | 2.75042 | ||||||||||||||||||
1.13 | −0.413206 | −2.92261 | −1.82926 | −1.23288 | 1.20764 | 5.16907 | 1.58227 | 5.54166 | 0.509432 | ||||||||||||||||||
1.14 | −0.187162 | 3.03513 | −1.96497 | 3.17183 | −0.568060 | −3.01076 | 0.742092 | 6.21199 | −0.593646 | ||||||||||||||||||
1.15 | 0.00234753 | 2.44529 | −1.99999 | 1.45917 | 0.00574040 | 4.30357 | −0.00939012 | 2.97943 | 0.00342546 | ||||||||||||||||||
1.16 | 0.262885 | 0.849379 | −1.93089 | −1.83441 | 0.223289 | −0.312976 | −1.03337 | −2.27856 | −0.482239 | ||||||||||||||||||
1.17 | 0.651423 | −0.506634 | −1.57565 | −2.90422 | −0.330033 | −3.71271 | −2.32926 | −2.74332 | −1.89188 | ||||||||||||||||||
1.18 | 0.800366 | −3.25822 | −1.35941 | 0.350077 | −2.60777 | −2.72656 | −2.68876 | 7.61603 | 0.280189 | ||||||||||||||||||
1.19 | 0.867375 | 0.595469 | −1.24766 | 3.84366 | 0.516495 | 1.26797 | −2.81694 | −2.64542 | 3.33390 | ||||||||||||||||||
1.20 | 1.13757 | −1.95057 | −0.705942 | 1.75766 | −2.21891 | 1.09174 | −3.07819 | 0.804735 | 1.99945 | ||||||||||||||||||
See all 31 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.f | ✓ | 31 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1441.2.a.f | ✓ | 31 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{31} - 6 T_{2}^{30} - 32 T_{2}^{29} + 248 T_{2}^{28} + 368 T_{2}^{27} - 4537 T_{2}^{26} + \cdots - 32 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1441))\).