Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8049,2,Mod(1,8049)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8049, 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("8049.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8049 = 3 \cdot 2683 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8049.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: | \(64.2715885869\) |
Analytic rank: | \(0\) |
Dimension: | \(119\) |
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 | −2.76674 | −1.00000 | 5.65483 | −0.357353 | 2.76674 | −2.67071 | −10.1119 | 1.00000 | 0.988701 | ||||||||||||||||||
1.2 | −2.73168 | −1.00000 | 5.46207 | −0.892345 | 2.73168 | −1.57860 | −9.45726 | 1.00000 | 2.43760 | ||||||||||||||||||
1.3 | −2.70718 | −1.00000 | 5.32884 | −1.26891 | 2.70718 | −3.95115 | −9.01180 | 1.00000 | 3.43519 | ||||||||||||||||||
1.4 | −2.67169 | −1.00000 | 5.13794 | 4.23533 | 2.67169 | 3.62248 | −8.38362 | 1.00000 | −11.3155 | ||||||||||||||||||
1.5 | −2.65333 | −1.00000 | 5.04016 | 2.95577 | 2.65333 | 2.19565 | −8.06655 | 1.00000 | −7.84263 | ||||||||||||||||||
1.6 | −2.62888 | −1.00000 | 4.91100 | 3.06865 | 2.62888 | −2.79906 | −7.65266 | 1.00000 | −8.06710 | ||||||||||||||||||
1.7 | −2.61270 | −1.00000 | 4.82618 | 1.51274 | 2.61270 | 3.06130 | −7.38394 | 1.00000 | −3.95232 | ||||||||||||||||||
1.8 | −2.60392 | −1.00000 | 4.78038 | 3.66153 | 2.60392 | −3.52018 | −7.23988 | 1.00000 | −9.53431 | ||||||||||||||||||
1.9 | −2.46430 | −1.00000 | 4.07278 | −0.321627 | 2.46430 | 1.46780 | −5.10794 | 1.00000 | 0.792584 | ||||||||||||||||||
1.10 | −2.39135 | −1.00000 | 3.71855 | −1.51157 | 2.39135 | 2.32148 | −4.10966 | 1.00000 | 3.61469 | ||||||||||||||||||
1.11 | −2.34737 | −1.00000 | 3.51014 | −3.81233 | 2.34737 | −0.357303 | −3.54486 | 1.00000 | 8.94895 | ||||||||||||||||||
1.12 | −2.34205 | −1.00000 | 3.48520 | 0.568796 | 2.34205 | −0.877328 | −3.47840 | 1.00000 | −1.33215 | ||||||||||||||||||
1.13 | −2.28998 | −1.00000 | 3.24400 | 0.0628594 | 2.28998 | −3.99468 | −2.84873 | 1.00000 | −0.143947 | ||||||||||||||||||
1.14 | −2.26557 | −1.00000 | 3.13283 | 0.270785 | 2.26557 | 1.54301 | −2.56650 | 1.00000 | −0.613482 | ||||||||||||||||||
1.15 | −2.25432 | −1.00000 | 3.08196 | −2.79786 | 2.25432 | −0.509718 | −2.43909 | 1.00000 | 6.30728 | ||||||||||||||||||
1.16 | −2.24285 | −1.00000 | 3.03036 | −3.65739 | 2.24285 | −4.78600 | −2.31094 | 1.00000 | 8.20296 | ||||||||||||||||||
1.17 | −2.22986 | −1.00000 | 2.97225 | 2.09159 | 2.22986 | 2.90452 | −2.16799 | 1.00000 | −4.66395 | ||||||||||||||||||
1.18 | −2.22780 | −1.00000 | 2.96308 | 4.08366 | 2.22780 | 2.32778 | −2.14554 | 1.00000 | −9.09757 | ||||||||||||||||||
1.19 | −2.21831 | −1.00000 | 2.92089 | 1.70107 | 2.21831 | 0.0884207 | −2.04282 | 1.00000 | −3.77350 | ||||||||||||||||||
1.20 | −2.08408 | −1.00000 | 2.34338 | −1.55135 | 2.08408 | 4.26949 | −0.715634 | 1.00000 | 3.23314 | ||||||||||||||||||
See next 80 embeddings (of 119 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(2683\) | \(-1\) |
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 | 8049.2.a.c | ✓ | 119 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8049.2.a.c | ✓ | 119 | 1.a | even | 1 | 1 | trivial |