Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6009,2,Mod(1,6009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6009, 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("6009.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6009 = 3 \cdot 2003 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6009.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: | \(47.9821065746\) |
Analytic rank: | \(1\) |
Dimension: | \(74\) |
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 | −2.66866 | −1.00000 | 5.12175 | 4.30387 | 2.66866 | −3.20120 | −8.33090 | 1.00000 | −11.4856 | ||||||||||||||||||
1.2 | −2.64890 | −1.00000 | 5.01665 | 1.26127 | 2.64890 | −1.41128 | −7.99079 | 1.00000 | −3.34096 | ||||||||||||||||||
1.3 | −2.63884 | −1.00000 | 4.96347 | −0.945135 | 2.63884 | −0.582601 | −7.82011 | 1.00000 | 2.49406 | ||||||||||||||||||
1.4 | −2.61467 | −1.00000 | 4.83652 | 2.19958 | 2.61467 | 2.37176 | −7.41659 | 1.00000 | −5.75118 | ||||||||||||||||||
1.5 | −2.47104 | −1.00000 | 4.10606 | 0.309716 | 2.47104 | 5.28779 | −5.20416 | 1.00000 | −0.765323 | ||||||||||||||||||
1.6 | −2.34730 | −1.00000 | 3.50981 | 0.307481 | 2.34730 | −0.174984 | −3.54398 | 1.00000 | −0.721749 | ||||||||||||||||||
1.7 | −2.32830 | −1.00000 | 3.42098 | −2.84103 | 2.32830 | −3.28939 | −3.30847 | 1.00000 | 6.61477 | ||||||||||||||||||
1.8 | −2.24634 | −1.00000 | 3.04604 | 0.438748 | 2.24634 | −3.83866 | −2.34975 | 1.00000 | −0.985577 | ||||||||||||||||||
1.9 | −2.22597 | −1.00000 | 2.95493 | 3.45290 | 2.22597 | 1.32160 | −2.12563 | 1.00000 | −7.68605 | ||||||||||||||||||
1.10 | −2.21160 | −1.00000 | 2.89120 | −2.88699 | 2.21160 | −1.76734 | −1.97097 | 1.00000 | 6.38487 | ||||||||||||||||||
1.11 | −2.15398 | −1.00000 | 2.63964 | 0.152687 | 2.15398 | −0.504417 | −1.37778 | 1.00000 | −0.328885 | ||||||||||||||||||
1.12 | −2.04145 | −1.00000 | 2.16752 | 0.190807 | 2.04145 | 0.816452 | −0.341978 | 1.00000 | −0.389523 | ||||||||||||||||||
1.13 | −1.99608 | −1.00000 | 1.98433 | −1.73451 | 1.99608 | 0.947777 | 0.0312835 | 1.00000 | 3.46221 | ||||||||||||||||||
1.14 | −1.91624 | −1.00000 | 1.67197 | 3.03343 | 1.91624 | 1.04097 | 0.628588 | 1.00000 | −5.81278 | ||||||||||||||||||
1.15 | −1.83090 | −1.00000 | 1.35220 | 3.40522 | 1.83090 | 2.98119 | 1.18606 | 1.00000 | −6.23462 | ||||||||||||||||||
1.16 | −1.78023 | −1.00000 | 1.16922 | −3.17382 | 1.78023 | 0.886559 | 1.47898 | 1.00000 | 5.65013 | ||||||||||||||||||
1.17 | −1.75582 | −1.00000 | 1.08292 | −2.00871 | 1.75582 | 2.72232 | 1.61024 | 1.00000 | 3.52694 | ||||||||||||||||||
1.18 | −1.72049 | −1.00000 | 0.960101 | 0.549541 | 1.72049 | −4.21154 | 1.78914 | 1.00000 | −0.945483 | ||||||||||||||||||
1.19 | −1.68808 | −1.00000 | 0.849604 | 3.75173 | 1.68808 | −4.24571 | 1.94196 | 1.00000 | −6.33322 | ||||||||||||||||||
1.20 | −1.52631 | −1.00000 | 0.329628 | −3.83581 | 1.52631 | 1.78836 | 2.54951 | 1.00000 | 5.85464 | ||||||||||||||||||
See all 74 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(2003\) | \(1\) |
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 | 6009.2.a.b | ✓ | 74 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6009.2.a.b | ✓ | 74 | 1.a | even | 1 | 1 | trivial |