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: | \(0\) |
Dimension: | \(92\) |
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 | −2.78023 | 1.00000 | 5.72970 | −1.51058 | −2.78023 | −3.00947 | −10.3695 | 1.00000 | 4.19978 | ||||||||||||||||||
1.2 | −2.66639 | 1.00000 | 5.10965 | 3.96550 | −2.66639 | 2.39934 | −8.29154 | 1.00000 | −10.5736 | ||||||||||||||||||
1.3 | −2.62844 | 1.00000 | 4.90869 | −1.09828 | −2.62844 | −4.21500 | −7.64532 | 1.00000 | 2.88675 | ||||||||||||||||||
1.4 | −2.60458 | 1.00000 | 4.78383 | 2.52259 | −2.60458 | −2.57820 | −7.25072 | 1.00000 | −6.57030 | ||||||||||||||||||
1.5 | −2.49751 | 1.00000 | 4.23755 | 1.03851 | −2.49751 | 3.59057 | −5.58831 | 1.00000 | −2.59368 | ||||||||||||||||||
1.6 | −2.48563 | 1.00000 | 4.17837 | 2.55956 | −2.48563 | 1.79588 | −5.41463 | 1.00000 | −6.36212 | ||||||||||||||||||
1.7 | −2.40248 | 1.00000 | 3.77189 | 0.0504655 | −2.40248 | −0.217870 | −4.25692 | 1.00000 | −0.121242 | ||||||||||||||||||
1.8 | −2.37626 | 1.00000 | 3.64660 | −0.379121 | −2.37626 | 2.10974 | −3.91276 | 1.00000 | 0.900889 | ||||||||||||||||||
1.9 | −2.31255 | 1.00000 | 3.34790 | −0.928530 | −2.31255 | −3.24971 | −3.11708 | 1.00000 | 2.14727 | ||||||||||||||||||
1.10 | −2.28351 | 1.00000 | 3.21443 | −3.38813 | −2.28351 | 0.924728 | −2.77317 | 1.00000 | 7.73684 | ||||||||||||||||||
1.11 | −2.23454 | 1.00000 | 2.99319 | 2.17127 | −2.23454 | −3.00622 | −2.21933 | 1.00000 | −4.85180 | ||||||||||||||||||
1.12 | −2.22934 | 1.00000 | 2.96995 | −1.56663 | −2.22934 | 2.33278 | −2.16235 | 1.00000 | 3.49254 | ||||||||||||||||||
1.13 | −2.22794 | 1.00000 | 2.96373 | 3.46015 | −2.22794 | 3.74184 | −2.14714 | 1.00000 | −7.70902 | ||||||||||||||||||
1.14 | −2.15336 | 1.00000 | 2.63696 | −3.93365 | −2.15336 | −2.88832 | −1.37159 | 1.00000 | 8.47056 | ||||||||||||||||||
1.15 | −1.90307 | 1.00000 | 1.62167 | −1.77542 | −1.90307 | −0.953587 | 0.719990 | 1.00000 | 3.37874 | ||||||||||||||||||
1.16 | −1.89797 | 1.00000 | 1.60229 | 4.27518 | −1.89797 | −0.912076 | 0.754847 | 1.00000 | −8.11416 | ||||||||||||||||||
1.17 | −1.84272 | 1.00000 | 1.39563 | −0.293700 | −1.84272 | 0.646833 | 1.11368 | 1.00000 | 0.541209 | ||||||||||||||||||
1.18 | −1.77532 | 1.00000 | 1.15178 | 0.669700 | −1.77532 | 2.59999 | 1.50587 | 1.00000 | −1.18893 | ||||||||||||||||||
1.19 | −1.73440 | 1.00000 | 1.00813 | 1.28297 | −1.73440 | −0.642029 | 1.72030 | 1.00000 | −2.22518 | ||||||||||||||||||
1.20 | −1.66735 | 1.00000 | 0.780065 | −0.964493 | −1.66735 | 4.83846 | 2.03406 | 1.00000 | 1.60815 | ||||||||||||||||||
See all 92 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.c | ✓ | 92 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6009.2.a.c | ✓ | 92 | 1.a | even | 1 | 1 | trivial |