Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8009,2,Mod(1,8009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8009, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8009.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8009 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8009.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: | \(63.9521869788\) |
Analytic rank: | \(1\) |
Dimension: | \(306\) |
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 | −2.77719 | −1.72357 | 5.71279 | −0.359137 | 4.78667 | −0.363859 | −10.3111 | −0.0293152 | 0.997391 | ||||||||||||||||||
1.2 | −2.76073 | 2.52985 | 5.62166 | 0.0311099 | −6.98425 | 0.584590 | −9.99844 | 3.40015 | −0.0858861 | ||||||||||||||||||
1.3 | −2.72646 | 1.03232 | 5.43357 | 3.59624 | −2.81456 | 2.40174 | −9.36147 | −1.93433 | −9.80500 | ||||||||||||||||||
1.4 | −2.72316 | −0.885304 | 5.41558 | 2.13071 | 2.41082 | −0.787494 | −9.30115 | −2.21624 | −5.80224 | ||||||||||||||||||
1.5 | −2.70598 | −0.529374 | 5.32233 | 3.62605 | 1.43248 | −4.04119 | −8.99016 | −2.71976 | −9.81202 | ||||||||||||||||||
1.6 | −2.66884 | 3.40767 | 5.12268 | 0.00216831 | −9.09450 | −2.11405 | −8.33393 | 8.61218 | −0.00578687 | ||||||||||||||||||
1.7 | −2.66627 | −0.991276 | 5.10897 | −2.13375 | 2.64301 | 1.36601 | −8.28935 | −2.01737 | 5.68914 | ||||||||||||||||||
1.8 | −2.64630 | 2.23042 | 5.00289 | −0.684253 | −5.90235 | 0.773793 | −7.94655 | 1.97477 | 1.81074 | ||||||||||||||||||
1.9 | −2.63809 | −1.56073 | 4.95951 | 2.27175 | 4.11735 | −0.895063 | −7.80745 | −0.564115 | −5.99307 | ||||||||||||||||||
1.10 | −2.63380 | −2.43471 | 4.93693 | −0.366932 | 6.41255 | 4.18826 | −7.73529 | 2.92781 | 0.966426 | ||||||||||||||||||
1.11 | −2.60787 | 1.96510 | 4.80101 | 3.32724 | −5.12474 | 0.944759 | −7.30469 | 0.861625 | −8.67702 | ||||||||||||||||||
1.12 | −2.59381 | 0.956385 | 4.72785 | −4.27828 | −2.48068 | 1.62544 | −7.07552 | −2.08533 | 11.0970 | ||||||||||||||||||
1.13 | −2.58891 | −2.43363 | 4.70248 | −1.53347 | 6.30046 | 1.06858 | −6.99648 | 2.92255 | 3.97002 | ||||||||||||||||||
1.14 | −2.58035 | 2.52546 | 4.65823 | 0.386293 | −6.51658 | −3.83630 | −6.85917 | 3.37795 | −0.996774 | ||||||||||||||||||
1.15 | −2.56583 | −0.0859559 | 4.58350 | 0.644754 | 0.220548 | −0.863168 | −6.62882 | −2.99261 | −1.65433 | ||||||||||||||||||
1.16 | −2.56252 | 1.99928 | 4.56650 | −3.64858 | −5.12319 | −3.78692 | −6.57671 | 0.997123 | 9.34955 | ||||||||||||||||||
1.17 | −2.55906 | 0.342269 | 4.54881 | −2.51856 | −0.875888 | −3.84856 | −6.52258 | −2.88285 | 6.44516 | ||||||||||||||||||
1.18 | −2.52963 | 0.577486 | 4.39902 | −1.57582 | −1.46082 | −2.02955 | −6.06864 | −2.66651 | 3.98624 | ||||||||||||||||||
1.19 | −2.52232 | −2.77220 | 4.36210 | 3.44597 | 6.99239 | −1.87628 | −5.95798 | 4.68511 | −8.69185 | ||||||||||||||||||
1.20 | −2.50471 | 3.07842 | 4.27357 | 4.01883 | −7.71055 | −2.99037 | −5.69464 | 6.47666 | −10.0660 | ||||||||||||||||||
See next 80 embeddings (of 306 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(8009\) | \(1\) |
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 | 8009.2.a.a | ✓ | 306 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8009.2.a.a | ✓ | 306 | 1.a | even | 1 | 1 | trivial |