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: | \(1\) |
Dimension: | \(104\) |
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 | −2.70998 | −1.00000 | 5.34399 | −2.58706 | 2.70998 | 1.21378 | −9.06215 | 1.00000 | 7.01089 | ||||||||||||||||||
1.2 | −2.69984 | −1.00000 | 5.28916 | 0.253804 | 2.69984 | 2.11001 | −8.88022 | 1.00000 | −0.685231 | ||||||||||||||||||
1.3 | −2.68757 | −1.00000 | 5.22302 | 2.72258 | 2.68757 | −1.36352 | −8.66207 | 1.00000 | −7.31712 | ||||||||||||||||||
1.4 | −2.67025 | −1.00000 | 5.13022 | −3.32329 | 2.67025 | −3.25210 | −8.35846 | 1.00000 | 8.87402 | ||||||||||||||||||
1.5 | −2.64591 | −1.00000 | 5.00085 | 1.12056 | 2.64591 | 4.16027 | −7.94000 | 1.00000 | −2.96491 | ||||||||||||||||||
1.6 | −2.57733 | −1.00000 | 4.64262 | 2.42154 | 2.57733 | −2.34868 | −6.81090 | 1.00000 | −6.24111 | ||||||||||||||||||
1.7 | −2.53392 | −1.00000 | 4.42075 | −2.15927 | 2.53392 | −0.210471 | −6.13399 | 1.00000 | 5.47142 | ||||||||||||||||||
1.8 | −2.52808 | −1.00000 | 4.39119 | −2.04731 | 2.52808 | 3.64200 | −6.04511 | 1.00000 | 5.17577 | ||||||||||||||||||
1.9 | −2.51614 | −1.00000 | 4.33096 | −3.38392 | 2.51614 | 2.24516 | −5.86503 | 1.00000 | 8.51441 | ||||||||||||||||||
1.10 | −2.49098 | −1.00000 | 4.20498 | −3.30007 | 2.49098 | −0.252788 | −5.49255 | 1.00000 | 8.22041 | ||||||||||||||||||
1.11 | −2.36016 | −1.00000 | 3.57036 | 3.05076 | 2.36016 | −4.76162 | −3.70631 | 1.00000 | −7.20028 | ||||||||||||||||||
1.12 | −2.28384 | −1.00000 | 3.21593 | −0.172989 | 2.28384 | 5.09025 | −2.77699 | 1.00000 | 0.395079 | ||||||||||||||||||
1.13 | −2.24892 | −1.00000 | 3.05765 | 3.28762 | 2.24892 | 3.22602 | −2.37858 | 1.00000 | −7.39360 | ||||||||||||||||||
1.14 | −2.22425 | −1.00000 | 2.94729 | −0.996267 | 2.22425 | −4.15468 | −2.10702 | 1.00000 | 2.21595 | ||||||||||||||||||
1.15 | −2.13514 | −1.00000 | 2.55881 | 1.37673 | 2.13514 | −0.308813 | −1.19313 | 1.00000 | −2.93951 | ||||||||||||||||||
1.16 | −2.12836 | −1.00000 | 2.52994 | 0.301462 | 2.12836 | −3.41779 | −1.12790 | 1.00000 | −0.641620 | ||||||||||||||||||
1.17 | −2.10025 | −1.00000 | 2.41107 | 2.84518 | 2.10025 | 2.53466 | −0.863342 | 1.00000 | −5.97560 | ||||||||||||||||||
1.18 | −2.00502 | −1.00000 | 2.02010 | −0.637068 | 2.00502 | −2.14824 | −0.0403022 | 1.00000 | 1.27733 | ||||||||||||||||||
1.19 | −1.98457 | −1.00000 | 1.93852 | 3.95408 | 1.98457 | −0.848310 | 0.122004 | 1.00000 | −7.84715 | ||||||||||||||||||
1.20 | −1.91933 | −1.00000 | 1.68384 | 0.443938 | 1.91933 | −2.95964 | 0.606807 | 1.00000 | −0.852066 | ||||||||||||||||||
See next 80 embeddings (of 104 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.b | ✓ | 104 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8049.2.a.b | ✓ | 104 | 1.a | even | 1 | 1 | trivial |