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: | \(129\) |
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 | −2.82087 | 1.00000 | 5.95730 | 1.28710 | −2.82087 | 2.58338 | −11.1630 | 1.00000 | −3.63075 | ||||||||||||||||||
1.2 | −2.77720 | 1.00000 | 5.71284 | 2.12496 | −2.77720 | −3.21138 | −10.3113 | 1.00000 | −5.90145 | ||||||||||||||||||
1.3 | −2.72806 | 1.00000 | 5.44232 | −3.98613 | −2.72806 | −2.69921 | −9.39088 | 1.00000 | 10.8744 | ||||||||||||||||||
1.4 | −2.70395 | 1.00000 | 5.31133 | 1.76368 | −2.70395 | −4.33108 | −8.95365 | 1.00000 | −4.76891 | ||||||||||||||||||
1.5 | −2.70347 | 1.00000 | 5.30878 | −2.52602 | −2.70347 | 3.06715 | −8.94519 | 1.00000 | 6.82903 | ||||||||||||||||||
1.6 | −2.67215 | 1.00000 | 5.14039 | −2.79285 | −2.67215 | 3.90889 | −8.39161 | 1.00000 | 7.46293 | ||||||||||||||||||
1.7 | −2.65892 | 1.00000 | 5.06986 | 3.58637 | −2.65892 | 4.24083 | −8.16251 | 1.00000 | −9.53588 | ||||||||||||||||||
1.8 | −2.65190 | 1.00000 | 5.03256 | −2.74220 | −2.65190 | 0.184989 | −8.04204 | 1.00000 | 7.27204 | ||||||||||||||||||
1.9 | −2.62578 | 1.00000 | 4.89474 | −0.512843 | −2.62578 | 1.48852 | −7.60096 | 1.00000 | 1.34662 | ||||||||||||||||||
1.10 | −2.58791 | 1.00000 | 4.69728 | 1.64365 | −2.58791 | −4.45906 | −6.98031 | 1.00000 | −4.25362 | ||||||||||||||||||
1.11 | −2.57732 | 1.00000 | 4.64257 | −1.35723 | −2.57732 | 3.16714 | −6.81075 | 1.00000 | 3.49801 | ||||||||||||||||||
1.12 | −2.46621 | 1.00000 | 4.08220 | 1.94142 | −2.46621 | −0.674366 | −5.13515 | 1.00000 | −4.78795 | ||||||||||||||||||
1.13 | −2.42396 | 1.00000 | 3.87556 | −3.49875 | −2.42396 | 3.65352 | −4.54627 | 1.00000 | 8.48081 | ||||||||||||||||||
1.14 | −2.31100 | 1.00000 | 3.34070 | 3.34170 | −2.31100 | 4.45674 | −3.09836 | 1.00000 | −7.72265 | ||||||||||||||||||
1.15 | −2.31049 | 1.00000 | 3.33834 | 3.79120 | −2.31049 | −2.69218 | −3.09222 | 1.00000 | −8.75950 | ||||||||||||||||||
1.16 | −2.30919 | 1.00000 | 3.33237 | −0.223778 | −2.30919 | 3.94364 | −3.07670 | 1.00000 | 0.516746 | ||||||||||||||||||
1.17 | −2.21866 | 1.00000 | 2.92244 | −3.24553 | −2.21866 | −2.52068 | −2.04658 | 1.00000 | 7.20072 | ||||||||||||||||||
1.18 | −2.20805 | 1.00000 | 2.87549 | −1.04969 | −2.20805 | −2.20196 | −1.93312 | 1.00000 | 2.31777 | ||||||||||||||||||
1.19 | −2.16763 | 1.00000 | 2.69860 | 2.25786 | −2.16763 | 0.531131 | −1.51431 | 1.00000 | −4.89419 | ||||||||||||||||||
1.20 | −2.16635 | 1.00000 | 2.69306 | 3.84293 | −2.16635 | 1.73846 | −1.50141 | 1.00000 | −8.32512 | ||||||||||||||||||
See next 80 embeddings (of 129 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.d | ✓ | 129 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8049.2.a.d | ✓ | 129 | 1.a | even | 1 | 1 | trivial |