Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8037,2,Mod(1,8037)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8037, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8037.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8037 = 3^{2} \cdot 19 \cdot 47 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8037.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: | \(64.1757681045\) |
Analytic rank: | \(1\) |
Dimension: | \(23\) |
Twist minimal: | no (minimal twist has level 2679) |
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 | −2.70026 | 0 | 5.29141 | −2.20508 | 0 | 0.381908 | −8.88768 | 0 | 5.95430 | ||||||||||||||||||
1.2 | −2.65726 | 0 | 5.06101 | 3.68922 | 0 | −1.66641 | −8.13390 | 0 | −9.80320 | ||||||||||||||||||
1.3 | −2.46332 | 0 | 4.06796 | −1.49096 | 0 | 0.898858 | −5.09404 | 0 | 3.67270 | ||||||||||||||||||
1.4 | −2.43689 | 0 | 3.93845 | −3.39444 | 0 | −2.76712 | −4.72380 | 0 | 8.27188 | ||||||||||||||||||
1.5 | −2.28716 | 0 | 3.23110 | 0.184427 | 0 | 3.46017 | −2.81571 | 0 | −0.421813 | ||||||||||||||||||
1.6 | −1.73912 | 0 | 1.02455 | −3.12749 | 0 | 2.42300 | 1.69643 | 0 | 5.43909 | ||||||||||||||||||
1.7 | −1.60510 | 0 | 0.576359 | 2.10169 | 0 | 1.01773 | 2.28509 | 0 | −3.37344 | ||||||||||||||||||
1.8 | −1.45453 | 0 | 0.115648 | −0.945028 | 0 | −4.13147 | 2.74084 | 0 | 1.37457 | ||||||||||||||||||
1.9 | −1.07420 | 0 | −0.846102 | 3.82265 | 0 | −2.96171 | 3.05727 | 0 | −4.10628 | ||||||||||||||||||
1.10 | −0.803522 | 0 | −1.35435 | −2.59980 | 0 | 4.06920 | 2.69530 | 0 | 2.08899 | ||||||||||||||||||
1.11 | −0.641477 | 0 | −1.58851 | −2.95137 | 0 | 1.16825 | 2.30195 | 0 | 1.89324 | ||||||||||||||||||
1.12 | −0.553572 | 0 | −1.69356 | 0.688386 | 0 | 3.33451 | 2.04465 | 0 | −0.381072 | ||||||||||||||||||
1.13 | 0.176164 | 0 | −1.96897 | −4.18334 | 0 | −4.24699 | −0.699188 | 0 | −0.736953 | ||||||||||||||||||
1.14 | 0.395557 | 0 | −1.84353 | 0.554972 | 0 | −0.565101 | −1.52034 | 0 | 0.219523 | ||||||||||||||||||
1.15 | 0.539644 | 0 | −1.70878 | 1.68281 | 0 | −4.41391 | −2.00142 | 0 | 0.908118 | ||||||||||||||||||
1.16 | 0.844930 | 0 | −1.28609 | 2.16056 | 0 | 4.75097 | −2.77652 | 0 | 1.82552 | ||||||||||||||||||
1.17 | 1.11388 | 0 | −0.759262 | −2.97800 | 0 | 1.49100 | −3.07350 | 0 | −3.31715 | ||||||||||||||||||
1.18 | 1.29424 | 0 | −0.324956 | 2.49851 | 0 | −3.56737 | −3.00904 | 0 | 3.23365 | ||||||||||||||||||
1.19 | 1.60314 | 0 | 0.570052 | −3.51993 | 0 | 3.73543 | −2.29240 | 0 | −5.64293 | ||||||||||||||||||
1.20 | 2.13087 | 0 | 2.54061 | 2.40433 | 0 | 0.918790 | 1.15198 | 0 | 5.12332 | ||||||||||||||||||
See all 23 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(19\) | \(-1\) |
\(47\) | \(1\) |
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 | 8037.2.a.p | 23 | |
3.b | odd | 2 | 1 | 2679.2.a.n | ✓ | 23 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2679.2.a.n | ✓ | 23 | 3.b | odd | 2 | 1 | |
8037.2.a.p | 23 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):
\( T_{2}^{23} + 5 T_{2}^{22} - 23 T_{2}^{21} - 146 T_{2}^{20} + 174 T_{2}^{19} + 1776 T_{2}^{18} - 135 T_{2}^{17} - 11712 T_{2}^{16} - 5445 T_{2}^{15} + 45722 T_{2}^{14} + 35387 T_{2}^{13} - 108725 T_{2}^{12} - 104907 T_{2}^{11} + \cdots - 276 \) |
\( T_{5}^{23} + 12 T_{5}^{22} - T_{5}^{21} - 544 T_{5}^{20} - 1571 T_{5}^{19} + 8829 T_{5}^{18} + 44230 T_{5}^{17} - 50758 T_{5}^{16} - 548297 T_{5}^{15} - 187286 T_{5}^{14} + 3667575 T_{5}^{13} + 4208544 T_{5}^{12} + \cdots + 137661 \) |