Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8619,2,Mod(1,8619)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8619, 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("8619.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8619 = 3 \cdot 13^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8619.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: | \(68.8230615021\) |
Analytic rank: | \(1\) |
Dimension: | \(21\) |
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 | −2.72531 | 1.00000 | 5.42733 | −1.92039 | −2.72531 | 1.85372 | −9.34055 | 1.00000 | 5.23366 | ||||||||||||||||||
1.2 | −2.71200 | 1.00000 | 5.35495 | −1.71865 | −2.71200 | −2.50513 | −9.09863 | 1.00000 | 4.66097 | ||||||||||||||||||
1.3 | −2.61367 | 1.00000 | 4.83127 | 2.60872 | −2.61367 | 2.00384 | −7.40001 | 1.00000 | −6.81834 | ||||||||||||||||||
1.4 | −2.55115 | 1.00000 | 4.50837 | −4.40886 | −2.55115 | −0.225599 | −6.39922 | 1.00000 | 11.2477 | ||||||||||||||||||
1.5 | −2.15254 | 1.00000 | 2.63345 | −4.25374 | −2.15254 | 5.21772 | −1.36353 | 1.00000 | 9.15637 | ||||||||||||||||||
1.6 | −2.02507 | 1.00000 | 2.10093 | −2.79919 | −2.02507 | −4.41587 | −0.204384 | 1.00000 | 5.66856 | ||||||||||||||||||
1.7 | −1.89317 | 1.00000 | 1.58410 | 2.58913 | −1.89317 | −0.953007 | 0.787374 | 1.00000 | −4.90167 | ||||||||||||||||||
1.8 | −1.70893 | 1.00000 | 0.920426 | −1.13927 | −1.70893 | −3.51731 | 1.84491 | 1.00000 | 1.94693 | ||||||||||||||||||
1.9 | −1.66488 | 1.00000 | 0.771826 | 0.938526 | −1.66488 | 2.88538 | 2.04476 | 1.00000 | −1.56253 | ||||||||||||||||||
1.10 | −0.634677 | 1.00000 | −1.59718 | −2.98066 | −0.634677 | 1.67204 | 2.28305 | 1.00000 | 1.89176 | ||||||||||||||||||
1.11 | −0.575048 | 1.00000 | −1.66932 | −1.38919 | −0.575048 | 4.30260 | 2.11004 | 1.00000 | 0.798850 | ||||||||||||||||||
1.12 | −0.296463 | 1.00000 | −1.91211 | −3.60484 | −0.296463 | −4.47069 | 1.15980 | 1.00000 | 1.06870 | ||||||||||||||||||
1.13 | 0.0282265 | 1.00000 | −1.99920 | 0.933793 | 0.0282265 | 4.39781 | −0.112883 | 1.00000 | 0.0263577 | ||||||||||||||||||
1.14 | 0.0929476 | 1.00000 | −1.99136 | −3.17289 | 0.0929476 | 0.513856 | −0.370988 | 1.00000 | −0.294913 | ||||||||||||||||||
1.15 | 0.886807 | 1.00000 | −1.21357 | 1.36817 | 0.886807 | −4.28444 | −2.84982 | 1.00000 | 1.21331 | ||||||||||||||||||
1.16 | 1.49792 | 1.00000 | 0.243764 | −3.93316 | 1.49792 | 4.20773 | −2.63070 | 1.00000 | −5.89156 | ||||||||||||||||||
1.17 | 1.66522 | 1.00000 | 0.772953 | 2.12863 | 1.66522 | 1.32842 | −2.04330 | 1.00000 | 3.54464 | ||||||||||||||||||
1.18 | 1.66910 | 1.00000 | 0.785901 | 2.71566 | 1.66910 | −2.36594 | −2.02645 | 1.00000 | 4.53271 | ||||||||||||||||||
1.19 | 1.76323 | 1.00000 | 1.10898 | −0.184346 | 1.76323 | 1.70842 | −1.57107 | 1.00000 | −0.325045 | ||||||||||||||||||
1.20 | 2.26272 | 1.00000 | 3.11992 | −4.00833 | 2.26272 | −1.84015 | 2.53407 | 1.00000 | −9.06973 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(13\) | \(-1\) |
\(17\) | \(1\) |
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 | 8619.2.a.bp | ✓ | 21 |
13.b | even | 2 | 1 | 8619.2.a.bq | yes | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8619.2.a.bp | ✓ | 21 | 1.a | even | 1 | 1 | trivial |
8619.2.a.bq | yes | 21 | 13.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):
\( T_{2}^{21} + 9 T_{2}^{20} + 5 T_{2}^{19} - 178 T_{2}^{18} - 448 T_{2}^{17} + 1188 T_{2}^{16} + 5228 T_{2}^{15} - 1844 T_{2}^{14} - 27758 T_{2}^{13} - 14479 T_{2}^{12} + 77160 T_{2}^{11} + 81690 T_{2}^{10} - 107543 T_{2}^{9} + \cdots - 13 \) |
\( T_{5}^{21} + 26 T_{5}^{20} + 257 T_{5}^{19} + 962 T_{5}^{18} - 1955 T_{5}^{17} - 29076 T_{5}^{16} - 68477 T_{5}^{15} + 179294 T_{5}^{14} + 1113521 T_{5}^{13} + 731744 T_{5}^{12} - 6017014 T_{5}^{11} + \cdots - 4794259 \) |
\( T_{7}^{21} - 5 T_{7}^{20} - 84 T_{7}^{19} + 429 T_{7}^{18} + 2829 T_{7}^{17} - 15091 T_{7}^{16} - 48262 T_{7}^{15} + 280993 T_{7}^{14} + 426193 T_{7}^{13} - 2991801 T_{7}^{12} - 1605566 T_{7}^{11} + 18446283 T_{7}^{10} + \cdots - 3109499 \) |