Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4009,2,Mod(1,4009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4009, 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("4009.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4009 = 19 \cdot 211 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4009.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: | \(32.0120261703\) |
Analytic rank: | \(1\) |
Dimension: | \(71\) |
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 | −2.82251 | 2.93474 | 5.96657 | 0.420997 | −8.28333 | −3.44749 | −11.1957 | 5.61269 | −1.18827 | ||||||||||||||||||
1.2 | −2.70369 | 1.93926 | 5.30992 | 1.68888 | −5.24315 | 4.83108 | −8.94900 | 0.760725 | −4.56619 | ||||||||||||||||||
1.3 | −2.69837 | 1.88273 | 5.28118 | −2.05530 | −5.08030 | 0.784756 | −8.85381 | 0.544674 | 5.54594 | ||||||||||||||||||
1.4 | −2.67657 | 0.00971531 | 5.16401 | 0.0508514 | −0.0260037 | 0.312050 | −8.46867 | −2.99991 | −0.136107 | ||||||||||||||||||
1.5 | −2.66661 | −1.96401 | 5.11081 | −2.76483 | 5.23725 | 2.50280 | −8.29531 | 0.857341 | 7.37272 | ||||||||||||||||||
1.6 | −2.61964 | −2.82043 | 4.86252 | −3.49446 | 7.38851 | 2.54132 | −7.49878 | 4.95481 | 9.15424 | ||||||||||||||||||
1.7 | −2.51184 | −2.51408 | 4.30932 | 3.86977 | 6.31496 | 2.24136 | −5.80064 | 3.32060 | −9.72023 | ||||||||||||||||||
1.8 | −2.48400 | 0.221625 | 4.17025 | −2.50863 | −0.550515 | 2.33062 | −5.39089 | −2.95088 | 6.23143 | ||||||||||||||||||
1.9 | −2.35519 | 0.815593 | 3.54692 | 1.05865 | −1.92088 | −4.06376 | −3.64328 | −2.33481 | −2.49333 | ||||||||||||||||||
1.10 | −2.30773 | −2.77619 | 3.32561 | 2.46750 | 6.40670 | −0.00709873 | −3.05915 | 4.70724 | −5.69433 | ||||||||||||||||||
1.11 | −2.28791 | −0.433636 | 3.23453 | 1.92947 | 0.992121 | 3.12765 | −2.82450 | −2.81196 | −4.41445 | ||||||||||||||||||
1.12 | −2.25185 | −1.69642 | 3.07082 | −1.12007 | 3.82008 | −4.02650 | −2.41131 | −0.122162 | 2.52222 | ||||||||||||||||||
1.13 | −2.08913 | 2.53881 | 2.36446 | 0.545075 | −5.30390 | 0.502260 | −0.761409 | 3.44555 | −1.13873 | ||||||||||||||||||
1.14 | −2.08320 | 2.50442 | 2.33972 | −4.26568 | −5.21720 | −3.89477 | −0.707698 | 3.27211 | 8.88625 | ||||||||||||||||||
1.15 | −1.98460 | 0.183292 | 1.93862 | −1.97615 | −0.363761 | −5.05020 | 0.121816 | −2.96640 | 3.92186 | ||||||||||||||||||
1.16 | −1.93894 | −1.61421 | 1.75950 | −0.710068 | 3.12986 | −1.51648 | 0.466323 | −0.394323 | 1.37678 | ||||||||||||||||||
1.17 | −1.89616 | 2.32800 | 1.59541 | −3.08512 | −4.41426 | 2.27635 | 0.767159 | 2.41961 | 5.84987 | ||||||||||||||||||
1.18 | −1.89239 | −0.431235 | 1.58115 | 3.97235 | 0.816065 | −2.66937 | 0.792636 | −2.81404 | −7.51724 | ||||||||||||||||||
1.19 | −1.74061 | −3.34117 | 1.02972 | −2.66559 | 5.81567 | 0.0455688 | 1.68888 | 8.16342 | 4.63975 | ||||||||||||||||||
1.20 | −1.69413 | −0.649307 | 0.870061 | −0.0469641 | 1.10001 | 3.97959 | 1.91426 | −2.57840 | 0.0795630 | ||||||||||||||||||
See all 71 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(19\) | \(-1\) |
\(211\) | \(-1\) |
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 | 4009.2.a.c | ✓ | 71 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4009.2.a.c | ✓ | 71 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{71} + 15 T_{2}^{70} + 7 T_{2}^{69} - 987 T_{2}^{68} - 3958 T_{2}^{67} + 26949 T_{2}^{66} + \cdots - 104841 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).