Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4033,2,Mod(1,4033)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, 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("4033.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4033 = 37 \cdot 109 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4033.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: | \(32.2036671352\) |
Analytic rank: | \(1\) |
Dimension: | \(79\) |
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 | −2.81086 | −2.68213 | 5.90091 | 2.99603 | 7.53909 | 4.86255 | −10.9649 | 4.19384 | −8.42140 | ||||||||||||||||||
1.2 | −2.77527 | 1.86757 | 5.70214 | 1.50597 | −5.18302 | −3.04702 | −10.2744 | 0.487817 | −4.17947 | ||||||||||||||||||
1.3 | −2.76342 | −1.75117 | 5.63647 | −3.34867 | 4.83922 | −0.176382 | −10.0491 | 0.0666133 | 9.25378 | ||||||||||||||||||
1.4 | −2.75248 | 1.04725 | 5.57617 | −2.70262 | −2.88255 | 3.33593 | −9.84336 | −1.90326 | 7.43892 | ||||||||||||||||||
1.5 | −2.66315 | 1.52081 | 5.09239 | 1.90830 | −4.05016 | −2.83326 | −8.23552 | −0.687131 | −5.08211 | ||||||||||||||||||
1.6 | −2.65601 | −2.08930 | 5.05439 | −0.388260 | 5.54922 | −5.15918 | −8.11249 | 1.36520 | 1.03122 | ||||||||||||||||||
1.7 | −2.60105 | 3.31359 | 4.76544 | −1.17110 | −8.61880 | 2.01510 | −7.19305 | 7.97988 | 3.04608 | ||||||||||||||||||
1.8 | −2.50943 | 1.55716 | 4.29722 | −3.66304 | −3.90758 | 4.73862 | −5.76470 | −0.575254 | 9.19213 | ||||||||||||||||||
1.9 | −2.50108 | −2.74712 | 4.25541 | 3.75941 | 6.87078 | −1.77571 | −5.64098 | 4.54668 | −9.40258 | ||||||||||||||||||
1.10 | −2.32027 | 1.35424 | 3.38367 | 2.79503 | −3.14221 | 4.55894 | −3.21050 | −1.16603 | −6.48525 | ||||||||||||||||||
1.11 | −2.27855 | −2.02568 | 3.19181 | −1.32967 | 4.61563 | 2.20899 | −2.71560 | 1.10340 | 3.02972 | ||||||||||||||||||
1.12 | −2.25482 | −0.982243 | 3.08420 | 2.20296 | 2.21478 | 0.164703 | −2.44467 | −2.03520 | −4.96728 | ||||||||||||||||||
1.13 | −2.24777 | −0.430015 | 3.05245 | −2.54923 | 0.966572 | −1.16281 | −2.36567 | −2.81509 | 5.73008 | ||||||||||||||||||
1.14 | −2.13414 | −1.44031 | 2.55454 | 3.76689 | 3.07382 | 1.36855 | −1.18346 | −0.925504 | −8.03907 | ||||||||||||||||||
1.15 | −2.08653 | 2.88703 | 2.35363 | −4.18679 | −6.02388 | −1.99666 | −0.737856 | 5.33492 | 8.73588 | ||||||||||||||||||
1.16 | −2.06826 | 0.566483 | 2.27770 | 0.827262 | −1.17163 | 0.908209 | −0.574346 | −2.67910 | −1.71099 | ||||||||||||||||||
1.17 | −1.97055 | −1.02182 | 1.88308 | −1.99471 | 2.01355 | −4.59826 | 0.230406 | −1.95588 | 3.93068 | ||||||||||||||||||
1.18 | −1.80025 | 1.41167 | 1.24090 | 0.213559 | −2.54136 | −2.54704 | 1.36657 | −1.00719 | −0.384459 | ||||||||||||||||||
1.19 | −1.79445 | −3.46032 | 1.22004 | 0.638539 | 6.20936 | −3.14453 | 1.39960 | 8.97380 | −1.14582 | ||||||||||||||||||
1.20 | −1.74629 | 2.61915 | 1.04954 | −0.0450252 | −4.57381 | −1.64580 | 1.65978 | 3.85995 | 0.0786272 | ||||||||||||||||||
See all 79 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(37\) | \(1\) |
\(109\) | \(1\) |
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 | 4033.2.a.d | ✓ | 79 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4033.2.a.d | ✓ | 79 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{79} + 11 T_{2}^{78} - 58 T_{2}^{77} - 1053 T_{2}^{76} + 480 T_{2}^{75} + 47487 T_{2}^{74} + \cdots - 62848 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\).