Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4001,2,Mod(1,4001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4001, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4001 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4001.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: | \(31.9481458487\) |
Analytic rank: | \(1\) |
Dimension: | \(149\) |
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 | −2.71370 | 0.987518 | 5.36416 | 1.22965 | −2.67983 | 4.88451 | −9.12930 | −2.02481 | −3.33689 | ||||||||||||||||||
1.2 | −2.68009 | 1.97360 | 5.18289 | 2.92338 | −5.28944 | −1.04102 | −8.53045 | 0.895108 | −7.83492 | ||||||||||||||||||
1.3 | −2.67250 | 0.814974 | 5.14227 | −1.39982 | −2.17802 | −2.73506 | −8.39771 | −2.33582 | 3.74102 | ||||||||||||||||||
1.4 | −2.66308 | −1.99633 | 5.09200 | 2.13614 | 5.31640 | 2.50209 | −8.23426 | 0.985342 | −5.68872 | ||||||||||||||||||
1.5 | −2.62536 | 1.08746 | 4.89252 | 0.733437 | −2.85498 | −4.12609 | −7.59390 | −1.81743 | −1.92554 | ||||||||||||||||||
1.6 | −2.59431 | −3.00898 | 4.73047 | −0.189685 | 7.80624 | 1.12756 | −7.08370 | 6.05397 | 0.492103 | ||||||||||||||||||
1.7 | −2.59150 | −1.41020 | 4.71587 | −1.43230 | 3.65455 | −1.66447 | −7.03818 | −1.01132 | 3.71180 | ||||||||||||||||||
1.8 | −2.58646 | −1.10071 | 4.68977 | −0.828720 | 2.84694 | −1.43853 | −6.95698 | −1.78844 | 2.14345 | ||||||||||||||||||
1.9 | −2.47526 | 2.78845 | 4.12689 | −0.919151 | −6.90213 | −1.88527 | −5.26460 | 4.77546 | 2.27513 | ||||||||||||||||||
1.10 | −2.46316 | 2.96188 | 4.06716 | −3.16778 | −7.29560 | 1.16261 | −5.09175 | 5.77276 | 7.80275 | ||||||||||||||||||
1.11 | −2.45256 | 0.529200 | 4.01507 | 2.50676 | −1.29790 | −0.602620 | −4.94208 | −2.71995 | −6.14798 | ||||||||||||||||||
1.12 | −2.44683 | −2.82076 | 3.98699 | 4.21697 | 6.90192 | −1.62820 | −4.86182 | 4.95666 | −10.3182 | ||||||||||||||||||
1.13 | −2.41976 | −1.68750 | 3.85525 | −0.342140 | 4.08335 | 2.33028 | −4.48927 | −0.152339 | 0.827898 | ||||||||||||||||||
1.14 | −2.31463 | 1.63767 | 3.35751 | −2.89581 | −3.79059 | −1.05145 | −3.14214 | −0.318053 | 6.70274 | ||||||||||||||||||
1.15 | −2.27915 | 2.08452 | 3.19451 | −0.864719 | −4.75093 | 2.39591 | −2.72247 | 1.34523 | 1.97082 | ||||||||||||||||||
1.16 | −2.27484 | −0.745234 | 3.17488 | 3.93220 | 1.69529 | 2.00217 | −2.67267 | −2.44463 | −8.94513 | ||||||||||||||||||
1.17 | −2.25752 | −0.289421 | 3.09639 | −1.87801 | 0.653373 | 3.41261 | −2.47511 | −2.91624 | 4.23965 | ||||||||||||||||||
1.18 | −2.24881 | −2.62223 | 3.05717 | −2.07631 | 5.89691 | −2.22773 | −2.37738 | 3.87609 | 4.66924 | ||||||||||||||||||
1.19 | −2.21148 | 1.63586 | 2.89063 | 1.70947 | −3.61768 | 0.528860 | −1.96961 | −0.323947 | −3.78045 | ||||||||||||||||||
1.20 | −2.16569 | −0.380840 | 2.69021 | −3.90103 | 0.824780 | −2.04947 | −1.49478 | −2.85496 | 8.44841 | ||||||||||||||||||
See next 80 embeddings (of 149 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(4001\) | \(1\) |
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 | 4001.2.a.a | ✓ | 149 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4001.2.a.a | ✓ | 149 | 1.a | even | 1 | 1 | trivial |