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: | \(0\) |
Dimension: | \(184\) |
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 | −2.80114 | −3.24231 | 5.84639 | −0.362239 | 9.08218 | 0.683027 | −10.7743 | 7.51260 | 1.01468 | ||||||||||||||||||
1.2 | −2.79357 | 0.695080 | 5.80404 | −1.43429 | −1.94176 | 3.16464 | −10.6269 | −2.51686 | 4.00680 | ||||||||||||||||||
1.3 | −2.77961 | −0.970175 | 5.72623 | 0.973772 | 2.69671 | 0.392591 | −10.3575 | −2.05876 | −2.70671 | ||||||||||||||||||
1.4 | −2.76351 | −0.782711 | 5.63697 | 4.06144 | 2.16303 | 1.66325 | −10.0508 | −2.38736 | −11.2238 | ||||||||||||||||||
1.5 | −2.73134 | 2.60660 | 5.46019 | −2.78013 | −7.11951 | −4.77010 | −9.45095 | 3.79438 | 7.59346 | ||||||||||||||||||
1.6 | −2.71583 | 2.95270 | 5.37572 | 2.81232 | −8.01903 | −1.71576 | −9.16786 | 5.71846 | −7.63778 | ||||||||||||||||||
1.7 | −2.69288 | −1.22667 | 5.25159 | 2.75099 | 3.30328 | −4.02180 | −8.75615 | −1.49527 | −7.40807 | ||||||||||||||||||
1.8 | −2.68105 | −2.59933 | 5.18802 | −3.48579 | 6.96892 | −5.16960 | −8.54725 | 3.75650 | 9.34557 | ||||||||||||||||||
1.9 | −2.66682 | 0.260886 | 5.11190 | −1.12952 | −0.695735 | −1.67133 | −8.29887 | −2.93194 | 3.01221 | ||||||||||||||||||
1.10 | −2.65266 | 3.05290 | 5.03661 | −1.94746 | −8.09830 | 2.48037 | −8.05510 | 6.32018 | 5.16596 | ||||||||||||||||||
1.11 | −2.64594 | −1.99415 | 5.00101 | −3.20075 | 5.27640 | 3.40947 | −7.94051 | 0.976617 | 8.46900 | ||||||||||||||||||
1.12 | −2.63285 | 1.55445 | 4.93191 | −4.32053 | −4.09264 | −0.850828 | −7.71930 | −0.583679 | 11.3753 | ||||||||||||||||||
1.13 | −2.61467 | 2.84813 | 4.83651 | 0.542328 | −7.44692 | 4.00529 | −7.41654 | 5.11183 | −1.41801 | ||||||||||||||||||
1.14 | −2.61059 | −0.259768 | 4.81518 | −4.09316 | 0.678147 | 2.08780 | −7.34928 | −2.93252 | 10.6856 | ||||||||||||||||||
1.15 | −2.55462 | 2.40673 | 4.52611 | 3.51168 | −6.14829 | 3.52515 | −6.45325 | 2.79234 | −8.97103 | ||||||||||||||||||
1.16 | −2.43211 | 0.645473 | 3.91518 | −1.15456 | −1.56986 | 1.52754 | −4.65795 | −2.58336 | 2.80801 | ||||||||||||||||||
1.17 | −2.42514 | −1.95794 | 3.88128 | 1.35530 | 4.74827 | 4.91337 | −4.56237 | 0.833523 | −3.28679 | ||||||||||||||||||
1.18 | −2.40164 | −1.91938 | 3.76787 | −2.41584 | 4.60966 | −1.95333 | −4.24578 | 0.684021 | 5.80197 | ||||||||||||||||||
1.19 | −2.38954 | 1.02374 | 3.70989 | −0.691971 | −2.44626 | 1.59189 | −4.08584 | −1.95196 | 1.65349 | ||||||||||||||||||
1.20 | −2.38517 | 0.919938 | 3.68905 | 3.44498 | −2.19421 | −3.87561 | −4.02868 | −2.15371 | −8.21687 | ||||||||||||||||||
See next 80 embeddings (of 184 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.b | ✓ | 184 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4001.2.a.b | ✓ | 184 | 1.a | even | 1 | 1 | trivial |