Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3997,2,Mod(1,3997)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3997, 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("3997.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3997 = 7 \cdot 571 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3997.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: | \(31.9162056879\) |
Analytic rank: | \(1\) |
Dimension: | \(73\) |
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 | −2.81169 | 1.31125 | 5.90557 | −2.86392 | −3.68682 | −1.00000 | −10.9812 | −1.28063 | 8.05245 | ||||||||||||||||||
1.2 | −2.77337 | 3.06572 | 5.69157 | 0.175455 | −8.50238 | −1.00000 | −10.2381 | 6.39867 | −0.486600 | ||||||||||||||||||
1.3 | −2.75858 | −3.27707 | 5.60979 | −4.00427 | 9.04009 | −1.00000 | −9.95791 | 7.73922 | 11.0461 | ||||||||||||||||||
1.4 | −2.74161 | −1.71177 | 5.51645 | −0.706027 | 4.69300 | −1.00000 | −9.64074 | −0.0698569 | 1.93565 | ||||||||||||||||||
1.5 | −2.71787 | 1.25778 | 5.38682 | 2.21902 | −3.41850 | −1.00000 | −9.20494 | −1.41798 | −6.03101 | ||||||||||||||||||
1.6 | −2.55174 | −3.37645 | 4.51140 | 1.25610 | 8.61583 | −1.00000 | −6.40844 | 8.40039 | −3.20524 | ||||||||||||||||||
1.7 | −2.50126 | −0.776277 | 4.25628 | 3.25093 | 1.94167 | −1.00000 | −5.64353 | −2.39739 | −8.13141 | ||||||||||||||||||
1.8 | −2.43180 | −0.486629 | 3.91367 | −3.05406 | 1.18339 | −1.00000 | −4.65368 | −2.76319 | 7.42687 | ||||||||||||||||||
1.9 | −2.40299 | −0.676782 | 3.77436 | 0.0679789 | 1.62630 | −1.00000 | −4.26378 | −2.54197 | −0.163353 | ||||||||||||||||||
1.10 | −2.32771 | 0.489476 | 3.41825 | 2.22055 | −1.13936 | −1.00000 | −3.30127 | −2.76041 | −5.16880 | ||||||||||||||||||
1.11 | −2.29416 | −2.50879 | 3.26315 | 2.90852 | 5.75556 | −1.00000 | −2.89787 | 3.29404 | −6.67259 | ||||||||||||||||||
1.12 | −2.23006 | −0.609547 | 2.97316 | −3.83808 | 1.35933 | −1.00000 | −2.17021 | −2.62845 | 8.55915 | ||||||||||||||||||
1.13 | −2.14277 | −1.83467 | 2.59148 | 4.00463 | 3.93129 | −1.00000 | −1.26740 | 0.366032 | −8.58101 | ||||||||||||||||||
1.14 | −2.13582 | 3.19362 | 2.56175 | −3.66797 | −6.82102 | −1.00000 | −1.19979 | 7.19923 | 7.83414 | ||||||||||||||||||
1.15 | −2.09474 | 2.52741 | 2.38792 | −0.118420 | −5.29425 | −1.00000 | −0.812600 | 3.38778 | 0.248059 | ||||||||||||||||||
1.16 | −2.04685 | −2.72535 | 2.18958 | −2.65600 | 5.57836 | −1.00000 | −0.388040 | 4.42751 | 5.43642 | ||||||||||||||||||
1.17 | −2.00689 | 2.49559 | 2.02762 | 1.90000 | −5.00838 | −1.00000 | −0.0554365 | 3.22797 | −3.81311 | ||||||||||||||||||
1.18 | −1.77539 | 0.578119 | 1.15201 | −1.24699 | −1.02639 | −1.00000 | 1.50551 | −2.66578 | 2.21390 | ||||||||||||||||||
1.19 | −1.73295 | 0.693780 | 1.00312 | −3.82645 | −1.20229 | −1.00000 | 1.72754 | −2.51867 | 6.63105 | ||||||||||||||||||
1.20 | −1.65693 | 2.05345 | 0.745418 | −0.611378 | −3.40242 | −1.00000 | 2.07876 | 1.21665 | 1.01301 | ||||||||||||||||||
See all 73 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \(1\) |
\(571\) | \(1\) |
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 | 3997.2.a.e | ✓ | 73 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3997.2.a.e | ✓ | 73 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{73} + 12 T_{2}^{72} - 38 T_{2}^{71} - 1003 T_{2}^{70} - 934 T_{2}^{69} + 38710 T_{2}^{68} + \cdots - 49664 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3997))\).