Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3017,2,Mod(1,3017)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3017, 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("3017.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3017 = 7 \cdot 431 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3017.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: | \(24.0908662898\) |
Analytic rank: | \(1\) |
Dimension: | \(58\) |
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 | −2.82277 | 2.54795 | 5.96803 | 1.78968 | −7.19226 | −1.00000 | −11.2008 | 3.49203 | −5.05186 | ||||||||||||||||||
1.2 | −2.65884 | −2.00133 | 5.06944 | −2.52098 | 5.32122 | −1.00000 | −8.16116 | 1.00532 | 6.70290 | ||||||||||||||||||
1.3 | −2.62895 | −1.39806 | 4.91138 | 4.02065 | 3.67543 | −1.00000 | −7.65387 | −1.04543 | −10.5701 | ||||||||||||||||||
1.4 | −2.47072 | 0.401523 | 4.10446 | 0.548603 | −0.992050 | −1.00000 | −5.19953 | −2.83878 | −1.35544 | ||||||||||||||||||
1.5 | −2.42550 | −0.868648 | 3.88305 | 1.07126 | 2.10690 | −1.00000 | −4.56734 | −2.24545 | −2.59835 | ||||||||||||||||||
1.6 | −2.41990 | 0.0673693 | 3.85589 | −1.26270 | −0.163027 | −1.00000 | −4.49106 | −2.99546 | 3.05559 | ||||||||||||||||||
1.7 | −2.39025 | −3.28358 | 3.71327 | −1.85162 | 7.84855 | −1.00000 | −4.09514 | 7.78187 | 4.42582 | ||||||||||||||||||
1.8 | −2.34905 | −2.72915 | 3.51804 | 2.43756 | 6.41091 | −1.00000 | −3.56595 | 4.44827 | −5.72595 | ||||||||||||||||||
1.9 | −2.30750 | 2.00264 | 3.32457 | −0.388574 | −4.62111 | −1.00000 | −3.05645 | 1.01058 | 0.896636 | ||||||||||||||||||
1.10 | −2.17837 | 2.89320 | 2.74530 | 0.964637 | −6.30247 | −1.00000 | −1.62354 | 5.37063 | −2.10134 | ||||||||||||||||||
1.11 | −2.11251 | −0.125672 | 2.46269 | −3.91996 | 0.265483 | −1.00000 | −0.977427 | −2.98421 | 8.28095 | ||||||||||||||||||
1.12 | −2.01352 | −0.341065 | 2.05426 | 4.04298 | 0.686741 | −1.00000 | −0.109247 | −2.88367 | −8.14062 | ||||||||||||||||||
1.13 | −1.85686 | 3.12959 | 1.44793 | −2.45478 | −5.81122 | −1.00000 | 1.02512 | 6.79436 | 4.55819 | ||||||||||||||||||
1.14 | −1.54727 | 0.564794 | 0.394030 | −4.24292 | −0.873887 | −1.00000 | 2.48486 | −2.68101 | 6.56493 | ||||||||||||||||||
1.15 | −1.51674 | −2.87905 | 0.300515 | 2.94363 | 4.36679 | −1.00000 | 2.57769 | 5.28893 | −4.46474 | ||||||||||||||||||
1.16 | −1.50816 | 1.01262 | 0.274561 | −1.95552 | −1.52720 | −1.00000 | 2.60225 | −1.97460 | 2.94924 | ||||||||||||||||||
1.17 | −1.30880 | 1.92914 | −0.287035 | 1.37472 | −2.52486 | −1.00000 | 2.99328 | 0.721564 | −1.79924 | ||||||||||||||||||
1.18 | −1.29755 | −3.44629 | −0.316352 | −0.579342 | 4.47174 | −1.00000 | 3.00559 | 8.87689 | 0.751728 | ||||||||||||||||||
1.19 | −1.20213 | 2.22527 | −0.554886 | 3.08793 | −2.67506 | −1.00000 | 3.07130 | 1.95184 | −3.71209 | ||||||||||||||||||
1.20 | −1.18895 | −1.81970 | −0.586406 | 1.72144 | 2.16352 | −1.00000 | 3.07510 | 0.311293 | −2.04670 | ||||||||||||||||||
See all 58 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \(1\) |
\(431\) | \(1\) |
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 | 3017.2.a.e | ✓ | 58 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3017.2.a.e | ✓ | 58 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{58} + 4 T_{2}^{57} - 77 T_{2}^{56} - 319 T_{2}^{55} + 2776 T_{2}^{54} + 11968 T_{2}^{53} + \cdots - 1563 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3017))\).