Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6002,2,Mod(1,6002)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6002, 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("6002.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6002 = 2 \cdot 3001 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6002.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: | \(47.9262112932\) |
Analytic rank: | \(0\) |
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 | 1.00000 | −3.35201 | 1.00000 | 1.77899 | −3.35201 | −4.18423 | 1.00000 | 8.23600 | 1.77899 | ||||||||||||||||||
1.2 | 1.00000 | −3.34863 | 1.00000 | −3.86481 | −3.34863 | −0.0235424 | 1.00000 | 8.21330 | −3.86481 | ||||||||||||||||||
1.3 | 1.00000 | −3.27666 | 1.00000 | 3.56247 | −3.27666 | 2.04525 | 1.00000 | 7.73650 | 3.56247 | ||||||||||||||||||
1.4 | 1.00000 | −3.13631 | 1.00000 | 1.68515 | −3.13631 | −2.68800 | 1.00000 | 6.83646 | 1.68515 | ||||||||||||||||||
1.5 | 1.00000 | −3.08764 | 1.00000 | −2.69580 | −3.08764 | −4.17911 | 1.00000 | 6.53354 | −2.69580 | ||||||||||||||||||
1.6 | 1.00000 | −2.95638 | 1.00000 | −0.704147 | −2.95638 | −0.285333 | 1.00000 | 5.74016 | −0.704147 | ||||||||||||||||||
1.7 | 1.00000 | −2.83087 | 1.00000 | 4.29043 | −2.83087 | 1.15444 | 1.00000 | 5.01383 | 4.29043 | ||||||||||||||||||
1.8 | 1.00000 | −2.78890 | 1.00000 | −1.90255 | −2.78890 | 1.43646 | 1.00000 | 4.77794 | −1.90255 | ||||||||||||||||||
1.9 | 1.00000 | −2.71638 | 1.00000 | −0.997028 | −2.71638 | 0.847005 | 1.00000 | 4.37871 | −0.997028 | ||||||||||||||||||
1.10 | 1.00000 | −2.68976 | 1.00000 | 1.26880 | −2.68976 | 4.90307 | 1.00000 | 4.23478 | 1.26880 | ||||||||||||||||||
1.11 | 1.00000 | −2.51885 | 1.00000 | −4.29358 | −2.51885 | 3.16827 | 1.00000 | 3.34462 | −4.29358 | ||||||||||||||||||
1.12 | 1.00000 | −2.46943 | 1.00000 | 4.39243 | −2.46943 | −4.67132 | 1.00000 | 3.09808 | 4.39243 | ||||||||||||||||||
1.13 | 1.00000 | −2.46614 | 1.00000 | 2.56513 | −2.46614 | 4.43338 | 1.00000 | 3.08187 | 2.56513 | ||||||||||||||||||
1.14 | 1.00000 | −2.38888 | 1.00000 | −1.13195 | −2.38888 | −2.56197 | 1.00000 | 2.70673 | −1.13195 | ||||||||||||||||||
1.15 | 1.00000 | −2.13218 | 1.00000 | −2.93673 | −2.13218 | 2.83180 | 1.00000 | 1.54618 | −2.93673 | ||||||||||||||||||
1.16 | 1.00000 | −2.06582 | 1.00000 | 0.633869 | −2.06582 | −3.21505 | 1.00000 | 1.26759 | 0.633869 | ||||||||||||||||||
1.17 | 1.00000 | −2.02789 | 1.00000 | 2.83925 | −2.02789 | −1.41231 | 1.00000 | 1.11233 | 2.83925 | ||||||||||||||||||
1.18 | 1.00000 | −1.97835 | 1.00000 | 0.563154 | −1.97835 | 3.00235 | 1.00000 | 0.913870 | 0.563154 | ||||||||||||||||||
1.19 | 1.00000 | −1.93711 | 1.00000 | −2.50082 | −1.93711 | −1.24216 | 1.00000 | 0.752399 | −2.50082 | ||||||||||||||||||
1.20 | 1.00000 | −1.66636 | 1.00000 | 2.51408 | −1.66636 | 2.07406 | 1.00000 | −0.223255 | 2.51408 | ||||||||||||||||||
See all 79 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3001\) | \(1\) |
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 | 6002.2.a.d | ✓ | 79 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6002.2.a.d | ✓ | 79 | 1.a | even | 1 | 1 | trivial |