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: | \(69\) |
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 | −3.40047 | 1.00000 | −0.810151 | 3.40047 | 1.45096 | −1.00000 | 8.56323 | 0.810151 | ||||||||||||||||||
1.2 | −1.00000 | −3.06382 | 1.00000 | −3.70565 | 3.06382 | 4.91357 | −1.00000 | 6.38702 | 3.70565 | ||||||||||||||||||
1.3 | −1.00000 | −2.95145 | 1.00000 | 1.54937 | 2.95145 | 1.05407 | −1.00000 | 5.71107 | −1.54937 | ||||||||||||||||||
1.4 | −1.00000 | −2.80010 | 1.00000 | 1.44998 | 2.80010 | −0.125244 | −1.00000 | 4.84055 | −1.44998 | ||||||||||||||||||
1.5 | −1.00000 | −2.79134 | 1.00000 | 2.80110 | 2.79134 | 2.82300 | −1.00000 | 4.79155 | −2.80110 | ||||||||||||||||||
1.6 | −1.00000 | −2.68289 | 1.00000 | −4.14764 | 2.68289 | −3.25782 | −1.00000 | 4.19789 | 4.14764 | ||||||||||||||||||
1.7 | −1.00000 | −2.66325 | 1.00000 | −0.772436 | 2.66325 | 4.62727 | −1.00000 | 4.09290 | 0.772436 | ||||||||||||||||||
1.8 | −1.00000 | −2.63767 | 1.00000 | 0.590977 | 2.63767 | −1.00146 | −1.00000 | 3.95729 | −0.590977 | ||||||||||||||||||
1.9 | −1.00000 | −2.45213 | 1.00000 | −3.61679 | 2.45213 | 0.0579543 | −1.00000 | 3.01293 | 3.61679 | ||||||||||||||||||
1.10 | −1.00000 | −2.40560 | 1.00000 | −3.03439 | 2.40560 | −1.41996 | −1.00000 | 2.78689 | 3.03439 | ||||||||||||||||||
1.11 | −1.00000 | −2.33178 | 1.00000 | −0.767442 | 2.33178 | −0.132499 | −1.00000 | 2.43720 | 0.767442 | ||||||||||||||||||
1.12 | −1.00000 | −2.31615 | 1.00000 | 2.10238 | 2.31615 | −3.46770 | −1.00000 | 2.36454 | −2.10238 | ||||||||||||||||||
1.13 | −1.00000 | −2.29088 | 1.00000 | 2.54707 | 2.29088 | 4.39407 | −1.00000 | 2.24813 | −2.54707 | ||||||||||||||||||
1.14 | −1.00000 | −2.19747 | 1.00000 | −2.90019 | 2.19747 | −0.534047 | −1.00000 | 1.82886 | 2.90019 | ||||||||||||||||||
1.15 | −1.00000 | −2.02205 | 1.00000 | 3.59068 | 2.02205 | 0.941965 | −1.00000 | 1.08869 | −3.59068 | ||||||||||||||||||
1.16 | −1.00000 | −1.90243 | 1.00000 | −0.692359 | 1.90243 | −4.50666 | −1.00000 | 0.619255 | 0.692359 | ||||||||||||||||||
1.17 | −1.00000 | −1.43088 | 1.00000 | 1.76140 | 1.43088 | −2.51027 | −1.00000 | −0.952590 | −1.76140 | ||||||||||||||||||
1.18 | −1.00000 | −1.36851 | 1.00000 | 0.212949 | 1.36851 | −1.32734 | −1.00000 | −1.12718 | −0.212949 | ||||||||||||||||||
1.19 | −1.00000 | −1.30456 | 1.00000 | −4.01452 | 1.30456 | 3.40012 | −1.00000 | −1.29813 | 4.01452 | ||||||||||||||||||
1.20 | −1.00000 | −1.29953 | 1.00000 | −2.92449 | 1.29953 | −2.47331 | −1.00000 | −1.31122 | 2.92449 | ||||||||||||||||||
See all 69 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.c | ✓ | 69 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6002.2.a.c | ✓ | 69 | 1.a | even | 1 | 1 | trivial |