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: | \(1\) |
Dimension: | \(47\) |
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 | −3.27359 | 1.00000 | 0.938843 | −3.27359 | 3.32454 | 1.00000 | 7.71641 | 0.938843 | ||||||||||||||||||
1.2 | 1.00000 | −3.25356 | 1.00000 | −1.47632 | −3.25356 | 0.915630 | 1.00000 | 7.58563 | −1.47632 | ||||||||||||||||||
1.3 | 1.00000 | −2.97652 | 1.00000 | 2.57540 | −2.97652 | −1.79126 | 1.00000 | 5.85967 | 2.57540 | ||||||||||||||||||
1.4 | 1.00000 | −2.82438 | 1.00000 | −1.67949 | −2.82438 | 4.44281 | 1.00000 | 4.97711 | −1.67949 | ||||||||||||||||||
1.5 | 1.00000 | −2.77545 | 1.00000 | 1.33751 | −2.77545 | −0.156144 | 1.00000 | 4.70313 | 1.33751 | ||||||||||||||||||
1.6 | 1.00000 | −2.69765 | 1.00000 | −1.67926 | −2.69765 | −4.07202 | 1.00000 | 4.27732 | −1.67926 | ||||||||||||||||||
1.7 | 1.00000 | −2.53917 | 1.00000 | 0.927459 | −2.53917 | −2.39755 | 1.00000 | 3.44739 | 0.927459 | ||||||||||||||||||
1.8 | 1.00000 | −2.32207 | 1.00000 | −3.20826 | −2.32207 | 3.53388 | 1.00000 | 2.39199 | −3.20826 | ||||||||||||||||||
1.9 | 1.00000 | −2.18587 | 1.00000 | 0.199820 | −2.18587 | 0.0442522 | 1.00000 | 1.77803 | 0.199820 | ||||||||||||||||||
1.10 | 1.00000 | −2.03423 | 1.00000 | −3.81435 | −2.03423 | −2.54300 | 1.00000 | 1.13810 | −3.81435 | ||||||||||||||||||
1.11 | 1.00000 | −1.93355 | 1.00000 | 2.56764 | −1.93355 | 0.413652 | 1.00000 | 0.738635 | 2.56764 | ||||||||||||||||||
1.12 | 1.00000 | −1.68417 | 1.00000 | −2.87245 | −1.68417 | −3.82153 | 1.00000 | −0.163575 | −2.87245 | ||||||||||||||||||
1.13 | 1.00000 | −1.63738 | 1.00000 | 3.71883 | −1.63738 | 2.15943 | 1.00000 | −0.318981 | 3.71883 | ||||||||||||||||||
1.14 | 1.00000 | −1.60528 | 1.00000 | 1.61064 | −1.60528 | −2.90280 | 1.00000 | −0.423077 | 1.61064 | ||||||||||||||||||
1.15 | 1.00000 | −1.49159 | 1.00000 | 1.25928 | −1.49159 | −0.926965 | 1.00000 | −0.775168 | 1.25928 | ||||||||||||||||||
1.16 | 1.00000 | −1.48674 | 1.00000 | 1.68773 | −1.48674 | 3.31084 | 1.00000 | −0.789607 | 1.68773 | ||||||||||||||||||
1.17 | 1.00000 | −1.41473 | 1.00000 | −1.78933 | −1.41473 | 1.98770 | 1.00000 | −0.998550 | −1.78933 | ||||||||||||||||||
1.18 | 1.00000 | −1.29467 | 1.00000 | −3.90714 | −1.29467 | −0.238571 | 1.00000 | −1.32383 | −3.90714 | ||||||||||||||||||
1.19 | 1.00000 | −0.985702 | 1.00000 | −1.57345 | −0.985702 | −1.72023 | 1.00000 | −2.02839 | −1.57345 | ||||||||||||||||||
1.20 | 1.00000 | −0.879988 | 1.00000 | −1.86923 | −0.879988 | 3.94056 | 1.00000 | −2.22562 | −1.86923 | ||||||||||||||||||
See all 47 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.a | ✓ | 47 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6002.2.a.a | ✓ | 47 | 1.a | even | 1 | 1 | trivial |