Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6047,2,Mod(1,6047)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6047, 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("6047.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6047 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6047.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: | \(48.2855381023\) |
Analytic rank: | \(1\) |
Dimension: | \(217\) |
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 | −2.79459 | −1.21209 | 5.80974 | −0.914621 | 3.38731 | −4.48023 | −10.6467 | −1.53083 | 2.55599 | ||||||||||||||||||
1.2 | −2.78935 | 0.603668 | 5.78045 | 3.11422 | −1.68384 | 0.146622 | −10.5450 | −2.63558 | −8.68663 | ||||||||||||||||||
1.3 | −2.77438 | −3.26218 | 5.69718 | 2.18536 | 9.05052 | 2.86186 | −10.2574 | 7.64181 | −6.06301 | ||||||||||||||||||
1.4 | −2.74777 | 2.83106 | 5.55022 | −0.268060 | −7.77911 | 0.663291 | −9.75518 | 5.01493 | 0.736566 | ||||||||||||||||||
1.5 | −2.70650 | 0.217632 | 5.32513 | 0.669687 | −0.589021 | −0.581450 | −8.99946 | −2.95264 | −1.81251 | ||||||||||||||||||
1.6 | −2.67338 | −1.23035 | 5.14695 | 0.526879 | 3.28919 | −0.672464 | −8.41297 | −1.48624 | −1.40855 | ||||||||||||||||||
1.7 | −2.65722 | 0.109936 | 5.06084 | 0.753739 | −0.292126 | 3.98656 | −8.13334 | −2.98791 | −2.00285 | ||||||||||||||||||
1.8 | −2.65451 | −2.99510 | 5.04641 | −2.08040 | 7.95052 | −1.11556 | −8.08670 | 5.97065 | 5.52243 | ||||||||||||||||||
1.9 | −2.63907 | −2.61338 | 4.96471 | 2.16033 | 6.89691 | −4.93155 | −7.82409 | 3.82978 | −5.70128 | ||||||||||||||||||
1.10 | −2.62503 | 1.59617 | 4.89080 | 2.83370 | −4.19001 | −4.08027 | −7.58846 | −0.452230 | −7.43856 | ||||||||||||||||||
1.11 | −2.59272 | 3.24215 | 4.72220 | −1.12445 | −8.40598 | −0.119273 | −7.05789 | 7.51153 | 2.91540 | ||||||||||||||||||
1.12 | −2.57504 | 1.64822 | 4.63081 | −3.44910 | −4.24422 | −0.596125 | −6.77442 | −0.283374 | 8.88154 | ||||||||||||||||||
1.13 | −2.54610 | −2.31345 | 4.48264 | −3.02578 | 5.89027 | −2.15804 | −6.32106 | 2.35204 | 7.70396 | ||||||||||||||||||
1.14 | −2.50842 | 0.797647 | 4.29216 | −3.82675 | −2.00083 | −2.73995 | −5.74970 | −2.36376 | 9.59909 | ||||||||||||||||||
1.15 | −2.50438 | 2.01082 | 4.27191 | 3.19046 | −5.03585 | 0.790850 | −5.68973 | 1.04339 | −7.99013 | ||||||||||||||||||
1.16 | −2.47362 | −1.15677 | 4.11880 | −1.98795 | 2.86140 | 3.96077 | −5.24112 | −1.66189 | 4.91743 | ||||||||||||||||||
1.17 | −2.46440 | −0.383920 | 4.07326 | 2.75346 | 0.946133 | −1.51513 | −5.10935 | −2.85261 | −6.78563 | ||||||||||||||||||
1.18 | −2.46358 | 2.70933 | 4.06925 | 0.207418 | −6.67465 | −2.27877 | −5.09776 | 4.34044 | −0.510992 | ||||||||||||||||||
1.19 | −2.45739 | 0.0299595 | 4.03876 | −2.30057 | −0.0736221 | −3.67314 | −5.01004 | −2.99910 | 5.65341 | ||||||||||||||||||
1.20 | −2.44753 | −2.68992 | 3.99041 | −1.41647 | 6.58366 | −0.890122 | −4.87158 | 4.23566 | 3.46686 | ||||||||||||||||||
See next 80 embeddings (of 217 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(6047\) | \(1\) |
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 | 6047.2.a.a | ✓ | 217 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6047.2.a.a | ✓ | 217 | 1.a | even | 1 | 1 | trivial |