Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6037,2,Mod(1,6037)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6037, 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("6037.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6037 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6037.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: | \(48.2056877002\) |
Analytic rank: | \(0\) |
Dimension: | \(259\) |
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 | −2.75717 | −1.09794 | 5.60201 | 2.84974 | 3.02722 | 3.79647 | −9.93137 | −1.79452 | −7.85722 | ||||||||||||||||||
1.2 | −2.66503 | 2.65532 | 5.10237 | −0.476820 | −7.07650 | −1.66384 | −8.26789 | 4.05073 | 1.27074 | ||||||||||||||||||
1.3 | −2.65823 | −1.47107 | 5.06620 | −0.123247 | 3.91044 | −2.04180 | −8.15068 | −0.835960 | 0.327620 | ||||||||||||||||||
1.4 | −2.65707 | 2.92115 | 5.06000 | 0.928477 | −7.76170 | 3.41739 | −8.13063 | 5.53314 | −2.46702 | ||||||||||||||||||
1.5 | −2.64523 | 1.84208 | 4.99727 | 3.41959 | −4.87275 | 2.32041 | −7.92847 | 0.393274 | −9.04562 | ||||||||||||||||||
1.6 | −2.61164 | 1.28286 | 4.82065 | −1.11903 | −3.35038 | −4.20141 | −7.36653 | −1.35426 | 2.92250 | ||||||||||||||||||
1.7 | −2.60200 | −2.50808 | 4.77039 | 0.421972 | 6.52603 | −1.06326 | −7.20855 | 3.29049 | −1.09797 | ||||||||||||||||||
1.8 | −2.60071 | −0.749599 | 4.76372 | −3.60076 | 1.94949 | −2.40322 | −7.18764 | −2.43810 | 9.36455 | ||||||||||||||||||
1.9 | −2.58809 | −2.19108 | 4.69820 | 0.221659 | 5.67071 | 0.704833 | −6.98319 | 1.80083 | −0.573673 | ||||||||||||||||||
1.10 | −2.58711 | −3.26786 | 4.69312 | 2.66521 | 8.45431 | 2.69955 | −6.96738 | 7.67894 | −6.89517 | ||||||||||||||||||
1.11 | −2.58398 | 0.626068 | 4.67694 | −1.39938 | −1.61774 | 0.699914 | −6.91715 | −2.60804 | 3.61596 | ||||||||||||||||||
1.12 | −2.56682 | 2.69132 | 4.58858 | 1.90137 | −6.90815 | −1.37100 | −6.64441 | 4.24322 | −4.88048 | ||||||||||||||||||
1.13 | −2.47379 | −1.46896 | 4.11964 | 2.18568 | 3.63389 | −2.37075 | −5.24354 | −0.842162 | −5.40693 | ||||||||||||||||||
1.14 | −2.46629 | 1.74062 | 4.08257 | −1.65001 | −4.29287 | 1.73775 | −5.13620 | 0.0297629 | 4.06940 | ||||||||||||||||||
1.15 | −2.46237 | −0.199276 | 4.06325 | −0.688498 | 0.490692 | 0.363231 | −5.08049 | −2.96029 | 1.69534 | ||||||||||||||||||
1.16 | −2.44806 | 0.638947 | 3.99301 | 2.71697 | −1.56418 | −2.58742 | −4.87901 | −2.59175 | −6.65132 | ||||||||||||||||||
1.17 | −2.42988 | 0.381455 | 3.90430 | −2.17078 | −0.926887 | −2.70764 | −4.62721 | −2.85449 | 5.27472 | ||||||||||||||||||
1.18 | −2.42952 | −2.16225 | 3.90258 | −1.13309 | 5.25323 | 1.20579 | −4.62237 | 1.67531 | 2.75288 | ||||||||||||||||||
1.19 | −2.38315 | −2.49497 | 3.67942 | −2.67299 | 5.94590 | −2.23204 | −4.00231 | 3.22488 | 6.37013 | ||||||||||||||||||
1.20 | −2.38263 | −0.344809 | 3.67692 | 0.273793 | 0.821552 | 0.727572 | −3.99549 | −2.88111 | −0.652346 | ||||||||||||||||||
See next 80 embeddings (of 259 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(6037\) | \(-1\) |
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 | 6037.2.a.b | ✓ | 259 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6037.2.a.b | ✓ | 259 | 1.a | even | 1 | 1 | trivial |