Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6001,2,Mod(1,6001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6001 = 17 \cdot 353 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6001.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: | \(47.9182262530\) |
Analytic rank: | \(0\) |
Dimension: | \(121\) |
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 | −2.80099 | 1.23587 | 5.84553 | −1.08657 | −3.46165 | −3.96716 | −10.7713 | −1.47263 | 3.04347 | ||||||||||||||||||
1.2 | −2.75847 | −2.11283 | 5.60915 | 3.70679 | 5.82818 | −2.88565 | −9.95572 | 1.46406 | −10.2251 | ||||||||||||||||||
1.3 | −2.69106 | 2.90048 | 5.24179 | 1.92081 | −7.80536 | −1.20204 | −8.72383 | 5.41280 | −5.16901 | ||||||||||||||||||
1.4 | −2.63511 | −2.54509 | 4.94380 | −1.75130 | 6.70660 | −0.500367 | −7.75725 | 3.47750 | 4.61488 | ||||||||||||||||||
1.5 | −2.61678 | −3.27387 | 4.84754 | 0.197154 | 8.56700 | −1.64878 | −7.45139 | 7.71823 | −0.515910 | ||||||||||||||||||
1.6 | −2.60140 | −0.594038 | 4.76726 | 2.91382 | 1.54533 | 0.0965456 | −7.19873 | −2.64712 | −7.57999 | ||||||||||||||||||
1.7 | −2.58940 | 0.995654 | 4.70501 | 0.868437 | −2.57815 | −0.888525 | −7.00437 | −2.00867 | −2.24873 | ||||||||||||||||||
1.8 | −2.54984 | 3.38084 | 4.50169 | −1.07763 | −8.62059 | 1.13594 | −6.37892 | 8.43005 | 2.74778 | ||||||||||||||||||
1.9 | −2.53200 | 0.341990 | 4.41103 | 2.95840 | −0.865919 | 4.74675 | −6.10474 | −2.88304 | −7.49068 | ||||||||||||||||||
1.10 | −2.48167 | 1.61831 | 4.15867 | −3.04222 | −4.01610 | 0.428830 | −5.35710 | −0.381079 | 7.54977 | ||||||||||||||||||
1.11 | −2.44940 | 0.192595 | 3.99956 | 1.46027 | −0.471742 | −3.96634 | −4.89773 | −2.96291 | −3.57679 | ||||||||||||||||||
1.12 | −2.29935 | −2.65633 | 3.28702 | −0.0208883 | 6.10783 | 4.05842 | −2.95931 | 4.05607 | 0.0480296 | ||||||||||||||||||
1.13 | −2.25146 | 1.98627 | 3.06909 | 4.11708 | −4.47201 | 2.76168 | −2.40701 | 0.945268 | −9.26946 | ||||||||||||||||||
1.14 | −2.24876 | −0.983386 | 3.05693 | −2.62112 | 2.21140 | 0.682123 | −2.37679 | −2.03295 | 5.89428 | ||||||||||||||||||
1.15 | −2.22966 | 1.54651 | 2.97139 | −1.67307 | −3.44820 | −3.79558 | −2.16587 | −0.608295 | 3.73037 | ||||||||||||||||||
1.16 | −2.18200 | −1.07317 | 2.76112 | −3.39433 | 2.34166 | −4.74202 | −1.66076 | −1.84830 | 7.40643 | ||||||||||||||||||
1.17 | −2.13704 | −0.352862 | 2.56694 | 0.777890 | 0.754080 | 0.156355 | −1.21157 | −2.87549 | −1.66238 | ||||||||||||||||||
1.18 | −2.08608 | 1.06648 | 2.35172 | −3.22907 | −2.22475 | 0.560788 | −0.733710 | −1.86263 | 6.73610 | ||||||||||||||||||
1.19 | −2.05839 | 1.11600 | 2.23697 | 1.51993 | −2.29716 | 0.698036 | −0.487784 | −1.75454 | −3.12862 | ||||||||||||||||||
1.20 | −2.05373 | −1.71490 | 2.21782 | −3.91336 | 3.52195 | 1.58894 | −0.447336 | −0.0591109 | 8.03700 | ||||||||||||||||||
See next 80 embeddings (of 121 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(17\) | \(1\) |
\(353\) | \(-1\) |
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 | 6001.2.a.c | ✓ | 121 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6001.2.a.c | ✓ | 121 | 1.a | even | 1 | 1 | trivial |