Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6033,2,Mod(1,6033)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6033, 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("6033.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6033 = 3 \cdot 2011 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6033.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: | \(48.1737475394\) |
Analytic rank: | \(0\) |
Dimension: | \(97\) |
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 | −2.77569 | 1.00000 | 5.70444 | −4.08383 | −2.77569 | 2.32651 | −10.2824 | 1.00000 | 11.3354 | ||||||||||||||||||
1.2 | −2.72701 | 1.00000 | 5.43660 | −1.98915 | −2.72701 | −1.33626 | −9.37164 | 1.00000 | 5.42444 | ||||||||||||||||||
1.3 | −2.72505 | 1.00000 | 5.42589 | −0.505520 | −2.72505 | −2.64822 | −9.33570 | 1.00000 | 1.37757 | ||||||||||||||||||
1.4 | −2.69536 | 1.00000 | 5.26496 | 2.52315 | −2.69536 | 0.986697 | −8.80025 | 1.00000 | −6.80079 | ||||||||||||||||||
1.5 | −2.65791 | 1.00000 | 5.06448 | 3.14919 | −2.65791 | 3.30721 | −8.14510 | 1.00000 | −8.37025 | ||||||||||||||||||
1.6 | −2.63042 | 1.00000 | 4.91910 | −0.782993 | −2.63042 | 4.49130 | −7.67844 | 1.00000 | 2.05960 | ||||||||||||||||||
1.7 | −2.53542 | 1.00000 | 4.42834 | −3.51748 | −2.53542 | −3.36027 | −6.15684 | 1.00000 | 8.91826 | ||||||||||||||||||
1.8 | −2.50001 | 1.00000 | 4.25007 | −1.07876 | −2.50001 | 3.84793 | −5.62521 | 1.00000 | 2.69691 | ||||||||||||||||||
1.9 | −2.41345 | 1.00000 | 3.82475 | 3.09815 | −2.41345 | 4.57422 | −4.40394 | 1.00000 | −7.47724 | ||||||||||||||||||
1.10 | −2.38207 | 1.00000 | 3.67426 | −3.40207 | −2.38207 | 2.20870 | −3.98822 | 1.00000 | 8.10396 | ||||||||||||||||||
1.11 | −2.37614 | 1.00000 | 3.64603 | 3.12222 | −2.37614 | −0.986181 | −3.91118 | 1.00000 | −7.41883 | ||||||||||||||||||
1.12 | −2.30219 | 1.00000 | 3.30008 | −3.13763 | −2.30219 | 1.81298 | −2.99304 | 1.00000 | 7.22342 | ||||||||||||||||||
1.13 | −2.19685 | 1.00000 | 2.82615 | 3.79567 | −2.19685 | −1.66772 | −1.81492 | 1.00000 | −8.33851 | ||||||||||||||||||
1.14 | −2.11811 | 1.00000 | 2.48641 | 0.996414 | −2.11811 | −4.64745 | −1.03027 | 1.00000 | −2.11052 | ||||||||||||||||||
1.15 | −2.06099 | 1.00000 | 2.24767 | −0.122069 | −2.06099 | −4.45814 | −0.510454 | 1.00000 | 0.251582 | ||||||||||||||||||
1.16 | −2.05972 | 1.00000 | 2.24246 | −1.93322 | −2.05972 | 2.16498 | −0.499401 | 1.00000 | 3.98191 | ||||||||||||||||||
1.17 | −1.96725 | 1.00000 | 1.87006 | −1.28401 | −1.96725 | 2.13094 | 0.255618 | 1.00000 | 2.52597 | ||||||||||||||||||
1.18 | −1.95525 | 1.00000 | 1.82299 | 1.82748 | −1.95525 | −0.999221 | 0.346105 | 1.00000 | −3.57318 | ||||||||||||||||||
1.19 | −1.92742 | 1.00000 | 1.71496 | −2.34894 | −1.92742 | −3.43132 | 0.549392 | 1.00000 | 4.52741 | ||||||||||||||||||
1.20 | −1.88583 | 1.00000 | 1.55635 | −4.37058 | −1.88583 | 3.62055 | 0.836652 | 1.00000 | 8.24216 | ||||||||||||||||||
See all 97 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(2011\) | \(1\) |
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 | 6033.2.a.e | ✓ | 97 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6033.2.a.e | ✓ | 97 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{97} - 12 T_{2}^{96} - 85 T_{2}^{95} + 1570 T_{2}^{94} + 1783 T_{2}^{93} - 97687 T_{2}^{92} + \cdots - 3597258368 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6033))\).