Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6021,2,Mod(1,6021)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6021, 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("6021.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6021 = 3^{3} \cdot 223 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6021.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: | \(48.0779270570\) |
Analytic rank: | \(0\) |
Dimension: | \(35\) |
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 | −2.73710 | 0 | 5.49174 | 0.00733490 | 0 | −3.12271 | −9.55725 | 0 | −0.0200764 | ||||||||||||||||||
1.2 | −2.45047 | 0 | 4.00478 | 1.31775 | 0 | −2.98449 | −4.91265 | 0 | −3.22911 | ||||||||||||||||||
1.3 | −2.31533 | 0 | 3.36077 | 2.18639 | 0 | 4.16397 | −3.15063 | 0 | −5.06222 | ||||||||||||||||||
1.4 | −2.30052 | 0 | 3.29241 | 0.673783 | 0 | 1.52445 | −2.97323 | 0 | −1.55005 | ||||||||||||||||||
1.5 | −2.13504 | 0 | 2.55839 | 2.06664 | 0 | −0.593247 | −1.19218 | 0 | −4.41235 | ||||||||||||||||||
1.6 | −2.10383 | 0 | 2.42609 | 2.21279 | 0 | −3.17552 | −0.896424 | 0 | −4.65533 | ||||||||||||||||||
1.7 | −2.08857 | 0 | 2.36212 | −1.86886 | 0 | −1.80880 | −0.756310 | 0 | 3.90323 | ||||||||||||||||||
1.8 | −1.74554 | 0 | 1.04691 | −3.98518 | 0 | −0.618122 | 1.66366 | 0 | 6.95629 | ||||||||||||||||||
1.9 | −1.41065 | 0 | −0.0100678 | 3.64343 | 0 | 2.96382 | 2.83550 | 0 | −5.13961 | ||||||||||||||||||
1.10 | −1.33519 | 0 | −0.217276 | 0.108401 | 0 | 4.69056 | 2.96048 | 0 | −0.144736 | ||||||||||||||||||
1.11 | −0.982520 | 0 | −1.03465 | −0.593429 | 0 | 1.54053 | 2.98161 | 0 | 0.583056 | ||||||||||||||||||
1.12 | −0.941194 | 0 | −1.11415 | 2.01775 | 0 | −3.14622 | 2.93102 | 0 | −1.89910 | ||||||||||||||||||
1.13 | −0.627854 | 0 | −1.60580 | −3.06245 | 0 | 0.00430391 | 2.26392 | 0 | 1.92277 | ||||||||||||||||||
1.14 | −0.563437 | 0 | −1.68254 | 3.58037 | 0 | 0.250597 | 2.07488 | 0 | −2.01732 | ||||||||||||||||||
1.15 | −0.544885 | 0 | −1.70310 | −3.52229 | 0 | −1.92545 | 2.01776 | 0 | 1.91924 | ||||||||||||||||||
1.16 | −0.384611 | 0 | −1.85207 | −1.68485 | 0 | −1.18684 | 1.48155 | 0 | 0.648011 | ||||||||||||||||||
1.17 | −0.0961107 | 0 | −1.99076 | 2.13805 | 0 | −1.72518 | 0.383555 | 0 | −0.205489 | ||||||||||||||||||
1.18 | 0.145112 | 0 | −1.97894 | 4.00700 | 0 | −4.35864 | −0.577391 | 0 | 0.581462 | ||||||||||||||||||
1.19 | 0.378423 | 0 | −1.85680 | 0.673616 | 0 | 3.71590 | −1.45950 | 0 | 0.254912 | ||||||||||||||||||
1.20 | 0.424438 | 0 | −1.81985 | −1.02884 | 0 | −4.02930 | −1.62129 | 0 | −0.436677 | ||||||||||||||||||
See all 35 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(223\) | \(-1\) |
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 | 6021.2.a.r | yes | 35 |
3.b | odd | 2 | 1 | 6021.2.a.q | ✓ | 35 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6021.2.a.q | ✓ | 35 | 3.b | odd | 2 | 1 | |
6021.2.a.r | yes | 35 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):
\( T_{2}^{35} - 4 T_{2}^{34} - 45 T_{2}^{33} + 191 T_{2}^{32} + 898 T_{2}^{31} - 4104 T_{2}^{30} - 10447 T_{2}^{29} + 52472 T_{2}^{28} + 78189 T_{2}^{27} - 444821 T_{2}^{26} - 390260 T_{2}^{25} + 2637421 T_{2}^{24} + 1288333 T_{2}^{23} + \cdots + 534 \) |
\( T_{5}^{35} - 10 T_{5}^{34} - 51 T_{5}^{33} + 805 T_{5}^{32} + 348 T_{5}^{31} - 28343 T_{5}^{30} + 35781 T_{5}^{29} + 572902 T_{5}^{28} - 1316105 T_{5}^{27} - 7307228 T_{5}^{26} + 23445918 T_{5}^{25} + \cdots + 4984161 \) |