Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6015,2,Mod(1,6015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6015, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6015 = 3 \cdot 5 \cdot 401 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6015.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: | \(48.0300168158\) |
Analytic rank: | \(0\) |
Dimension: | \(39\) |
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 | −2.73278 | 1.00000 | 5.46807 | −1.00000 | −2.73278 | −1.71200 | −9.47746 | 1.00000 | 2.73278 | ||||||||||||||||||
1.2 | −2.72880 | 1.00000 | 5.44634 | −1.00000 | −2.72880 | 4.68686 | −9.40437 | 1.00000 | 2.72880 | ||||||||||||||||||
1.3 | −2.61259 | 1.00000 | 4.82563 | −1.00000 | −2.61259 | 1.06107 | −7.38220 | 1.00000 | 2.61259 | ||||||||||||||||||
1.4 | −2.40618 | 1.00000 | 3.78972 | −1.00000 | −2.40618 | −5.00600 | −4.30638 | 1.00000 | 2.40618 | ||||||||||||||||||
1.5 | −2.39683 | 1.00000 | 3.74481 | −1.00000 | −2.39683 | 3.91052 | −4.18201 | 1.00000 | 2.39683 | ||||||||||||||||||
1.6 | −2.30681 | 1.00000 | 3.32135 | −1.00000 | −2.30681 | 0.182032 | −3.04810 | 1.00000 | 2.30681 | ||||||||||||||||||
1.7 | −2.17202 | 1.00000 | 2.71767 | −1.00000 | −2.17202 | −0.722026 | −1.55879 | 1.00000 | 2.17202 | ||||||||||||||||||
1.8 | −2.05828 | 1.00000 | 2.23653 | −1.00000 | −2.05828 | 4.53201 | −0.486836 | 1.00000 | 2.05828 | ||||||||||||||||||
1.9 | −1.76442 | 1.00000 | 1.11320 | −1.00000 | −1.76442 | −1.91400 | 1.56470 | 1.00000 | 1.76442 | ||||||||||||||||||
1.10 | −1.60401 | 1.00000 | 0.572848 | −1.00000 | −1.60401 | 0.763953 | 2.28917 | 1.00000 | 1.60401 | ||||||||||||||||||
1.11 | −1.46868 | 1.00000 | 0.157024 | −1.00000 | −1.46868 | −1.36125 | 2.70674 | 1.00000 | 1.46868 | ||||||||||||||||||
1.12 | −1.45937 | 1.00000 | 0.129770 | −1.00000 | −1.45937 | 4.22534 | 2.72936 | 1.00000 | 1.45937 | ||||||||||||||||||
1.13 | −1.39292 | 1.00000 | −0.0597773 | −1.00000 | −1.39292 | 0.772686 | 2.86910 | 1.00000 | 1.39292 | ||||||||||||||||||
1.14 | −1.06568 | 1.00000 | −0.864332 | −1.00000 | −1.06568 | −3.38476 | 3.05245 | 1.00000 | 1.06568 | ||||||||||||||||||
1.15 | −0.832555 | 1.00000 | −1.30685 | −1.00000 | −0.832555 | 2.56306 | 2.75314 | 1.00000 | 0.832555 | ||||||||||||||||||
1.16 | −0.749262 | 1.00000 | −1.43861 | −1.00000 | −0.749262 | −2.59303 | 2.57642 | 1.00000 | 0.749262 | ||||||||||||||||||
1.17 | −0.693654 | 1.00000 | −1.51884 | −1.00000 | −0.693654 | 5.04357 | 2.44086 | 1.00000 | 0.693654 | ||||||||||||||||||
1.18 | −0.407003 | 1.00000 | −1.83435 | −1.00000 | −0.407003 | −1.58531 | 1.56059 | 1.00000 | 0.407003 | ||||||||||||||||||
1.19 | −0.369183 | 1.00000 | −1.86370 | −1.00000 | −0.369183 | −1.65507 | 1.42641 | 1.00000 | 0.369183 | ||||||||||||||||||
1.20 | 0.104331 | 1.00000 | −1.98912 | −1.00000 | 0.104331 | −3.27326 | −0.416187 | 1.00000 | −0.104331 | ||||||||||||||||||
See all 39 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(5\) | \(1\) |
\(401\) | \(1\) |
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 | 6015.2.a.h | ✓ | 39 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6015.2.a.h | ✓ | 39 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{39} - 63 T_{2}^{37} - T_{2}^{36} + 1820 T_{2}^{35} + 58 T_{2}^{34} - 31968 T_{2}^{33} + \cdots + 22239 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\).