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: | \(1\) |
Dimension: | \(36\) |
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 | −2.82805 | −1.00000 | 5.99789 | −1.00000 | 2.82805 | 0.186142 | −11.3062 | 1.00000 | 2.82805 | ||||||||||||||||||
1.2 | −2.70250 | −1.00000 | 5.30351 | −1.00000 | 2.70250 | −3.42053 | −8.92774 | 1.00000 | 2.70250 | ||||||||||||||||||
1.3 | −2.65304 | −1.00000 | 5.03861 | −1.00000 | 2.65304 | −2.89459 | −8.06155 | 1.00000 | 2.65304 | ||||||||||||||||||
1.4 | −2.61575 | −1.00000 | 4.84213 | −1.00000 | 2.61575 | 3.80178 | −7.43431 | 1.00000 | 2.61575 | ||||||||||||||||||
1.5 | −2.58666 | −1.00000 | 4.69083 | −1.00000 | 2.58666 | 2.48059 | −6.96028 | 1.00000 | 2.58666 | ||||||||||||||||||
1.6 | −2.45396 | −1.00000 | 4.02194 | −1.00000 | 2.45396 | 3.31635 | −4.96176 | 1.00000 | 2.45396 | ||||||||||||||||||
1.7 | −2.15201 | −1.00000 | 2.63116 | −1.00000 | 2.15201 | −1.35819 | −1.35826 | 1.00000 | 2.15201 | ||||||||||||||||||
1.8 | −2.03595 | −1.00000 | 2.14509 | −1.00000 | 2.03595 | −4.31879 | −0.295397 | 1.00000 | 2.03595 | ||||||||||||||||||
1.9 | −1.98597 | −1.00000 | 1.94407 | −1.00000 | 1.98597 | −2.98220 | 0.111071 | 1.00000 | 1.98597 | ||||||||||||||||||
1.10 | −1.98559 | −1.00000 | 1.94257 | −1.00000 | 1.98559 | 4.82942 | 0.114041 | 1.00000 | 1.98559 | ||||||||||||||||||
1.11 | −1.50384 | −1.00000 | 0.261530 | −1.00000 | 1.50384 | 2.03402 | 2.61438 | 1.00000 | 1.50384 | ||||||||||||||||||
1.12 | −1.46470 | −1.00000 | 0.145344 | −1.00000 | 1.46470 | 1.42914 | 2.71651 | 1.00000 | 1.46470 | ||||||||||||||||||
1.13 | −1.30040 | −1.00000 | −0.308957 | −1.00000 | 1.30040 | −4.16384 | 3.00257 | 1.00000 | 1.30040 | ||||||||||||||||||
1.14 | −1.26310 | −1.00000 | −0.404578 | −1.00000 | 1.26310 | −1.43647 | 3.03722 | 1.00000 | 1.26310 | ||||||||||||||||||
1.15 | −0.886874 | −1.00000 | −1.21345 | −1.00000 | 0.886874 | 0.962079 | 2.84993 | 1.00000 | 0.886874 | ||||||||||||||||||
1.16 | −0.823088 | −1.00000 | −1.32253 | −1.00000 | 0.823088 | 2.12642 | 2.73473 | 1.00000 | 0.823088 | ||||||||||||||||||
1.17 | −0.518532 | −1.00000 | −1.73112 | −1.00000 | 0.518532 | 3.44981 | 1.93471 | 1.00000 | 0.518532 | ||||||||||||||||||
1.18 | −0.409079 | −1.00000 | −1.83265 | −1.00000 | 0.409079 | 2.40403 | 1.56786 | 1.00000 | 0.409079 | ||||||||||||||||||
1.19 | −0.139206 | −1.00000 | −1.98062 | −1.00000 | 0.139206 | −3.46594 | 0.554125 | 1.00000 | 0.139206 | ||||||||||||||||||
1.20 | 0.278732 | −1.00000 | −1.92231 | −1.00000 | −0.278732 | −3.61104 | −1.09327 | 1.00000 | −0.278732 | ||||||||||||||||||
See all 36 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.f | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6015.2.a.f | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 7 T_{2}^{35} - 34 T_{2}^{34} - 335 T_{2}^{33} + 344 T_{2}^{32} + 7174 T_{2}^{31} + \cdots - 9576 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\).