Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6014,2,Mod(1,6014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6014, 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("6014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6014 = 2 \cdot 31 \cdot 97 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6014.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: | \(48.0220317756\) |
Analytic rank: | \(1\) |
Dimension: | \(22\) |
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 | −1.00000 | −2.92127 | 1.00000 | −1.99624 | 2.92127 | −0.468229 | −1.00000 | 5.53384 | 1.99624 | ||||||||||||||||||
1.2 | −1.00000 | −2.80047 | 1.00000 | 0.364133 | 2.80047 | 1.53058 | −1.00000 | 4.84261 | −0.364133 | ||||||||||||||||||
1.3 | −1.00000 | −2.51447 | 1.00000 | 3.35437 | 2.51447 | 0.494792 | −1.00000 | 3.32256 | −3.35437 | ||||||||||||||||||
1.4 | −1.00000 | −2.06526 | 1.00000 | 1.84106 | 2.06526 | 0.807248 | −1.00000 | 1.26528 | −1.84106 | ||||||||||||||||||
1.5 | −1.00000 | −1.79599 | 1.00000 | −3.70171 | 1.79599 | −3.49561 | −1.00000 | 0.225587 | 3.70171 | ||||||||||||||||||
1.6 | −1.00000 | −1.60979 | 1.00000 | 1.84866 | 1.60979 | −2.27059 | −1.00000 | −0.408562 | −1.84866 | ||||||||||||||||||
1.7 | −1.00000 | −1.34015 | 1.00000 | 0.146848 | 1.34015 | 4.48511 | −1.00000 | −1.20399 | −0.146848 | ||||||||||||||||||
1.8 | −1.00000 | −1.04488 | 1.00000 | −0.722746 | 1.04488 | −0.912399 | −1.00000 | −1.90823 | 0.722746 | ||||||||||||||||||
1.9 | −1.00000 | −0.916242 | 1.00000 | −2.81390 | 0.916242 | 0.508180 | −1.00000 | −2.16050 | 2.81390 | ||||||||||||||||||
1.10 | −1.00000 | −0.531212 | 1.00000 | 1.86385 | 0.531212 | −4.06404 | −1.00000 | −2.71781 | −1.86385 | ||||||||||||||||||
1.11 | −1.00000 | 0.0685928 | 1.00000 | −2.24786 | −0.0685928 | −4.29331 | −1.00000 | −2.99530 | 2.24786 | ||||||||||||||||||
1.12 | −1.00000 | 0.191955 | 1.00000 | −1.50666 | −0.191955 | −2.08203 | −1.00000 | −2.96315 | 1.50666 | ||||||||||||||||||
1.13 | −1.00000 | 0.253397 | 1.00000 | −0.480250 | −0.253397 | 2.09789 | −1.00000 | −2.93579 | 0.480250 | ||||||||||||||||||
1.14 | −1.00000 | 0.813553 | 1.00000 | 3.21035 | −0.813553 | −2.44817 | −1.00000 | −2.33813 | −3.21035 | ||||||||||||||||||
1.15 | −1.00000 | 1.04707 | 1.00000 | 1.01417 | −1.04707 | 3.78577 | −1.00000 | −1.90364 | −1.01417 | ||||||||||||||||||
1.16 | −1.00000 | 1.29017 | 1.00000 | 3.44095 | −1.29017 | −0.471801 | −1.00000 | −1.33547 | −3.44095 | ||||||||||||||||||
1.17 | −1.00000 | 1.47323 | 1.00000 | −3.13161 | −1.47323 | 0.874706 | −1.00000 | −0.829602 | 3.13161 | ||||||||||||||||||
1.18 | −1.00000 | 2.07834 | 1.00000 | −0.312935 | −2.07834 | 0.578926 | −1.00000 | 1.31950 | 0.312935 | ||||||||||||||||||
1.19 | −1.00000 | 2.17019 | 1.00000 | 1.18841 | −2.17019 | −1.11869 | −1.00000 | 1.70972 | −1.18841 | ||||||||||||||||||
1.20 | −1.00000 | 2.25454 | 1.00000 | −2.47204 | −2.25454 | 0.953481 | −1.00000 | 2.08293 | 2.47204 | ||||||||||||||||||
See all 22 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(31\) | \(-1\) |
\(97\) | \(-1\) |
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 | 6014.2.a.f | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6014.2.a.f | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{22} - 37 T_{3}^{20} - T_{3}^{19} + 574 T_{3}^{18} + 28 T_{3}^{17} - 4877 T_{3}^{16} - 318 T_{3}^{15} + \cdots + 40 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6014))\).