Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6004,2,Mod(1,6004)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6004, 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("6004.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6004 = 2^{2} \cdot 19 \cdot 79 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6004.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: | \(47.9421813736\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | 0 | −3.35141 | 0 | −3.29178 | 0 | −3.21633 | 0 | 8.23192 | 0 | ||||||||||||||||||
1.2 | 0 | −3.19734 | 0 | 3.86202 | 0 | 0.0940904 | 0 | 7.22301 | 0 | ||||||||||||||||||
1.3 | 0 | −3.17665 | 0 | 0.661665 | 0 | 4.43999 | 0 | 7.09108 | 0 | ||||||||||||||||||
1.4 | 0 | −2.94455 | 0 | 3.51674 | 0 | −3.98269 | 0 | 5.67040 | 0 | ||||||||||||||||||
1.5 | 0 | −2.58747 | 0 | −3.96262 | 0 | 2.03599 | 0 | 3.69501 | 0 | ||||||||||||||||||
1.6 | 0 | −2.48790 | 0 | −1.86011 | 0 | −0.813337 | 0 | 3.18963 | 0 | ||||||||||||||||||
1.7 | 0 | −2.24876 | 0 | −0.730072 | 0 | −4.54869 | 0 | 2.05693 | 0 | ||||||||||||||||||
1.8 | 0 | −1.95303 | 0 | 2.03258 | 0 | 1.63097 | 0 | 0.814318 | 0 | ||||||||||||||||||
1.9 | 0 | −1.93761 | 0 | 0.795901 | 0 | 1.65858 | 0 | 0.754317 | 0 | ||||||||||||||||||
1.10 | 0 | −1.53751 | 0 | 2.83612 | 0 | −2.00746 | 0 | −0.636048 | 0 | ||||||||||||||||||
1.11 | 0 | −1.34737 | 0 | −0.232989 | 0 | 3.14804 | 0 | −1.18461 | 0 | ||||||||||||||||||
1.12 | 0 | −1.33751 | 0 | −1.75116 | 0 | −1.54762 | 0 | −1.21107 | 0 | ||||||||||||||||||
1.13 | 0 | 1.34688 | 0 | 3.43943 | 0 | 0.297446 | 0 | −1.18593 | 0 | ||||||||||||||||||
1.14 | 0 | 1.70311 | 0 | −2.13957 | 0 | −3.96784 | 0 | −0.0994128 | 0 | ||||||||||||||||||
1.15 | 0 | 1.84886 | 0 | 0.871764 | 0 | −3.24231 | 0 | 0.418286 | 0 | ||||||||||||||||||
1.16 | 0 | 1.98454 | 0 | 1.21429 | 0 | 4.02832 | 0 | 0.938380 | 0 | ||||||||||||||||||
1.17 | 0 | 2.05400 | 0 | −2.46181 | 0 | 3.02591 | 0 | 1.21890 | 0 | ||||||||||||||||||
1.18 | 0 | 2.22506 | 0 | −3.09603 | 0 | −0.873879 | 0 | 1.95088 | 0 | ||||||||||||||||||
1.19 | 0 | 2.59829 | 0 | 0.131830 | 0 | 4.07797 | 0 | 3.75113 | 0 | ||||||||||||||||||
1.20 | 0 | 2.68661 | 0 | 2.87584 | 0 | 2.53518 | 0 | 4.21786 | 0 | ||||||||||||||||||
See all 24 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(19\) | \(-1\) |
\(79\) | \(-1\) |
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 | 6004.2.a.e | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6004.2.a.e | ✓ | 24 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):
\( T_{3}^{24} - T_{3}^{23} - 73 T_{3}^{22} + 70 T_{3}^{21} + 2383 T_{3}^{20} - 2172 T_{3}^{19} - 45944 T_{3}^{18} + 39380 T_{3}^{17} + 582024 T_{3}^{16} - 462924 T_{3}^{15} - 5098571 T_{3}^{14} + 3699519 T_{3}^{13} + \cdots + 409318924 \) |
\( T_{5}^{24} - 9 T_{5}^{23} - 32 T_{5}^{22} + 495 T_{5}^{21} - 80 T_{5}^{20} - 11125 T_{5}^{19} + 16668 T_{5}^{18} + 132059 T_{5}^{17} - 308391 T_{5}^{16} - 883223 T_{5}^{15} + 2801155 T_{5}^{14} + 3156093 T_{5}^{13} + \cdots + 205761 \) |