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: | \(1\) |
Dimension: | \(27\) |
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 | −3.35817 | 0 | −0.581328 | 0 | 3.25261 | 0 | 8.27730 | 0 | ||||||||||||||||||
1.2 | 0 | −2.81775 | 0 | −0.243389 | 0 | 0.929492 | 0 | 4.93969 | 0 | ||||||||||||||||||
1.3 | 0 | −2.74782 | 0 | −0.322226 | 0 | −1.10048 | 0 | 4.55049 | 0 | ||||||||||||||||||
1.4 | 0 | −2.67388 | 0 | 3.55353 | 0 | 1.65788 | 0 | 4.14966 | 0 | ||||||||||||||||||
1.5 | 0 | −2.14389 | 0 | 2.12568 | 0 | −4.35707 | 0 | 1.59626 | 0 | ||||||||||||||||||
1.6 | 0 | −2.00223 | 0 | −2.46425 | 0 | 4.06602 | 0 | 1.00893 | 0 | ||||||||||||||||||
1.7 | 0 | −1.94547 | 0 | −1.68966 | 0 | −1.54452 | 0 | 0.784840 | 0 | ||||||||||||||||||
1.8 | 0 | −1.68125 | 0 | −3.32036 | 0 | 3.49635 | 0 | −0.173410 | 0 | ||||||||||||||||||
1.9 | 0 | −1.59467 | 0 | 2.11777 | 0 | 0.323115 | 0 | −0.457036 | 0 | ||||||||||||||||||
1.10 | 0 | −1.57318 | 0 | −4.10446 | 0 | −3.99696 | 0 | −0.525101 | 0 | ||||||||||||||||||
1.11 | 0 | −0.907672 | 0 | 1.31635 | 0 | −2.29510 | 0 | −2.17613 | 0 | ||||||||||||||||||
1.12 | 0 | −0.604994 | 0 | 2.44978 | 0 | 2.35442 | 0 | −2.63398 | 0 | ||||||||||||||||||
1.13 | 0 | −0.338800 | 0 | −3.08469 | 0 | −4.48412 | 0 | −2.88521 | 0 | ||||||||||||||||||
1.14 | 0 | −0.338332 | 0 | 0.413944 | 0 | −1.61145 | 0 | −2.88553 | 0 | ||||||||||||||||||
1.15 | 0 | 0.267543 | 0 | 3.98483 | 0 | −3.08625 | 0 | −2.92842 | 0 | ||||||||||||||||||
1.16 | 0 | 0.314419 | 0 | −4.24837 | 0 | 1.07745 | 0 | −2.90114 | 0 | ||||||||||||||||||
1.17 | 0 | 0.551494 | 0 | −2.33245 | 0 | 1.71388 | 0 | −2.69585 | 0 | ||||||||||||||||||
1.18 | 0 | 0.739341 | 0 | 2.08289 | 0 | 4.12124 | 0 | −2.45337 | 0 | ||||||||||||||||||
1.19 | 0 | 1.21945 | 0 | 1.36200 | 0 | −2.61900 | 0 | −1.51294 | 0 | ||||||||||||||||||
1.20 | 0 | 1.41351 | 0 | −0.774385 | 0 | −1.47209 | 0 | −1.00198 | 0 | ||||||||||||||||||
See all 27 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.g | ✓ | 27 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6004.2.a.g | ✓ | 27 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):
\( T_{3}^{27} + 4 T_{3}^{26} - 42 T_{3}^{25} - 179 T_{3}^{24} + 739 T_{3}^{23} + 3445 T_{3}^{22} - 7046 T_{3}^{21} - 37466 T_{3}^{20} + 38976 T_{3}^{19} + 254518 T_{3}^{18} - 120597 T_{3}^{17} - 1128252 T_{3}^{16} + 151681 T_{3}^{15} + \cdots - 2560 \) |
\( T_{5}^{27} + 10 T_{5}^{26} - 24 T_{5}^{25} - 523 T_{5}^{24} - 440 T_{5}^{23} + 11116 T_{5}^{22} + 22580 T_{5}^{21} - 126256 T_{5}^{20} - 351707 T_{5}^{19} + 851553 T_{5}^{18} + 2946572 T_{5}^{17} - 3585634 T_{5}^{16} + \cdots + 108441 \) |