Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4003,2,Mod(1,4003)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4003, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4003.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4003 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4003.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: | \(31.9641159291\) |
Analytic rank: | \(1\) |
Dimension: | \(152\) |
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 | −2.81629 | 1.48630 | 5.93147 | 2.03150 | −4.18584 | −2.55919 | −11.0722 | −0.790918 | −5.72129 | ||||||||||||||||||
1.2 | −2.78616 | −2.78181 | 5.76271 | −2.35951 | 7.75059 | 2.45659 | −10.4835 | 4.73848 | 6.57398 | ||||||||||||||||||
1.3 | −2.74603 | −0.470182 | 5.54067 | −1.60967 | 1.29113 | −2.65994 | −9.72280 | −2.77893 | 4.42020 | ||||||||||||||||||
1.4 | −2.71591 | −1.21901 | 5.37616 | 2.34595 | 3.31073 | 3.38769 | −9.16936 | −1.51401 | −6.37139 | ||||||||||||||||||
1.5 | −2.71344 | −2.64650 | 5.36273 | −1.98460 | 7.18110 | −4.86510 | −9.12456 | 4.00395 | 5.38509 | ||||||||||||||||||
1.6 | −2.70039 | 3.13054 | 5.29213 | −0.910228 | −8.45368 | 0.335007 | −8.89004 | 6.80026 | 2.45797 | ||||||||||||||||||
1.7 | −2.68690 | 0.420690 | 5.21946 | −4.00007 | −1.13035 | −1.97783 | −8.65037 | −2.82302 | 10.7478 | ||||||||||||||||||
1.8 | −2.64207 | 1.02501 | 4.98054 | 2.32673 | −2.70814 | −0.493904 | −7.87479 | −1.94936 | −6.14739 | ||||||||||||||||||
1.9 | −2.60205 | 1.48054 | 4.77066 | 1.31197 | −3.85243 | 3.44490 | −7.20939 | −0.808005 | −3.41380 | ||||||||||||||||||
1.10 | −2.59568 | 2.00123 | 4.73753 | −2.55331 | −5.19456 | 4.30357 | −7.10575 | 1.00494 | 6.62756 | ||||||||||||||||||
1.11 | −2.57418 | 1.68195 | 4.62641 | −3.90988 | −4.32965 | 1.64141 | −6.76085 | −0.171031 | 10.0647 | ||||||||||||||||||
1.12 | −2.54789 | −3.05680 | 4.49172 | 1.69745 | 7.78837 | −1.95167 | −6.34863 | 6.34402 | −4.32491 | ||||||||||||||||||
1.13 | −2.45287 | −0.154053 | 4.01656 | −1.18015 | 0.377872 | 4.09823 | −4.94635 | −2.97627 | 2.89476 | ||||||||||||||||||
1.14 | −2.44543 | −1.15615 | 3.98012 | −3.94364 | 2.82729 | 3.67660 | −4.84225 | −1.66331 | 9.64388 | ||||||||||||||||||
1.15 | −2.43441 | 1.77953 | 3.92633 | −0.360928 | −4.33209 | −1.53438 | −4.68946 | 0.166714 | 0.878646 | ||||||||||||||||||
1.16 | −2.42958 | −1.44088 | 3.90288 | 0.473210 | 3.50074 | −4.22120 | −4.62321 | −0.923870 | −1.14970 | ||||||||||||||||||
1.17 | −2.41802 | 1.98760 | 3.84680 | 1.18517 | −4.80604 | −3.98093 | −4.46560 | 0.950540 | −2.86577 | ||||||||||||||||||
1.18 | −2.37503 | −2.00580 | 3.64079 | 0.380635 | 4.76385 | 3.69515 | −3.89692 | 1.02324 | −0.904022 | ||||||||||||||||||
1.19 | −2.36790 | −2.62074 | 3.60697 | 3.23023 | 6.20567 | 1.16162 | −3.80516 | 3.86830 | −7.64888 | ||||||||||||||||||
1.20 | −2.36704 | −1.77609 | 3.60290 | −2.33381 | 4.20407 | 1.88575 | −3.79413 | 0.154480 | 5.52424 | ||||||||||||||||||
See next 80 embeddings (of 152 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(4003\) | \(1\) |
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 | 4003.2.a.b | ✓ | 152 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4003.2.a.b | ✓ | 152 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{152} + 22 T_{2}^{151} + 21 T_{2}^{150} - 3036 T_{2}^{149} - 18712 T_{2}^{148} + 177403 T_{2}^{147} + \cdots - 67141 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4003))\).