Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4007,2,Mod(1,4007)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4007, 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("4007.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4007 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4007.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: | \(31.9960560899\) |
Analytic rank: | \(0\) |
Dimension: | \(195\) |
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 | −2.80768 | 0.834495 | 5.88309 | 4.09159 | −2.34300 | 3.13441 | −10.9025 | −2.30362 | −11.4879 | ||||||||||||||||||
1.2 | −2.77582 | −0.198171 | 5.70518 | −0.442867 | 0.550088 | 0.719726 | −10.2849 | −2.96073 | 1.22932 | ||||||||||||||||||
1.3 | −2.77052 | −1.23937 | 5.67576 | −3.30641 | 3.43369 | 0.994092 | −10.1837 | −1.46397 | 9.16047 | ||||||||||||||||||
1.4 | −2.74088 | 3.29634 | 5.51241 | −3.80383 | −9.03487 | −0.926403 | −9.62710 | 7.86587 | 10.4258 | ||||||||||||||||||
1.5 | −2.73169 | 2.16378 | 5.46210 | −1.55160 | −5.91076 | 4.97440 | −9.45738 | 1.68193 | 4.23848 | ||||||||||||||||||
1.6 | −2.71610 | −2.73922 | 5.37720 | −0.158886 | 7.44001 | 4.34515 | −9.17282 | 4.50335 | 0.431551 | ||||||||||||||||||
1.7 | −2.69070 | −2.66626 | 5.23985 | 2.66315 | 7.17410 | −3.38912 | −8.71746 | 4.10894 | −7.16574 | ||||||||||||||||||
1.8 | −2.63607 | 1.85738 | 4.94884 | −0.474491 | −4.89618 | −2.15797 | −7.77333 | 0.449870 | 1.25079 | ||||||||||||||||||
1.9 | −2.60343 | 1.83156 | 4.77782 | −3.48115 | −4.76834 | −4.50968 | −7.23185 | 0.354625 | 9.06293 | ||||||||||||||||||
1.10 | −2.59764 | −1.72984 | 4.74771 | 1.17522 | 4.49349 | 2.46697 | −7.13754 | −0.00766352 | −3.05279 | ||||||||||||||||||
1.11 | −2.59160 | −1.40196 | 4.71637 | −3.26317 | 3.63331 | −1.83814 | −7.03972 | −1.03451 | 8.45682 | ||||||||||||||||||
1.12 | −2.58803 | 0.217741 | 4.69789 | −3.50376 | −0.563520 | 3.48169 | −6.98221 | −2.95259 | 9.06783 | ||||||||||||||||||
1.13 | −2.55145 | −0.786154 | 4.50991 | 3.16082 | 2.00584 | −3.53434 | −6.40393 | −2.38196 | −8.06468 | ||||||||||||||||||
1.14 | −2.54911 | 2.95187 | 4.49797 | 1.50040 | −7.52466 | 0.00462218 | −6.36761 | 5.71356 | −3.82468 | ||||||||||||||||||
1.15 | −2.50867 | 2.54308 | 4.29344 | 3.25074 | −6.37975 | 2.35749 | −5.75350 | 3.46723 | −8.15505 | ||||||||||||||||||
1.16 | −2.47249 | 0.496095 | 4.11323 | 2.63577 | −1.22659 | 3.34809 | −5.22495 | −2.75389 | −6.51692 | ||||||||||||||||||
1.17 | −2.43669 | 0.943453 | 3.93744 | −0.444210 | −2.29890 | −0.206330 | −4.72093 | −2.10990 | 1.08240 | ||||||||||||||||||
1.18 | −2.42566 | 0.883893 | 3.88381 | −3.90163 | −2.14402 | 3.11093 | −4.56949 | −2.21873 | 9.46401 | ||||||||||||||||||
1.19 | −2.41162 | −0.257694 | 3.81590 | 2.26401 | 0.621460 | −3.03056 | −4.37927 | −2.93359 | −5.45992 | ||||||||||||||||||
1.20 | −2.39027 | −2.93715 | 3.71341 | −2.06612 | 7.02059 | −0.622431 | −4.09551 | 5.62685 | 4.93859 | ||||||||||||||||||
See next 80 embeddings (of 195 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(4007\) | \(-1\) |
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 | 4007.2.a.b | ✓ | 195 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4007.2.a.b | ✓ | 195 | 1.a | even | 1 | 1 | trivial |