Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1003,6,Mod(1,1003)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1003, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1003.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1003 = 17 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 1003.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: | \(160.864971272\) |
Analytic rank: | \(0\) |
Dimension: | \(100\) |
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 | −11.0579 | −11.4049 | 90.2781 | 52.1219 | 126.115 | −53.9097 | −644.436 | −112.928 | −576.361 | ||||||||||||||||||
1.2 | −11.0372 | 11.4506 | 89.8190 | 107.836 | −126.382 | −170.958 | −638.158 | −111.883 | −1190.20 | ||||||||||||||||||
1.3 | −10.8798 | 27.3463 | 86.3711 | 30.4368 | −297.524 | 91.8048 | −591.550 | 504.821 | −331.148 | ||||||||||||||||||
1.4 | −10.6829 | −2.44660 | 82.1240 | 3.37371 | 26.1367 | 100.320 | −535.468 | −237.014 | −36.0409 | ||||||||||||||||||
1.5 | −10.4782 | −2.88738 | 77.7924 | −75.5324 | 30.2545 | −43.6540 | −479.821 | −234.663 | 791.442 | ||||||||||||||||||
1.6 | −10.0929 | −24.7282 | 69.8675 | 20.2324 | 249.580 | −225.563 | −382.195 | 368.482 | −204.205 | ||||||||||||||||||
1.7 | −9.96834 | 28.0559 | 67.3679 | −85.7673 | −279.671 | 39.1762 | −352.559 | 544.135 | 854.958 | ||||||||||||||||||
1.8 | −9.86265 | −29.0101 | 65.2718 | −27.8967 | 286.116 | 106.432 | −328.148 | 598.586 | 275.136 | ||||||||||||||||||
1.9 | −9.41592 | 22.5666 | 56.6595 | −60.2165 | −212.485 | −93.2886 | −232.192 | 266.253 | 566.994 | ||||||||||||||||||
1.10 | −9.36177 | 7.91970 | 55.6428 | 15.2707 | −74.1424 | −60.9674 | −221.338 | −180.278 | −142.961 | ||||||||||||||||||
1.11 | −9.28879 | 6.79931 | 54.2816 | 55.3186 | −63.1573 | −109.817 | −206.969 | −196.769 | −513.843 | ||||||||||||||||||
1.12 | −9.24292 | −25.2981 | 53.4317 | 16.6075 | 233.829 | 77.1380 | −198.091 | 396.995 | −153.502 | ||||||||||||||||||
1.13 | −8.95289 | 10.5913 | 48.1542 | −18.1795 | −94.8223 | 196.308 | −144.627 | −130.825 | 162.759 | ||||||||||||||||||
1.14 | −8.88996 | 18.7065 | 47.0315 | 107.329 | −166.300 | 114.464 | −133.629 | 106.932 | −954.147 | ||||||||||||||||||
1.15 | −8.25067 | −17.5967 | 36.0736 | −93.5213 | 145.184 | 115.074 | −33.6097 | 66.6434 | 771.613 | ||||||||||||||||||
1.16 | −8.21422 | 13.1577 | 35.4733 | −55.6754 | −108.080 | 30.1229 | −28.5308 | −69.8746 | 457.330 | ||||||||||||||||||
1.17 | −8.03714 | −10.5980 | 32.5955 | −79.0989 | 85.1778 | −104.629 | −4.78643 | −130.682 | 635.729 | ||||||||||||||||||
1.18 | −7.89514 | −10.2115 | 30.3332 | 56.9036 | 80.6212 | 230.167 | 13.1594 | −138.725 | −449.262 | ||||||||||||||||||
1.19 | −7.79360 | −16.6480 | 28.7401 | −72.0478 | 129.748 | −223.681 | 25.4060 | 34.1567 | 561.511 | ||||||||||||||||||
1.20 | −7.61836 | 18.9618 | 26.0394 | 46.8788 | −144.458 | −223.496 | 45.4097 | 116.548 | −357.139 | ||||||||||||||||||
See all 100 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(17\) | \(1\) |
\(59\) | \(-1\) |
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 | 1003.6.a.d | ✓ | 100 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1003.6.a.d | ✓ | 100 | 1.a | even | 1 | 1 | trivial |