Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [859,2,Mod(1,859)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(859, 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("859.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 859 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 859.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: | \(6.85914953363\) |
Analytic rank: | \(0\) |
Dimension: | \(42\) |
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 | −2.63139 | −1.16127 | 4.92422 | −1.01415 | 3.05576 | −3.81074 | −7.69476 | −1.65144 | 2.66863 | ||||||||||||||||||
1.2 | −2.50833 | −2.75356 | 4.29173 | 2.56198 | 6.90685 | −3.49225 | −5.74842 | 4.58211 | −6.42630 | ||||||||||||||||||
1.3 | −2.44167 | 2.53203 | 3.96174 | 2.18158 | −6.18236 | 2.64770 | −4.78991 | 3.41115 | −5.32670 | ||||||||||||||||||
1.4 | −2.39243 | 0.929930 | 3.72373 | 3.81463 | −2.22480 | −1.17880 | −4.12391 | −2.13523 | −9.12624 | ||||||||||||||||||
1.5 | −2.19990 | 3.00238 | 2.83955 | −0.915767 | −6.60494 | 0.370418 | −1.84693 | 6.01431 | 2.01459 | ||||||||||||||||||
1.6 | −2.13988 | −3.02925 | 2.57907 | −1.54899 | 6.48223 | 1.60061 | −1.23914 | 6.17638 | 3.31464 | ||||||||||||||||||
1.7 | −1.95205 | −0.352561 | 1.81051 | 0.588410 | 0.688218 | 4.42108 | 0.369892 | −2.87570 | −1.14861 | ||||||||||||||||||
1.8 | −1.86825 | 0.805919 | 1.49036 | −1.84284 | −1.50566 | −3.09719 | 0.952140 | −2.35049 | 3.44289 | ||||||||||||||||||
1.9 | −1.66068 | −0.868667 | 0.757869 | −0.869159 | 1.44258 | 0.868049 | 2.06279 | −2.24542 | 1.44340 | ||||||||||||||||||
1.10 | −1.48888 | 0.831203 | 0.216773 | −4.09720 | −1.23756 | −3.04547 | 2.65502 | −2.30910 | 6.10025 | ||||||||||||||||||
1.11 | −1.30000 | 1.81233 | −0.310001 | 2.95498 | −2.35603 | 2.52751 | 3.00300 | 0.284549 | −3.84147 | ||||||||||||||||||
1.12 | −1.26489 | −3.36123 | −0.400048 | 0.315487 | 4.25160 | −3.03106 | 3.03580 | 8.29788 | −0.399057 | ||||||||||||||||||
1.13 | −1.24119 | 2.63551 | −0.459453 | 3.14475 | −3.27117 | −0.940601 | 3.05264 | 3.94593 | −3.90322 | ||||||||||||||||||
1.14 | −1.06408 | −2.69968 | −0.867728 | 3.52398 | 2.87269 | 2.11982 | 3.05150 | 4.28829 | −3.74981 | ||||||||||||||||||
1.15 | −0.618382 | −0.410722 | −1.61760 | 3.41550 | 0.253983 | −5.12644 | 2.23706 | −2.83131 | −2.11208 | ||||||||||||||||||
1.16 | −0.353447 | −0.654623 | −1.87507 | 1.93614 | 0.231375 | 4.76193 | 1.36963 | −2.57147 | −0.684323 | ||||||||||||||||||
1.17 | −0.327352 | 2.93975 | −1.89284 | −0.866451 | −0.962334 | −3.05348 | 1.27433 | 5.64214 | 0.283635 | ||||||||||||||||||
1.18 | −0.140862 | −1.32420 | −1.98016 | −1.96304 | 0.186529 | −3.48893 | 0.560653 | −1.24650 | 0.276518 | ||||||||||||||||||
1.19 | −0.0951469 | 3.39816 | −1.99095 | 1.26109 | −0.323325 | 4.51330 | 0.379726 | 8.54752 | −0.119989 | ||||||||||||||||||
1.20 | −0.0467311 | 1.76352 | −1.99782 | −2.63357 | −0.0824112 | 2.66028 | 0.186822 | 0.110005 | 0.123069 | ||||||||||||||||||
See all 42 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(859\) | \(-1\) |
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 | 859.2.a.b | ✓ | 42 |
3.b | odd | 2 | 1 | 7731.2.a.f | 42 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
859.2.a.b | ✓ | 42 | 1.a | even | 1 | 1 | trivial |
7731.2.a.f | 42 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} - 10 T_{2}^{41} - 16 T_{2}^{40} + 471 T_{2}^{39} - 651 T_{2}^{38} - 9662 T_{2}^{37} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(859))\).