Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [61,2,Mod(4,61)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 61.k (of order \(30\), degree \(8\), minimal) |
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: | no |
Analytic conductor: | \(0.487087452330\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −2.46695 | − | 0.259287i | −1.14152 | − | 0.829364i | 4.06232 | + | 0.863472i | −2.05628 | + | 2.28373i | 2.60103 | + | 2.34198i | −1.70327 | + | 3.82562i | −5.07938 | − | 1.65039i | −0.311823 | − | 0.959694i | 5.66488 | − | 5.10068i |
4.2 | −1.34984 | − | 0.141873i | 2.48935 | + | 1.80862i | −0.154366 | − | 0.0328114i | −1.79809 | + | 1.99699i | −3.10361 | − | 2.79451i | 1.21824 | − | 2.73622i | 2.78540 | + | 0.905030i | 1.99870 | + | 6.15137i | 2.71045 | − | 2.44050i |
4.3 | −0.269420 | − | 0.0283172i | −2.58882 | − | 1.88089i | −1.88451 | − | 0.400565i | 0.677902 | − | 0.752887i | 0.644220 | + | 0.580058i | 0.612497 | − | 1.37569i | 1.01167 | + | 0.328713i | 2.23721 | + | 6.88542i | −0.203960 | + | 0.183647i |
4.4 | 0.560519 | + | 0.0589129i | 0.925257 | + | 0.672238i | −1.64558 | − | 0.349780i | 1.41041 | − | 1.56642i | 0.479020 | + | 0.431312i | −1.58545 | + | 3.56099i | −1.97382 | − | 0.641332i | −0.522855 | − | 1.60918i | 0.882846 | − | 0.794918i |
4.5 | 1.54754 | + | 0.162653i | −0.184257 | − | 0.133870i | 0.412130 | + | 0.0876009i | −1.91658 | + | 2.12858i | −0.263370 | − | 0.237140i | 1.50074 | − | 3.37071i | −2.33627 | − | 0.759101i | −0.911022 | − | 2.80384i | −3.31220 | + | 2.98232i |
5.1 | −0.844477 | − | 1.89673i | −1.24277 | − | 0.902925i | −1.54617 | + | 1.71720i | −1.72334 | − | 0.366308i | −0.663112 | + | 3.11970i | 3.03851 | − | 0.319360i | 0.613541 | + | 0.199352i | −0.197847 | − | 0.608912i | 0.760538 | + | 3.57805i |
5.2 | −0.785340 | − | 1.76390i | 1.97012 | + | 1.43138i | −1.15633 | + | 1.28424i | 1.14861 | + | 0.244144i | 0.977595 | − | 4.59922i | −4.17120 | + | 0.438411i | −0.499279 | − | 0.162225i | 0.905491 | + | 2.78681i | −0.471401 | − | 2.21777i |
5.3 | 0.0302402 | + | 0.0679206i | −1.68685 | − | 1.22557i | 1.33456 | − | 1.48218i | 2.11663 | + | 0.449905i | 0.0322307 | − | 0.151633i | −1.79799 | + | 0.188976i | 0.282447 | + | 0.0917726i | 0.416398 | + | 1.28154i | 0.0334497 | + | 0.157368i |
5.4 | 0.320214 | + | 0.719213i | 1.32575 | + | 0.963215i | 0.923531 | − | 1.02569i | −4.10692 | − | 0.872953i | −0.268232 | + | 1.26193i | −1.68211 | + | 0.176797i | 2.53090 | + | 0.822340i | −0.0972162 | − | 0.299201i | −0.687255 | − | 3.23328i |
5.5 | 0.948493 | + | 2.13035i | −0.866253 | − | 0.629370i | −2.30049 | + | 2.55496i | −0.488412 | − | 0.103815i | 0.519143 | − | 2.44238i | 2.37906 | − | 0.250049i | −3.18931 | − | 1.03627i | −0.572763 | − | 1.76278i | −0.242093 | − | 1.13896i |
19.1 | −0.564484 | − | 2.65569i | −0.0581327 | + | 0.178914i | −4.90695 | + | 2.18472i | 0.269940 | − | 2.56831i | 0.507955 | + | 0.0533882i | 0.301292 | + | 0.271284i | 5.38013 | + | 7.40511i | 2.39842 | + | 1.74255i | −6.97300 | + | 0.732892i |
19.2 | −0.240153 | − | 1.12983i | 0.155231 | − | 0.477753i | 0.608249 | − | 0.270810i | −0.167336 | + | 1.59210i | −0.577058 | − | 0.0606513i | −2.62622 | − | 2.36466i | −1.80991 | − | 2.49113i | 2.22290 | + | 1.61503i | 1.83898 | − | 0.193285i |
19.3 | −0.187984 | − | 0.884393i | −1.04731 | + | 3.22329i | 1.08028 | − | 0.480971i | 0.0125746 | − | 0.119639i | 3.04753 | + | 0.320308i | 2.28414 | + | 2.05665i | −1.69133 | − | 2.32792i | −6.86567 | − | 4.98820i | −0.108172 | + | 0.0113693i |
19.4 | 0.422863 | + | 1.98941i | −0.443559 | + | 1.36514i | −1.95187 | + | 0.869027i | 0.366719 | − | 3.48909i | −2.90338 | − | 0.305158i | −1.27054 | − | 1.14400i | −0.163282 | − | 0.224738i | 0.760202 | + | 0.552319i | 7.09633 | − | 0.745854i |
19.5 | 0.483303 | + | 2.27376i | 0.893770 | − | 2.75074i | −3.10932 | + | 1.38436i | −0.281187 | + | 2.67531i | 6.68649 | + | 0.702779i | −2.18490 | − | 1.96729i | −1.91775 | − | 2.63956i | −4.34071 | − | 3.15371i | −6.21892 | + | 0.653635i |
36.1 | −1.93342 | − | 1.74086i | −0.935224 | − | 2.87832i | 0.498468 | + | 4.74261i | 0.269216 | + | 0.119863i | −3.20258 | + | 7.19311i | −0.0232368 | − | 0.109320i | 4.23402 | − | 5.82763i | −4.98305 | + | 3.62040i | −0.311844 | − | 0.700413i |
36.2 | −1.38021 | − | 1.24274i | 0.674596 | + | 2.07619i | 0.151502 | + | 1.44145i | 2.73442 | + | 1.21744i | 1.64910 | − | 3.70393i | −0.687078 | − | 3.23245i | −0.601086 | + | 0.827323i | −1.42845 | + | 1.03783i | −2.26110 | − | 5.07851i |
36.3 | −0.144683 | − | 0.130273i | −0.391997 | − | 1.20644i | −0.205095 | − | 1.95135i | 0.401532 | + | 0.178774i | −0.100452 | + | 0.225618i | 0.419277 | + | 1.97254i | −0.453406 | + | 0.624060i | 1.12521 | − | 0.817512i | −0.0348054 | − | 0.0781742i |
36.4 | 0.480752 | + | 0.432871i | 0.788261 | + | 2.42602i | −0.165312 | − | 1.57284i | −2.30675 | − | 1.02703i | −0.671195 | + | 1.50753i | 0.155097 | + | 0.729673i | 1.36186 | − | 1.87444i | −2.83715 | + | 2.06131i | −0.664404 | − | 1.49228i |
36.5 | 1.87303 | + | 1.68649i | −0.635636 | − | 1.95629i | 0.454959 | + | 4.32865i | −3.56305 | − | 1.58637i | 2.10868 | − | 4.73618i | 0.323145 | + | 1.52028i | −3.48513 | + | 4.79687i | −0.995971 | + | 0.723615i | −3.99832 | − | 8.98038i |
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
61.k | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 61.2.k.a | ✓ | 40 |
3.b | odd | 2 | 1 | 549.2.bs.e | 40 | ||
4.b | odd | 2 | 1 | 976.2.cl.c | 40 | ||
61.k | even | 30 | 1 | inner | 61.2.k.a | ✓ | 40 |
61.l | odd | 60 | 2 | 3721.2.a.m | 40 | ||
183.v | odd | 30 | 1 | 549.2.bs.e | 40 | ||
244.v | odd | 30 | 1 | 976.2.cl.c | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
61.2.k.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
61.2.k.a | ✓ | 40 | 61.k | even | 30 | 1 | inner |
549.2.bs.e | 40 | 3.b | odd | 2 | 1 | ||
549.2.bs.e | 40 | 183.v | odd | 30 | 1 | ||
976.2.cl.c | 40 | 4.b | odd | 2 | 1 | ||
976.2.cl.c | 40 | 244.v | odd | 30 | 1 | ||
3721.2.a.m | 40 | 61.l | odd | 60 | 2 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(61, [\chi])\).