Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [59,2,Mod(3,59)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(59, base_ring=CyclotomicField(58))
chi = DirichletCharacter(H, H._module([50]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("59.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 59 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 59.c (of order \(29\), degree \(28\), 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: | \(0.471117371926\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{29})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{29}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −2.18221 | + | 1.00960i | −1.39423 | − | 0.469770i | 2.44798 | − | 2.88198i | −1.93515 | − | 1.16434i | 3.51678 | − | 0.382473i | 0.168810 | − | 3.11351i | −1.14585 | + | 4.12697i | −0.665090 | − | 0.505588i | 5.39841 | + | 0.587112i |
3.2 | −1.11469 | + | 0.515711i | 0.0468625 | + | 0.0157898i | −0.318193 | + | 0.374606i | 2.81017 | + | 1.69082i | −0.0603802 | + | 0.00656675i | −0.144587 | + | 2.66676i | 0.818660 | − | 2.94855i | −2.38633 | − | 1.81404i | −4.00445 | − | 0.435510i |
3.3 | −0.554859 | + | 0.256705i | 2.89290 | + | 0.974731i | −1.05280 | + | 1.23945i | −2.81882 | − | 1.69603i | −1.85537 | + | 0.201783i | 0.0467597 | − | 0.862432i | 0.593097 | − | 2.13614i | 5.03048 | + | 3.82407i | 1.99943 | + | 0.217451i |
3.4 | 1.29680 | − | 0.599963i | −0.263634 | − | 0.0888288i | 0.0269577 | − | 0.0317371i | −0.768757 | − | 0.462546i | −0.395175 | + | 0.0429778i | 0.00331163 | − | 0.0610794i | −0.748603 | + | 2.69623i | −2.32667 | − | 1.76869i | −1.27443 | − | 0.138603i |
4.1 | −1.95520 | − | 0.212641i | −0.330952 | + | 0.389626i | 1.82437 | + | 0.401574i | 3.02411 | + | 2.29887i | 0.729929 | − | 0.691425i | 0.790338 | − | 1.98360i | 0.245940 | + | 0.0828667i | 0.443066 | + | 2.70258i | −5.42392 | − | 5.13781i |
4.2 | −1.36929 | − | 0.148919i | −1.47847 | + | 1.74059i | −0.100466 | − | 0.0221143i | −3.18119 | − | 2.41828i | 2.28366 | − | 2.16320i | −1.14976 | + | 2.88569i | 2.74480 | + | 0.924830i | −0.358432 | − | 2.18634i | 3.99584 | + | 3.78507i |
4.3 | 0.670646 | + | 0.0729372i | 0.865386 | − | 1.01881i | −1.50879 | − | 0.332111i | 0.331060 | + | 0.251665i | 0.654677 | − | 0.620143i | −0.928345 | + | 2.32997i | −2.26622 | − | 0.763578i | 0.196262 | + | 1.19715i | 0.203668 | + | 0.192925i |
4.4 | 1.67136 | + | 0.181772i | −1.22878 | + | 1.44663i | 0.807178 | + | 0.177673i | −0.547988 | − | 0.416570i | −2.31670 | + | 2.19450i | 1.70727 | − | 4.28492i | −1.86963 | − | 0.629953i | −0.0974987 | − | 0.594716i | −0.840167 | − | 0.795848i |
5.1 | −2.17862 | − | 0.734061i | 1.07303 | + | 0.645618i | 2.61533 | + | 1.98812i | 0.194454 | − | 0.488043i | −1.86379 | − | 2.19422i | 3.85587 | − | 1.78392i | −1.65811 | − | 2.44553i | −0.670664 | − | 1.26501i | −0.781893 | + | 0.920516i |
5.2 | −0.588366 | − | 0.198243i | −1.87280 | − | 1.12683i | −1.28531 | − | 0.977069i | 0.787871 | − | 1.97741i | 0.878507 | + | 1.03426i | 0.478175 | − | 0.221227i | 1.25938 | + | 1.85745i | 0.832424 | + | 1.57012i | −0.855565 | + | 1.00725i |
5.3 | 0.0887205 | + | 0.0298934i | 1.79944 | + | 1.08269i | −1.58521 | − | 1.20504i | −0.0780955 | + | 0.196005i | 0.127282 | + | 0.149848i | −1.67564 | + | 0.775234i | −0.209696 | − | 0.309278i | 0.660556 | + | 1.24594i | −0.0127879 | + | 0.0150551i |
5.4 | 1.82982 | + | 0.616538i | −1.38105 | − | 0.830948i | 1.37594 | + | 1.04596i | −0.455265 | + | 1.14263i | −2.01476 | − | 2.37196i | −0.696774 | + | 0.322362i | −0.294344 | − | 0.434125i | −0.188410 | − | 0.355378i | −1.53753 | + | 1.81012i |
7.1 | −1.04598 | − | 1.54270i | 0.0724858 | − | 1.33692i | −0.545581 | + | 1.36930i | −0.914971 | + | 0.423310i | −2.13829 | + | 1.28657i | 0.293352 | − | 1.05656i | −0.957486 | + | 0.210759i | 1.20031 | + | 0.130541i | 1.61008 | + | 0.968753i |
7.2 | −0.804656 | − | 1.18678i | −0.142015 | + | 2.61930i | −0.0206970 | + | 0.0519455i | 3.22031 | − | 1.48987i | 3.22281 | − | 1.93910i | −0.416708 | + | 1.50085i | −2.72235 | + | 0.599234i | −3.85817 | − | 0.419601i | −4.35939 | − | 2.62296i |
7.3 | 0.461940 | + | 0.681311i | −0.107820 | + | 1.98863i | 0.489481 | − | 1.22850i | −2.53352 | + | 1.17213i | −1.40468 | + | 0.845168i | 0.992982 | − | 3.57640i | 2.67091 | − | 0.587912i | −0.960605 | − | 0.104472i | −1.96892 | − | 1.18466i |
7.4 | 1.32002 | + | 1.94688i | 0.160846 | − | 2.96664i | −1.30762 | + | 3.28188i | −2.86423 | + | 1.32513i | 5.98801 | − | 3.60287i | 0.00132718 | − | 0.00478009i | −3.52113 | + | 0.775059i | −5.79266 | − | 0.629990i | −6.36072 | − | 3.82712i |
9.1 | −1.42070 | + | 1.67258i | 1.16204 | + | 0.883363i | −0.455567 | − | 2.77884i | 0.717227 | + | 1.35283i | −3.12842 | + | 0.688617i | −2.70417 | − | 0.294096i | 1.53427 | + | 0.923141i | −0.232569 | − | 0.837638i | −3.28169 | − | 0.722355i |
9.2 | −1.20997 | + | 1.42449i | −2.64227 | − | 2.00860i | −0.241575 | − | 1.47354i | −1.05767 | − | 1.99497i | 6.05831 | − | 1.33354i | −1.16042 | − | 0.126203i | −0.811609 | − | 0.488329i | 2.14453 | + | 7.72389i | 4.12157 | + | 0.907225i |
9.3 | 0.0212776 | − | 0.0250500i | 0.632796 | + | 0.481039i | 0.323389 | + | 1.97259i | −0.938493 | − | 1.77019i | 0.0255144 | − | 0.00561614i | 0.117902 | + | 0.0128227i | 0.112619 | + | 0.0677605i | −0.633553 | − | 2.28185i | −0.0643120 | − | 0.0141561i |
9.4 | 1.12380 | − | 1.32304i | −1.60754 | − | 1.22202i | −0.163940 | − | 0.999990i | 0.789401 | + | 1.48897i | −3.42332 | + | 0.753530i | −1.11689 | − | 0.121470i | 1.46757 | + | 0.883010i | 0.288265 | + | 1.03824i | 2.85708 | + | 0.628892i |
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
59.c | even | 29 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 59.2.c.a | ✓ | 112 |
3.b | odd | 2 | 1 | 531.2.i.a | 112 | ||
4.b | odd | 2 | 1 | 944.2.m.c | 112 | ||
59.c | even | 29 | 1 | inner | 59.2.c.a | ✓ | 112 |
59.c | even | 29 | 1 | 3481.2.a.p | 56 | ||
59.d | odd | 58 | 1 | 3481.2.a.q | 56 | ||
177.h | odd | 58 | 1 | 531.2.i.a | 112 | ||
236.h | odd | 58 | 1 | 944.2.m.c | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
59.2.c.a | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
59.2.c.a | ✓ | 112 | 59.c | even | 29 | 1 | inner |
531.2.i.a | 112 | 3.b | odd | 2 | 1 | ||
531.2.i.a | 112 | 177.h | odd | 58 | 1 | ||
944.2.m.c | 112 | 4.b | odd | 2 | 1 | ||
944.2.m.c | 112 | 236.h | odd | 58 | 1 | ||
3481.2.a.p | 56 | 59.c | even | 29 | 1 | ||
3481.2.a.q | 56 | 59.d | odd | 58 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(59, [\chi])\).