Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,2,Mod(44,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 4, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.44");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 273.cd (of order \(12\), degree \(4\), 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: | \(2.17991597518\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
44.1 | −2.69539 | + | 0.722228i | 1.57685 | + | 0.716628i | 5.01147 | − | 2.89338i | 1.51996 | − | 0.407273i | −4.76779 | − | 0.792751i | −0.780863 | + | 2.52790i | −7.47188 | + | 7.47188i | 1.97289 | + | 2.26003i | −3.80275 | + | 2.19552i |
44.2 | −2.42391 | + | 0.649484i | −1.28483 | − | 1.16155i | 3.72144 | − | 2.14858i | 3.58252 | − | 0.959933i | 3.86873 | + | 1.98102i | 1.55291 | − | 2.14207i | −4.07612 | + | 4.07612i | 0.301593 | + | 2.98480i | −8.06023 | + | 4.65358i |
44.3 | −2.41247 | + | 0.646418i | 0.872737 | − | 1.49611i | 3.67009 | − | 2.11892i | −1.51647 | + | 0.406338i | −1.13834 | + | 4.17346i | −1.79039 | − | 1.94795i | −3.95215 | + | 3.95215i | −1.47666 | − | 2.61141i | 3.39578 | − | 1.96055i |
44.4 | −2.19229 | + | 0.587422i | −1.51343 | + | 0.842335i | 2.72902 | − | 1.57560i | 0.959762 | − | 0.257167i | 2.82307 | − | 2.73567i | −2.15459 | + | 1.53549i | −1.84753 | + | 1.84753i | 1.58094 | − | 2.54963i | −1.95301 | + | 1.12757i |
44.5 | −2.10280 | + | 0.563443i | −0.755319 | − | 1.55868i | 2.37223 | − | 1.36961i | −2.37528 | + | 0.636456i | 2.46651 | + | 2.85201i | 1.72404 | + | 2.00691i | −1.13792 | + | 1.13792i | −1.85899 | + | 2.35461i | 4.63613 | − | 2.67667i |
44.6 | −2.00566 | + | 0.537415i | −0.123517 | + | 1.72764i | 2.00181 | − | 1.15574i | 0.878767 | − | 0.235465i | −0.680727 | − | 3.53144i | −1.67646 | − | 2.04682i | −0.457340 | + | 0.457340i | −2.96949 | − | 0.426786i | −1.63596 | + | 0.944525i |
44.7 | −1.87036 | + | 0.501163i | 1.45895 | + | 0.933521i | 1.51505 | − | 0.874714i | −1.33892 | + | 0.358762i | −3.19662 | − | 1.01485i | 1.90555 | − | 1.83545i | 0.343083 | − | 0.343083i | 1.25708 | + | 2.72392i | 2.32447 | − | 1.34203i |
44.8 | −1.64181 | + | 0.439921i | 1.02109 | − | 1.39906i | 0.769949 | − | 0.444530i | 3.49677 | − | 0.936956i | −1.06096 | + | 2.74619i | 0.612738 | + | 2.57382i | 1.33522 | − | 1.33522i | −0.914735 | − | 2.85714i | −5.32883 | + | 3.07660i |
44.9 | −1.55291 | + | 0.416100i | −1.66835 | + | 0.465425i | 0.506326 | − | 0.292327i | 0.0914679 | − | 0.0245087i | 2.39712 | − | 1.41696i | 2.58235 | + | 0.575745i | 1.60897 | − | 1.60897i | 2.56676 | − | 1.55298i | −0.131843 | + | 0.0761195i |
44.10 | −1.27346 | + | 0.341222i | 1.73165 | − | 0.0374536i | −0.226792 | + | 0.130939i | −3.02659 | + | 0.810973i | −2.19240 | + | 0.638570i | −0.631669 | + | 2.56924i | 2.10860 | − | 2.10860i | 2.99719 | − | 0.129713i | 3.57751 | − | 2.06548i |
44.11 | −1.25001 | + | 0.334940i | −1.60871 | − | 0.641920i | −0.281700 | + | 0.162639i | −3.70264 | + | 0.992121i | 2.22591 | + | 0.263589i | −1.90960 | − | 1.83124i | 2.12780 | − | 2.12780i | 2.17588 | + | 2.06532i | 4.29606 | − | 2.48033i |
44.12 | −0.623254 | + | 0.167000i | −1.63214 | − | 0.579750i | −1.37149 | + | 0.791833i | 2.98988 | − | 0.801137i | 1.11406 | + | 0.0887629i | −2.60516 | + | 0.461652i | 1.63506 | − | 1.63506i | 2.32778 | + | 1.89247i | −1.72966 | + | 0.998622i |
44.13 | −0.578797 | + | 0.155088i | 0.590016 | − | 1.62846i | −1.42110 | + | 0.820471i | −0.609213 | + | 0.163238i | −0.0889444 | + | 1.03405i | 2.12540 | − | 1.57565i | 1.54270 | − | 1.54270i | −2.30376 | − | 1.92163i | 0.327294 | − | 0.188963i |
44.14 | −0.246180 | + | 0.0659636i | −0.753906 | + | 1.55937i | −1.67580 | + | 0.967522i | −2.71744 | + | 0.728135i | 0.0827348 | − | 0.433615i | 1.41179 | − | 2.23760i | 0.709158 | − | 0.709158i | −1.86325 | − | 2.35123i | 0.620947 | − | 0.358504i |
44.15 | 0.246180 | − | 0.0659636i | 1.72740 | + | 0.126782i | −1.67580 | + | 0.967522i | 2.71744 | − | 0.728135i | 0.433615 | − | 0.0827348i | 1.41179 | − | 2.23760i | −0.709158 | + | 0.709158i | 2.96785 | + | 0.438007i | 0.620947 | − | 0.358504i |
44.16 | 0.578797 | − | 0.155088i | −1.70530 | − | 0.303261i | −1.42110 | + | 0.820471i | 0.609213 | − | 0.163238i | −1.03405 | + | 0.0889444i | 2.12540 | − | 1.57565i | −1.54270 | + | 1.54270i | 2.81607 | + | 1.03430i | 0.327294 | − | 0.188963i |
44.17 | 0.623254 | − | 0.167000i | 0.313993 | − | 1.70335i | −1.37149 | + | 0.791833i | −2.98988 | + | 0.801137i | −0.0887629 | − | 1.11406i | −2.60516 | + | 0.461652i | −1.63506 | + | 1.63506i | −2.80282 | − | 1.06968i | −1.72966 | + | 0.998622i |
44.18 | 1.25001 | − | 0.334940i | 0.248434 | − | 1.71414i | −0.281700 | + | 0.162639i | 3.70264 | − | 0.992121i | −0.263589 | − | 2.22591i | −1.90960 | − | 1.83124i | −2.12780 | + | 2.12780i | −2.87656 | − | 0.851703i | 4.29606 | − | 2.48033i |
44.19 | 1.27346 | − | 0.341222i | −0.898259 | + | 1.48092i | −0.226792 | + | 0.130939i | 3.02659 | − | 0.810973i | −0.638570 | + | 2.19240i | −0.631669 | + | 2.56924i | −2.10860 | + | 2.10860i | −1.38626 | − | 2.66050i | 3.57751 | − | 2.06548i |
44.20 | 1.55291 | − | 0.416100i | 1.23724 | − | 1.21212i | 0.506326 | − | 0.292327i | −0.0914679 | + | 0.0245087i | 1.41696 | − | 2.39712i | 2.58235 | + | 0.575745i | −1.60897 | + | 1.60897i | 0.0615393 | − | 2.99937i | −0.131843 | + | 0.0761195i |
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
13.d | odd | 4 | 1 | inner |
21.h | odd | 6 | 1 | inner |
39.f | even | 4 | 1 | inner |
91.z | odd | 12 | 1 | inner |
273.cd | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 273.2.cd.e | ✓ | 112 |
3.b | odd | 2 | 1 | inner | 273.2.cd.e | ✓ | 112 |
7.c | even | 3 | 1 | inner | 273.2.cd.e | ✓ | 112 |
13.d | odd | 4 | 1 | inner | 273.2.cd.e | ✓ | 112 |
21.h | odd | 6 | 1 | inner | 273.2.cd.e | ✓ | 112 |
39.f | even | 4 | 1 | inner | 273.2.cd.e | ✓ | 112 |
91.z | odd | 12 | 1 | inner | 273.2.cd.e | ✓ | 112 |
273.cd | even | 12 | 1 | inner | 273.2.cd.e | ✓ | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.cd.e | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
273.2.cd.e | ✓ | 112 | 3.b | odd | 2 | 1 | inner |
273.2.cd.e | ✓ | 112 | 7.c | even | 3 | 1 | inner |
273.2.cd.e | ✓ | 112 | 13.d | odd | 4 | 1 | inner |
273.2.cd.e | ✓ | 112 | 21.h | odd | 6 | 1 | inner |
273.2.cd.e | ✓ | 112 | 39.f | even | 4 | 1 | inner |
273.2.cd.e | ✓ | 112 | 91.z | odd | 12 | 1 | inner |
273.2.cd.e | ✓ | 112 | 273.cd | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):
\( T_{2}^{112} - 242 T_{2}^{108} + 33605 T_{2}^{104} - 3152030 T_{2}^{100} + 221630758 T_{2}^{96} - 12146123884 T_{2}^{92} + 535268243529 T_{2}^{88} - 19244548977438 T_{2}^{84} + \cdots + 42\!\cdots\!41 \)
|
\( T_{5}^{112} - 882 T_{5}^{108} + 456457 T_{5}^{104} - 157971354 T_{5}^{100} + 40834781064 T_{5}^{96} - 8078915475000 T_{5}^{92} + \cdots + 27\!\cdots\!76 \)
|
\( T_{19}^{56} - 20 T_{19}^{55} + 200 T_{19}^{54} - 1956 T_{19}^{53} + 15541 T_{19}^{52} - 71276 T_{19}^{51} + 230288 T_{19}^{50} + 87122 T_{19}^{49} - 13087904 T_{19}^{48} + 87101224 T_{19}^{47} + \cdots + 88\!\cdots\!25 \)
|