Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [360,2,Mod(59,360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 5, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("360.59");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 360.bd (of order \(6\), degree \(2\), 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: | \(2.87461447277\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(64\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
59.1 | −1.41125 | + | 0.0914777i | 1.32971 | + | 1.10989i | 1.98326 | − | 0.258196i | −1.67067 | + | 1.48622i | −1.97809 | − | 1.44470i | −2.22153 | − | 3.84780i | −2.77527 | + | 0.545804i | 0.536278 | + | 2.95168i | 2.22179 | − | 2.25026i |
59.2 | −1.40842 | + | 0.127848i | −0.679246 | − | 1.59331i | 1.96731 | − | 0.360128i | −2.06621 | − | 0.854849i | 1.16037 | + | 2.15721i | 1.70997 | + | 2.96175i | −2.72476 | + | 0.758729i | −2.07725 | + | 2.16449i | 3.01939 | + | 0.939829i |
59.3 | −1.40586 | − | 0.153453i | 1.71600 | − | 0.235231i | 1.95290 | + | 0.431469i | 2.23606 | − | 0.00425117i | −2.44856 | + | 0.0673764i | 1.34736 | + | 2.33369i | −2.67931 | − | 0.906266i | 2.88933 | − | 0.807315i | −3.14425 | − | 0.337155i |
59.4 | −1.39143 | − | 0.252849i | −0.573395 | + | 1.63439i | 1.87213 | + | 0.703642i | −0.0677451 | − | 2.23504i | 1.21109 | − | 2.12915i | −0.0861243 | − | 0.149172i | −2.42702 | − | 1.45243i | −2.34244 | − | 1.87430i | −0.470866 | + | 3.12702i |
59.5 | −1.38690 | + | 0.276612i | −1.70706 | − | 0.293175i | 1.84697 | − | 0.767264i | 2.22251 | + | 0.245870i | 2.44861 | − | 0.0655888i | 0.193132 | + | 0.334514i | −2.34933 | + | 1.57501i | 2.82810 | + | 1.00093i | −3.15040 | + | 0.273776i |
59.6 | −1.37971 | + | 0.310484i | −0.0381767 | + | 1.73163i | 1.80720 | − | 0.856756i | 0.289425 | + | 2.21726i | −0.484971 | − | 2.40100i | 2.06690 | + | 3.57998i | −2.22740 | + | 1.74318i | −2.99709 | − | 0.132216i | −1.08775 | − | 2.96931i |
59.7 | −1.37227 | − | 0.341873i | 1.16746 | − | 1.27946i | 1.76625 | + | 0.938283i | −0.996511 | + | 2.00174i | −2.03949 | + | 1.35665i | −0.0301683 | − | 0.0522531i | −2.10299 | − | 1.89141i | −0.274059 | − | 2.98746i | 2.05182 | − | 2.40625i |
59.8 | −1.29321 | + | 0.572373i | 1.41725 | − | 0.995694i | 1.34478 | − | 1.48040i | −0.776540 | − | 2.09690i | −1.26289 | + | 2.09884i | −0.720108 | − | 1.24726i | −0.891741 | + | 2.68418i | 1.01719 | − | 2.82229i | 2.20444 | + | 2.26726i |
59.9 | −1.27900 | − | 0.603460i | −1.13393 | + | 1.30928i | 1.27167 | + | 1.54365i | 1.63855 | + | 1.52157i | 2.24039 | − | 0.990283i | −2.19953 | − | 3.80970i | −0.694935 | − | 2.74173i | −0.428416 | − | 2.96925i | −1.17749 | − | 2.93488i |
59.10 | −1.24345 | − | 0.673664i | 1.46348 | + | 0.926406i | 1.09235 | + | 1.67534i | −1.40051 | − | 1.74315i | −1.19568 | − | 2.13784i | 0.618137 | + | 1.07065i | −0.229672 | − | 2.81909i | 1.28354 | + | 2.71155i | 0.567178 | + | 3.11100i |
59.11 | −1.24141 | − | 0.677426i | −1.65025 | + | 0.525984i | 1.08219 | + | 1.68192i | −1.85825 | + | 1.24374i | 2.40495 | + | 0.464965i | 1.64197 | + | 2.84397i | −0.204056 | − | 2.82106i | 2.44668 | − | 1.73602i | 3.14939 | − | 0.285165i |
59.12 | −1.17126 | − | 0.792553i | −0.681584 | − | 1.59231i | 0.743721 | + | 1.85658i | 1.18672 | + | 1.89518i | −0.463672 | + | 2.40520i | −0.113645 | − | 0.196839i | 0.600342 | − | 2.76398i | −2.07089 | + | 2.17058i | 0.112070 | − | 3.16029i |
59.13 | −1.14229 | + | 0.833765i | −1.41725 | + | 0.995694i | 0.609671 | − | 1.90481i | −2.20424 | + | 0.375946i | 0.788739 | − | 2.31903i | −0.720108 | − | 1.24726i | 0.891741 | + | 2.68418i | 1.01719 | − | 2.82229i | 2.20444 | − | 2.26726i |
59.14 | −1.02286 | − | 0.976600i | −1.56092 | − | 0.750684i | 0.0925056 | + | 1.99786i | 1.04334 | − | 1.97774i | 0.863494 | + | 2.29224i | 0.753496 | + | 1.30509i | 1.85649 | − | 2.13388i | 1.87295 | + | 2.34352i | −2.99865 | + | 1.00404i |
59.15 | −0.958742 | + | 1.03962i | 0.0381767 | − | 1.73163i | −0.161627 | − | 1.99346i | 2.06491 | − | 0.857979i | 1.76364 | + | 1.69988i | 2.06690 | + | 3.57998i | 2.22740 | + | 1.74318i | −2.99709 | − | 0.132216i | −1.08775 | + | 2.96931i |
59.16 | −0.933002 | + | 1.06278i | 1.70706 | + | 0.293175i | −0.259016 | − | 1.98316i | 1.32418 | + | 1.80181i | −1.90427 | + | 1.54070i | 0.193132 | + | 0.334514i | 2.34933 | + | 1.57501i | 2.82810 | + | 1.00093i | −3.15040 | − | 0.273776i |
59.17 | −0.912258 | − | 1.08064i | 1.72276 | − | 0.179111i | −0.335569 | + | 1.97165i | 1.97813 | − | 1.04260i | −1.76516 | − | 1.69829i | −2.24901 | − | 3.89541i | 2.43677 | − | 1.43602i | 2.93584 | − | 0.617133i | −2.93124 | − | 1.18653i |
59.18 | −0.903164 | − | 1.08825i | 0.662421 | − | 1.60037i | −0.368591 | + | 1.96574i | −1.94669 | − | 1.10018i | −2.33989 | + | 0.724518i | −1.20226 | − | 2.08238i | 2.47212 | − | 1.37427i | −2.12240 | − | 2.12024i | 0.560906 | + | 3.11214i |
59.19 | −0.814931 | + | 1.15581i | 0.679246 | + | 1.59331i | −0.671775 | − | 1.88380i | −1.77343 | − | 1.36197i | −2.39509 | − | 0.513357i | 1.70997 | + | 2.96175i | 2.72476 | + | 0.758729i | −2.07725 | + | 2.16449i | 3.01939 | − | 0.939829i |
59.20 | −0.784848 | + | 1.17644i | −1.32971 | − | 1.10989i | −0.768027 | − | 1.84665i | 0.451766 | − | 2.18996i | 2.34935 | − | 0.693233i | −2.22153 | − | 3.84780i | 2.77527 | + | 0.545804i | 0.536278 | + | 2.95168i | 2.22179 | + | 2.25026i |
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
40.e | odd | 2 | 1 | inner |
45.h | odd | 6 | 1 | inner |
72.l | even | 6 | 1 | inner |
360.bd | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 360.2.bd.b | ✓ | 128 |
5.b | even | 2 | 1 | inner | 360.2.bd.b | ✓ | 128 |
8.d | odd | 2 | 1 | inner | 360.2.bd.b | ✓ | 128 |
9.d | odd | 6 | 1 | inner | 360.2.bd.b | ✓ | 128 |
40.e | odd | 2 | 1 | inner | 360.2.bd.b | ✓ | 128 |
45.h | odd | 6 | 1 | inner | 360.2.bd.b | ✓ | 128 |
72.l | even | 6 | 1 | inner | 360.2.bd.b | ✓ | 128 |
360.bd | even | 6 | 1 | inner | 360.2.bd.b | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
360.2.bd.b | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
360.2.bd.b | ✓ | 128 | 5.b | even | 2 | 1 | inner |
360.2.bd.b | ✓ | 128 | 8.d | odd | 2 | 1 | inner |
360.2.bd.b | ✓ | 128 | 9.d | odd | 6 | 1 | inner |
360.2.bd.b | ✓ | 128 | 40.e | odd | 2 | 1 | inner |
360.2.bd.b | ✓ | 128 | 45.h | odd | 6 | 1 | inner |
360.2.bd.b | ✓ | 128 | 72.l | even | 6 | 1 | inner |
360.2.bd.b | ✓ | 128 | 360.bd | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{64} + 125 T_{7}^{62} + 8745 T_{7}^{60} + 420748 T_{7}^{58} + 15371411 T_{7}^{56} + \cdots + 850305600 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).