Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [572,2,Mod(463,572)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(572, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("572.463");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 572 = 2^{2} \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 572.j (of order \(4\), 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: | \(4.56744299562\) |
Analytic rank: | \(0\) |
Dimension: | \(140\) |
Relative dimension: | \(70\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
463.1 | −1.40964 | − | 0.113659i | − | 0.946143i | 1.97416 | + | 0.320437i | −1.09559 | − | 1.09559i | −0.107538 | + | 1.33372i | 0.184005 | + | 0.184005i | −2.74644 | − | 0.676083i | 2.10481 | 1.41986 | + | 1.66891i | |||
463.2 | −1.40874 | − | 0.124340i | − | 2.77989i | 1.96908 | + | 0.350325i | −0.0767607 | − | 0.0767607i | −0.345652 | + | 3.91613i | 2.31083 | + | 2.31083i | −2.73035 | − | 0.738352i | −4.72779 | 0.0985912 | + | 0.117680i | |||
463.3 | −1.39906 | + | 0.206499i | 2.87009i | 1.91472 | − | 0.577808i | 0.956348 | + | 0.956348i | −0.592670 | − | 4.01541i | 3.01968 | + | 3.01968i | −2.55948 | + | 1.20377i | −5.23740 | −1.53547 | − | 1.14050i | ||||
463.4 | −1.38865 | + | 0.267664i | − | 2.20360i | 1.85671 | − | 0.743385i | −2.49116 | − | 2.49116i | 0.589826 | + | 3.06004i | −2.07384 | − | 2.07384i | −2.37935 | + | 1.52928i | −1.85587 | 4.12615 | + | 2.79256i | |||
463.5 | −1.37854 | + | 0.315622i | − | 0.826677i | 1.80077 | − | 0.870198i | 1.70772 | + | 1.70772i | 0.260917 | + | 1.13961i | 0.257567 | + | 0.257567i | −2.20778 | + | 1.76797i | 2.31661 | −2.89317 | − | 1.81518i | |||
463.6 | −1.35126 | − | 0.417264i | 0.449813i | 1.65178 | + | 1.12766i | −1.24390 | − | 1.24390i | 0.187691 | − | 0.607812i | −2.23254 | − | 2.23254i | −1.76145 | − | 2.21299i | 2.79767 | 1.16179 | + | 2.19986i | ||||
463.7 | −1.34707 | − | 0.430589i | − | 1.83416i | 1.62919 | + | 1.16007i | 3.05786 | + | 3.05786i | −0.789770 | + | 2.47074i | 0.449413 | + | 0.449413i | −1.69511 | − | 2.26420i | −0.364149 | −2.80247 | − | 5.43583i | |||
463.8 | −1.34567 | + | 0.434926i | 0.845393i | 1.62168 | − | 1.17054i | 1.03191 | + | 1.03191i | −0.367683 | − | 1.13762i | −2.79241 | − | 2.79241i | −1.67316 | + | 2.28047i | 2.28531 | −1.83742 | − | 0.939812i | ||||
463.9 | −1.31145 | + | 0.529251i | 2.04664i | 1.43979 | − | 1.38817i | −1.50803 | − | 1.50803i | −1.08318 | − | 2.68406i | 0.828927 | + | 0.828927i | −1.15351 | + | 2.58252i | −1.18872 | 2.77583 | + | 1.17957i | ||||
463.10 | −1.28898 | − | 0.581825i | 3.13637i | 1.32296 | + | 1.49993i | 2.34094 | + | 2.34094i | 1.82482 | − | 4.04273i | −2.63806 | − | 2.63806i | −0.832580 | − | 2.70311i | −6.83681 | −1.65542 | − | 4.37946i | ||||
463.11 | −1.22033 | − | 0.714703i | − | 0.940195i | 0.978400 | + | 1.74434i | −2.27069 | − | 2.27069i | −0.671960 | + | 1.14735i | 2.86451 | + | 2.86451i | 0.0527177 | − | 2.82794i | 2.11603 | 1.14812 | + | 4.39385i | |||
463.12 | −1.19924 | − | 0.749544i | 2.85994i | 0.876368 | + | 1.79777i | −2.69418 | − | 2.69418i | 2.14365 | − | 3.42977i | 1.37814 | + | 1.37814i | 0.296529 | − | 2.81284i | −5.17927 | 1.21157 | + | 5.25038i | ||||
463.13 | −1.16815 | + | 0.797142i | − | 3.05966i | 0.729128 | − | 1.86236i | 2.32382 | + | 2.32382i | 2.43899 | + | 3.57413i | −2.18069 | − | 2.18069i | 0.632835 | + | 2.75672i | −6.36154 | −4.56698 | − | 0.862145i | |||
463.14 | −1.11823 | − | 0.865774i | − | 0.913063i | 0.500870 | + | 1.93627i | 1.64493 | + | 1.64493i | −0.790506 | + | 1.02101i | −3.07399 | − | 3.07399i | 1.11628 | − | 2.59883i | 2.16632 | −0.415270 | − | 3.26355i | |||
463.15 | −1.11565 | + | 0.869100i | − | 0.202427i | 0.489330 | − | 1.93922i | −1.00276 | − | 1.00276i | 0.175929 | + | 0.225836i | 0.926254 | + | 0.926254i | 1.13945 | + | 2.58875i | 2.95902 | 1.99022 | + | 0.247225i | |||
463.16 | −1.09876 | + | 0.890354i | 3.11188i | 0.414539 | − | 1.95657i | −0.115988 | − | 0.115988i | −2.77067 | − | 3.41920i | −2.35663 | − | 2.35663i | 1.28656 | + | 2.51888i | −6.68380 | 0.230714 | + | 0.0241724i | ||||
463.17 | −1.03582 | + | 0.962851i | 0.112664i | 0.145835 | − | 1.99468i | 2.00381 | + | 2.00381i | −0.108479 | − | 0.116699i | 3.01400 | + | 3.01400i | 1.76952 | + | 2.20654i | 2.98731 | −4.00495 | − | 0.146211i | ||||
463.18 | −1.01828 | − | 0.981374i | 1.60363i | 0.0738084 | + | 1.99864i | 0.228079 | + | 0.228079i | 1.57376 | − | 1.63295i | 0.445735 | + | 0.445735i | 1.88625 | − | 2.10762i | 0.428375 | −0.00841849 | − | 0.456080i | ||||
463.19 | −0.981374 | − | 1.01828i | − | 1.60363i | −0.0738084 | + | 1.99864i | 0.228079 | + | 0.228079i | −1.63295 | + | 1.57376i | −0.445735 | − | 0.445735i | 2.10762 | − | 1.88625i | 0.428375 | 0.00841849 | − | 0.456080i | |||
463.20 | −0.865774 | − | 1.11823i | 0.913063i | −0.500870 | + | 1.93627i | 1.64493 | + | 1.64493i | 1.02101 | − | 0.790506i | 3.07399 | + | 3.07399i | 2.59883 | − | 1.11628i | 2.16632 | 0.415270 | − | 3.26355i | ||||
See next 80 embeddings (of 140 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
13.d | odd | 4 | 1 | inner |
52.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 572.2.j.a | ✓ | 140 |
4.b | odd | 2 | 1 | inner | 572.2.j.a | ✓ | 140 |
13.d | odd | 4 | 1 | inner | 572.2.j.a | ✓ | 140 |
52.f | even | 4 | 1 | inner | 572.2.j.a | ✓ | 140 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
572.2.j.a | ✓ | 140 | 1.a | even | 1 | 1 | trivial |
572.2.j.a | ✓ | 140 | 4.b | odd | 2 | 1 | inner |
572.2.j.a | ✓ | 140 | 13.d | odd | 4 | 1 | inner |
572.2.j.a | ✓ | 140 | 52.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(572, [\chi])\).