Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [920,2,Mod(323,920)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(920, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("920.323");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 920 = 2^{3} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 920.v (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: | \(7.34623698596\) |
Analytic rank: | \(0\) |
Dimension: | \(264\) |
Relative dimension: | \(132\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
323.1 | −1.41404 | − | 0.0224370i | −1.94106 | − | 1.94106i | 1.99899 | + | 0.0634534i | 1.41681 | + | 1.72992i | 2.70118 | + | 2.78828i | −1.58571 | − | 1.58571i | −2.82522 | − | 0.134577i | 4.53543i | −1.96461 | − | 2.47796i | ||
323.2 | −1.41346 | − | 0.0460389i | −0.830909 | − | 0.830909i | 1.99576 | + | 0.130149i | −2.06164 | − | 0.865819i | 1.13621 | + | 1.21271i | −3.35989 | − | 3.35989i | −2.81494 | − | 0.275843i | − | 1.61918i | 2.87419 | + | 1.31872i | |
323.3 | −1.41336 | − | 0.0490551i | 1.66418 | + | 1.66418i | 1.99519 | + | 0.138665i | 0.502259 | + | 2.17893i | −2.27046 | − | 2.43373i | 2.07765 | + | 2.07765i | −2.81312 | − | 0.293859i | 2.53901i | −0.602987 | − | 3.10426i | ||
323.4 | −1.40971 | − | 0.112742i | 2.42474 | + | 2.42474i | 1.97458 | + | 0.317866i | −1.74367 | + | 1.39986i | −3.14481 | − | 3.69155i | −2.58602 | − | 2.58602i | −2.74775 | − | 0.670717i | 8.75870i | 2.61590 | − | 1.77682i | ||
323.5 | −1.40871 | − | 0.124592i | 1.11995 | + | 1.11995i | 1.96895 | + | 0.351030i | 2.04565 | − | 0.902959i | −1.43816 | − | 1.71723i | 2.08392 | + | 2.08392i | −2.72996 | − | 0.739818i | − | 0.491408i | −2.99423 | + | 1.01714i | |
323.6 | −1.40681 | − | 0.144563i | 1.37916 | + | 1.37916i | 1.95820 | + | 0.406744i | −0.158443 | − | 2.23045i | −1.74084 | − | 2.13959i | −0.711602 | − | 0.711602i | −2.69601 | − | 0.855293i | 0.804167i | −0.0995414 | + | 3.16071i | ||
323.7 | −1.39818 | + | 0.212355i | −1.45207 | − | 1.45207i | 1.90981 | − | 0.593822i | 2.18123 | − | 0.492171i | 2.33860 | + | 1.72190i | 2.01979 | + | 2.01979i | −2.54416 | + | 1.23583i | 1.21700i | −2.94524 | + | 1.15134i | ||
323.8 | −1.39655 | − | 0.222850i | −1.16052 | − | 1.16052i | 1.90068 | + | 0.622439i | −0.00501944 | − | 2.23606i | 1.36209 | + | 1.87933i | 0.119412 | + | 0.119412i | −2.51567 | − | 1.29283i | − | 0.306410i | −0.491296 | + | 3.12388i | |
323.9 | −1.38863 | + | 0.267761i | −0.700567 | − | 0.700567i | 1.85661 | − | 0.743644i | −0.735317 | − | 2.11171i | 1.16042 | + | 0.785246i | 3.06000 | + | 3.06000i | −2.37903 | + | 1.52978i | − | 2.01841i | 1.58652 | + | 2.73550i | |
323.10 | −1.37226 | − | 0.341893i | −0.562073 | − | 0.562073i | 1.76622 | + | 0.938334i | −0.819638 | + | 2.08043i | 0.579144 | + | 0.963482i | 0.0338316 | + | 0.0338316i | −2.10291 | − | 1.89150i | − | 2.36815i | 1.83604 | − | 2.57467i | |
323.11 | −1.36777 | + | 0.359447i | 0.488759 | + | 0.488759i | 1.74160 | − | 0.983282i | 1.73112 | − | 1.41535i | −0.844193 | − | 0.492828i | −2.86140 | − | 2.86140i | −2.02867 | + | 1.97092i | − | 2.52223i | −1.85903 | + | 2.55812i | |
323.12 | −1.36421 | + | 0.372744i | −2.34634 | − | 2.34634i | 1.72212 | − | 1.01700i | −2.20858 | + | 0.349547i | 4.07547 | + | 2.32631i | 1.42785 | + | 1.42785i | −1.97025 | + | 2.02931i | 8.01059i | 2.88267 | − | 1.30009i | ||
323.13 | −1.36028 | − | 0.386815i | 0.210032 | + | 0.210032i | 1.70075 | + | 1.05236i | −2.13300 | + | 0.671043i | −0.204460 | − | 0.366946i | 1.84573 | + | 1.84573i | −1.90644 | − | 2.08938i | − | 2.91177i | 3.16106 | − | 0.0877327i | |
323.14 | −1.33716 | + | 0.460449i | −1.13667 | − | 1.13667i | 1.57597 | − | 1.23139i | −1.32806 | + | 1.79896i | 2.04328 | + | 0.996526i | 0.174949 | + | 0.174949i | −1.54033 | + | 2.37221i | − | 0.415965i | 0.947486 | − | 3.01700i | |
323.15 | −1.29776 | − | 0.561986i | 1.12321 | + | 1.12321i | 1.36834 | + | 1.45864i | −1.82601 | − | 1.29062i | −0.826426 | − | 2.08888i | −1.78171 | − | 1.78171i | −0.956042 | − | 2.66195i | − | 0.476796i | 1.64440 | + | 2.70110i | |
323.16 | −1.28220 | − | 0.596623i | −0.209415 | − | 0.209415i | 1.28808 | + | 1.52998i | 2.22712 | − | 0.199831i | 0.143571 | + | 0.393455i | 0.716662 | + | 0.716662i | −0.738761 | − | 2.73024i | − | 2.91229i | −2.97484 | − | 1.07253i | |
323.17 | −1.27843 | + | 0.604658i | 0.523906 | + | 0.523906i | 1.26878 | − | 1.54603i | −1.86638 | − | 1.23152i | −0.986562 | − | 0.352994i | 1.76116 | + | 1.76116i | −0.687226 | + | 2.74367i | − | 2.45105i | 3.13068 | + | 0.445889i | |
323.18 | −1.27201 | + | 0.618053i | 2.18353 | + | 2.18353i | 1.23602 | − | 1.57234i | 2.23354 | + | 0.106286i | −4.12701 | − | 1.42794i | −0.781712 | − | 0.781712i | −0.600445 | + | 2.76396i | 6.53562i | −2.90678 | + | 1.24525i | ||
323.19 | −1.26157 | − | 0.639090i | −1.73633 | − | 1.73633i | 1.18313 | + | 1.61252i | 1.09980 | + | 1.94690i | 1.08083 | + | 3.30017i | 3.40058 | + | 3.40058i | −0.462060 | − | 2.79043i | 3.02966i | −0.143237 | − | 3.15903i | ||
323.20 | −1.25601 | + | 0.649962i | 1.82618 | + | 1.82618i | 1.15510 | − | 1.63271i | −1.45456 | − | 1.69831i | −3.48065 | − | 1.10675i | −0.606946 | − | 0.606946i | −0.389612 | + | 2.80146i | 3.66990i | 2.93077 | + | 1.18768i | ||
See next 80 embeddings (of 264 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
8.d | odd | 2 | 1 | inner |
40.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 920.2.v.a | ✓ | 264 |
5.c | odd | 4 | 1 | inner | 920.2.v.a | ✓ | 264 |
8.d | odd | 2 | 1 | inner | 920.2.v.a | ✓ | 264 |
40.k | even | 4 | 1 | inner | 920.2.v.a | ✓ | 264 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
920.2.v.a | ✓ | 264 | 1.a | even | 1 | 1 | trivial |
920.2.v.a | ✓ | 264 | 5.c | odd | 4 | 1 | inner |
920.2.v.a | ✓ | 264 | 8.d | odd | 2 | 1 | inner |
920.2.v.a | ✓ | 264 | 40.k | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(920, [\chi])\).