Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,2,Mod(31,230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 230.g (of order \(11\), degree \(10\), 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: | \(1.83655924649\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −0.142315 | + | 0.989821i | −0.844960 | − | 1.85020i | −0.959493 | − | 0.281733i | 0.654861 | + | 0.755750i | 1.95162 | − | 0.573048i | 1.31877 | + | 0.847525i | 0.415415 | − | 0.909632i | −0.744718 | + | 0.859450i | −0.841254 | + | 0.540641i |
31.2 | −0.142315 | + | 0.989821i | −0.0488649 | − | 0.106999i | −0.959493 | − | 0.281733i | 0.654861 | + | 0.755750i | 0.112864 | − | 0.0331400i | 1.43095 | + | 0.919619i | 0.415415 | − | 0.909632i | 1.95552 | − | 2.25679i | −0.841254 | + | 0.540641i |
31.3 | −0.142315 | + | 0.989821i | 1.30924 | + | 2.86684i | −0.959493 | − | 0.281733i | 0.654861 | + | 0.755750i | −3.02398 | + | 0.887921i | −0.493060 | − | 0.316870i | 0.415415 | − | 0.909632i | −4.54006 | + | 5.23951i | −0.841254 | + | 0.540641i |
41.1 | −0.959493 | − | 0.281733i | −1.80420 | + | 2.08215i | 0.841254 | + | 0.540641i | 0.142315 | − | 0.989821i | 2.31772 | − | 1.48951i | 1.84799 | + | 4.04654i | −0.654861 | − | 0.755750i | −0.653295 | − | 4.54377i | −0.415415 | + | 0.909632i |
41.2 | −0.959493 | − | 0.281733i | −0.858654 | + | 0.990939i | 0.841254 | + | 0.540641i | 0.142315 | − | 0.989821i | 1.10305 | − | 0.708888i | −1.50220 | − | 3.28936i | −0.654861 | − | 0.755750i | 0.182270 | + | 1.26772i | −0.415415 | + | 0.909632i |
41.3 | −0.959493 | − | 0.281733i | 2.00799 | − | 2.31734i | 0.841254 | + | 0.540641i | 0.142315 | − | 0.989821i | −2.57952 | + | 1.65776i | 0.414762 | + | 0.908202i | −0.654861 | − | 0.755750i | −0.911115 | − | 6.33694i | −0.415415 | + | 0.909632i |
71.1 | 0.841254 | + | 0.540641i | −0.369448 | − | 2.56957i | 0.415415 | + | 0.909632i | 0.959493 | + | 0.281733i | 1.07841 | − | 2.36139i | 1.04688 | − | 1.20816i | −0.142315 | + | 0.989821i | −3.58770 | + | 1.05344i | 0.654861 | + | 0.755750i |
71.2 | 0.841254 | + | 0.540641i | 0.00806642 | + | 0.0561032i | 0.415415 | + | 0.909632i | 0.959493 | + | 0.281733i | −0.0235458 | + | 0.0515580i | −3.27862 | + | 3.78373i | −0.142315 | + | 0.989821i | 2.87540 | − | 0.844293i | 0.654861 | + | 0.755750i |
71.3 | 0.841254 | + | 0.540641i | 0.219067 | + | 1.52364i | 0.415415 | + | 0.909632i | 0.959493 | + | 0.281733i | −0.639452 | + | 1.40020i | 2.43457 | − | 2.80964i | −0.142315 | + | 0.989821i | 0.604987 | − | 0.177640i | 0.654861 | + | 0.755750i |
81.1 | 0.841254 | − | 0.540641i | −0.369448 | + | 2.56957i | 0.415415 | − | 0.909632i | 0.959493 | − | 0.281733i | 1.07841 | + | 2.36139i | 1.04688 | + | 1.20816i | −0.142315 | − | 0.989821i | −3.58770 | − | 1.05344i | 0.654861 | − | 0.755750i |
81.2 | 0.841254 | − | 0.540641i | 0.00806642 | − | 0.0561032i | 0.415415 | − | 0.909632i | 0.959493 | − | 0.281733i | −0.0235458 | − | 0.0515580i | −3.27862 | − | 3.78373i | −0.142315 | − | 0.989821i | 2.87540 | + | 0.844293i | 0.654861 | − | 0.755750i |
81.3 | 0.841254 | − | 0.540641i | 0.219067 | − | 1.52364i | 0.415415 | − | 0.909632i | 0.959493 | − | 0.281733i | −0.639452 | − | 1.40020i | 2.43457 | + | 2.80964i | −0.142315 | − | 0.989821i | 0.604987 | + | 0.177640i | 0.654861 | − | 0.755750i |
101.1 | −0.959493 | + | 0.281733i | −1.80420 | − | 2.08215i | 0.841254 | − | 0.540641i | 0.142315 | + | 0.989821i | 2.31772 | + | 1.48951i | 1.84799 | − | 4.04654i | −0.654861 | + | 0.755750i | −0.653295 | + | 4.54377i | −0.415415 | − | 0.909632i |
101.2 | −0.959493 | + | 0.281733i | −0.858654 | − | 0.990939i | 0.841254 | − | 0.540641i | 0.142315 | + | 0.989821i | 1.10305 | + | 0.708888i | −1.50220 | + | 3.28936i | −0.654861 | + | 0.755750i | 0.182270 | − | 1.26772i | −0.415415 | − | 0.909632i |
101.3 | −0.959493 | + | 0.281733i | 2.00799 | + | 2.31734i | 0.841254 | − | 0.540641i | 0.142315 | + | 0.989821i | −2.57952 | − | 1.65776i | 0.414762 | − | 0.908202i | −0.654861 | + | 0.755750i | −0.911115 | + | 6.33694i | −0.415415 | − | 0.909632i |
121.1 | 0.415415 | − | 0.909632i | −3.22429 | − | 0.946737i | −0.654861 | − | 0.755750i | −0.841254 | + | 0.540641i | −2.20060 | + | 2.53963i | −0.324589 | + | 2.25757i | −0.959493 | + | 0.281733i | 6.97598 | + | 4.48319i | 0.142315 | + | 0.989821i |
121.2 | 0.415415 | − | 0.909632i | −0.732525 | − | 0.215089i | −0.654861 | − | 0.755750i | −0.841254 | + | 0.540641i | −0.499953 | + | 0.576977i | 0.575516 | − | 4.00280i | −0.959493 | + | 0.281733i | −2.03343 | − | 1.30681i | 0.142315 | + | 0.989821i |
121.3 | 0.415415 | − | 0.909632i | 2.99732 | + | 0.880093i | −0.654861 | − | 0.755750i | −0.841254 | + | 0.540641i | 2.04569 | − | 2.36086i | −0.352734 | + | 2.45332i | −0.959493 | + | 0.281733i | 5.68562 | + | 3.65393i | 0.142315 | + | 0.989821i |
131.1 | −0.654861 | + | 0.755750i | −0.925739 | + | 0.594936i | −0.142315 | − | 0.989821i | −0.415415 | − | 0.909632i | 0.156607 | − | 1.08923i | 3.86758 | − | 1.13562i | 0.841254 | + | 0.540641i | −0.743202 | + | 1.62739i | 0.959493 | + | 0.281733i |
131.2 | −0.654861 | + | 0.755750i | −0.578669 | + | 0.371888i | −0.142315 | − | 0.989821i | −0.415415 | − | 0.909632i | 0.0978934 | − | 0.680863i | −2.88525 | + | 0.847187i | 0.841254 | + | 0.540641i | −1.04969 | + | 2.29850i | 0.959493 | + | 0.281733i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 230.2.g.d | ✓ | 30 |
23.c | even | 11 | 1 | inner | 230.2.g.d | ✓ | 30 |
23.c | even | 11 | 1 | 5290.2.a.bk | 15 | ||
23.d | odd | 22 | 1 | 5290.2.a.bl | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.2.g.d | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
230.2.g.d | ✓ | 30 | 23.c | even | 11 | 1 | inner |
5290.2.a.bk | 15 | 23.c | even | 11 | 1 | ||
5290.2.a.bl | 15 | 23.d | odd | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} + T_{3}^{29} - T_{3}^{28} + 18 T_{3}^{27} + 52 T_{3}^{26} + 39 T_{3}^{25} + 16 T_{3}^{24} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(230, [\chi])\).