Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [460,2,Mod(11,460)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 0, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("460.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 460.q (of order \(22\), 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: | \(3.67311849298\) |
Analytic rank: | \(0\) |
Dimension: | \(480\) |
Relative dimension: | \(48\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −1.41401 | − | 0.0241273i | 0.0682998 | + | 0.00982003i | 1.99884 | + | 0.0682323i | 0.281733 | + | 0.959493i | −0.0963395 | − | 0.0155335i | 0.126885 | + | 0.146433i | −2.82472 | − | 0.144707i | −2.87391 | − | 0.843856i | −0.375222 | − | 1.36353i |
11.2 | −1.41004 | + | 0.108605i | −2.31335 | − | 0.332609i | 1.97641 | − | 0.306274i | −0.281733 | − | 0.959493i | 3.29803 | + | 0.217751i | −1.52622 | − | 1.76136i | −2.75355 | + | 0.646505i | 2.36248 | + | 0.693685i | 0.501459 | + | 1.32232i |
11.3 | −1.39626 | − | 0.224613i | 1.77069 | + | 0.254587i | 1.89910 | + | 0.627237i | 0.281733 | + | 0.959493i | −2.41517 | − | 0.753191i | −0.375831 | − | 0.433732i | −2.51075 | − | 1.30235i | 0.192056 | + | 0.0563926i | −0.177858 | − | 1.40298i |
11.4 | −1.31872 | − | 0.510869i | 0.994266 | + | 0.142954i | 1.47803 | + | 1.34738i | −0.281733 | − | 0.959493i | −1.23812 | − | 0.696455i | −3.13881 | − | 3.62238i | −1.26076 | − | 2.53189i | −1.91035 | − | 0.560929i | −0.118649 | + | 1.40923i |
11.5 | −1.30815 | − | 0.537340i | −0.501133 | − | 0.0720521i | 1.42253 | + | 1.40585i | −0.281733 | − | 0.959493i | 0.616843 | + | 0.363534i | 1.51820 | + | 1.75210i | −1.10547 | − | 2.60345i | −2.63254 | − | 0.772982i | −0.147024 | + | 1.40655i |
11.6 | −1.27312 | + | 0.615774i | −0.608389 | − | 0.0874731i | 1.24165 | − | 1.56790i | −0.281733 | − | 0.959493i | 0.828413 | − | 0.263266i | 0.613272 | + | 0.707753i | −0.615286 | + | 2.76069i | −2.51599 | − | 0.738762i | 0.949508 | + | 1.04806i |
11.7 | −1.27090 | + | 0.620337i | −1.04773 | − | 0.150641i | 1.23036 | − | 1.57677i | 0.281733 | + | 0.959493i | 1.42501 | − | 0.458497i | 3.09996 | + | 3.57755i | −0.585539 | + | 2.76715i | −1.80343 | − | 0.529535i | −0.953262 | − | 1.04465i |
11.8 | −1.19302 | + | 0.759408i | 3.06496 | + | 0.440674i | 0.846600 | − | 1.81198i | 0.281733 | + | 0.959493i | −3.99121 | + | 1.80182i | 1.05960 | + | 1.22284i | 0.366018 | + | 2.80464i | 6.32130 | + | 1.85610i | −1.06476 | − | 0.930746i |
11.9 | −1.19089 | − | 0.762752i | −2.15085 | − | 0.309246i | 0.836418 | + | 1.81670i | 0.281733 | + | 0.959493i | 2.32554 | + | 2.00885i | −2.11824 | − | 2.44458i | 0.389615 | − | 2.80146i | 1.65206 | + | 0.485089i | 0.396344 | − | 1.35754i |
11.10 | −1.18638 | + | 0.769741i | −3.06496 | − | 0.440674i | 0.814997 | − | 1.82641i | 0.281733 | + | 0.959493i | 3.97541 | − | 1.83642i | −1.05960 | − | 1.22284i | 0.438968 | + | 2.79416i | 6.32130 | + | 1.85610i | −1.07280 | − | 0.921463i |
11.11 | −1.18308 | − | 0.774803i | 3.11182 | + | 0.447412i | 0.799362 | + | 1.83331i | −0.281733 | − | 0.959493i | −3.33488 | − | 2.94037i | 1.26361 | + | 1.45829i | 0.474742 | − | 2.78830i | 6.60476 | + | 1.93933i | −0.410105 | + | 1.35345i |
11.12 | −1.09223 | + | 0.898352i | 1.04773 | + | 0.150641i | 0.385926 | − | 1.96241i | 0.281733 | + | 0.959493i | −1.27969 | + | 0.776697i | −3.09996 | − | 3.57755i | 1.34142 | + | 2.49010i | −1.80343 | − | 0.529535i | −1.16968 | − | 0.794890i |
11.13 | −1.08900 | + | 0.902265i | 0.608389 | + | 0.0874731i | 0.371836 | − | 1.96513i | −0.281733 | − | 0.959493i | −0.741459 | + | 0.453670i | −0.613272 | − | 0.707753i | 1.36814 | + | 2.47552i | −2.51599 | − | 0.738762i | 1.17252 | + | 0.790689i |
11.14 | −0.745947 | − | 1.20148i | 0.934803 | + | 0.134404i | −0.887125 | + | 1.79249i | 0.281733 | + | 0.959493i | −0.535829 | − | 1.22341i | 2.39561 | + | 2.76468i | 2.81539 | − | 0.271235i | −2.02269 | − | 0.593915i | 0.942657 | − | 1.05423i |
11.15 | −0.684541 | + | 1.23750i | 2.31335 | + | 0.332609i | −1.06281 | − | 1.69424i | −0.281733 | − | 0.959493i | −1.99519 | + | 2.63508i | 1.52622 | + | 1.76136i | 2.82415 | − | 0.155447i | 2.36248 | + | 0.693685i | 1.38023 | + | 0.308169i |
11.16 | −0.636057 | − | 1.26310i | −3.26795 | − | 0.469861i | −1.19086 | + | 1.60681i | −0.281733 | − | 0.959493i | 1.48512 | + | 4.42662i | 0.616457 | + | 0.711429i | 2.78703 | + | 0.482157i | 7.58027 | + | 2.22577i | −1.03274 | + | 0.966150i |
11.17 | −0.565453 | + | 1.29625i | −0.0682998 | − | 0.00982003i | −1.36053 | − | 1.46594i | 0.281733 | + | 0.959493i | 0.0513495 | − | 0.0829808i | −0.126885 | − | 0.146433i | 2.66953 | − | 0.934662i | −2.87391 | − | 0.843856i | −1.40305 | − | 0.177353i |
11.18 | −0.520455 | − | 1.31496i | −1.91127 | − | 0.274799i | −1.45825 | + | 1.36876i | 0.281733 | + | 0.959493i | 0.633379 | + | 2.65627i | 1.03497 | + | 1.19442i | 2.55882 | + | 1.20517i | 0.698960 | + | 0.205233i | 1.11507 | − | 0.869841i |
11.19 | −0.497467 | − | 1.32383i | 3.04838 | + | 0.438291i | −1.50505 | + | 1.31712i | 0.281733 | + | 0.959493i | −0.936246 | − | 4.25357i | −1.74188 | − | 2.01024i | 2.49236 | + | 1.33721i | 6.22203 | + | 1.82695i | 1.13005 | − | 0.850282i |
11.20 | −0.375713 | + | 1.36339i | −1.77069 | − | 0.254587i | −1.71768 | − | 1.02449i | 0.281733 | + | 0.959493i | 1.01237 | − | 2.31850i | 0.375831 | + | 0.433732i | 2.04214 | − | 1.95696i | 0.192056 | + | 0.0563926i | −1.41402 | + | 0.0236179i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
92.h | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 460.2.q.a | ✓ | 480 |
4.b | odd | 2 | 1 | inner | 460.2.q.a | ✓ | 480 |
23.d | odd | 22 | 1 | inner | 460.2.q.a | ✓ | 480 |
92.h | even | 22 | 1 | inner | 460.2.q.a | ✓ | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
460.2.q.a | ✓ | 480 | 1.a | even | 1 | 1 | trivial |
460.2.q.a | ✓ | 480 | 4.b | odd | 2 | 1 | inner |
460.2.q.a | ✓ | 480 | 23.d | odd | 22 | 1 | inner |
460.2.q.a | ✓ | 480 | 92.h | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(460, [\chi])\).