Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [552,2,Mod(13,552)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(552, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 11, 0, 14]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("552.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 552 = 2^{3} \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 552.bb (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: | \(4.40774219157\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −1.40928 | − | 0.118034i | 0.540641 | − | 0.841254i | 1.97214 | + | 0.332685i | −3.96495 | − | 1.81073i | −0.861210 | + | 1.12175i | −1.61967 | − | 0.475578i | −2.74002 | − | 0.701625i | −0.415415 | − | 0.909632i | 5.37400 | + | 3.01983i |
13.2 | −1.39148 | − | 0.252579i | −0.540641 | + | 0.841254i | 1.87241 | + | 0.702914i | −2.13541 | − | 0.975208i | 0.964771 | − | 1.03403i | 0.595694 | + | 0.174912i | −2.42787 | − | 1.45102i | −0.415415 | − | 0.909632i | 2.72505 | + | 1.89634i |
13.3 | −1.37613 | − | 0.325998i | −0.540641 | + | 0.841254i | 1.78745 | + | 0.897229i | 0.902350 | + | 0.412090i | 1.01824 | − | 0.981424i | −3.01070 | − | 0.884022i | −2.16726 | − | 1.81741i | −0.415415 | − | 0.909632i | −1.10741 | − | 0.861252i |
13.4 | −1.37160 | + | 0.344553i | 0.540641 | − | 0.841254i | 1.76257 | − | 0.945176i | 2.26933 | + | 1.03637i | −0.451686 | + | 1.34014i | −2.49497 | − | 0.732589i | −2.09187 | + | 1.90370i | −0.415415 | − | 0.909632i | −3.46970 | − | 0.639578i |
13.5 | −1.36486 | + | 0.370358i | 0.540641 | − | 0.841254i | 1.72567 | − | 1.01097i | 0.808939 | + | 0.369430i | −0.426333 | + | 1.34842i | 0.759802 | + | 0.223098i | −1.98087 | + | 2.01895i | −0.415415 | − | 0.909632i | −1.24091 | − | 0.204623i |
13.6 | −1.32689 | − | 0.489239i | 0.540641 | − | 0.841254i | 1.52129 | + | 1.29834i | 3.08337 | + | 1.40813i | −1.12895 | + | 0.851751i | 2.20998 | + | 0.648908i | −1.38339 | − | 2.46703i | −0.415415 | − | 0.909632i | −3.40240 | − | 3.37694i |
13.7 | −1.30419 | − | 0.546886i | −0.540641 | + | 0.841254i | 1.40183 | + | 1.42649i | −0.190010 | − | 0.0867748i | 1.16517 | − | 0.801487i | 2.41838 | + | 0.710100i | −1.04813 | − | 2.62706i | −0.415415 | − | 0.909632i | 0.200354 | + | 0.217085i |
13.8 | −1.28649 | + | 0.587327i | −0.540641 | + | 0.841254i | 1.31009 | − | 1.51118i | 2.37560 | + | 1.08490i | 0.201436 | − | 1.39979i | 1.76843 | + | 0.519257i | −0.797864 | + | 2.71356i | −0.415415 | − | 0.909632i | −3.69337 | 0.000455346i | |
13.9 | −1.23417 | − | 0.690530i | 0.540641 | − | 0.841254i | 1.04634 | + | 1.70446i | 0.282332 | + | 0.128937i | −1.24815 | + | 0.664919i | −2.03550 | − | 0.597678i | −0.114374 | − | 2.82611i | −0.415415 | − | 0.909632i | −0.259410 | − | 0.354088i |
13.10 | −1.22386 | + | 0.708629i | −0.540641 | + | 0.841254i | 0.995689 | − | 1.73453i | −2.18961 | − | 0.999963i | 0.0655344 | − | 1.41269i | 2.12232 | + | 0.623169i | 0.0105513 | + | 2.82841i | −0.415415 | − | 0.909632i | 3.38839 | − | 0.327805i |
13.11 | −1.11564 | + | 0.869107i | 0.540641 | − | 0.841254i | 0.489306 | − | 1.93922i | −0.756543 | − | 0.345502i | 0.127979 | + | 1.40841i | −3.66239 | − | 1.07537i | 1.13950 | + | 2.58873i | −0.415415 | − | 0.909632i | 1.14431 | − | 0.272061i |
13.12 | −1.08248 | + | 0.910071i | 0.540641 | − | 0.841254i | 0.343541 | − | 1.97027i | −2.04961 | − | 0.936025i | 0.180366 | + | 1.40266i | 3.54305 | + | 1.04033i | 1.42121 | + | 2.44544i | −0.415415 | − | 0.909632i | 3.07052 | − | 0.852058i |
13.13 | −1.02595 | − | 0.973358i | −0.540641 | + | 0.841254i | 0.105149 | + | 1.99723i | 2.63955 | + | 1.20544i | 1.37351 | − | 0.336848i | 3.10238 | + | 0.910940i | 1.83615 | − | 2.15141i | −0.415415 | − | 0.909632i | −1.53472 | − | 3.80595i |
13.14 | −0.832989 | + | 1.14286i | −0.540641 | + | 0.841254i | −0.612258 | − | 1.90398i | 2.21313 | + | 1.01070i | −0.511087 | − | 1.31863i | −3.93606 | − | 1.15573i | 2.68599 | + | 0.886269i | −0.415415 | − | 0.909632i | −2.99860 | + | 1.68739i |
13.15 | −0.817442 | − | 1.15403i | 0.540641 | − | 0.841254i | −0.663576 | + | 1.88671i | −2.63955 | − | 1.20544i | −1.41278 | + | 0.0637600i | 3.10238 | + | 0.910940i | 2.71975 | − | 0.776488i | −0.415415 | − | 0.909632i | 0.766563 | + | 4.03151i |
13.16 | −0.731481 | + | 1.21035i | 0.540641 | − | 0.841254i | −0.929871 | − | 1.77069i | 3.52460 | + | 1.60963i | 0.622739 | + | 1.26972i | 4.53885 | + | 1.33273i | 2.82333 | + | 0.169761i | −0.415415 | − | 0.909632i | −4.52639 | + | 3.08857i |
13.17 | −0.682311 | + | 1.23873i | −0.540641 | + | 0.841254i | −1.06890 | − | 1.69040i | −3.63898 | − | 1.66187i | −0.673201 | − | 1.24370i | −2.20249 | − | 0.646708i | 2.82327 | − | 0.170707i | −0.415415 | − | 0.909632i | 4.54152 | − | 3.37381i |
13.18 | −0.507861 | − | 1.31988i | −0.540641 | + | 0.841254i | −1.48415 | + | 1.34063i | −0.282332 | − | 0.128937i | 1.38492 | + | 0.286340i | −2.03550 | − | 0.597678i | 2.52321 | + | 1.27805i | −0.415415 | − | 0.909632i | −0.0267952 | + | 0.438126i |
13.19 | −0.464427 | + | 1.33578i | −0.540641 | + | 0.841254i | −1.56862 | − | 1.24074i | 0.798595 | + | 0.364706i | −0.872641 | − | 1.11288i | 2.23854 | + | 0.657294i | 2.38587 | − | 1.51909i | −0.415415 | − | 0.909632i | −0.858056 | + | 0.897368i |
13.20 | −0.355714 | − | 1.36875i | 0.540641 | − | 0.841254i | −1.74694 | + | 0.973765i | 0.190010 | + | 0.0867748i | −1.34378 | − | 0.440755i | 2.41838 | + | 0.710100i | 1.95425 | + | 2.04473i | −0.415415 | − | 0.909632i | 0.0511834 | − | 0.290943i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
23.c | even | 11 | 1 | inner |
184.p | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 552.2.bb.a | ✓ | 480 |
8.b | even | 2 | 1 | inner | 552.2.bb.a | ✓ | 480 |
23.c | even | 11 | 1 | inner | 552.2.bb.a | ✓ | 480 |
184.p | even | 22 | 1 | inner | 552.2.bb.a | ✓ | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
552.2.bb.a | ✓ | 480 | 1.a | even | 1 | 1 | trivial |
552.2.bb.a | ✓ | 480 | 8.b | even | 2 | 1 | inner |
552.2.bb.a | ✓ | 480 | 23.c | even | 11 | 1 | inner |
552.2.bb.a | ✓ | 480 | 184.p | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(552, [\chi])\).