Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [667,2,Mod(144,667)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(667, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([18, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("667.144");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 667 = 23 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 667.m (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: | \(5.32602181482\) |
Analytic rank: | \(0\) |
Dimension: | \(580\) |
Relative dimension: | \(58\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
144.1 | −2.54580 | − | 1.16263i | 0.749755 | − | 2.55343i | 3.81966 | + | 4.40812i | 1.79747 | − | 1.15516i | −4.87741 | + | 5.62883i | 0.569562 | − | 3.96139i | −3.02210 | − | 10.2923i | −3.43412 | − | 2.20697i | −5.91902 | + | 0.851026i |
144.2 | −2.51111 | − | 1.14678i | −0.217225 | + | 0.739800i | 3.68082 | + | 4.24789i | −3.19853 | + | 2.05557i | 1.39387 | − | 1.60861i | 0.0962455 | − | 0.669402i | −2.81603 | − | 9.59051i | 2.02364 | + | 1.30052i | 10.3891 | − | 1.49373i |
144.3 | −2.48132 | − | 1.13318i | −0.842284 | + | 2.86855i | 3.56314 | + | 4.11209i | 0.774486 | − | 0.497732i | 5.34057 | − | 6.16335i | −0.176461 | + | 1.22731i | −2.64453 | − | 9.00644i | −4.99540 | − | 3.21035i | −2.48577 | + | 0.357400i |
144.4 | −2.18166 | − | 0.996331i | 0.556006 | − | 1.89358i | 2.45725 | + | 2.83582i | −0.910707 | + | 0.585276i | −3.09965 | + | 3.57719i | −0.300085 | + | 2.08714i | −1.18406 | − | 4.03253i | −0.752750 | − | 0.483763i | 2.56998 | − | 0.369508i |
144.5 | −2.17561 | − | 0.993569i | 0.00167375 | − | 0.00570027i | 2.43639 | + | 2.81174i | −0.262814 | + | 0.168900i | −0.00930505 | + | 0.0107386i | 0.234084 | − | 1.62809i | −1.15931 | − | 3.94824i | 2.52373 | + | 1.62190i | 0.739596 | − | 0.106338i |
144.6 | −2.07058 | − | 0.945601i | 0.343410 | − | 1.16955i | 2.08341 | + | 2.40438i | 2.22541 | − | 1.43018i | −1.81698 | + | 2.09691i | −0.319932 | + | 2.22518i | −0.757665 | − | 2.58037i | 1.27385 | + | 0.818656i | −5.96025 | + | 0.856955i |
144.7 | −2.06913 | − | 0.944942i | −0.381900 | + | 1.30063i | 2.07868 | + | 2.39892i | 1.42266 | − | 0.914288i | 2.01922 | − | 2.33031i | 0.459303 | − | 3.19452i | −0.752512 | − | 2.56282i | 0.977964 | + | 0.628499i | −3.80762 | + | 0.547453i |
144.8 | −2.04490 | − | 0.933876i | 0.949127 | − | 3.23243i | 1.99978 | + | 2.30787i | −1.38377 | + | 0.889294i | −4.95956 | + | 5.72364i | −0.177723 | + | 1.23609i | −0.667394 | − | 2.27294i | −7.02400 | − | 4.51405i | 3.66016 | − | 0.526252i |
144.9 | −1.99987 | − | 0.913310i | −0.304515 | + | 1.03708i | 1.85562 | + | 2.14150i | −1.61721 | + | 1.03932i | 1.55617 | − | 1.79591i | −0.658031 | + | 4.57671i | −0.516342 | − | 1.75850i | 1.54095 | + | 0.990307i | 4.18342 | − | 0.601485i |
144.10 | −1.89057 | − | 0.863396i | 0.546078 | − | 1.85977i | 1.51910 | + | 1.75313i | −3.48014 | + | 2.23655i | −2.63812 | + | 3.04456i | 0.478945 | − | 3.33114i | −0.187216 | − | 0.637599i | −0.636791 | − | 0.409241i | 8.51049 | − | 1.22362i |
144.11 | −1.80723 | − | 0.825333i | −0.788540 | + | 2.68552i | 1.27518 | + | 1.47163i | −0.498625 | + | 0.320447i | 3.64152 | − | 4.20254i | −0.0489859 | + | 0.340704i | 0.0295275 | + | 0.100561i | −4.06647 | − | 2.61336i | 1.16561 | − | 0.167589i |
144.12 | −1.66220 | − | 0.759101i | 0.672795 | − | 2.29133i | 0.876951 | + | 1.01205i | 2.47820 | − | 1.59264i | −2.85767 | + | 3.29793i | −0.417854 | + | 2.90624i | 0.340223 | + | 1.15869i | −2.27378 | − | 1.46127i | −5.32824 | + | 0.766085i |
144.13 | −1.54248 | − | 0.704427i | −0.706327 | + | 2.40553i | 0.573306 | + | 0.661630i | 2.50374 | − | 1.60906i | 2.78402 | − | 3.21293i | −0.197489 | + | 1.37356i | 0.537236 | + | 1.82966i | −2.76392 | − | 1.77626i | −4.99544 | + | 0.718235i |
144.14 | −1.48898 | − | 0.679994i | −0.410781 | + | 1.39899i | 0.444947 | + | 0.513496i | −2.75771 | + | 1.77227i | 1.56295 | − | 1.80374i | 0.318439 | − | 2.21479i | 0.608995 | + | 2.07405i | 0.735326 | + | 0.472566i | 5.31131 | − | 0.763651i |
144.15 | −1.44351 | − | 0.659230i | 0.294625 | − | 1.00340i | 0.339420 | + | 0.391712i | 0.450432 | − | 0.289475i | −1.08677 | + | 1.25420i | 0.597006 | − | 4.15227i | 0.662444 | + | 2.25608i | 1.60375 | + | 1.03067i | −0.841035 | + | 0.120923i |
144.16 | −1.24405 | − | 0.568139i | 0.489921 | − | 1.66852i | −0.0848397 | − | 0.0979102i | −2.34079 | + | 1.50433i | −1.55744 | + | 1.79738i | −0.277740 | + | 1.93172i | 0.820537 | + | 2.79449i | −0.0201691 | − | 0.0129619i | 3.76673 | − | 0.541574i |
144.17 | −1.15704 | − | 0.528401i | −0.256259 | + | 0.872739i | −0.250194 | − | 0.288739i | 1.57616 | − | 1.01293i | 0.757658 | − | 0.874384i | −0.622254 | + | 4.32787i | 0.853632 | + | 2.90720i | 1.82776 | + | 1.17463i | −2.35891 | + | 0.339160i |
144.18 | −1.11723 | − | 0.510223i | −0.0884349 | + | 0.301182i | −0.321839 | − | 0.371422i | 2.81111 | − | 1.80659i | 0.252472 | − | 0.291369i | 0.132526 | − | 0.921742i | 0.862123 | + | 2.93612i | 2.44087 | + | 1.56865i | −4.06243 | + | 0.584089i |
144.19 | −1.02135 | − | 0.466436i | 0.271355 | − | 0.924149i | −0.484125 | − | 0.558710i | −0.771899 | + | 0.496069i | −0.708204 | + | 0.817311i | 0.190973 | − | 1.32825i | 0.866528 | + | 2.95112i | 1.74334 | + | 1.12038i | 1.01976 | − | 0.146620i |
144.20 | −0.941279 | − | 0.429868i | −0.504321 | + | 1.71756i | −0.608501 | − | 0.702248i | −1.58182 | + | 1.01657i | 1.21303 | − | 1.39991i | −0.00146717 | + | 0.0102044i | 0.853965 | + | 2.90834i | −0.171912 | − | 0.110481i | 1.92592 | − | 0.276906i |
See next 80 embeddings (of 580 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
29.b | even | 2 | 1 | inner |
667.m | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 667.2.m.a | ✓ | 580 |
23.c | even | 11 | 1 | inner | 667.2.m.a | ✓ | 580 |
29.b | even | 2 | 1 | inner | 667.2.m.a | ✓ | 580 |
667.m | even | 22 | 1 | inner | 667.2.m.a | ✓ | 580 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
667.2.m.a | ✓ | 580 | 1.a | even | 1 | 1 | trivial |
667.2.m.a | ✓ | 580 | 23.c | even | 11 | 1 | inner |
667.2.m.a | ✓ | 580 | 29.b | even | 2 | 1 | inner |
667.2.m.a | ✓ | 580 | 667.m | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(667, [\chi])\).