Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [861,2,Mod(206,861)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(861, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("861.206");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 861 = 3 \cdot 7 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 861.s (of order \(6\), 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: | \(6.87511961403\) |
Analytic rank: | \(0\) |
Dimension: | \(106\) |
Relative dimension: | \(53\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
206.1 | −2.41357 | + | 1.39347i | −1.72865 | + | 0.108517i | 2.88354 | − | 4.99444i | 0.0887849 | + | 0.153780i | 4.02099 | − | 2.67074i | 1.75882 | + | 1.97650i | 10.4987i | 2.97645 | − | 0.375176i | −0.428577 | − | 0.247439i | ||
206.2 | −2.31389 | + | 1.33592i | 1.10471 | + | 1.33402i | 2.56938 | − | 4.45031i | −1.13816 | − | 1.97134i | −4.33832 | − | 1.61098i | −2.63605 | − | 0.226330i | 8.38632i | −0.559244 | + | 2.94741i | 5.26713 | + | 3.04098i | ||
206.3 | −2.27082 | + | 1.31106i | −0.573901 | − | 1.63421i | 2.43775 | − | 4.22230i | 1.02980 | + | 1.78367i | 3.44577 | + | 2.95857i | −2.05759 | + | 1.66323i | 7.53988i | −2.34128 | + | 1.87575i | −4.67700 | − | 2.70027i | ||
206.4 | −2.15448 | + | 1.24389i | −1.08687 | + | 1.34860i | 2.09453 | − | 3.62783i | −1.86942 | − | 3.23792i | 0.664120 | − | 4.25748i | 1.06496 | − | 2.42195i | 5.44590i | −0.637447 | − | 2.93149i | 8.05525 | + | 4.65070i | ||
206.5 | −2.13361 | + | 1.23184i | 1.71477 | − | 0.244047i | 2.03486 | − | 3.52448i | 2.02087 | + | 3.50024i | −3.35803 | + | 2.63302i | −2.59807 | − | 0.500056i | 5.09914i | 2.88088 | − | 0.836968i | −8.62348 | − | 4.97877i | ||
206.6 | −2.03071 | + | 1.17243i | −1.00636 | − | 1.40969i | 1.74919 | − | 3.02968i | −0.645695 | − | 1.11838i | 3.69640 | + | 1.68279i | 2.64245 | − | 0.132132i | 3.51348i | −0.974466 | + | 2.83733i | 2.62244 | + | 1.51407i | ||
206.7 | −1.94516 | + | 1.12304i | 1.64499 | − | 0.542238i | 1.52243 | − | 2.63692i | −2.05023 | − | 3.55111i | −2.59080 | + | 2.90212i | 2.47054 | + | 0.946801i | 2.34682i | 2.41196 | − | 1.78395i | 7.97605 | + | 4.60497i | ||
206.8 | −1.84438 | + | 1.06485i | 0.697137 | − | 1.58556i | 1.26783 | − | 2.19594i | 1.12887 | + | 1.95526i | 0.402604 | + | 3.66673i | 2.05168 | + | 1.67051i | 1.14079i | −2.02800 | − | 2.21070i | −4.16413 | − | 2.40416i | ||
206.9 | −1.70805 | + | 0.986142i | −1.18189 | + | 1.26615i | 0.944954 | − | 1.63671i | 1.73937 | + | 3.01268i | 0.770120 | − | 3.32816i | 2.64574 | − | 0.00746536i | − | 0.217133i | −0.206274 | − | 2.99290i | −5.94186 | − | 3.43054i | |
206.10 | −1.59373 | + | 0.920138i | 0.561910 | + | 1.63837i | 0.693309 | − | 1.20085i | 0.958800 | + | 1.66069i | −2.40306 | − | 2.09408i | −1.44254 | − | 2.21790i | − | 1.12879i | −2.36851 | + | 1.84123i | −3.05613 | − | 1.76446i | |
206.11 | −1.51321 | + | 0.873651i | −1.66589 | − | 0.474148i | 0.526532 | − | 0.911981i | −1.16247 | − | 2.01346i | 2.93508 | − | 0.737920i | −2.59569 | − | 0.512229i | − | 1.65458i | 2.55037 | + | 1.57976i | 3.51813 | + | 2.03119i | |
206.12 | −1.49330 | + | 0.862157i | 1.14383 | + | 1.30063i | 0.486628 | − | 0.842865i | −0.428510 | − | 0.742200i | −2.82944 | − | 0.956067i | −0.337814 | + | 2.62410i | − | 1.77043i | −0.383285 | + | 2.97541i | 1.27979 | + | 0.738885i | |
206.13 | −1.44529 | + | 0.834437i | 0.474014 | − | 1.66593i | 0.392570 | − | 0.679952i | 1.92742 | + | 3.33839i | 0.705023 | + | 2.80328i | −1.41003 | − | 2.23871i | − | 2.02745i | −2.55062 | − | 1.57935i | −5.57136 | − | 3.21663i | |
206.14 | −1.44370 | + | 0.833520i | 1.51624 | + | 0.837276i | 0.389510 | − | 0.674651i | 0.984574 | + | 1.70533i | −2.88687 | + | 0.0550382i | 0.232807 | + | 2.63549i | − | 2.03542i | 1.59794 | + | 2.53901i | −2.84285 | − | 1.64132i | |
206.15 | −1.28760 | + | 0.743398i | −0.267036 | − | 1.71134i | 0.105281 | − | 0.182352i | −1.62942 | − | 2.82224i | 1.61604 | + | 2.00502i | −0.433422 | − | 2.61001i | − | 2.66053i | −2.85738 | + | 0.913979i | 4.19609 | + | 2.42262i | |
206.16 | −1.27500 | + | 0.736123i | 1.57218 | − | 0.726806i | 0.0837529 | − | 0.145064i | −0.207914 | − | 0.360118i | −1.46951 | + | 2.08400i | 2.35740 | − | 1.20110i | − | 2.69788i | 1.94350 | − | 2.28534i | 0.530182 | + | 0.306101i | |
206.17 | −1.21231 | + | 0.699930i | −1.63502 | + | 0.571591i | −0.0201967 | + | 0.0349817i | −1.06109 | − | 1.83786i | 1.58208 | − | 1.83735i | 1.41763 | − | 2.23390i | − | 2.85626i | 2.34657 | − | 1.86912i | 2.57275 | + | 1.48538i | |
206.18 | −1.09044 | + | 0.629568i | −0.568895 | + | 1.63596i | −0.207288 | + | 0.359033i | −0.914757 | − | 1.58441i | −0.409598 | − | 2.14208i | −2.62661 | + | 0.317703i | − | 3.04028i | −2.35272 | − | 1.86138i | 1.99498 | + | 1.15180i | |
206.19 | −0.977958 | + | 0.564624i | 0.835884 | − | 1.51700i | −0.362399 | + | 0.627693i | −0.0432421 | − | 0.0748975i | 0.0390773 | + | 1.95553i | −2.31142 | + | 1.28737i | − | 3.07697i | −1.60260 | − | 2.53608i | 0.0845779 | + | 0.0488311i | |
206.20 | −0.651390 | + | 0.376080i | −1.69203 | − | 0.370192i | −0.717127 | + | 1.24210i | 0.355703 | + | 0.616096i | 1.24139 | − | 0.395199i | 1.69688 | + | 2.02992i | − | 2.58311i | 2.72592 | + | 1.25275i | −0.463404 | − | 0.267546i | |
See next 80 embeddings (of 106 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 861.2.s.a | ✓ | 106 |
3.b | odd | 2 | 1 | 861.2.s.b | yes | 106 | |
7.d | odd | 6 | 1 | 861.2.s.b | yes | 106 | |
21.g | even | 6 | 1 | inner | 861.2.s.a | ✓ | 106 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
861.2.s.a | ✓ | 106 | 1.a | even | 1 | 1 | trivial |
861.2.s.a | ✓ | 106 | 21.g | even | 6 | 1 | inner |
861.2.s.b | yes | 106 | 3.b | odd | 2 | 1 | |
861.2.s.b | yes | 106 | 7.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{106} - 79 T_{2}^{104} + 3315 T_{2}^{102} - 96054 T_{2}^{100} - 30 T_{2}^{99} + 2134915 T_{2}^{98} + 2370 T_{2}^{97} - 38486201 T_{2}^{96} - 95778 T_{2}^{95} + 582396323 T_{2}^{94} + 2621442 T_{2}^{93} + \cdots + 407027712 \)
acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\).