Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [171,3,Mod(23,171)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(171, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([15, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("171.23");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 171 = 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 171.bf (of order \(18\), degree \(6\), 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: | \(4.65941252056\) |
Analytic rank: | \(0\) |
Dimension: | \(228\) |
Relative dimension: | \(38\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −3.65933 | + | 0.645238i | −0.459116 | − | 2.96466i | 9.21558 | − | 3.35420i | 5.91829 | − | 1.04355i | 3.59297 | + | 10.5524i | 10.9088 | −18.6867 | + | 10.7888i | −8.57843 | + | 2.72225i | −20.9836 | + | 7.63741i | ||
23.2 | −3.64108 | + | 0.642021i | −2.57388 | − | 1.54115i | 9.08651 | − | 3.30722i | −6.42488 | + | 1.13288i | 10.3612 | + | 3.95898i | −7.72654 | −18.1538 | + | 10.4811i | 4.24969 | + | 7.93348i | 22.6662 | − | 8.24981i | ||
23.3 | −3.62609 | + | 0.639378i | 1.79662 | + | 2.40253i | 8.98096 | − | 3.26880i | −1.84348 | + | 0.325055i | −8.05084 | − | 7.56307i | −2.47175 | −17.7208 | + | 10.2311i | −2.54429 | + | 8.63288i | 6.47679 | − | 2.35736i | ||
23.4 | −3.46435 | + | 0.610858i | −2.11681 | + | 2.12582i | 7.86980 | − | 2.86437i | 9.17839 | − | 1.61840i | 6.03480 | − | 8.65765i | −9.01954 | −13.3280 | + | 7.69493i | −0.0382204 | − | 8.99992i | −30.8085 | + | 11.2134i | ||
23.5 | −3.28012 | + | 0.578374i | −1.93034 | + | 2.29647i | 6.66591 | − | 2.42619i | −4.81965 | + | 0.849835i | 5.00352 | − | 8.64917i | 11.2987 | −8.92377 | + | 5.15214i | −1.54759 | − | 8.86594i | 15.3175 | − | 5.57512i | ||
23.6 | −3.09884 | + | 0.546410i | 2.53490 | − | 1.60445i | 5.54550 | − | 2.01840i | −6.17011 | + | 1.08796i | −6.97858 | + | 6.35704i | 5.25884 | −5.18146 | + | 2.99152i | 3.85147 | − | 8.13426i | 18.5257 | − | 6.74281i | ||
23.7 | −2.74457 | + | 0.483941i | 0.979111 | − | 2.83573i | 3.53968 | − | 1.28834i | 1.23762 | − | 0.218226i | −1.31491 | + | 8.25668i | −12.5962 | 0.562706 | − | 0.324878i | −7.08268 | − | 5.55298i | −3.29113 | + | 1.19787i | ||
23.8 | −2.73684 | + | 0.482579i | 2.99746 | − | 0.123538i | 3.49866 | − | 1.27341i | 7.74651 | − | 1.36592i | −8.14395 | + | 1.78461i | 2.71206 | 0.666199 | − | 0.384630i | 8.96948 | − | 0.740598i | −20.5418 | + | 7.47661i | ||
23.9 | −2.35710 | + | 0.415620i | −2.93680 | + | 0.612531i | 1.62440 | − | 0.591235i | 1.35703 | − | 0.239281i | 6.66775 | − | 2.66439i | 3.53164 | 4.70804 | − | 2.71819i | 8.24961 | − | 3.59776i | −3.09920 | + | 1.12802i | ||
23.10 | −2.31732 | + | 0.408605i | 2.50810 | + | 1.64603i | 1.44422 | − | 0.525655i | 1.25775 | − | 0.221775i | −6.48464 | − | 2.78955i | −0.0673877 | 5.01931 | − | 2.89790i | 3.58115 | + | 8.25684i | −2.82399 | + | 1.02785i | ||
23.11 | −1.99638 | + | 0.352015i | −2.18466 | − | 2.05603i | 0.102834 | − | 0.0374285i | 2.67211 | − | 0.471166i | 5.08516 | + | 3.33557i | 3.51086 | 6.83022 | − | 3.94343i | 0.545494 | + | 8.98345i | −5.16869 | + | 1.88125i | ||
23.12 | −1.98725 | + | 0.350406i | −0.0711634 | + | 2.99916i | 0.0676156 | − | 0.0246101i | −1.58861 | + | 0.280114i | −0.909503 | − | 5.98501i | −6.99408 | 6.86449 | − | 3.96322i | −8.98987 | − | 0.426860i | 3.05881 | − | 1.11332i | ||
23.13 | −1.52687 | + | 0.269229i | −0.833307 | − | 2.88194i | −1.49991 | + | 0.545922i | −7.11297 | + | 1.25421i | 2.04826 | + | 4.17602i | 2.04957 | 7.51404 | − | 4.33823i | −7.61120 | + | 4.80309i | 10.5229 | − | 3.83004i | ||
23.14 | −1.24893 | + | 0.220221i | −2.79191 | + | 1.09782i | −2.24743 | + | 0.817998i | −8.14879 | + | 1.43685i | 3.24515 | − | 1.98595i | −8.61467 | 7.01993 | − | 4.05296i | 6.58956 | − | 6.13006i | 9.86088 | − | 3.58907i | ||
23.15 | −1.21478 | + | 0.214198i | −0.319844 | + | 2.98290i | −2.32897 | + | 0.847674i | 7.14776 | − | 1.26034i | −0.250393 | − | 3.69207i | 8.34631 | 6.92064 | − | 3.99563i | −8.79540 | − | 1.90813i | −8.41298 | + | 3.06207i | ||
23.16 | −0.817290 | + | 0.144110i | 1.58887 | + | 2.54470i | −3.11157 | + | 1.13252i | −8.08346 | + | 1.42533i | −1.66529 | − | 1.85078i | 10.6388 | 5.25470 | − | 3.03380i | −3.95097 | + | 8.08640i | 6.40113 | − | 2.32982i | ||
23.17 | −0.624400 | + | 0.110099i | 2.97879 | − | 0.356077i | −3.38102 | + | 1.23059i | −2.37342 | + | 0.418498i | −1.82075 | + | 0.550296i | −10.8393 | 4.17197 | − | 2.40869i | 8.74642 | − | 2.12136i | 1.43589 | − | 0.522620i | ||
23.18 | −0.461247 | + | 0.0813303i | 2.96881 | − | 0.431476i | −3.55264 | + | 1.29305i | −1.75481 | + | 0.309421i | −1.33426 | + | 0.440471i | 3.68677 | 3.15593 | − | 1.82208i | 8.62766 | − | 2.56194i | 0.784238 | − | 0.285439i | ||
23.19 | −0.452066 | + | 0.0797114i | 1.39632 | − | 2.65524i | −3.56076 | + | 1.29601i | 6.95951 | − | 1.22715i | −0.419578 | + | 1.31165i | 2.58194 | 3.09655 | − | 1.78780i | −5.10056 | − | 7.41514i | −3.04834 | + | 1.10950i | ||
23.20 | −0.295197 | + | 0.0520513i | −2.80082 | − | 1.07489i | −3.67434 | + | 1.33735i | 7.64615 | − | 1.34822i | 0.882745 | + | 0.171520i | −11.2077 | 2.05341 | − | 1.18554i | 6.68920 | + | 6.02118i | −2.18695 | + | 0.795983i | ||
See next 80 embeddings (of 228 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
171.bf | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 171.3.bf.a | yes | 228 |
9.d | odd | 6 | 1 | 171.3.z.a | ✓ | 228 | |
19.e | even | 9 | 1 | 171.3.z.a | ✓ | 228 | |
171.bf | odd | 18 | 1 | inner | 171.3.bf.a | yes | 228 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
171.3.z.a | ✓ | 228 | 9.d | odd | 6 | 1 | |
171.3.z.a | ✓ | 228 | 19.e | even | 9 | 1 | |
171.3.bf.a | yes | 228 | 1.a | even | 1 | 1 | trivial |
171.3.bf.a | yes | 228 | 171.bf | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(171, [\chi])\).