Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [414,3,Mod(29,414)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(66))
chi = DirichletCharacter(H, H._module([11, 54]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("414.29");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 414.o (of order \(66\), degree \(20\), 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: | \(11.2806829445\) |
Analytic rank: | \(0\) |
Dimension: | \(960\) |
Relative dimension: | \(48\) over \(\Q(\zeta_{66})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{66}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | −1.15198 | + | 0.820324i | −2.99334 | − | 0.199741i | 0.654136 | − | 1.89000i | −4.08143 | + | 7.91688i | 3.61214 | − | 2.22541i | 8.43255 | + | 6.63143i | 0.796860 | + | 2.71386i | 8.92021 | + | 1.19579i | −1.79266 | − | 12.4682i |
29.2 | −1.15198 | + | 0.820324i | −2.92929 | + | 0.647508i | 0.654136 | − | 1.89000i | −0.971524 | + | 1.88449i | 2.84333 | − | 3.14889i | −7.01983 | − | 5.52046i | 0.796860 | + | 2.71386i | 8.16147 | − | 3.79348i | −0.426716 | − | 2.96787i |
29.3 | −1.15198 | + | 0.820324i | −2.84434 | − | 0.953805i | 0.654136 | − | 1.89000i | 0.224329 | − | 0.435137i | 4.05906 | − | 1.23451i | −1.83852 | − | 1.44583i | 0.796860 | + | 2.71386i | 7.18051 | + | 5.42589i | 0.0985302 | + | 0.685293i |
29.4 | −1.15198 | + | 0.820324i | −2.73798 | − | 1.22617i | 0.654136 | − | 1.89000i | 2.56559 | − | 4.97655i | 4.15996 | − | 0.833504i | 8.26502 | + | 6.49968i | 0.796860 | + | 2.71386i | 5.99303 | + | 6.71443i | 1.12687 | + | 7.83753i |
29.5 | −1.15198 | + | 0.820324i | −2.68304 | + | 1.34212i | 0.654136 | − | 1.89000i | 3.75061 | − | 7.27517i | 1.98985 | − | 3.74706i | 2.16342 | + | 1.70134i | 0.796860 | + | 2.71386i | 5.39744 | − | 7.20192i | 1.64735 | + | 11.4576i |
29.6 | −1.15198 | + | 0.820324i | −2.48619 | + | 1.67895i | 0.654136 | − | 1.89000i | −2.13326 | + | 4.13794i | 1.48677 | − | 3.97360i | 1.87456 | + | 1.47417i | 0.796860 | + | 2.71386i | 3.36227 | − | 8.34836i | −0.936975 | − | 6.51680i |
29.7 | −1.15198 | + | 0.820324i | −1.88417 | + | 2.33451i | 0.654136 | − | 1.89000i | 0.426441 | − | 0.827180i | 0.255479 | − | 4.23494i | 3.49911 | + | 2.75173i | 0.796860 | + | 2.71386i | −1.89983 | − | 8.79720i | 0.187303 | + | 1.30272i |
29.8 | −1.15198 | + | 0.820324i | −1.57511 | − | 2.55324i | 0.654136 | − | 1.89000i | −3.77717 | + | 7.32668i | 3.90899 | + | 1.64919i | −7.64250 | − | 6.01013i | 0.796860 | + | 2.71386i | −4.03806 | + | 8.04326i | −1.65902 | − | 11.5387i |
29.9 | −1.15198 | + | 0.820324i | −1.39829 | − | 2.65420i | 0.654136 | − | 1.89000i | 1.56148 | − | 3.02884i | 3.78812 | + | 1.91054i | 0.462983 | + | 0.364094i | 0.796860 | + | 2.71386i | −5.08956 | + | 7.42269i | 0.685837 | + | 4.77010i |
29.10 | −1.15198 | + | 0.820324i | −0.995486 | − | 2.83002i | 0.654136 | − | 1.89000i | 2.41189 | − | 4.67841i | 3.46832 | + | 2.44352i | −4.90570 | − | 3.85789i | 0.796860 | + | 2.71386i | −7.01802 | + | 5.63449i | 1.05936 | + | 7.36799i |
29.11 | −1.15198 | + | 0.820324i | −0.858721 | + | 2.87447i | 0.654136 | − | 1.89000i | 1.27146 | − | 2.46629i | −1.36877 | − | 4.01578i | −9.21164 | − | 7.24412i | 0.796860 | + | 2.71386i | −7.52520 | − | 4.93674i | 0.558454 | + | 3.88413i |
29.12 | −1.15198 | + | 0.820324i | −0.211823 | + | 2.99251i | 0.654136 | − | 1.89000i | −2.84452 | + | 5.51760i | −2.21081 | − | 3.62109i | 1.88035 | + | 1.47873i | 0.796860 | + | 2.71386i | −8.91026 | − | 1.26777i | −1.24938 | − | 8.68961i |
29.13 | −1.15198 | + | 0.820324i | −0.00945685 | − | 2.99999i | 0.654136 | − | 1.89000i | −2.36088 | + | 4.57947i | 2.47185 | + | 3.44818i | 7.63159 | + | 6.00155i | 0.796860 | + | 2.71386i | −8.99982 | + | 0.0567408i | −1.03695 | − | 7.21217i |
29.14 | −1.15198 | + | 0.820324i | 0.355720 | + | 2.97884i | 0.654136 | − | 1.89000i | 1.78394 | − | 3.46036i | −2.85340 | − | 3.13977i | 10.3427 | + | 8.13360i | 0.796860 | + | 2.71386i | −8.74693 | + | 2.11926i | 0.783547 | + | 5.44969i |
29.15 | −1.15198 | + | 0.820324i | 1.32781 | + | 2.69015i | 0.654136 | − | 1.89000i | 3.22219 | − | 6.25018i | −3.73641 | − | 2.00978i | −4.61888 | − | 3.63233i | 0.796860 | + | 2.71386i | −5.47384 | + | 7.14402i | 1.41526 | + | 9.84335i |
29.16 | −1.15198 | + | 0.820324i | 1.35985 | − | 2.67410i | 0.654136 | − | 1.89000i | −2.17040 | + | 4.20998i | 0.627101 | + | 4.19604i | −1.31557 | − | 1.03458i | 0.796860 | + | 2.71386i | −5.30161 | − | 7.27275i | −0.953288 | − | 6.63026i |
29.17 | −1.15198 | + | 0.820324i | 1.38285 | − | 2.66228i | 0.654136 | − | 1.89000i | 1.47947 | − | 2.86978i | 0.590910 | + | 4.20129i | −8.32828 | − | 6.54943i | 0.796860 | + | 2.71386i | −5.17545 | − | 7.36306i | 0.649819 | + | 4.51959i |
29.18 | −1.15198 | + | 0.820324i | 1.68468 | + | 2.48231i | 0.654136 | − | 1.89000i | −3.43268 | + | 6.65847i | −3.97702 | − | 1.47760i | 0.376015 | + | 0.295701i | 0.796860 | + | 2.71386i | −3.32371 | + | 8.36379i | −1.50771 | − | 10.4864i |
29.19 | −1.15198 | + | 0.820324i | 1.93664 | − | 2.29116i | 0.654136 | − | 1.89000i | 0.257284 | − | 0.499061i | −0.351483 | + | 4.22806i | 2.18749 | + | 1.72026i | 0.796860 | + | 2.71386i | −1.49885 | − | 8.87431i | 0.113005 | + | 0.785967i |
29.20 | −1.15198 | + | 0.820324i | 2.48787 | + | 1.67646i | 0.654136 | − | 1.89000i | 0.670722 | − | 1.30102i | −4.24122 | + | 0.109603i | 1.82494 | + | 1.43515i | 0.796860 | + | 2.71386i | 3.37897 | + | 8.34162i | 0.294596 | + | 2.04896i |
See next 80 embeddings (of 960 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
23.c | even | 11 | 1 | inner |
207.n | odd | 66 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 414.3.o.a | ✓ | 960 |
9.d | odd | 6 | 1 | inner | 414.3.o.a | ✓ | 960 |
23.c | even | 11 | 1 | inner | 414.3.o.a | ✓ | 960 |
207.n | odd | 66 | 1 | inner | 414.3.o.a | ✓ | 960 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
414.3.o.a | ✓ | 960 | 1.a | even | 1 | 1 | trivial |
414.3.o.a | ✓ | 960 | 9.d | odd | 6 | 1 | inner |
414.3.o.a | ✓ | 960 | 23.c | even | 11 | 1 | inner |
414.3.o.a | ✓ | 960 | 207.n | odd | 66 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(414, [\chi])\).