Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(361,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.361");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.do (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: | \(6.53974792554\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(18\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
361.1 | −2.33665 | − | 1.34906i | 0 | 2.63995 | + | 4.57253i | −0.933651 | + | 0.539043i | 0 | 2.27537 | + | 1.35008i | − | 8.84961i | 0 | 2.90882 | |||||||||
361.2 | −2.27495 | − | 1.31344i | 0 | 2.45026 | + | 4.24397i | 2.99000 | − | 1.72628i | 0 | −1.10870 | − | 2.40225i | − | 7.61930i | 0 | −9.06945 | |||||||||
361.3 | −1.94159 | − | 1.12098i | 0 | 1.51319 | + | 2.62092i | −0.550805 | + | 0.318007i | 0 | −1.65674 | + | 2.06282i | − | 2.30110i | 0 | 1.42592 | |||||||||
361.4 | −1.54694 | − | 0.893127i | 0 | 0.595353 | + | 1.03118i | 0.972470 | − | 0.561456i | 0 | −0.0853404 | − | 2.64437i | 1.44561i | 0 | −2.00581 | ||||||||||
361.5 | −1.48056 | − | 0.854802i | 0 | 0.461374 | + | 0.799123i | −1.84265 | + | 1.06385i | 0 | 2.64237 | + | 0.133694i | 1.84167i | 0 | 3.63754 | ||||||||||
361.6 | −1.26493 | − | 0.730310i | 0 | 0.0667051 | + | 0.115537i | 2.51646 | − | 1.45288i | 0 | −1.58711 | + | 2.11686i | 2.72638i | 0 | −4.24420 | ||||||||||
361.7 | −1.00552 | − | 0.580538i | 0 | −0.325951 | − | 0.564563i | −3.43209 | + | 1.98152i | 0 | −2.43773 | − | 1.02834i | 3.07906i | 0 | 4.60139 | ||||||||||
361.8 | −0.356535 | − | 0.205846i | 0 | −0.915255 | − | 1.58527i | 1.54520 | − | 0.892120i | 0 | 0.355695 | + | 2.62173i | 1.57699i | 0 | −0.734557 | ||||||||||
361.9 | −0.146841 | − | 0.0847787i | 0 | −0.985625 | − | 1.70715i | 2.65412 | − | 1.53236i | 0 | 2.60218 | − | 0.478169i | 0.673355i | 0 | −0.519645 | ||||||||||
361.10 | 0.146841 | + | 0.0847787i | 0 | −0.985625 | − | 1.70715i | −2.65412 | + | 1.53236i | 0 | 2.60218 | − | 0.478169i | − | 0.673355i | 0 | −0.519645 | |||||||||
361.11 | 0.356535 | + | 0.205846i | 0 | −0.915255 | − | 1.58527i | −1.54520 | + | 0.892120i | 0 | 0.355695 | + | 2.62173i | − | 1.57699i | 0 | −0.734557 | |||||||||
361.12 | 1.00552 | + | 0.580538i | 0 | −0.325951 | − | 0.564563i | 3.43209 | − | 1.98152i | 0 | −2.43773 | − | 1.02834i | − | 3.07906i | 0 | 4.60139 | |||||||||
361.13 | 1.26493 | + | 0.730310i | 0 | 0.0667051 | + | 0.115537i | −2.51646 | + | 1.45288i | 0 | −1.58711 | + | 2.11686i | − | 2.72638i | 0 | −4.24420 | |||||||||
361.14 | 1.48056 | + | 0.854802i | 0 | 0.461374 | + | 0.799123i | 1.84265 | − | 1.06385i | 0 | 2.64237 | + | 0.133694i | − | 1.84167i | 0 | 3.63754 | |||||||||
361.15 | 1.54694 | + | 0.893127i | 0 | 0.595353 | + | 1.03118i | −0.972470 | + | 0.561456i | 0 | −0.0853404 | − | 2.64437i | − | 1.44561i | 0 | −2.00581 | |||||||||
361.16 | 1.94159 | + | 1.12098i | 0 | 1.51319 | + | 2.62092i | 0.550805 | − | 0.318007i | 0 | −1.65674 | + | 2.06282i | 2.30110i | 0 | 1.42592 | ||||||||||
361.17 | 2.27495 | + | 1.31344i | 0 | 2.45026 | + | 4.24397i | −2.99000 | + | 1.72628i | 0 | −1.10870 | − | 2.40225i | 7.61930i | 0 | −9.06945 | ||||||||||
361.18 | 2.33665 | + | 1.34906i | 0 | 2.63995 | + | 4.57253i | 0.933651 | − | 0.539043i | 0 | 2.27537 | + | 1.35008i | 8.84961i | 0 | 2.90882 | ||||||||||
667.1 | −2.33665 | + | 1.34906i | 0 | 2.63995 | − | 4.57253i | −0.933651 | − | 0.539043i | 0 | 2.27537 | − | 1.35008i | 8.84961i | 0 | 2.90882 | ||||||||||
667.2 | −2.27495 | + | 1.31344i | 0 | 2.45026 | − | 4.24397i | 2.99000 | + | 1.72628i | 0 | −1.10870 | + | 2.40225i | 7.61930i | 0 | −9.06945 | ||||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
91.u | even | 6 | 1 | inner |
273.x | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.do.h | yes | 36 |
3.b | odd | 2 | 1 | inner | 819.2.do.h | yes | 36 |
7.c | even | 3 | 1 | 819.2.bm.h | ✓ | 36 | |
13.e | even | 6 | 1 | 819.2.bm.h | ✓ | 36 | |
21.h | odd | 6 | 1 | 819.2.bm.h | ✓ | 36 | |
39.h | odd | 6 | 1 | 819.2.bm.h | ✓ | 36 | |
91.u | even | 6 | 1 | inner | 819.2.do.h | yes | 36 |
273.x | odd | 6 | 1 | inner | 819.2.do.h | yes | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
819.2.bm.h | ✓ | 36 | 7.c | even | 3 | 1 | |
819.2.bm.h | ✓ | 36 | 13.e | even | 6 | 1 | |
819.2.bm.h | ✓ | 36 | 21.h | odd | 6 | 1 | |
819.2.bm.h | ✓ | 36 | 39.h | odd | 6 | 1 | |
819.2.do.h | yes | 36 | 1.a | even | 1 | 1 | trivial |
819.2.do.h | yes | 36 | 3.b | odd | 2 | 1 | inner |
819.2.do.h | yes | 36 | 91.u | even | 6 | 1 | inner |
819.2.do.h | yes | 36 | 273.x | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 29 T_{2}^{34} + 496 T_{2}^{32} - 5647 T_{2}^{30} + 47934 T_{2}^{28} - 310100 T_{2}^{26} + \cdots + 1089 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).