Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(370,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 6, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.370");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.fm (of order \(12\), degree \(4\), 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: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 91) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
370.1 | −0.612835 | − | 2.28713i | 0 | −3.12335 | + | 1.80327i | −2.25527 | − | 2.25527i | 0 | −2.60590 | − | 0.457468i | 2.68981 | + | 2.68981i | 0 | −3.77599 | + | 6.54020i | ||||||
370.2 | −0.612835 | − | 2.28713i | 0 | −3.12335 | + | 1.80327i | 2.25527 | + | 2.25527i | 0 | 2.48551 | + | 0.906771i | 2.68981 | + | 2.68981i | 0 | 3.77599 | − | 6.54020i | ||||||
370.3 | −0.218036 | − | 0.813723i | 0 | 1.11745 | − | 0.645157i | −1.01306 | − | 1.01306i | 0 | 2.43848 | + | 1.02656i | −1.96000 | − | 1.96000i | 0 | −0.603466 | + | 1.04523i | ||||||
370.4 | −0.218036 | − | 0.813723i | 0 | 1.11745 | − | 0.645157i | 1.01306 | + | 1.01306i | 0 | −2.62506 | − | 0.330214i | −1.96000 | − | 1.96000i | 0 | 0.603466 | − | 1.04523i | ||||||
370.5 | 0.357317 | + | 1.33353i | 0 | 0.0814361 | − | 0.0470171i | −2.02925 | − | 2.02925i | 0 | −2.32989 | + | 1.25364i | 2.04421 | + | 2.04421i | 0 | 1.98097 | − | 3.43114i | ||||||
370.6 | 0.357317 | + | 1.33353i | 0 | 0.0814361 | − | 0.0470171i | 2.02925 | + | 2.02925i | 0 | 1.39092 | + | 2.25063i | 2.04421 | + | 2.04421i | 0 | −1.98097 | + | 3.43114i | ||||||
370.7 | 0.607529 | + | 2.26733i | 0 | −3.03963 | + | 1.75493i | −1.12678 | − | 1.12678i | 0 | −0.437995 | − | 2.60925i | −2.50608 | − | 2.50608i | 0 | 1.87023 | − | 3.23933i | ||||||
370.8 | 0.607529 | + | 2.26733i | 0 | −3.03963 | + | 1.75493i | 1.12678 | + | 1.12678i | 0 | 1.68394 | − | 2.04067i | −2.50608 | − | 2.50608i | 0 | −1.87023 | + | 3.23933i | ||||||
496.1 | −1.61897 | − | 0.433802i | 0 | 0.700831 | + | 0.404625i | −1.42145 | − | 1.42145i | 0 | −1.11484 | − | 2.39940i | 1.41124 | + | 1.41124i | 0 | 1.68466 | + | 2.91792i | ||||||
496.2 | −1.61897 | − | 0.433802i | 0 | 0.700831 | + | 0.404625i | 1.42145 | + | 1.42145i | 0 | 0.234216 | + | 2.63536i | 1.41124 | + | 1.41124i | 0 | −1.68466 | − | 2.91792i | ||||||
496.3 | 0.112775 | + | 0.0302180i | 0 | −1.72025 | − | 0.993184i | −1.24841 | − | 1.24841i | 0 | −2.63771 | − | 0.206123i | −0.329103 | − | 0.329103i | 0 | −0.103066 | − | 0.178515i | ||||||
496.4 | 0.112775 | + | 0.0302180i | 0 | −1.72025 | − | 0.993184i | 1.24841 | + | 1.24841i | 0 | −2.18126 | + | 1.49736i | −0.329103 | − | 0.329103i | 0 | 0.103066 | + | 0.178515i | ||||||
496.5 | 0.892257 | + | 0.239080i | 0 | −0.993087 | − | 0.573359i | −2.80028 | − | 2.80028i | 0 | 2.62900 | + | 0.297264i | −2.05537 | − | 2.05537i | 0 | −1.82908 | − | 3.16806i | ||||||
496.6 | 0.892257 | + | 0.239080i | 0 | −0.993087 | − | 0.573359i | 2.80028 | + | 2.80028i | 0 | 2.12815 | − | 1.57194i | −2.05537 | − | 2.05537i | 0 | 1.82908 | + | 3.16806i | ||||||
496.7 | 2.47996 | + | 0.664504i | 0 | 3.97660 | + | 2.29589i | −0.281686 | − | 0.281686i | 0 | 1.14426 | + | 2.38551i | 4.70528 | + | 4.70528i | 0 | −0.511390 | − | 0.885754i | ||||||
496.8 | 2.47996 | + | 0.664504i | 0 | 3.97660 | + | 2.29589i | 0.281686 | + | 0.281686i | 0 | −0.201802 | − | 2.63804i | 4.70528 | + | 4.70528i | 0 | 0.511390 | + | 0.885754i | ||||||
622.1 | −0.612835 | + | 2.28713i | 0 | −3.12335 | − | 1.80327i | −2.25527 | + | 2.25527i | 0 | −2.60590 | + | 0.457468i | 2.68981 | − | 2.68981i | 0 | −3.77599 | − | 6.54020i | ||||||
622.2 | −0.612835 | + | 2.28713i | 0 | −3.12335 | − | 1.80327i | 2.25527 | − | 2.25527i | 0 | 2.48551 | − | 0.906771i | 2.68981 | − | 2.68981i | 0 | 3.77599 | + | 6.54020i | ||||||
622.3 | −0.218036 | + | 0.813723i | 0 | 1.11745 | + | 0.645157i | −1.01306 | + | 1.01306i | 0 | 2.43848 | − | 1.02656i | −1.96000 | + | 1.96000i | 0 | −0.603466 | − | 1.04523i | ||||||
622.4 | −0.218036 | + | 0.813723i | 0 | 1.11745 | + | 0.645157i | 1.01306 | − | 1.01306i | 0 | −2.62506 | + | 0.330214i | −1.96000 | + | 1.96000i | 0 | 0.603466 | + | 1.04523i | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
13.f | odd | 12 | 1 | inner |
91.bc | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.fm.g | 32 | |
3.b | odd | 2 | 1 | 91.2.bc.a | ✓ | 32 | |
7.b | odd | 2 | 1 | inner | 819.2.fm.g | 32 | |
13.f | odd | 12 | 1 | inner | 819.2.fm.g | 32 | |
21.c | even | 2 | 1 | 91.2.bc.a | ✓ | 32 | |
21.g | even | 6 | 1 | 637.2.x.b | 32 | ||
21.g | even | 6 | 1 | 637.2.bb.b | 32 | ||
21.h | odd | 6 | 1 | 637.2.x.b | 32 | ||
21.h | odd | 6 | 1 | 637.2.bb.b | 32 | ||
39.k | even | 12 | 1 | 91.2.bc.a | ✓ | 32 | |
91.bc | even | 12 | 1 | inner | 819.2.fm.g | 32 | |
273.bs | odd | 12 | 1 | 637.2.x.b | 32 | ||
273.bv | even | 12 | 1 | 637.2.x.b | 32 | ||
273.bw | even | 12 | 1 | 637.2.bb.b | 32 | ||
273.ca | odd | 12 | 1 | 91.2.bc.a | ✓ | 32 | |
273.ch | odd | 12 | 1 | 637.2.bb.b | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.bc.a | ✓ | 32 | 3.b | odd | 2 | 1 | |
91.2.bc.a | ✓ | 32 | 21.c | even | 2 | 1 | |
91.2.bc.a | ✓ | 32 | 39.k | even | 12 | 1 | |
91.2.bc.a | ✓ | 32 | 273.ca | odd | 12 | 1 | |
637.2.x.b | 32 | 21.g | even | 6 | 1 | ||
637.2.x.b | 32 | 21.h | odd | 6 | 1 | ||
637.2.x.b | 32 | 273.bs | odd | 12 | 1 | ||
637.2.x.b | 32 | 273.bv | even | 12 | 1 | ||
637.2.bb.b | 32 | 21.g | even | 6 | 1 | ||
637.2.bb.b | 32 | 21.h | odd | 6 | 1 | ||
637.2.bb.b | 32 | 273.bw | even | 12 | 1 | ||
637.2.bb.b | 32 | 273.ch | odd | 12 | 1 | ||
819.2.fm.g | 32 | 1.a | even | 1 | 1 | trivial | |
819.2.fm.g | 32 | 7.b | odd | 2 | 1 | inner | |
819.2.fm.g | 32 | 13.f | odd | 12 | 1 | inner | |
819.2.fm.g | 32 | 91.bc | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):
\( T_{2}^{16} - 4 T_{2}^{15} + 11 T_{2}^{14} - 28 T_{2}^{13} + 31 T_{2}^{12} - 10 T_{2}^{11} - 68 T_{2}^{10} + \cdots + 9 \) |
\( T_{5}^{32} + 454 T_{5}^{28} + 64945 T_{5}^{24} + 3728042 T_{5}^{20} + 87250612 T_{5}^{16} + \cdots + 187388721 \) |
\( T_{19}^{32} + 204 T_{19}^{30} + 17159 T_{19}^{28} + 670548 T_{19}^{26} + 7775888 T_{19}^{24} + \cdots + 55\!\cdots\!76 \) |