Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(136,819)]
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, 2, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.136");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.et (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: | \(28\) |
Relative dimension: | \(7\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
136.1 | −1.74384 | + | 1.74384i | 0 | − | 4.08193i | 0.638637 | − | 2.38343i | 0 | −2.04541 | + | 1.67818i | 3.63054 | + | 3.63054i | 0 | 3.04263 | + | 5.26998i | |||||||
136.2 | −1.14693 | + | 1.14693i | 0 | − | 0.630890i | −0.395109 | + | 1.47457i | 0 | 0.0531605 | − | 2.64522i | −1.57027 | − | 1.57027i | 0 | −1.23806 | − | 2.14439i | |||||||
136.3 | −0.490988 | + | 0.490988i | 0 | 1.51786i | 0.00962681 | − | 0.0359277i | 0 | −0.176775 | + | 2.63984i | −1.72723 | − | 1.72723i | 0 | 0.0129135 | + | 0.0223668i | ||||||||
136.4 | 0.193244 | − | 0.193244i | 0 | 1.92531i | 0.383199 | − | 1.43012i | 0 | −2.15474 | − | 1.53528i | 0.758543 | + | 0.758543i | 0 | −0.202311 | − | 0.350412i | ||||||||
136.5 | 0.270646 | − | 0.270646i | 0 | 1.85350i | −0.959617 | + | 3.58134i | 0 | 1.30385 | + | 2.30217i | 1.04293 | + | 1.04293i | 0 | 0.709559 | + | 1.22899i | ||||||||
136.6 | 0.984398 | − | 0.984398i | 0 | 0.0619199i | 0.172312 | − | 0.643078i | 0 | −2.46519 | − | 0.960657i | 2.02975 | + | 2.02975i | 0 | −0.463421 | − | 0.802669i | ||||||||
136.7 | 1.56744 | − | 1.56744i | 0 | − | 2.91373i | 0.784926 | − | 2.92938i | 0 | 2.25305 | + | 1.38700i | −1.43221 | − | 1.43221i | 0 | −3.36131 | − | 5.82195i | |||||||
145.1 | −1.51485 | + | 1.51485i | 0 | − | 2.58954i | 1.34505 | − | 0.360406i | 0 | −0.246373 | + | 2.63426i | 0.893066 | + | 0.893066i | 0 | −1.49159 | + | 2.58351i | |||||||
145.2 | −1.28453 | + | 1.28453i | 0 | − | 1.30006i | −1.07541 | + | 0.288156i | 0 | −0.978346 | − | 2.45822i | −0.899098 | − | 0.899098i | 0 | 1.01126 | − | 1.75155i | |||||||
145.3 | −0.203761 | + | 0.203761i | 0 | 1.91696i | −0.499383 | + | 0.133809i | 0 | −2.60732 | − | 0.449339i | −0.798123 | − | 0.798123i | 0 | 0.0744896 | − | 0.129020i | ||||||||
145.4 | 0.347096 | − | 0.347096i | 0 | 1.75905i | 3.47544 | − | 0.931242i | 0 | 0.701045 | + | 2.55118i | 1.30475 | + | 1.30475i | 0 | 0.883082 | − | 1.52954i | ||||||||
145.5 | 0.876516 | − | 0.876516i | 0 | 0.463441i | 2.51660 | − | 0.674321i | 0 | 2.20101 | − | 1.46818i | 2.15924 | + | 2.15924i | 0 | 1.61479 | − | 2.79689i | ||||||||
145.6 | 1.42500 | − | 1.42500i | 0 | − | 2.06123i | −3.16920 | + | 0.849184i | 0 | −0.111396 | + | 2.64341i | −0.0872533 | − | 0.0872533i | 0 | −3.30601 | + | 5.72618i | |||||||
145.7 | 1.72056 | − | 1.72056i | 0 | − | 3.92067i | −0.227080 | + | 0.0608458i | 0 | 1.27342 | − | 2.31914i | −3.30464 | − | 3.30464i | 0 | −0.286016 | + | 0.495394i | |||||||
271.1 | −1.74384 | − | 1.74384i | 0 | 4.08193i | 0.638637 | + | 2.38343i | 0 | −2.04541 | − | 1.67818i | 3.63054 | − | 3.63054i | 0 | 3.04263 | − | 5.26998i | ||||||||
271.2 | −1.14693 | − | 1.14693i | 0 | 0.630890i | −0.395109 | − | 1.47457i | 0 | 0.0531605 | + | 2.64522i | −1.57027 | + | 1.57027i | 0 | −1.23806 | + | 2.14439i | ||||||||
271.3 | −0.490988 | − | 0.490988i | 0 | − | 1.51786i | 0.00962681 | + | 0.0359277i | 0 | −0.176775 | − | 2.63984i | −1.72723 | + | 1.72723i | 0 | 0.0129135 | − | 0.0223668i | |||||||
271.4 | 0.193244 | + | 0.193244i | 0 | − | 1.92531i | 0.383199 | + | 1.43012i | 0 | −2.15474 | + | 1.53528i | 0.758543 | − | 0.758543i | 0 | −0.202311 | + | 0.350412i | |||||||
271.5 | 0.270646 | + | 0.270646i | 0 | − | 1.85350i | −0.959617 | − | 3.58134i | 0 | 1.30385 | − | 2.30217i | 1.04293 | − | 1.04293i | 0 | 0.709559 | − | 1.22899i | |||||||
271.6 | 0.984398 | + | 0.984398i | 0 | − | 0.0619199i | 0.172312 | + | 0.643078i | 0 | −2.46519 | + | 0.960657i | 2.02975 | − | 2.02975i | 0 | −0.463421 | + | 0.802669i | |||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.ba | 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.et.b | 28 | |
3.b | odd | 2 | 1 | 91.2.ba.a | yes | 28 | |
7.d | odd | 6 | 1 | 819.2.gh.b | 28 | ||
13.f | odd | 12 | 1 | 819.2.gh.b | 28 | ||
21.c | even | 2 | 1 | 637.2.bb.a | 28 | ||
21.g | even | 6 | 1 | 91.2.w.a | ✓ | 28 | |
21.g | even | 6 | 1 | 637.2.bd.a | 28 | ||
21.h | odd | 6 | 1 | 637.2.x.a | 28 | ||
21.h | odd | 6 | 1 | 637.2.bd.b | 28 | ||
39.k | even | 12 | 1 | 91.2.w.a | ✓ | 28 | |
91.ba | even | 12 | 1 | inner | 819.2.et.b | 28 | |
273.bs | odd | 12 | 1 | 91.2.ba.a | yes | 28 | |
273.bv | even | 12 | 1 | 637.2.bb.a | 28 | ||
273.bw | even | 12 | 1 | 637.2.bd.a | 28 | ||
273.ca | odd | 12 | 1 | 637.2.x.a | 28 | ||
273.ch | odd | 12 | 1 | 637.2.bd.b | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.w.a | ✓ | 28 | 21.g | even | 6 | 1 | |
91.2.w.a | ✓ | 28 | 39.k | even | 12 | 1 | |
91.2.ba.a | yes | 28 | 3.b | odd | 2 | 1 | |
91.2.ba.a | yes | 28 | 273.bs | odd | 12 | 1 | |
637.2.x.a | 28 | 21.h | odd | 6 | 1 | ||
637.2.x.a | 28 | 273.ca | odd | 12 | 1 | ||
637.2.bb.a | 28 | 21.c | even | 2 | 1 | ||
637.2.bb.a | 28 | 273.bv | even | 12 | 1 | ||
637.2.bd.a | 28 | 21.g | even | 6 | 1 | ||
637.2.bd.a | 28 | 273.bw | even | 12 | 1 | ||
637.2.bd.b | 28 | 21.h | odd | 6 | 1 | ||
637.2.bd.b | 28 | 273.ch | odd | 12 | 1 | ||
819.2.et.b | 28 | 1.a | even | 1 | 1 | trivial | |
819.2.et.b | 28 | 91.ba | even | 12 | 1 | inner | |
819.2.gh.b | 28 | 7.d | odd | 6 | 1 | ||
819.2.gh.b | 28 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 2 T_{2}^{27} + 2 T_{2}^{26} + 77 T_{2}^{24} - 152 T_{2}^{23} + 150 T_{2}^{22} - 6 T_{2}^{21} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).