Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(19,819)]
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, 10, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.gh (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: | \(36\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 273) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −2.45222 | − | 0.657070i | 0 | 3.84958 | + | 2.22256i | 2.20549 | − | 0.590961i | 0 | 2.58986 | + | 0.540936i | −4.38935 | − | 4.38935i | 0 | −5.79666 | ||||||||
19.2 | −2.02536 | − | 0.542694i | 0 | 2.07553 | + | 1.19831i | −1.89270 | + | 0.507149i | 0 | −1.04247 | + | 2.43172i | −0.588043 | − | 0.588043i | 0 | 4.10864 | ||||||||
19.3 | −1.64664 | − | 0.441217i | 0 | 0.784712 | + | 0.453054i | −1.35835 | + | 0.363968i | 0 | 0.501823 | − | 2.59772i | 1.31861 | + | 1.31861i | 0 | 2.39731 | ||||||||
19.4 | −0.562494 | − | 0.150720i | 0 | −1.43837 | − | 0.830442i | 0.672922 | − | 0.180309i | 0 | −2.49458 | − | 0.881516i | 1.50746 | + | 1.50746i | 0 | −0.405691 | ||||||||
19.5 | 0.511829 | + | 0.137144i | 0 | −1.48889 | − | 0.859611i | 2.03763 | − | 0.545981i | 0 | 0.917679 | + | 2.48150i | −1.39354 | − | 1.39354i | 0 | 1.11780 | ||||||||
19.6 | 0.759625 | + | 0.203541i | 0 | −1.19645 | − | 0.690770i | −2.02645 | + | 0.542987i | 0 | 1.65876 | − | 2.06120i | −1.88042 | − | 1.88042i | 0 | −1.64986 | ||||||||
19.7 | 1.50246 | + | 0.402582i | 0 | 0.363252 | + | 0.209723i | −2.78312 | + | 0.745735i | 0 | 0.599883 | + | 2.57685i | −1.73841 | − | 1.73841i | 0 | −4.48173 | ||||||||
19.8 | 1.53002 | + | 0.409967i | 0 | 0.440828 | + | 0.254512i | 3.63307 | − | 0.973479i | 0 | −1.31124 | − | 2.29797i | −1.66997 | − | 1.66997i | 0 | 5.95776 | ||||||||
19.9 | 2.38279 | + | 0.638467i | 0 | 3.53801 | + | 2.04267i | −0.488495 | + | 0.130892i | 0 | −2.15177 | + | 1.53945i | 3.63751 | + | 3.63751i | 0 | −1.24755 | ||||||||
262.1 | −0.705851 | − | 2.63427i | 0 | −4.70911 | + | 2.71881i | 0.914933 | − | 3.41458i | 0 | 2.62078 | − | 0.362619i | 6.62917 | + | 6.62917i | 0 | −9.64073 | ||||||||
262.2 | −0.569735 | − | 2.12628i | 0 | −2.46442 | + | 1.42283i | −0.837395 | + | 3.12520i | 0 | 0.780325 | − | 2.52806i | 1.31631 | + | 1.31631i | 0 | 7.12215 | ||||||||
262.3 | −0.255728 | − | 0.954388i | 0 | 0.886590 | − | 0.511873i | 0.244406 | − | 0.912136i | 0 | 2.46785 | − | 0.953787i | −2.11257 | − | 2.11257i | 0 | −0.933034 | ||||||||
262.4 | −0.157691 | − | 0.588511i | 0 | 1.41057 | − | 0.814394i | −0.529856 | + | 1.97745i | 0 | −2.23905 | − | 1.40948i | −1.56335 | − | 1.56335i | 0 | 1.24730 | ||||||||
262.5 | −0.0407083 | − | 0.151925i | 0 | 1.71063 | − | 0.987631i | 0.570893 | − | 2.13060i | 0 | 0.961303 | + | 2.46493i | −0.442117 | − | 0.442117i | 0 | −0.346933 | ||||||||
262.6 | 0.273028 | + | 1.01896i | 0 | 0.768324 | − | 0.443592i | −1.01943 | + | 3.80456i | 0 | 1.44977 | + | 2.21318i | 2.15363 | + | 2.15363i | 0 | −4.15502 | ||||||||
262.7 | 0.339011 | + | 1.26521i | 0 | 0.246231 | − | 0.142161i | −0.109857 | + | 0.409991i | 0 | −2.64485 | − | 0.0688957i | 2.11573 | + | 2.11573i | 0 | −0.555967 | ||||||||
262.8 | 0.478662 | + | 1.78639i | 0 | −1.23003 | + | 0.710156i | 0.0199621 | − | 0.0744995i | 0 | 1.85948 | − | 1.88211i | 0.758075 | + | 0.758075i | 0 | 0.142640 | ||||||||
262.9 | 0.639011 | + | 2.38482i | 0 | −3.54699 | + | 2.04786i | 0.746344 | − | 2.78539i | 0 | −2.52355 | + | 0.794791i | −3.65872 | − | 3.65872i | 0 | 7.11959 | ||||||||
388.1 | −2.45222 | + | 0.657070i | 0 | 3.84958 | − | 2.22256i | 2.20549 | + | 0.590961i | 0 | 2.58986 | − | 0.540936i | −4.38935 | + | 4.38935i | 0 | −5.79666 | ||||||||
388.2 | −2.02536 | + | 0.542694i | 0 | 2.07553 | − | 1.19831i | −1.89270 | − | 0.507149i | 0 | −1.04247 | − | 2.43172i | −0.588043 | + | 0.588043i | 0 | 4.10864 | ||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.w | 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.gh.c | 36 | |
3.b | odd | 2 | 1 | 273.2.cg.a | yes | 36 | |
7.d | odd | 6 | 1 | 819.2.et.c | 36 | ||
13.f | odd | 12 | 1 | 819.2.et.c | 36 | ||
21.g | even | 6 | 1 | 273.2.bt.a | ✓ | 36 | |
39.k | even | 12 | 1 | 273.2.bt.a | ✓ | 36 | |
91.w | even | 12 | 1 | inner | 819.2.gh.c | 36 | |
273.ch | odd | 12 | 1 | 273.2.cg.a | yes | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.bt.a | ✓ | 36 | 21.g | even | 6 | 1 | |
273.2.bt.a | ✓ | 36 | 39.k | even | 12 | 1 | |
273.2.cg.a | yes | 36 | 3.b | odd | 2 | 1 | |
273.2.cg.a | yes | 36 | 273.ch | odd | 12 | 1 | |
819.2.et.c | 36 | 7.d | odd | 6 | 1 | ||
819.2.et.c | 36 | 13.f | odd | 12 | 1 | ||
819.2.gh.c | 36 | 1.a | even | 1 | 1 | trivial | |
819.2.gh.c | 36 | 91.w | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 63 T_{2}^{32} + 38 T_{2}^{31} - 224 T_{2}^{29} + 3079 T_{2}^{28} - 1750 T_{2}^{27} + \cdots + 2304 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).