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: | \(40\) |
Relative dimension: | \(10\) 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 | −2.39126 | − | 0.640737i | 0 | 3.57554 | + | 2.06434i | 1.06950 | − | 0.286571i | 0 | 0.327682 | − | 2.62538i | −3.72630 | − | 3.72630i | 0 | −2.74106 | ||||||||
19.2 | −2.18205 | − | 0.584679i | 0 | 2.68745 | + | 1.55160i | −2.27456 | + | 0.609466i | 0 | −2.33957 | + | 1.23548i | −1.76223 | − | 1.76223i | 0 | 5.31955 | ||||||||
19.3 | −1.41061 | − | 0.377973i | 0 | 0.114915 | + | 0.0663464i | −1.70489 | + | 0.456824i | 0 | 1.96341 | + | 1.77342i | 1.92826 | + | 1.92826i | 0 | 2.57761 | ||||||||
19.4 | −1.07087 | − | 0.286939i | 0 | −0.667621 | − | 0.385451i | 3.71307 | − | 0.994915i | 0 | −1.35644 | + | 2.27158i | 2.17220 | + | 2.17220i | 0 | −4.26170 | ||||||||
19.5 | −0.446344 | − | 0.119598i | 0 | −1.54713 | − | 0.893237i | 2.09830 | − | 0.562238i | 0 | −1.26927 | − | 2.32141i | 1.23722 | + | 1.23722i | 0 | −1.00381 | ||||||||
19.6 | −0.251431 | − | 0.0673706i | 0 | −1.67337 | − | 0.966122i | −3.35960 | + | 0.900201i | 0 | 0.422545 | − | 2.61179i | 0.723769 | + | 0.723769i | 0 | 0.905353 | ||||||||
19.7 | 1.46932 | + | 0.393703i | 0 | 0.271846 | + | 0.156950i | 3.59085 | − | 0.962166i | 0 | 2.64173 | − | 0.145891i | −1.81360 | − | 1.81360i | 0 | 5.65491 | ||||||||
19.8 | 1.49615 | + | 0.400893i | 0 | 0.345704 | + | 0.199592i | −0.481371 | + | 0.128983i | 0 | −2.58563 | − | 0.560827i | −1.75331 | − | 1.75331i | 0 | −0.771912 | ||||||||
19.9 | 2.14623 | + | 0.575080i | 0 | 2.54352 | + | 1.46850i | −3.44337 | + | 0.922649i | 0 | 2.25660 | + | 1.38122i | 1.47218 | + | 1.47218i | 0 | −7.92086 | ||||||||
19.10 | 2.64088 | + | 0.707621i | 0 | 4.74145 | + | 2.73748i | 0.792066 | − | 0.212233i | 0 | −1.19502 | − | 2.36049i | 6.71797 | + | 6.71797i | 0 | 2.24193 | ||||||||
262.1 | −0.680470 | − | 2.53955i | 0 | −4.25421 | + | 2.45617i | −0.134776 | + | 0.502992i | 0 | −2.56445 | − | 0.650845i | 5.41427 | + | 5.41427i | 0 | 1.36908 | ||||||||
262.2 | −0.568195 | − | 2.12053i | 0 | −2.44176 | + | 1.40975i | 0.248247 | − | 0.926472i | 0 | 1.82268 | + | 1.91776i | 1.27214 | + | 1.27214i | 0 | −2.10567 | ||||||||
262.3 | −0.323834 | − | 1.20856i | 0 | 0.376291 | − | 0.217252i | −0.986163 | + | 3.68041i | 0 | −1.80936 | + | 1.93034i | −2.15388 | − | 2.15388i | 0 | 4.76737 | ||||||||
262.4 | −0.306419 | − | 1.14357i | 0 | 0.518187 | − | 0.299176i | 0.424345 | − | 1.58368i | 0 | 1.65856 | − | 2.06135i | −2.17522 | − | 2.17522i | 0 | −1.94108 | ||||||||
262.5 | −0.170536 | − | 0.636449i | 0 | 1.35607 | − | 0.782925i | 1.02345 | − | 3.81958i | 0 | −2.47383 | + | 0.938166i | −1.66138 | − | 1.66138i | 0 | −2.60550 | ||||||||
262.6 | 0.0881329 | + | 0.328916i | 0 | 1.63163 | − | 0.942023i | −0.552877 | + | 2.06336i | 0 | 2.31170 | + | 1.28688i | 0.935214 | + | 0.935214i | 0 | −0.727401 | ||||||||
262.7 | 0.185934 | + | 0.693915i | 0 | 1.28510 | − | 0.741955i | −0.295002 | + | 1.10096i | 0 | −0.650764 | − | 2.56447i | 1.76976 | + | 1.76976i | 0 | −0.818827 | ||||||||
262.8 | 0.449968 | + | 1.67930i | 0 | −0.885540 | + | 0.511267i | 0.524353 | − | 1.95691i | 0 | −0.616155 | + | 2.57300i | 1.20163 | + | 1.20163i | 0 | 3.52219 | ||||||||
262.9 | 0.604879 | + | 2.25744i | 0 | −2.99811 | + | 1.73096i | −0.721604 | + | 2.69306i | 0 | −1.83932 | + | 1.90181i | −2.41591 | − | 2.41591i | 0 | −6.51591 | ||||||||
262.10 | 0.720539 | + | 2.68909i | 0 | −4.97997 | + | 2.87519i | 0.470023 | − | 1.75415i | 0 | 1.29491 | − | 2.30721i | −7.38279 | − | 7.38279i | 0 | 5.05574 | ||||||||
See all 40 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.d | 40 | |
3.b | odd | 2 | 1 | 273.2.cg.b | yes | 40 | |
7.d | odd | 6 | 1 | 819.2.et.d | 40 | ||
13.f | odd | 12 | 1 | 819.2.et.d | 40 | ||
21.g | even | 6 | 1 | 273.2.bt.b | ✓ | 40 | |
39.k | even | 12 | 1 | 273.2.bt.b | ✓ | 40 | |
91.w | even | 12 | 1 | inner | 819.2.gh.d | 40 | |
273.ch | odd | 12 | 1 | 273.2.cg.b | yes | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.bt.b | ✓ | 40 | 21.g | even | 6 | 1 | |
273.2.bt.b | ✓ | 40 | 39.k | even | 12 | 1 | |
273.2.cg.b | yes | 40 | 3.b | odd | 2 | 1 | |
273.2.cg.b | yes | 40 | 273.ch | odd | 12 | 1 | |
819.2.et.d | 40 | 7.d | odd | 6 | 1 | ||
819.2.et.d | 40 | 13.f | odd | 12 | 1 | ||
819.2.gh.d | 40 | 1.a | even | 1 | 1 | trivial | |
819.2.gh.d | 40 | 91.w | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{40} - 84 T_{2}^{36} - 34 T_{2}^{35} + 208 T_{2}^{33} + 5191 T_{2}^{32} + 2828 T_{2}^{31} - 1657 T_{2}^{30} - 8230 T_{2}^{29} - 141264 T_{2}^{28} - 109820 T_{2}^{27} + 56586 T_{2}^{26} + 355194 T_{2}^{25} + \cdots + 59049 \)
acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).