Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(311,819)]
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([5, 1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.311");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.bt (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: | \(216\) |
Relative dimension: | \(108\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
311.1 | −1.40095 | + | 2.42652i | −0.702444 | − | 1.58322i | −2.92535 | − | 5.06685i | 0.894814i | 4.82580 | + | 0.513515i | −2.64130 | + | 0.153392i | 10.7893 | −2.01314 | + | 2.22424i | −2.17129 | − | 1.25359i | ||||
311.2 | −1.35333 | + | 2.34403i | −1.40179 | + | 1.01734i | −2.66299 | − | 4.61243i | 3.44110i | −0.487594 | − | 4.66264i | 1.60270 | − | 2.10508i | 9.00228 | 0.930039 | − | 2.85220i | −8.06605 | − | 4.65693i | ||||
311.3 | −1.32765 | + | 2.29956i | 1.59935 | − | 0.664889i | −2.52533 | − | 4.37399i | − | 0.104647i | −0.594429 | + | 4.56055i | 0.0645307 | + | 2.64496i | 8.10041 | 2.11585 | − | 2.12678i | 0.240642 | + | 0.138935i | |||
311.4 | −1.30953 | + | 2.26817i | −0.0246295 | − | 1.73188i | −2.42973 | − | 4.20841i | − | 3.59259i | 3.96044 | + | 2.21208i | 2.64514 | + | 0.0566951i | 7.48906 | −2.99879 | + | 0.0853104i | 8.14860 | + | 4.70460i | |||
311.5 | −1.28708 | + | 2.22930i | 1.35648 | + | 1.07702i | −2.31317 | − | 4.00654i | 3.96056i | −4.14690 | + | 1.63777i | −2.07605 | − | 1.64013i | 6.76067 | 0.680051 | + | 2.92191i | −8.82926 | − | 5.09757i | ||||
311.6 | −1.28180 | + | 2.22015i | −0.598913 | + | 1.62521i | −2.28603 | − | 3.95952i | − | 1.44047i | −2.84051 | − | 3.41287i | −2.34947 | + | 1.21655i | 6.59374 | −2.28261 | − | 1.94672i | 3.19805 | + | 1.84640i | |||
311.7 | −1.27852 | + | 2.21445i | 1.12633 | + | 1.31582i | −2.26920 | − | 3.93038i | 0.464206i | −4.35385 | + | 0.811922i | 0.656915 | + | 2.56290i | 6.49078 | −0.462745 | + | 2.96410i | −1.02796 | − | 0.593494i | ||||
311.8 | −1.27410 | + | 2.20681i | −1.62396 | + | 0.602296i | −2.24667 | − | 3.89135i | − | 2.25662i | 0.739935 | − | 4.35115i | 1.17180 | + | 2.37211i | 6.35354 | 2.27448 | − | 1.95621i | 4.97992 | + | 2.87516i | |||
311.9 | −1.25337 | + | 2.17089i | 1.59163 | − | 0.683159i | −2.14186 | − | 3.70980i | 0.964453i | −0.511832 | + | 4.31151i | 2.17734 | − | 1.50306i | 5.72465 | 2.06659 | − | 2.17468i | −2.09373 | − | 1.20881i | ||||
311.10 | −1.25223 | + | 2.16893i | −1.69656 | − | 0.348837i | −2.13616 | − | 3.69995i | − | 1.97217i | 2.88109 | − | 3.24289i | −0.596159 | − | 2.57771i | 5.69096 | 2.75662 | + | 1.18365i | 4.27749 | + | 2.46961i | |||
311.11 | −1.14095 | + | 1.97618i | 1.26485 | − | 1.18328i | −1.60353 | − | 2.77740i | − | 3.02648i | 0.895260 | + | 3.84964i | −1.99344 | − | 1.73960i | 2.75440 | 0.199675 | − | 2.99335i | 5.98088 | + | 3.45306i | |||
311.12 | −1.13737 | + | 1.96999i | −0.0777012 | − | 1.73031i | −1.58724 | − | 2.74918i | 2.13073i | 3.49706 | + | 1.81494i | 2.18018 | + | 1.49894i | 2.67163 | −2.98793 | + | 0.268894i | −4.19751 | − | 2.42343i | ||||
311.13 | −1.11525 | + | 1.93167i | −0.889365 | + | 1.48628i | −1.48756 | − | 2.57654i | − | 0.411732i | −1.87914 | − | 3.37553i | 2.63344 | − | 0.254908i | 2.17503 | −1.41806 | − | 2.64369i | 0.795331 | + | 0.459185i | |||
311.14 | −1.10144 | + | 1.90775i | 1.58497 | + | 0.698481i | −1.42634 | − | 2.47050i | − | 3.02757i | −3.07828 | + | 2.25439i | −2.52718 | + | 0.783189i | 1.87836 | 2.02425 | + | 2.21414i | 5.77586 | + | 3.33469i | |||
311.15 | −1.09691 | + | 1.89991i | 0.772711 | − | 1.55013i | −1.40643 | − | 2.43601i | 2.63971i | 2.09752 | + | 3.16844i | −2.34387 | − | 1.22731i | 1.78327 | −1.80584 | − | 2.39561i | −5.01521 | − | 2.89553i | ||||
311.16 | −1.07304 | + | 1.85856i | −1.08685 | − | 1.34861i | −1.30283 | − | 2.25657i | 2.74290i | 3.67271 | − | 0.572861i | 1.75443 | − | 1.98040i | 1.29979 | −0.637512 | + | 2.93148i | −5.09784 | − | 2.94324i | ||||
311.17 | −1.02705 | + | 1.77891i | 1.58175 | + | 0.705730i | −1.10968 | − | 1.92202i | − | 0.337826i | −2.87997 | + | 2.08897i | 0.801180 | − | 2.52153i | 0.450575 | 2.00389 | + | 2.23258i | 0.600961 | + | 0.346965i | |||
311.18 | −1.02089 | + | 1.76823i | −1.72010 | + | 0.203153i | −1.08442 | − | 1.87827i | 3.06373i | 1.39680 | − | 3.24891i | −1.72300 | + | 2.00780i | 0.344720 | 2.91746 | − | 0.698885i | −5.41737 | − | 3.12772i | ||||
311.19 | −1.00192 | + | 1.73537i | −1.09620 | − | 1.34102i | −1.00767 | − | 1.74534i | − | 3.71228i | 3.42547 | − | 0.558718i | −0.907306 | + | 2.48532i | 0.0307381 | −0.596692 | + | 2.94006i | 6.44217 | + | 3.71939i | |||
311.20 | −0.988721 | + | 1.71252i | 0.00938524 | + | 1.73203i | −0.955139 | − | 1.65435i | 0.999327i | −2.97540 | − | 1.69642i | −1.75401 | − | 1.98077i | −0.177420 | −2.99982 | + | 0.0325109i | −1.71136 | − | 0.988056i | ||||
See next 80 embeddings (of 216 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
63.s | even | 6 | 1 | inner |
819.bt | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.bt.a | ✓ | 216 |
7.d | odd | 6 | 1 | 819.2.eg.a | yes | 216 | |
9.d | odd | 6 | 1 | 819.2.eg.a | yes | 216 | |
13.b | even | 2 | 1 | inner | 819.2.bt.a | ✓ | 216 |
63.s | even | 6 | 1 | inner | 819.2.bt.a | ✓ | 216 |
91.s | odd | 6 | 1 | 819.2.eg.a | yes | 216 | |
117.n | odd | 6 | 1 | 819.2.eg.a | yes | 216 | |
819.bt | even | 6 | 1 | inner | 819.2.bt.a | ✓ | 216 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
819.2.bt.a | ✓ | 216 | 1.a | even | 1 | 1 | trivial |
819.2.bt.a | ✓ | 216 | 13.b | even | 2 | 1 | inner |
819.2.bt.a | ✓ | 216 | 63.s | even | 6 | 1 | inner |
819.2.bt.a | ✓ | 216 | 819.bt | even | 6 | 1 | inner |
819.2.eg.a | yes | 216 | 7.d | odd | 6 | 1 | |
819.2.eg.a | yes | 216 | 9.d | odd | 6 | 1 | |
819.2.eg.a | yes | 216 | 91.s | odd | 6 | 1 | |
819.2.eg.a | yes | 216 | 117.n | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).