Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(269,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.269");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.bb (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: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(76\) |
Relative dimension: | \(38\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
269.1 | − | 1.00000i | −1.68782 | − | 0.388948i | −1.00000 | −0.386492 | − | 0.669424i | −0.388948 | + | 1.68782i | 1.88739 | + | 1.85412i | 1.00000i | 2.69744 | + | 1.31294i | −0.669424 | + | 0.386492i | |||||
269.2 | − | 1.00000i | −1.59766 | + | 0.668950i | −1.00000 | −0.356048 | − | 0.616693i | 0.668950 | + | 1.59766i | 2.34738 | − | 1.22057i | 1.00000i | 2.10501 | − | 2.13750i | −0.616693 | + | 0.356048i | |||||
269.3 | − | 1.00000i | −1.54137 | + | 0.790045i | −1.00000 | 0.886718 | + | 1.53584i | 0.790045 | + | 1.54137i | −1.67305 | + | 2.04961i | 1.00000i | 1.75166 | − | 2.43551i | 1.53584 | − | 0.886718i | |||||
269.4 | − | 1.00000i | −1.48241 | − | 0.895807i | −1.00000 | 0.699535 | + | 1.21163i | −0.895807 | + | 1.48241i | −2.51159 | + | 0.831830i | 1.00000i | 1.39506 | + | 2.65590i | 1.21163 | − | 0.699535i | |||||
269.5 | − | 1.00000i | −1.42260 | − | 0.988029i | −1.00000 | 1.41378 | + | 2.44875i | −0.988029 | + | 1.42260i | 0.383892 | − | 2.61775i | 1.00000i | 1.04760 | + | 2.81115i | 2.44875 | − | 1.41378i | |||||
269.6 | − | 1.00000i | −0.926039 | + | 1.46371i | −1.00000 | −2.14723 | − | 3.71911i | 1.46371 | + | 0.926039i | −0.786623 | + | 2.52611i | 1.00000i | −1.28490 | − | 2.71091i | −3.71911 | + | 2.14723i | |||||
269.7 | − | 1.00000i | −0.834971 | + | 1.51751i | −1.00000 | 0.277881 | + | 0.481304i | 1.51751 | + | 0.834971i | −1.71510 | − | 2.01455i | 1.00000i | −1.60565 | − | 2.53415i | 0.481304 | − | 0.277881i | |||||
269.8 | − | 1.00000i | −0.592023 | + | 1.62773i | −1.00000 | 2.12998 | + | 3.68924i | 1.62773 | + | 0.592023i | 2.63074 | + | 0.281454i | 1.00000i | −2.29902 | − | 1.92731i | 3.68924 | − | 2.12998i | |||||
269.9 | − | 1.00000i | −0.459307 | − | 1.67004i | −1.00000 | −1.40279 | − | 2.42970i | −1.67004 | + | 0.459307i | 1.21733 | + | 2.34907i | 1.00000i | −2.57807 | + | 1.53412i | −2.42970 | + | 1.40279i | |||||
269.10 | − | 1.00000i | −0.215340 | − | 1.71861i | −1.00000 | −1.07511 | − | 1.86215i | −1.71861 | + | 0.215340i | −2.48199 | − | 0.916357i | 1.00000i | −2.90726 | + | 0.740172i | −1.86215 | + | 1.07511i | |||||
269.11 | − | 1.00000i | 0.201730 | − | 1.72026i | −1.00000 | 1.75377 | + | 3.03762i | −1.72026 | − | 0.201730i | 1.27169 | + | 2.32009i | 1.00000i | −2.91861 | − | 0.694059i | 3.03762 | − | 1.75377i | |||||
269.12 | − | 1.00000i | 0.691481 | + | 1.58803i | −1.00000 | −0.349437 | − | 0.605242i | 1.58803 | − | 0.691481i | 2.58515 | − | 0.563029i | 1.00000i | −2.04371 | + | 2.19619i | −0.605242 | + | 0.349437i | |||||
269.13 | − | 1.00000i | 1.03468 | − | 1.38904i | −1.00000 | 0.587127 | + | 1.01693i | −1.38904 | − | 1.03468i | −0.859975 | − | 2.50209i | 1.00000i | −0.858887 | − | 2.87442i | 1.01693 | − | 0.587127i | |||||
269.14 | − | 1.00000i | 1.15993 | + | 1.28630i | −1.00000 | −0.0272832 | − | 0.0472559i | 1.28630 | − | 1.15993i | 0.114956 | + | 2.64325i | 1.00000i | −0.309116 | + | 2.98403i | −0.0472559 | + | 0.0272832i | |||||
269.15 | − | 1.00000i | 1.32763 | + | 1.11238i | −1.00000 | −1.76347 | − | 3.05441i | 1.11238 | − | 1.32763i | −2.64100 | + | 0.158422i | 1.00000i | 0.525225 | + | 2.95367i | −3.05441 | + | 1.76347i | |||||
269.16 | − | 1.00000i | 1.34890 | − | 1.08650i | −1.00000 | −0.279399 | − | 0.483934i | −1.08650 | − | 1.34890i | 2.15416 | + | 1.53609i | 1.00000i | 0.639049 | − | 2.93115i | −0.483934 | + | 0.279399i | |||||
269.17 | − | 1.00000i | 1.63536 | − | 0.570612i | −1.00000 | −1.99252 | − | 3.45115i | −0.570612 | − | 1.63536i | 2.09752 | − | 1.61258i | 1.00000i | 2.34880 | − | 1.86631i | −3.45115 | + | 1.99252i | |||||
269.18 | − | 1.00000i | 1.63815 | + | 0.562548i | −1.00000 | 0.604752 | + | 1.04746i | 0.562548 | − | 1.63815i | 0.374319 | − | 2.61914i | 1.00000i | 2.36708 | + | 1.84308i | 1.04746 | − | 0.604752i | |||||
269.19 | − | 1.00000i | 1.72167 | − | 0.189348i | −1.00000 | 1.42622 | + | 2.47029i | −0.189348 | − | 1.72167i | −1.89518 | + | 1.84615i | 1.00000i | 2.92829 | − | 0.651988i | 2.47029 | − | 1.42622i | |||||
269.20 | 1.00000i | −1.73168 | − | 0.0356471i | −1.00000 | −0.277881 | − | 0.481304i | 0.0356471 | − | 1.73168i | −1.71510 | − | 2.01455i | − | 1.00000i | 2.99746 | + | 0.123459i | 0.481304 | − | 0.277881i | |||||
See all 76 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
91.v | odd | 6 | 1 | inner |
273.r | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.bb.a | yes | 76 |
3.b | odd | 2 | 1 | inner | 546.2.bb.a | yes | 76 |
7.d | odd | 6 | 1 | 546.2.u.a | ✓ | 76 | |
13.c | even | 3 | 1 | 546.2.u.a | ✓ | 76 | |
21.g | even | 6 | 1 | 546.2.u.a | ✓ | 76 | |
39.i | odd | 6 | 1 | 546.2.u.a | ✓ | 76 | |
91.v | odd | 6 | 1 | inner | 546.2.bb.a | yes | 76 |
273.r | even | 6 | 1 | inner | 546.2.bb.a | yes | 76 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.u.a | ✓ | 76 | 7.d | odd | 6 | 1 | |
546.2.u.a | ✓ | 76 | 13.c | even | 3 | 1 | |
546.2.u.a | ✓ | 76 | 21.g | even | 6 | 1 | |
546.2.u.a | ✓ | 76 | 39.i | odd | 6 | 1 | |
546.2.bb.a | yes | 76 | 1.a | even | 1 | 1 | trivial |
546.2.bb.a | yes | 76 | 3.b | odd | 2 | 1 | inner |
546.2.bb.a | yes | 76 | 91.v | odd | 6 | 1 | inner |
546.2.bb.a | yes | 76 | 273.r | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(546, [\chi])\).