Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(419,546)]
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, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.419");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.bq (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: | \(64\) |
Relative dimension: | \(32\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
419.1 | −0.866025 | + | 0.500000i | −1.73204 | − | 0.00466741i | 0.500000 | − | 0.866025i | 0.915374 | 1.50233 | − | 0.861980i | −2.64454 | + | 0.0800636i | 1.00000i | 2.99996 | + | 0.0161683i | −0.792737 | + | 0.457687i | ||||
419.2 | −0.866025 | + | 0.500000i | −1.73041 | + | 0.0753636i | 0.500000 | − | 0.866025i | −2.24165 | 1.46090 | − | 0.930472i | −0.301604 | − | 2.62850i | 1.00000i | 2.98864 | − | 0.260820i | 1.94133 | − | 1.12083i | ||||
419.3 | −0.866025 | + | 0.500000i | −1.69217 | + | 0.369559i | 0.500000 | − | 0.866025i | −3.03607 | 1.28068 | − | 1.16613i | 2.25111 | + | 1.39014i | 1.00000i | 2.72685 | − | 1.25071i | 2.62932 | − | 1.51804i | ||||
419.4 | −0.866025 | + | 0.500000i | −1.66375 | − | 0.481611i | 0.500000 | − | 0.866025i | 4.01418 | 1.68165 | − | 0.414785i | 1.77617 | − | 1.96093i | 1.00000i | 2.53610 | + | 1.60256i | −3.47639 | + | 2.00709i | ||||
419.5 | −0.866025 | + | 0.500000i | −1.22965 | + | 1.21982i | 0.500000 | − | 0.866025i | 1.88020 | 0.454992 | − | 1.67122i | −1.36777 | + | 2.26478i | 1.00000i | 0.0240567 | − | 2.99990i | −1.62830 | + | 0.940099i | ||||
419.6 | −0.866025 | + | 0.500000i | −0.998793 | − | 1.41507i | 0.500000 | − | 0.866025i | 1.30549 | 1.57251 | + | 0.726086i | 2.26007 | + | 1.37552i | 1.00000i | −1.00482 | + | 2.82672i | −1.13058 | + | 0.652743i | ||||
419.7 | −0.866025 | + | 0.500000i | −0.344709 | + | 1.69740i | 0.500000 | − | 0.866025i | 1.64213 | −0.550175 | − | 1.64235i | −1.55066 | − | 2.14370i | 1.00000i | −2.76235 | − | 1.17022i | −1.42213 | + | 0.821065i | ||||
419.8 | −0.866025 | + | 0.500000i | −0.211712 | + | 1.71906i | 0.500000 | − | 0.866025i | −0.932541 | −0.676184 | − | 1.59461i | 2.18650 | − | 1.48970i | 1.00000i | −2.91036 | − | 0.727891i | 0.807604 | − | 0.466271i | ||||
419.9 | −0.866025 | + | 0.500000i | 0.211712 | − | 1.71906i | 0.500000 | − | 0.866025i | 0.932541 | 0.676184 | + | 1.59461i | 0.196871 | − | 2.63842i | 1.00000i | −2.91036 | − | 0.727891i | −0.807604 | + | 0.466271i | ||||
419.10 | −0.866025 | + | 0.500000i | 0.344709 | − | 1.69740i | 0.500000 | − | 0.866025i | −1.64213 | 0.550175 | + | 1.64235i | 2.63183 | + | 0.271064i | 1.00000i | −2.76235 | − | 1.17022i | 1.42213 | − | 0.821065i | ||||
419.11 | −0.866025 | + | 0.500000i | 0.998793 | + | 1.41507i | 0.500000 | − | 0.866025i | −1.30549 | −1.57251 | − | 0.726086i | −2.32128 | − | 1.26952i | 1.00000i | −1.00482 | + | 2.82672i | 1.13058 | − | 0.652743i | ||||
419.12 | −0.866025 | + | 0.500000i | 1.22965 | − | 1.21982i | 0.500000 | − | 0.866025i | −1.88020 | −0.454992 | + | 1.67122i | −1.27747 | + | 2.31691i | 1.00000i | 0.0240567 | − | 2.99990i | 1.62830 | − | 0.940099i | ||||
419.13 | −0.866025 | + | 0.500000i | 1.66375 | + | 0.481611i | 0.500000 | − | 0.866025i | −4.01418 | −1.68165 | + | 0.414785i | 0.810131 | − | 2.51867i | 1.00000i | 2.53610 | + | 1.60256i | 3.47639 | − | 2.00709i | ||||
419.14 | −0.866025 | + | 0.500000i | 1.69217 | − | 0.369559i | 0.500000 | − | 0.866025i | 3.03607 | −1.28068 | + | 1.16613i | −2.32945 | − | 1.25445i | 1.00000i | 2.72685 | − | 1.25071i | −2.62932 | + | 1.51804i | ||||
419.15 | −0.866025 | + | 0.500000i | 1.73041 | − | 0.0753636i | 0.500000 | − | 0.866025i | 2.24165 | −1.46090 | + | 0.930472i | 2.42715 | − | 1.05306i | 1.00000i | 2.98864 | − | 0.260820i | −1.94133 | + | 1.12083i | ||||
419.16 | −0.866025 | + | 0.500000i | 1.73204 | + | 0.00466741i | 0.500000 | − | 0.866025i | −0.915374 | −1.50233 | + | 0.861980i | 1.25293 | + | 2.33027i | 1.00000i | 2.99996 | + | 0.0161683i | 0.792737 | − | 0.457687i | ||||
419.17 | 0.866025 | − | 0.500000i | −1.67122 | + | 0.454992i | 0.500000 | − | 0.866025i | −1.88020 | −1.21982 | + | 1.22965i | −1.36777 | + | 2.26478i | − | 1.00000i | 2.58596 | − | 1.52079i | −1.62830 | + | 0.940099i | |||
419.18 | 0.866025 | − | 0.500000i | −1.64235 | − | 0.550175i | 0.500000 | − | 0.866025i | −1.64213 | −1.69740 | + | 0.344709i | −1.55066 | − | 2.14370i | − | 1.00000i | 2.39462 | + | 1.80716i | −1.42213 | + | 0.821065i | |||
419.19 | 0.866025 | − | 0.500000i | −1.59461 | − | 0.676184i | 0.500000 | − | 0.866025i | 0.932541 | −1.71906 | + | 0.211712i | 2.18650 | − | 1.48970i | − | 1.00000i | 2.08555 | + | 2.15650i | 0.807604 | − | 0.466271i | |||
419.20 | 0.866025 | − | 0.500000i | −1.16613 | + | 1.28068i | 0.500000 | − | 0.866025i | 3.03607 | −0.369559 | + | 1.69217i | 2.25111 | + | 1.39014i | − | 1.00000i | −0.280280 | − | 2.98688i | 2.62932 | − | 1.51804i | |||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
21.c | even | 2 | 1 | inner |
39.i | odd | 6 | 1 | inner |
91.n | odd | 6 | 1 | inner |
273.bn | 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.bq.c | ✓ | 64 |
3.b | odd | 2 | 1 | inner | 546.2.bq.c | ✓ | 64 |
7.b | odd | 2 | 1 | inner | 546.2.bq.c | ✓ | 64 |
13.c | even | 3 | 1 | inner | 546.2.bq.c | ✓ | 64 |
21.c | even | 2 | 1 | inner | 546.2.bq.c | ✓ | 64 |
39.i | odd | 6 | 1 | inner | 546.2.bq.c | ✓ | 64 |
91.n | odd | 6 | 1 | inner | 546.2.bq.c | ✓ | 64 |
273.bn | even | 6 | 1 | inner | 546.2.bq.c | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.bq.c | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
546.2.bq.c | ✓ | 64 | 3.b | odd | 2 | 1 | inner |
546.2.bq.c | ✓ | 64 | 7.b | odd | 2 | 1 | inner |
546.2.bq.c | ✓ | 64 | 13.c | even | 3 | 1 | inner |
546.2.bq.c | ✓ | 64 | 21.c | even | 2 | 1 | inner |
546.2.bq.c | ✓ | 64 | 39.i | odd | 6 | 1 | inner |
546.2.bq.c | ✓ | 64 | 91.n | odd | 6 | 1 | inner |
546.2.bq.c | ✓ | 64 | 273.bn | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\):
\( T_{5}^{16} - 40T_{5}^{14} + 603T_{5}^{12} - 4508T_{5}^{10} + 18451T_{5}^{8} - 42568T_{5}^{6} + 54189T_{5}^{4} - 34932T_{5}^{2} + 8836 \) |
\( T_{61}^{32} - 242 T_{61}^{30} + 46172 T_{61}^{28} - 2648516 T_{61}^{26} + 110141918 T_{61}^{24} + \cdots + 35153041 \) |