Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(17,546)]
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, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.bi (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: | \(34\) |
Relative dimension: | \(17\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −1.00000 | −1.66749 | + | 0.468501i | 1.00000 | −1.58996 | + | 0.917964i | 1.66749 | − | 0.468501i | −0.364289 | − | 2.62055i | −1.00000 | 2.56101 | − | 1.56244i | 1.58996 | − | 0.917964i | ||||||
17.2 | −1.00000 | −1.40963 | − | 1.00646i | 1.00000 | −1.41302 | + | 0.815806i | 1.40963 | + | 1.00646i | 2.62951 | + | 0.292738i | −1.00000 | 0.974089 | + | 2.83745i | 1.41302 | − | 0.815806i | ||||||
17.3 | −1.00000 | −1.36360 | − | 1.06799i | 1.00000 | 2.60318 | − | 1.50295i | 1.36360 | + | 1.06799i | −2.60776 | − | 0.446759i | −1.00000 | 0.718787 | + | 2.91262i | −2.60318 | + | 1.50295i | ||||||
17.4 | −1.00000 | −1.31411 | + | 1.12832i | 1.00000 | 1.62172 | − | 0.936303i | 1.31411 | − | 1.12832i | 2.04281 | − | 1.68135i | −1.00000 | 0.453768 | − | 2.96548i | −1.62172 | + | 0.936303i | ||||||
17.5 | −1.00000 | −0.733819 | + | 1.56892i | 1.00000 | −3.72094 | + | 2.14828i | 0.733819 | − | 1.56892i | 1.54641 | + | 2.14677i | −1.00000 | −1.92302 | − | 2.30261i | 3.72094 | − | 2.14828i | ||||||
17.6 | −1.00000 | −0.646635 | − | 1.60682i | 1.00000 | −2.88000 | + | 1.66277i | 0.646635 | + | 1.60682i | −2.37914 | − | 1.15746i | −1.00000 | −2.16373 | + | 2.07805i | 2.88000 | − | 1.66277i | ||||||
17.7 | −1.00000 | −0.565041 | − | 1.63729i | 1.00000 | 1.26448 | − | 0.730045i | 0.565041 | + | 1.63729i | −1.08820 | + | 2.41160i | −1.00000 | −2.36146 | + | 1.85028i | −1.26448 | + | 0.730045i | ||||||
17.8 | −1.00000 | −0.436593 | + | 1.67612i | 1.00000 | −0.567570 | + | 0.327687i | 0.436593 | − | 1.67612i | −2.37289 | + | 1.17020i | −1.00000 | −2.61877 | − | 1.46357i | 0.567570 | − | 0.327687i | ||||||
17.9 | −1.00000 | −0.0248275 | − | 1.73187i | 1.00000 | −0.511132 | + | 0.295102i | 0.0248275 | + | 1.73187i | 2.62812 | − | 0.304939i | −1.00000 | −2.99877 | + | 0.0859963i | 0.511132 | − | 0.295102i | ||||||
17.10 | −1.00000 | 0.787258 | + | 1.54280i | 1.00000 | −3.27919 | + | 1.89324i | −0.787258 | − | 1.54280i | 0.475130 | − | 2.60274i | −1.00000 | −1.76045 | + | 2.42916i | 3.27919 | − | 1.89324i | ||||||
17.11 | −1.00000 | 0.805192 | − | 1.53351i | 1.00000 | 2.64914 | − | 1.52948i | −0.805192 | + | 1.53351i | 0.969634 | − | 2.46167i | −1.00000 | −1.70333 | − | 2.46955i | −2.64914 | + | 1.52948i | ||||||
17.12 | −1.00000 | 0.841952 | + | 1.51364i | 1.00000 | 2.84717 | − | 1.64381i | −0.841952 | − | 1.51364i | 1.87202 | + | 1.86964i | −1.00000 | −1.58223 | + | 2.54883i | −2.84717 | + | 1.64381i | ||||||
17.13 | −1.00000 | 0.929420 | − | 1.46157i | 1.00000 | 1.09866 | − | 0.634311i | −0.929420 | + | 1.46157i | 0.151485 | + | 2.64141i | −1.00000 | −1.27236 | − | 2.71682i | −1.09866 | + | 0.634311i | ||||||
17.14 | −1.00000 | 1.21751 | − | 1.23193i | 1.00000 | −1.57344 | + | 0.908426i | −1.21751 | + | 1.23193i | −2.47008 | − | 0.947995i | −1.00000 | −0.0353243 | − | 2.99979i | 1.57344 | − | 0.908426i | ||||||
17.15 | −1.00000 | 1.66045 | + | 0.492861i | 1.00000 | 1.80315 | − | 1.04105i | −1.66045 | − | 0.492861i | −0.800654 | − | 2.52170i | −1.00000 | 2.51418 | + | 1.63674i | −1.80315 | + | 1.04105i | ||||||
17.16 | −1.00000 | 1.69024 | + | 0.378264i | 1.00000 | −0.870413 | + | 0.502533i | −1.69024 | − | 0.378264i | 2.64571 | − | 0.0151415i | −1.00000 | 2.71383 | + | 1.27872i | 0.870413 | − | 0.502533i | ||||||
17.17 | −1.00000 | 1.72971 | − | 0.0900624i | 1.00000 | −1.98183 | + | 1.14421i | −1.72971 | + | 0.0900624i | −0.877809 | + | 2.49589i | −1.00000 | 2.98378 | − | 0.311563i | 1.98183 | − | 1.14421i | ||||||
257.1 | −1.00000 | −1.66749 | − | 0.468501i | 1.00000 | −1.58996 | − | 0.917964i | 1.66749 | + | 0.468501i | −0.364289 | + | 2.62055i | −1.00000 | 2.56101 | + | 1.56244i | 1.58996 | + | 0.917964i | ||||||
257.2 | −1.00000 | −1.40963 | + | 1.00646i | 1.00000 | −1.41302 | − | 0.815806i | 1.40963 | − | 1.00646i | 2.62951 | − | 0.292738i | −1.00000 | 0.974089 | − | 2.83745i | 1.41302 | + | 0.815806i | ||||||
257.3 | −1.00000 | −1.36360 | + | 1.06799i | 1.00000 | 2.60318 | + | 1.50295i | 1.36360 | − | 1.06799i | −2.60776 | + | 0.446759i | −1.00000 | 0.718787 | − | 2.91262i | −2.60318 | − | 1.50295i | ||||||
See all 34 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
273.br | 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.bi.e | ✓ | 34 |
3.b | odd | 2 | 1 | 546.2.bi.f | yes | 34 | |
7.d | odd | 6 | 1 | 546.2.bn.f | yes | 34 | |
13.e | even | 6 | 1 | 546.2.bn.e | yes | 34 | |
21.g | even | 6 | 1 | 546.2.bn.e | yes | 34 | |
39.h | odd | 6 | 1 | 546.2.bn.f | yes | 34 | |
91.l | odd | 6 | 1 | 546.2.bi.f | yes | 34 | |
273.br | even | 6 | 1 | inner | 546.2.bi.e | ✓ | 34 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.bi.e | ✓ | 34 | 1.a | even | 1 | 1 | trivial |
546.2.bi.e | ✓ | 34 | 273.br | even | 6 | 1 | inner |
546.2.bi.f | yes | 34 | 3.b | odd | 2 | 1 | |
546.2.bi.f | yes | 34 | 91.l | odd | 6 | 1 | |
546.2.bn.e | yes | 34 | 13.e | even | 6 | 1 | |
546.2.bn.e | yes | 34 | 21.g | even | 6 | 1 | |
546.2.bn.f | yes | 34 | 7.d | odd | 6 | 1 | |
546.2.bn.f | yes | 34 | 39.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{34} + 9 T_{5}^{33} - 10 T_{5}^{32} - 333 T_{5}^{31} - 234 T_{5}^{30} + 7575 T_{5}^{29} + \cdots + 3270763083 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).