Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [549,2,Mod(109,549)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(549, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("549.109");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 549 = 3^{2} \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 549.s (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: | \(4.38378707097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.542936601.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 2x^{6} - 4x^{5} + x^{4} - 8x^{3} + 8x^{2} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 61) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - x^{7} + 2x^{6} - 4x^{5} + x^{4} - 8x^{3} + 8x^{2} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} + 7\nu^{3} + 6\nu^{2} + 8\nu - 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 2\nu^{5} - 4\nu^{4} + \nu^{3} - 8\nu^{2} + 8\nu - 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{5} + \nu^{3} - 7\nu^{2} - 2\nu - 16 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} + 4\nu^{5} + \nu^{3} - 10\nu^{2} - 8\nu - 16 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} + \nu^{6} + 6\nu^{5} + 3\nu^{3} - 20\nu^{2} - 4\nu - 32 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{5} - \nu^{4} + \nu^{3} - 5\nu^{2} + \nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{3} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{4} + \beta_{2} - \beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} - 3\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 2\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + 3\beta_{5} - 6\beta_{4} - 2\beta_{3} + 4\beta _1 - 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4\beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} - 4\beta_{3} - \beta_{2} - 3\beta _1 - 1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/549\mathbb{Z}\right)^\times\).
| \(n\) | \(245\) | \(307\) |
| \(\chi(n)\) | \(1\) | \(1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
−2.15074 | − | 1.24173i | 0 | 2.08380 | + | 3.60925i | −1.17683 | + | 2.03833i | 0 | −2.32076 | − | 1.33989i | − | 5.38317i | 0 | 5.06212 | − | 2.92262i | |||||||||||||||||||||||||||||||
| 109.2 | −1.00909 | − | 0.582599i | 0 | −0.321156 | − | 0.556259i | 0.788634 | − | 1.36595i | 0 | −0.327706 | − | 0.189201i | 3.07882i | 0 | −1.59161 | + | 0.918915i | |||||||||||||||||||||||||||||||||
| 109.3 | 0.754161 | + | 0.435415i | 0 | −0.620827 | − | 1.07530i | −1.84552 | + | 3.19653i | 0 | 2.17067 | + | 1.25323i | − | 2.82293i | 0 | −2.78363 | + | 1.60713i | ||||||||||||||||||||||||||||||||
| 109.4 | 2.40567 | + | 1.38892i | 0 | 2.85818 | + | 4.95052i | −0.766288 | + | 1.32725i | 0 | −1.02220 | − | 0.590167i | 10.3234i | 0 | −3.68688 | + | 2.12862i | |||||||||||||||||||||||||||||||||
| 136.1 | −2.15074 | + | 1.24173i | 0 | 2.08380 | − | 3.60925i | −1.17683 | − | 2.03833i | 0 | −2.32076 | + | 1.33989i | 5.38317i | 0 | 5.06212 | + | 2.92262i | |||||||||||||||||||||||||||||||||
| 136.2 | −1.00909 | + | 0.582599i | 0 | −0.321156 | + | 0.556259i | 0.788634 | + | 1.36595i | 0 | −0.327706 | + | 0.189201i | − | 3.07882i | 0 | −1.59161 | − | 0.918915i | ||||||||||||||||||||||||||||||||
| 136.3 | 0.754161 | − | 0.435415i | 0 | −0.620827 | + | 1.07530i | −1.84552 | − | 3.19653i | 0 | 2.17067 | − | 1.25323i | 2.82293i | 0 | −2.78363 | − | 1.60713i | |||||||||||||||||||||||||||||||||
| 136.4 | 2.40567 | − | 1.38892i | 0 | 2.85818 | − | 4.95052i | −0.766288 | − | 1.32725i | 0 | −1.02220 | + | 0.590167i | − | 10.3234i | 0 | −3.68688 | − | 2.12862i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 61.f | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 549.2.s.j | 8 | |
| 3.b | odd | 2 | 1 | 61.2.f.b | ✓ | 8 | |
| 12.b | even | 2 | 1 | 976.2.ba.c | 8 | ||
| 61.f | even | 6 | 1 | inner | 549.2.s.j | 8 | |
| 183.i | odd | 6 | 1 | 61.2.f.b | ✓ | 8 | |
| 183.o | even | 12 | 2 | 3721.2.a.i | 8 | ||
| 732.r | even | 6 | 1 | 976.2.ba.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 61.2.f.b | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 61.2.f.b | ✓ | 8 | 183.i | odd | 6 | 1 | |
| 549.2.s.j | 8 | 1.a | even | 1 | 1 | trivial | |
| 549.2.s.j | 8 | 61.f | even | 6 | 1 | inner | |
| 976.2.ba.c | 8 | 12.b | even | 2 | 1 | ||
| 976.2.ba.c | 8 | 732.r | even | 6 | 1 | ||
| 3721.2.a.i | 8 | 183.o | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(549, [\chi])\):
|
\( T_{2}^{8} - 8T_{2}^{6} + 57T_{2}^{4} + 24T_{2}^{3} - 53T_{2}^{2} - 21T_{2} + 49 \)
|
|
\( T_{7}^{8} + 3T_{7}^{7} - 3T_{7}^{6} - 18T_{7}^{5} + 21T_{7}^{4} + 108T_{7}^{3} + 126T_{7}^{2} + 54T_{7} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 8 T^{6} + \cdots + 49 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 6 T^{7} + \cdots + 441 \)
|
| $7$ |
\( T^{8} + 3 T^{7} + \cdots + 9 \)
|
| $11$ |
\( T^{8} + 19 T^{6} + \cdots + 1 \)
|
| $13$ |
\( T^{8} + 3 T^{7} + \cdots + 3969 \)
|
| $17$ |
\( T^{8} + 6 T^{7} + \cdots + 2401 \)
|
| $19$ |
\( T^{8} - 3 T^{7} + \cdots + 441 \)
|
| $23$ |
\( T^{8} + 94 T^{6} + \cdots + 43681 \)
|
| $29$ |
\( T^{8} - 50 T^{6} + \cdots + 38809 \)
|
| $31$ |
\( T^{8} + 3 T^{7} + \cdots + 47961 \)
|
| $37$ |
\( T^{8} + 120 T^{6} + \cdots + 16641 \)
|
| $41$ |
\( (T^{4} - 12 T^{3} + \cdots - 147)^{2} \)
|
| $43$ |
\( T^{8} - 90 T^{6} + \cdots + 4761 \)
|
| $47$ |
\( T^{8} + 12 T^{7} + \cdots + 5625 \)
|
| $53$ |
\( T^{8} + 223 T^{6} + \cdots + 1907161 \)
|
| $59$ |
\( T^{8} - 3 T^{7} + \cdots + 1274641 \)
|
| $61$ |
\( T^{8} + 8 T^{7} + \cdots + 13845841 \)
|
| $67$ |
\( T^{8} + 15 T^{7} + \cdots + 1447209 \)
|
| $71$ |
\( T^{8} + 15 T^{7} + \cdots + 3721 \)
|
| $73$ |
\( T^{8} + 16 T^{7} + \cdots + 49 \)
|
| $79$ |
\( T^{8} - 42 T^{7} + \cdots + 2350089 \)
|
| $83$ |
\( T^{8} - 42 T^{7} + \cdots + 126900225 \)
|
| $89$ |
\( T^{8} + 220 T^{6} + \cdots + 2455489 \)
|
| $97$ |
\( T^{8} - 3 T^{7} + \cdots + 1199025 \)
|
| show more | |
| show less |