Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [891,2,Mod(298,891)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(891, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("891.298");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 891 = 3^{4} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 891.e (of order \(3\), degree \(2\), not 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: | \(7.11467082010\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.2101707.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 6x^{4} - 4x^{3} - 12x^{2} - 18x + 31 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 6x^{4} - 4x^{3} - 12x^{2} - 18x + 31 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 7\nu^{4} - 5\nu^{3} + 19\nu^{2} + 45\nu + 183 ) / 100 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{5} + \nu^{4} - 35\nu^{3} + 33\nu^{2} + 165\nu + 81 ) / 100 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\nu^{5} + 19\nu^{4} + 135\nu^{3} + 77\nu^{2} - 65\nu - 361 ) / 100 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 37\nu^{5} + 59\nu^{4} + 285\nu^{3} + 297\nu^{2} - 165\nu - 971 ) / 100 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 43\nu^{5} + 51\nu^{4} + 315\nu^{3} + 183\nu^{2} - 185\nu - 1069 ) / 100 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{5} - \beta_{4} - 2\beta_{3} + 2\beta_{2} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{4} - 2\beta_{3} - \beta_{2} + 4\beta _1 - 5 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{5} + 10\beta_{3} - 2\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4\beta_{5} + \beta_{4} - 10\beta_{3} + 10\beta_{2} - 31\beta _1 + 65 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 103\beta_{5} - 14\beta_{4} - 208\beta_{3} + \beta_{2} + 68\beta _1 + 89 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).
| \(n\) | \(244\) | \(650\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 298.1 |
|
−1.26446 | + | 2.19011i | 0 | −2.19771 | − | 3.80655i | 0.264459 | + | 0.458056i | 0 | −0.0667460 | + | 0.115607i | 6.05784 | 0 | −1.33759 | ||||||||||||||||||||||||||||
| 298.2 | 0.0837246 | − | 0.145015i | 0 | 0.985980 | + | 1.70777i | −1.08372 | − | 1.87707i | 0 | −1.90226 | + | 3.29480i | 0.665102 | 0 | −0.362938 | |||||||||||||||||||||||||||||
| 298.3 | 1.18073 | − | 2.04509i | 0 | −1.78827 | − | 3.09737i | −2.18073 | − | 3.77714i | 0 | 1.96900 | − | 3.41041i | −3.72294 | 0 | −10.2995 | |||||||||||||||||||||||||||||
| 595.1 | −1.26446 | − | 2.19011i | 0 | −2.19771 | + | 3.80655i | 0.264459 | − | 0.458056i | 0 | −0.0667460 | − | 0.115607i | 6.05784 | 0 | −1.33759 | |||||||||||||||||||||||||||||
| 595.2 | 0.0837246 | + | 0.145015i | 0 | 0.985980 | − | 1.70777i | −1.08372 | + | 1.87707i | 0 | −1.90226 | − | 3.29480i | 0.665102 | 0 | −0.362938 | |||||||||||||||||||||||||||||
| 595.3 | 1.18073 | + | 2.04509i | 0 | −1.78827 | + | 3.09737i | −2.18073 | + | 3.77714i | 0 | 1.96900 | + | 3.41041i | −3.72294 | 0 | −10.2995 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 891.2.e.r | 6 | |
| 3.b | odd | 2 | 1 | 891.2.e.s | 6 | ||
| 9.c | even | 3 | 1 | 891.2.a.n | yes | 3 | |
| 9.c | even | 3 | 1 | inner | 891.2.e.r | 6 | |
| 9.d | odd | 6 | 1 | 891.2.a.m | ✓ | 3 | |
| 9.d | odd | 6 | 1 | 891.2.e.s | 6 | ||
| 99.g | even | 6 | 1 | 9801.2.a.bf | 3 | ||
| 99.h | odd | 6 | 1 | 9801.2.a.bg | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 891.2.a.m | ✓ | 3 | 9.d | odd | 6 | 1 | |
| 891.2.a.n | yes | 3 | 9.c | even | 3 | 1 | |
| 891.2.e.r | 6 | 1.a | even | 1 | 1 | trivial | |
| 891.2.e.r | 6 | 9.c | even | 3 | 1 | inner | |
| 891.2.e.s | 6 | 3.b | odd | 2 | 1 | ||
| 891.2.e.s | 6 | 9.d | odd | 6 | 1 | ||
| 9801.2.a.bf | 3 | 99.g | even | 6 | 1 | ||
| 9801.2.a.bg | 3 | 99.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\):
|
\( T_{2}^{6} + 6T_{2}^{4} - 2T_{2}^{3} + 36T_{2}^{2} - 6T_{2} + 1 \)
|
|
\( T_{5}^{6} + 6T_{5}^{5} + 30T_{5}^{4} + 46T_{5}^{3} + 66T_{5}^{2} - 30T_{5} + 25 \)
|
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\( T_{7}^{6} + 15T_{7}^{4} + 4T_{7}^{3} + 225T_{7}^{2} + 30T_{7} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 6 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 6 T^{5} + \cdots + 25 \)
|
| $7$ |
\( T^{6} + 15 T^{4} + \cdots + 4 \)
|
| $11$ |
\( (T^{2} + T + 1)^{3} \)
|
| $13$ |
\( T^{6} + 6 T^{4} + \cdots + 1 \)
|
| $17$ |
\( (T^{3} - 9 T^{2} + 6 T + 20)^{2} \)
|
| $19$ |
\( (T^{3} + 3 T^{2} - 27 T - 90)^{2} \)
|
| $23$ |
\( T^{6} + 21 T^{4} + \cdots + 256 \)
|
| $29$ |
\( T^{6} + 18 T^{5} + \cdots + 25281 \)
|
| $31$ |
\( T^{6} - 15 T^{5} + \cdots + 2500 \)
|
| $37$ |
\( (T^{3} - 9 T^{2} + \cdots + 237)^{2} \)
|
| $41$ |
\( T^{6} + 3 T^{5} + \cdots + 676 \)
|
| $43$ |
\( T^{6} + 6 T^{5} + \cdots + 576 \)
|
| $47$ |
\( T^{6} + 9 T^{5} + \cdots + 15376 \)
|
| $53$ |
\( (T^{3} + 3 T^{2} + \cdots - 150)^{2} \)
|
| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 643204 \)
|
| $61$ |
\( T^{6} + 3 T^{5} + \cdots + 144 \)
|
| $67$ |
\( T^{6} - 21 T^{5} + \cdots + 209764 \)
|
| $71$ |
\( (T^{3} + 3 T^{2} + \cdots + 120)^{2} \)
|
| $73$ |
\( (T^{3} + 3 T^{2} + \cdots - 967)^{2} \)
|
| $79$ |
\( T^{6} - 3 T^{5} + \cdots + 85264 \)
|
| $83$ |
\( (T^{2} + 6 T + 36)^{3} \)
|
| $89$ |
\( (T^{3} - 36 T^{2} + \cdots - 1569)^{2} \)
|
| $97$ |
\( T^{6} - 6 T^{5} + \cdots + 1024 \)
|
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