Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [396,2,Mod(133,396)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(396, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("396.133");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 396.i (of order \(3\), 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: | \(3.16207592004\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.5206055409.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + x^{6} + 9x^{5} - 23x^{4} + 27x^{3} + 9x^{2} - 81x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} + x^{6} + 9x^{5} - 23x^{4} + 27x^{3} + 9x^{2} - 81x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{6} + 8\nu^{5} - 15\nu^{4} + 14\nu^{3} + 30\nu^{2} - 99\nu + 54 ) / 81 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} + \nu^{5} + 9\nu^{4} - 23\nu^{3} + 27\nu^{2} + 9\nu - 81 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{7} + 6\nu^{6} + 5\nu^{5} - 15\nu^{4} + 56\nu^{3} + 3\nu^{2} - 99\nu + 135 ) / 81 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{7} + 12\nu^{6} - 14\nu^{5} - 21\nu^{4} + 70\nu^{3} - 93\nu^{2} + 45\nu + 108 ) / 81 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{7} - 3\nu^{6} + 2\nu^{5} + 3\nu^{4} - 19\nu^{3} + 30\nu^{2} - 18\nu - 27 ) / 27 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7\nu^{7} - 6\nu^{6} - 29\nu^{5} + 78\nu^{4} - 71\nu^{3} + 6\nu^{2} + 315\nu - 540 ) / 81 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} - 3\beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - 2\beta _1 + 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} - 3\beta_{6} - 8\beta_{5} + 2\beta_{4} - 4\beta_{3} - 5\beta_{2} + 4\beta _1 - 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( -4\beta_{7} + 7\beta_{6} + 2\beta_{5} + 9\beta_{4} + 3\beta_{3} - 23\beta_{2} + \beta _1 - 13 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3\beta_{7} + 15\beta_{6} - 9\beta_{5} + 3\beta_{4} - 24\beta_{3} - 9\beta_{2} + 4\beta _1 - 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/396\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(199\) | \(353\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 133.1 |
|
0 | −1.70133 | − | 0.324778i | 0 | 0.0693986 | + | 0.120202i | 0 | −0.0625314 | + | 0.108308i | 0 | 2.78904 | + | 1.10511i | 0 | ||||||||||||||||||||||||||||||||||
| 133.2 | 0 | −1.35516 | + | 1.07868i | 0 | 1.11174 | + | 1.92559i | 0 | 2.36833 | − | 4.10206i | 0 | 0.672901 | − | 2.92356i | 0 | |||||||||||||||||||||||||||||||||||
| 133.3 | 0 | −0.172469 | − | 1.72344i | 0 | −1.90631 | − | 3.30183i | 0 | −2.48509 | + | 4.30430i | 0 | −2.94051 | + | 0.594481i | 0 | |||||||||||||||||||||||||||||||||||
| 133.4 | 0 | 1.72895 | + | 0.103515i | 0 | −1.27483 | − | 2.20807i | 0 | 0.679294 | − | 1.17657i | 0 | 2.97857 | + | 0.357947i | 0 | |||||||||||||||||||||||||||||||||||
| 265.1 | 0 | −1.70133 | + | 0.324778i | 0 | 0.0693986 | − | 0.120202i | 0 | −0.0625314 | − | 0.108308i | 0 | 2.78904 | − | 1.10511i | 0 | |||||||||||||||||||||||||||||||||||
| 265.2 | 0 | −1.35516 | − | 1.07868i | 0 | 1.11174 | − | 1.92559i | 0 | 2.36833 | + | 4.10206i | 0 | 0.672901 | + | 2.92356i | 0 | |||||||||||||||||||||||||||||||||||
| 265.3 | 0 | −0.172469 | + | 1.72344i | 0 | −1.90631 | + | 3.30183i | 0 | −2.48509 | − | 4.30430i | 0 | −2.94051 | − | 0.594481i | 0 | |||||||||||||||||||||||||||||||||||
| 265.4 | 0 | 1.72895 | − | 0.103515i | 0 | −1.27483 | + | 2.20807i | 0 | 0.679294 | + | 1.17657i | 0 | 2.97857 | − | 0.357947i | 0 | |||||||||||||||||||||||||||||||||||
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 | 396.2.i.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1188.2.i.e | 8 | ||
| 9.c | even | 3 | 1 | inner | 396.2.i.e | ✓ | 8 |
| 9.c | even | 3 | 1 | 3564.2.a.n | 4 | ||
| 9.d | odd | 6 | 1 | 1188.2.i.e | 8 | ||
| 9.d | odd | 6 | 1 | 3564.2.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 396.2.i.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 396.2.i.e | ✓ | 8 | 9.c | even | 3 | 1 | inner |
| 1188.2.i.e | 8 | 3.b | odd | 2 | 1 | ||
| 1188.2.i.e | 8 | 9.d | odd | 6 | 1 | ||
| 3564.2.a.m | 4 | 9.d | odd | 6 | 1 | ||
| 3564.2.a.n | 4 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 4T_{5}^{7} + 21T_{5}^{6} + 22T_{5}^{5} + 106T_{5}^{4} + 81T_{5}^{3} + 456T_{5}^{2} - 63T_{5} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(396, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 3 T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} + 4 T^{7} + \cdots + 9 \)
|
| $7$ |
\( T^{8} - T^{7} + \cdots + 16 \)
|
| $11$ |
\( (T^{2} - T + 1)^{4} \)
|
| $13$ |
\( T^{8} + T^{7} + \cdots + 36 \)
|
| $17$ |
\( (T^{4} - T^{3} - 32 T^{2} + \cdots - 42)^{2} \)
|
| $19$ |
\( (T^{4} - 9 T^{3} + 22 T^{2} + \cdots - 2)^{2} \)
|
| $23$ |
\( T^{8} - T^{7} + \cdots + 21609 \)
|
| $29$ |
\( T^{8} + 51 T^{6} + \cdots + 2916 \)
|
| $31$ |
\( T^{8} + 13 T^{7} + \cdots + 1327104 \)
|
| $37$ |
\( (T^{4} + 14 T^{3} + \cdots - 162)^{2} \)
|
| $41$ |
\( T^{8} - 12 T^{7} + \cdots + 14017536 \)
|
| $43$ |
\( T^{8} - 3 T^{7} + \cdots + 1032256 \)
|
| $47$ |
\( T^{8} - 3 T^{7} + \cdots + 9801 \)
|
| $53$ |
\( (T^{4} + 2 T^{3} + \cdots + 333)^{2} \)
|
| $59$ |
\( T^{8} + 9 T^{7} + \cdots + 746496 \)
|
| $61$ |
\( T^{8} + 15 T^{7} + \cdots + 3136 \)
|
| $67$ |
\( T^{8} + 30 T^{7} + \cdots + 259081 \)
|
| $71$ |
\( (T^{4} - 6 T^{3} + \cdots + 6408)^{2} \)
|
| $73$ |
\( (T^{4} - 22 T^{3} + \cdots - 9834)^{2} \)
|
| $79$ |
\( T^{8} + 41 T^{7} + \cdots + 49758916 \)
|
| $83$ |
\( T^{8} + 20 T^{7} + \cdots + 121749156 \)
|
| $89$ |
\( (T^{4} + 25 T^{3} + \cdots + 264)^{2} \)
|
| $97$ |
\( T^{8} + 3 T^{7} + \cdots + 68644 \)
|
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