Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [936,2,Mod(469,936)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(936, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("936.469");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 936.g (of order \(2\), degree \(1\), 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: | \(7.47399762919\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.399424.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 104) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 6\nu + 8 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 3\nu^{3} + 3\nu^{2} - 2\nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 6\nu - 12 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 2\nu^{4} - 5\nu^{3} + 10\nu^{2} - 2\nu + 12 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - 3\beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} + 7\beta _1 + 3 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{5} + 5\beta_{4} + 3\beta_{3} + 3\beta_{2} - \beta _1 + 1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) | \(469\) | \(703\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 469.1 |
|
−0.671462 | − | 1.24464i | 0 | −1.09828 | + | 1.67146i | − | 1.00000i | 0 | −0.146365 | 2.81783 | + | 0.244644i | 0 | −1.24464 | + | 0.671462i | |||||||||||||||||||||||||||
| 469.2 | −0.671462 | + | 1.24464i | 0 | −1.09828 | − | 1.67146i | 1.00000i | 0 | −0.146365 | 2.81783 | − | 0.244644i | 0 | −1.24464 | − | 0.671462i | |||||||||||||||||||||||||||||
| 469.3 | 0.264658 | − | 1.38923i | 0 | −1.85991 | − | 0.735342i | 1.00000i | 0 | 3.24914 | −1.51380 | + | 2.38923i | 0 | 1.38923 | + | 0.264658i | |||||||||||||||||||||||||||||
| 469.4 | 0.264658 | + | 1.38923i | 0 | −1.85991 | + | 0.735342i | − | 1.00000i | 0 | 3.24914 | −1.51380 | − | 2.38923i | 0 | 1.38923 | − | 0.264658i | ||||||||||||||||||||||||||||
| 469.5 | 1.40680 | − | 0.144584i | 0 | 1.95819 | − | 0.406803i | − | 1.00000i | 0 | −2.10278 | 2.69597 | − | 0.855416i | 0 | −0.144584 | − | 1.40680i | ||||||||||||||||||||||||||||
| 469.6 | 1.40680 | + | 0.144584i | 0 | 1.95819 | + | 0.406803i | 1.00000i | 0 | −2.10278 | 2.69597 | + | 0.855416i | 0 | −0.144584 | + | 1.40680i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 936.2.g.c | 6 | |
| 3.b | odd | 2 | 1 | 104.2.b.c | ✓ | 6 | |
| 4.b | odd | 2 | 1 | 3744.2.g.c | 6 | ||
| 8.b | even | 2 | 1 | inner | 936.2.g.c | 6 | |
| 8.d | odd | 2 | 1 | 3744.2.g.c | 6 | ||
| 12.b | even | 2 | 1 | 416.2.b.c | 6 | ||
| 24.f | even | 2 | 1 | 416.2.b.c | 6 | ||
| 24.h | odd | 2 | 1 | 104.2.b.c | ✓ | 6 | |
| 48.i | odd | 4 | 1 | 3328.2.a.be | 3 | ||
| 48.i | odd | 4 | 1 | 3328.2.a.bh | 3 | ||
| 48.k | even | 4 | 1 | 3328.2.a.bf | 3 | ||
| 48.k | even | 4 | 1 | 3328.2.a.bg | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.2.b.c | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 104.2.b.c | ✓ | 6 | 24.h | odd | 2 | 1 | |
| 416.2.b.c | 6 | 12.b | even | 2 | 1 | ||
| 416.2.b.c | 6 | 24.f | even | 2 | 1 | ||
| 936.2.g.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 936.2.g.c | 6 | 8.b | even | 2 | 1 | inner | |
| 3328.2.a.be | 3 | 48.i | odd | 4 | 1 | ||
| 3328.2.a.bf | 3 | 48.k | even | 4 | 1 | ||
| 3328.2.a.bg | 3 | 48.k | even | 4 | 1 | ||
| 3328.2.a.bh | 3 | 48.i | odd | 4 | 1 | ||
| 3744.2.g.c | 6 | 4.b | odd | 2 | 1 | ||
| 3744.2.g.c | 6 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots + 8 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} + 1)^{3} \)
|
| $7$ |
\( (T^{3} - T^{2} - 7 T - 1)^{2} \)
|
| $11$ |
\( T^{6} + 40 T^{4} + \cdots + 256 \)
|
| $13$ |
\( (T^{2} + 1)^{3} \)
|
| $17$ |
\( (T + 1)^{6} \)
|
| $19$ |
\( (T^{2} + 16)^{3} \)
|
| $23$ |
\( (T^{3} - 8 T^{2} + 4 T + 32)^{2} \)
|
| $29$ |
\( T^{6} + 136 T^{4} + \cdots + 65536 \)
|
| $31$ |
\( (T^{3} - 16 T^{2} + \cdots - 64)^{2} \)
|
| $37$ |
\( T^{6} + 59 T^{4} + \cdots + 121 \)
|
| $41$ |
\( (T^{3} - 10 T^{2} + \cdots + 352)^{2} \)
|
| $43$ |
\( T^{6} + 47 T^{4} + \cdots + 1 \)
|
| $47$ |
\( (T^{3} - 5 T^{2} + \cdots + 227)^{2} \)
|
| $53$ |
\( T^{6} + 104 T^{4} + \cdots + 4096 \)
|
| $59$ |
\( T^{6} + 260 T^{4} + \cdots + 369664 \)
|
| $61$ |
\( T^{6} + 324 T^{4} + \cdots + 1048576 \)
|
| $67$ |
\( T^{6} + 220 T^{4} + \cdots + 23104 \)
|
| $71$ |
\( (T^{3} - 9 T^{2} + \cdots + 271)^{2} \)
|
| $73$ |
\( (T^{3} + 6 T^{2} - 16 T - 64)^{2} \)
|
| $79$ |
\( (T^{3} + 6 T^{2} + \cdots - 1448)^{2} \)
|
| $83$ |
\( T^{6} + 92 T^{4} + \cdots + 7744 \)
|
| $89$ |
\( (T^{3} + 8 T^{2} + \cdots - 128)^{2} \)
|
| $97$ |
\( (T^{3} - 8 T^{2} + \cdots + 848)^{2} \)
|
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