Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,2,Mod(49,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.q (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: | \(7.28235666434\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.954288.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 57) |
| 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
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} + \nu^{3} - 9\nu^{2} + 21\nu + 9 ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} + 8\nu^{4} - 2\nu^{3} - 9\nu^{2} + 12\nu - 18 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{5} - 2\nu^{4} + 14\nu^{3} + 18\nu^{2} + 24\nu + 45 ) / 27 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10\nu^{5} + 5\nu^{4} - 8\nu^{3} + 36\nu^{2} - 6\nu - 153 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 2\beta_{3} + 3\beta_{2} + 4\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - \beta_{3} + 3\beta_{2} - 2\beta_1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 7\beta_{4} + 2\beta_{3} - 24\beta_{2} + 4\beta _1 + 9 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 8\beta_{3} - 6\beta_{2} - 2\beta _1 + 18 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7\beta_{5} - 2\beta_{4} + 2\beta_{3} - 33\beta_{2} + 22\beta _1 + 81 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | 0.500000 | − | 0.866025i | 0 | −1.66044 | + | 2.87597i | 0 | −2.32088 | 0 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||
| 49.2 | 0 | 0.500000 | − | 0.866025i | 0 | −0.675970 | + | 1.17081i | 0 | −0.351939 | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||
| 49.3 | 0 | 0.500000 | − | 0.866025i | 0 | 1.33641 | − | 2.31473i | 0 | 3.67282 | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||
| 577.1 | 0 | 0.500000 | + | 0.866025i | 0 | −1.66044 | − | 2.87597i | 0 | −2.32088 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||
| 577.2 | 0 | 0.500000 | + | 0.866025i | 0 | −0.675970 | − | 1.17081i | 0 | −0.351939 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||
| 577.3 | 0 | 0.500000 | + | 0.866025i | 0 | 1.33641 | + | 2.31473i | 0 | 3.67282 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 912.2.q.l | 6 | |
| 3.b | odd | 2 | 1 | 2736.2.s.z | 6 | ||
| 4.b | odd | 2 | 1 | 57.2.e.b | ✓ | 6 | |
| 12.b | even | 2 | 1 | 171.2.f.b | 6 | ||
| 19.c | even | 3 | 1 | inner | 912.2.q.l | 6 | |
| 57.h | odd | 6 | 1 | 2736.2.s.z | 6 | ||
| 76.f | even | 6 | 1 | 1083.2.a.o | 3 | ||
| 76.g | odd | 6 | 1 | 57.2.e.b | ✓ | 6 | |
| 76.g | odd | 6 | 1 | 1083.2.a.l | 3 | ||
| 228.m | even | 6 | 1 | 171.2.f.b | 6 | ||
| 228.m | even | 6 | 1 | 3249.2.a.y | 3 | ||
| 228.n | odd | 6 | 1 | 3249.2.a.t | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.2.e.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 57.2.e.b | ✓ | 6 | 76.g | odd | 6 | 1 | |
| 171.2.f.b | 6 | 12.b | even | 2 | 1 | ||
| 171.2.f.b | 6 | 228.m | even | 6 | 1 | ||
| 912.2.q.l | 6 | 1.a | even | 1 | 1 | trivial | |
| 912.2.q.l | 6 | 19.c | even | 3 | 1 | inner | |
| 1083.2.a.l | 3 | 76.g | odd | 6 | 1 | ||
| 1083.2.a.o | 3 | 76.f | even | 6 | 1 | ||
| 2736.2.s.z | 6 | 3.b | odd | 2 | 1 | ||
| 2736.2.s.z | 6 | 57.h | odd | 6 | 1 | ||
| 3249.2.a.t | 3 | 228.n | odd | 6 | 1 | ||
| 3249.2.a.y | 3 | 228.m | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):
|
\( T_{5}^{6} + 2T_{5}^{5} + 12T_{5}^{4} + 8T_{5}^{3} + 88T_{5}^{2} + 96T_{5} + 144 \)
|
|
\( T_{7}^{3} - T_{7}^{2} - 9T_{7} - 3 \)
|
|
\( T_{13}^{6} - T_{13}^{5} + 22T_{13}^{4} + 27T_{13}^{3} + 438T_{13}^{2} + 63T_{13} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} - T + 1)^{3} \)
|
| $5$ |
\( T^{6} + 2 T^{5} + \cdots + 144 \)
|
| $7$ |
\( (T^{3} - T^{2} - 9 T - 3)^{2} \)
|
| $11$ |
\( (T^{3} - 24 T + 36)^{2} \)
|
| $13$ |
\( T^{6} - T^{5} + 22 T^{4} + \cdots + 9 \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} + 4 T^{5} + \cdots + 6859 \)
|
| $23$ |
\( T^{6} - 14 T^{5} + \cdots + 24336 \)
|
| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 9216 \)
|
| $31$ |
\( (T^{3} + 15 T^{2} + \cdots - 31)^{2} \)
|
| $37$ |
\( (T + 1)^{6} \)
|
| $41$ |
\( T^{6} - 4 T^{5} + \cdots + 9216 \)
|
| $43$ |
\( T^{6} + 3 T^{5} + \cdots + 169 \)
|
| $47$ |
\( (T^{2} + 6 T + 36)^{3} \)
|
| $53$ |
\( T^{6} - 6 T^{5} + \cdots + 104976 \)
|
| $59$ |
\( T^{6} + 24 T^{4} + \cdots + 1296 \)
|
| $61$ |
\( T^{6} + 13 T^{5} + \cdots + 5329 \)
|
| $67$ |
\( T^{6} - 9 T^{5} + \cdots + 292681 \)
|
| $71$ |
\( T^{6} + 18 T^{5} + \cdots + 419904 \)
|
| $73$ |
\( T^{6} + 19 T^{5} + \cdots + 961 \)
|
| $79$ |
\( T^{6} - 11 T^{5} + \cdots + 29241 \)
|
| $83$ |
\( (T^{3} - 4 T^{2} + \cdots + 192)^{2} \)
|
| $89$ |
\( T^{6} - 16 T^{5} + \cdots + 11664 \)
|
| $97$ |
\( T^{6} - 2 T^{5} + \cdots + 2096704 \)
|
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