Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,6,Mod(1,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.a (trivial) |
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: | yes |
| Analytic conductor: | \(146.270043669\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 4327x^{2} + 78705x - 258666 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 228) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 4327x^{2} + 78705x - 258666 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 5\nu^{3} + 196\nu^{2} - 19991\nu - 140614 ) / 1732 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 31\nu^{2} - 11908\nu + 103034 ) / 866 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -13\nu^{3} + 10\nu^{2} + 61849\nu - 762282 ) / 1732 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} - \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 57\beta_{2} + 71\beta _1 + 12986 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3959\beta_{3} + 14229\beta_{2} - 4703\beta _1 - 332318 ) / 6 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 9.00000 | 0 | −31.7936 | 0 | 136.490 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | −21.2328 | 0 | −102.241 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | 74.6969 | 0 | −227.637 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.00000 | 0 | 95.3295 | 0 | 226.388 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(19\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 912.6.a.s | 4 | |
| 4.b | odd | 2 | 1 | 228.6.a.c | ✓ | 4 | |
| 12.b | even | 2 | 1 | 684.6.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 228.6.a.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 684.6.a.c | 4 | 12.b | even | 2 | 1 | ||
| 912.6.a.s | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 117T_{5}^{3} - 1220T_{5}^{2} + 262812T_{5} + 4807024 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} - 117 T^{3} + \cdots + 4807024 \)
|
| $7$ |
\( T^{4} - 33 T^{3} + \cdots + 719152128 \)
|
| $11$ |
\( T^{4} + \cdots - 2947580480 \)
|
| $13$ |
\( T^{4} + \cdots + 34302938368 \)
|
| $17$ |
\( T^{4} + \cdots + 2211826294584 \)
|
| $19$ |
\( (T - 361)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 6466156229824 \)
|
| $29$ |
\( T^{4} + \cdots + 41401237109728 \)
|
| $31$ |
\( T^{4} + \cdots - 1260880553600 \)
|
| $37$ |
\( T^{4} + \cdots - 39\!\cdots\!04 \)
|
| $41$ |
\( T^{4} + \cdots - 54\!\cdots\!32 \)
|
| $43$ |
\( T^{4} + \cdots + 28\!\cdots\!56 \)
|
| $47$ |
\( T^{4} + \cdots + 63\!\cdots\!28 \)
|
| $53$ |
\( T^{4} + \cdots + 52\!\cdots\!16 \)
|
| $59$ |
\( T^{4} + \cdots - 16\!\cdots\!56 \)
|
| $61$ |
\( T^{4} + \cdots + 23\!\cdots\!12 \)
|
| $67$ |
\( T^{4} + \cdots - 30\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots - 20\!\cdots\!80 \)
|
| $73$ |
\( T^{4} + \cdots + 51\!\cdots\!84 \)
|
| $79$ |
\( T^{4} + \cdots + 33\!\cdots\!80 \)
|
| $83$ |
\( T^{4} + \cdots - 96\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots + 80\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots + 39\!\cdots\!44 \)
|
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