Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1444,2,Mod(1,1444)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1444, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1444.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1444 = 2^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1444.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: | \(11.5303980519\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.20319417.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 9x^{4} + 19x^{3} + 27x^{2} - 27x - 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 76) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 9x^{4} + 19x^{3} + 27x^{2} - 27x - 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 6\nu^{2} + 10\nu + 9 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 6\nu^{4} + 37\nu^{2} - 12\nu - 45 ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 6\nu^{3} + 10\nu^{2} + 9\nu + 3 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{5} - 15\nu^{4} - 18\nu^{3} + 76\nu^{2} + 6\nu - 72 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} - 2\beta_{4} - 6\beta_{3} - 3\beta_{2} + 6\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 15\beta_{5} - 12\beta_{4} - 24\beta_{3} - 6\beta_{2} + 14\beta _1 + 30 \)
|
| \(\nu^{5}\) | \(=\) |
\( 53\beta_{5} - 35\beta_{4} - 98\beta_{3} - 36\beta_{2} + 59\beta _1 + 77 \)
|
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 | −2.69196 | 0 | −1.28220 | 0 | −3.34467 | 0 | 4.24666 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.08888 | 0 | −3.20035 | 0 | −0.556791 | 0 | −1.81434 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.486320 | 0 | 2.79337 | 0 | −3.36570 | 0 | −2.76349 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.81258 | 0 | 0.282205 | 0 | 1.15987 | 0 | 0.285432 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.43618 | 0 | 2.20035 | 0 | 2.96827 | 0 | 2.93496 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.01841 | 0 | −3.79337 | 0 | 0.139023 | 0 | 6.11079 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(1\) |
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 | 1444.2.a.h | 6 | |
| 4.b | odd | 2 | 1 | 5776.2.a.bw | 6 | ||
| 19.b | odd | 2 | 1 | 1444.2.a.g | 6 | ||
| 19.c | even | 3 | 2 | 1444.2.e.g | 12 | ||
| 19.d | odd | 6 | 2 | 1444.2.e.h | 12 | ||
| 19.e | even | 9 | 2 | 76.2.i.a | ✓ | 12 | |
| 57.l | odd | 18 | 2 | 684.2.bo.c | 12 | ||
| 76.d | even | 2 | 1 | 5776.2.a.by | 6 | ||
| 76.l | odd | 18 | 2 | 304.2.u.e | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.2.i.a | ✓ | 12 | 19.e | even | 9 | 2 | |
| 304.2.u.e | 12 | 76.l | odd | 18 | 2 | ||
| 684.2.bo.c | 12 | 57.l | odd | 18 | 2 | ||
| 1444.2.a.g | 6 | 19.b | odd | 2 | 1 | ||
| 1444.2.a.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 1444.2.e.g | 12 | 19.c | even | 3 | 2 | ||
| 1444.2.e.h | 12 | 19.d | odd | 6 | 2 | ||
| 5776.2.a.bw | 6 | 4.b | odd | 2 | 1 | ||
| 5776.2.a.by | 6 | 76.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 3T_{3}^{5} - 9T_{3}^{4} + 27T_{3}^{3} + 15T_{3}^{2} - 39T_{3} - 19 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1444))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots - 19 \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 27 \)
|
| $7$ |
\( T^{6} + 3 T^{5} + \cdots - 3 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots - 27 \)
|
| $13$ |
\( T^{6} - 12 T^{5} + \cdots + 361 \)
|
| $17$ |
\( T^{6} + 6 T^{5} + \cdots + 2997 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 75 T^{4} + \cdots - 12312 \)
|
| $29$ |
\( T^{6} - 21 T^{5} + \cdots - 513 \)
|
| $31$ |
\( T^{6} + 6 T^{5} + \cdots - 361 \)
|
| $37$ |
\( T^{6} + 6 T^{5} + \cdots + 11096 \)
|
| $41$ |
\( T^{6} - 36 T^{5} + \cdots + 18981 \)
|
| $43$ |
\( T^{6} + 18 T^{5} + \cdots + 1097 \)
|
| $47$ |
\( T^{6} - 30 T^{5} + \cdots + 27 \)
|
| $53$ |
\( T^{6} - 18 T^{5} + \cdots - 9747 \)
|
| $59$ |
\( T^{6} - 21 T^{5} + \cdots - 112347 \)
|
| $61$ |
\( T^{6} - 9 T^{5} + \cdots + 37657 \)
|
| $67$ |
\( T^{6} - 18 T^{5} + \cdots + 77976 \)
|
| $71$ |
\( T^{6} + 12 T^{5} + \cdots - 4104 \)
|
| $73$ |
\( T^{6} + 24 T^{5} + \cdots + 192 \)
|
| $79$ |
\( T^{6} - 9 T^{5} + \cdots + 39349 \)
|
| $83$ |
\( T^{6} + 3 T^{5} + \cdots - 109269 \)
|
| $89$ |
\( T^{6} - 45 T^{5} + \cdots - 87723 \)
|
| $97$ |
\( T^{6} - 15 T^{5} + \cdots - 130321 \)
|
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