Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3344,2,Mod(1,3344)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("3344.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3344 = 2^{4} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3344.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: | \(26.7019744359\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.106392688.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1672) |
| 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
\( x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 10\nu^{2} + 11\nu - 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 12\nu^{2} + 9\nu - 12 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} + 5\nu^{3} - 24\nu^{2} - 5\nu + 22 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 8\beta_{4} - 7\beta_{3} + \beta_{2} + 10\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + 13\beta_{4} - 9\beta_{3} + 9\beta_{2} + 41\beta _1 + 20 \)
|
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 | −1.80207 | 0 | 0.702208 | 0 | 1.70221 | 0 | 0.247456 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.50058 | 0 | −2.88388 | 0 | −1.88388 | 0 | −0.748274 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.0278351 | 0 | −0.122408 | 0 | 0.877592 | 0 | −2.99923 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.87046 | 0 | 2.83301 | 0 | 3.83301 | 0 | 0.498611 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.25613 | 0 | −4.20604 | 0 | −3.20604 | 0 | 2.09011 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.14822 | 0 | 0.677107 | 0 | 1.67711 | 0 | 6.91132 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
| \(19\) | \(1\) |
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 | 3344.2.a.z | 6 | |
| 4.b | odd | 2 | 1 | 1672.2.a.i | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1672.2.a.i | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 3344.2.a.z | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):
|
\( T_{3}^{6} - 4T_{3}^{5} - 4T_{3}^{4} + 24T_{3}^{3} + 2T_{3}^{2} - 36T_{3} + 1 \)
|
|
\( T_{5}^{6} + 3T_{5}^{5} - 13T_{5}^{4} - 23T_{5}^{3} + 41T_{5}^{2} - 11T_{5} - 2 \)
|
|
\( T_{7}^{6} - 3T_{7}^{5} - 13T_{7}^{4} + 39T_{7}^{3} + 17T_{7}^{2} - 101T_{7} + 58 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 1 \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 2 \)
|
| $7$ |
\( T^{6} - 3 T^{5} + \cdots + 58 \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} + 3 T^{5} + \cdots - 212 \)
|
| $17$ |
\( T^{6} + T^{5} + \cdots - 3232 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} - 4 T^{5} + \cdots - 2096 \)
|
| $29$ |
\( T^{6} + 7 T^{5} + \cdots - 844 \)
|
| $31$ |
\( T^{6} - T^{5} - 71 T^{4} + \cdots - 2 \)
|
| $37$ |
\( T^{6} - 5 T^{5} + \cdots + 13856 \)
|
| $41$ |
\( T^{6} + 6 T^{5} + \cdots - 151432 \)
|
| $43$ |
\( T^{6} - 16 T^{5} + \cdots - 152 \)
|
| $47$ |
\( T^{6} - 4 T^{5} + \cdots - 8192 \)
|
| $53$ |
\( T^{6} + 11 T^{5} + \cdots - 40736 \)
|
| $59$ |
\( T^{6} - 18 T^{5} + \cdots - 6736 \)
|
| $61$ |
\( T^{6} - 16 T^{5} + \cdots + 1024 \)
|
| $67$ |
\( T^{6} - 18 T^{5} + \cdots + 13891 \)
|
| $71$ |
\( T^{6} - 7 T^{5} + \cdots - 34 \)
|
| $73$ |
\( T^{6} + 21 T^{5} + \cdots + 99328 \)
|
| $79$ |
\( T^{6} - 22 T^{5} + \cdots + 256 \)
|
| $83$ |
\( T^{6} - 16 T^{5} + \cdots + 122336 \)
|
| $89$ |
\( T^{6} + 13 T^{5} + \cdots + 38912 \)
|
| $97$ |
\( T^{6} + 15 T^{5} + \cdots + 424064 \)
|
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