Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [322,4,Mod(1,322)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("322.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 322 = 2 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 322.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: | \(18.9986150218\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3368.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 15x + 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 15x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 2\nu - 11 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 2\nu - 9 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 10 \)
|
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 |
|
2.00000 | −8.34310 | 4.00000 | 1.16019 | −16.6862 | −7.00000 | 8.00000 | 42.6073 | 2.32039 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −1.61873 | 4.00000 | 3.31608 | −3.23745 | −7.00000 | 8.00000 | −24.3797 | 6.63217 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 2.96183 | 4.00000 | −12.4763 | 5.92365 | −7.00000 | 8.00000 | −18.2276 | −24.9526 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(23\) | \( +1 \) |
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 | 322.4.a.e | ✓ | 3 |
| 7.b | odd | 2 | 1 | 2254.4.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.4.a.e | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2254.4.a.i | 3 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 7T_{3}^{2} - 16T_{3} - 40 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(322))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} + 7 T^{2} + \cdots - 40 \)
|
| $5$ |
\( T^{3} + 8 T^{2} + \cdots + 48 \)
|
| $7$ |
\( (T + 7)^{3} \)
|
| $11$ |
\( T^{3} - 18 T^{2} + \cdots + 31936 \)
|
| $13$ |
\( T^{3} + 63 T^{2} + \cdots - 49392 \)
|
| $17$ |
\( T^{3} + 112 T^{2} + \cdots + 1024 \)
|
| $19$ |
\( T^{3} + 128 T^{2} + \cdots - 60720 \)
|
| $23$ |
\( (T + 23)^{3} \)
|
| $29$ |
\( T^{3} + 103 T^{2} + \cdots - 4980 \)
|
| $31$ |
\( T^{3} + 347 T^{2} + \cdots + 1015524 \)
|
| $37$ |
\( T^{3} + 152 T^{2} + \cdots + 2206896 \)
|
| $41$ |
\( T^{3} + 387 T^{2} + \cdots - 125700 \)
|
| $43$ |
\( T^{3} - 100 T^{2} + \cdots - 2497344 \)
|
| $47$ |
\( T^{3} + 403 T^{2} + \cdots - 27319676 \)
|
| $53$ |
\( T^{3} - 44 T^{2} + \cdots - 370688 \)
|
| $59$ |
\( T^{3} - 156 T^{2} + \cdots + 7954272 \)
|
| $61$ |
\( T^{3} + 682 T^{2} + \cdots - 68522744 \)
|
| $67$ |
\( T^{3} + 104 T^{2} + \cdots + 41876480 \)
|
| $71$ |
\( T^{3} - 207 T^{2} + \cdots + 9533808 \)
|
| $73$ |
\( T^{3} + 1249 T^{2} + \cdots + 59609428 \)
|
| $79$ |
\( T^{3} - 512 T^{2} + \cdots + 14557696 \)
|
| $83$ |
\( T^{3} + 156 T^{2} + \cdots + 578259232 \)
|
| $89$ |
\( T^{3} + 4 T^{2} + \cdots + 177485408 \)
|
| $97$ |
\( T^{3} - 390 T^{2} + \cdots + 27861584 \)
|
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