Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [213,2,Mod(1,213)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(213, 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("213.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 213 = 3 \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 213.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: | \(1.70081356305\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2225.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + 2x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 5x^{2} + 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 2 \)
|
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 |
|
−1.43828 | −1.00000 | 0.0686587 | −3.94523 | 1.43828 | 0.493058 | 2.77782 | 1.00000 | 5.67435 | ||||||||||||||||||||||||||||||
| 1.2 | −0.138564 | −1.00000 | −1.98080 | 0.703671 | 0.138564 | 3.84224 | 0.551597 | 1.00000 | −0.0975037 | |||||||||||||||||||||||||||||||
| 1.3 | 1.82025 | −1.00000 | 1.31331 | 1.32719 | −1.82025 | 2.50694 | −1.24995 | 1.00000 | 2.41582 | |||||||||||||||||||||||||||||||
| 1.4 | 2.75660 | −1.00000 | 5.59883 | −1.08564 | −2.75660 | −0.842236 | 9.92054 | 1.00000 | −2.99267 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(71\) | \( -1 \) |
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 | 213.2.a.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 639.2.a.i | 4 | ||
| 4.b | odd | 2 | 1 | 3408.2.a.w | 4 | ||
| 5.b | even | 2 | 1 | 5325.2.a.w | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 213.2.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 639.2.a.i | 4 | 3.b | odd | 2 | 1 | ||
| 3408.2.a.w | 4 | 4.b | odd | 2 | 1 | ||
| 5325.2.a.w | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 3T_{2}^{3} - 2T_{2}^{2} + 7T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(213))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{4} \)
|
| $5$ |
\( T^{4} + 3 T^{3} + \cdots + 4 \)
|
| $7$ |
\( T^{4} - 6 T^{3} + \cdots - 4 \)
|
| $11$ |
\( T^{4} - 2 T^{3} + \cdots - 16 \)
|
| $13$ |
\( T^{4} - 5 T^{3} + \cdots + 4 \)
|
| $17$ |
\( T^{4} + 8 T^{3} + \cdots - 604 \)
|
| $19$ |
\( T^{4} - 8 T^{3} + \cdots - 304 \)
|
| $23$ |
\( T^{4} + T^{3} + \cdots - 64 \)
|
| $29$ |
\( T^{4} + 5 T^{3} + \cdots - 1076 \)
|
| $31$ |
\( T^{4} - 2 T^{3} + \cdots + 2096 \)
|
| $37$ |
\( T^{4} - 19 T^{3} + \cdots + 284 \)
|
| $41$ |
\( T^{4} + 19 T^{3} + \cdots + 244 \)
|
| $43$ |
\( T^{4} - 25 T^{3} + \cdots + 400 \)
|
| $47$ |
\( T^{4} - 7 T^{3} + \cdots - 496 \)
|
| $53$ |
\( T^{4} + 5 T^{3} + \cdots + 524 \)
|
| $59$ |
\( T^{4} - 10 T^{3} + \cdots - 1936 \)
|
| $61$ |
\( T^{4} - 2 T^{3} + \cdots + 604 \)
|
| $67$ |
\( T^{4} - 35 T^{3} + \cdots + 3284 \)
|
| $71$ |
\( (T - 1)^{4} \)
|
| $73$ |
\( T^{4} - 2 T^{3} + \cdots - 656 \)
|
| $79$ |
\( T^{4} + T^{3} + \cdots - 656 \)
|
| $83$ |
\( T^{4} - 18 T^{3} + \cdots - 11216 \)
|
| $89$ |
\( T^{4} + 16 T^{3} + \cdots - 3644 \)
|
| $97$ |
\( T^{4} + T^{3} + \cdots + 76 \)
|
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