Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [51,8,Mod(1,51)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(51, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("51.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 51 = 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 51.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: | \(15.9316362997\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 298x^{2} + 300x + 6120 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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
\( x^{4} - x^{3} - 298x^{2} + 300x + 6120 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 11\nu^{2} + 244\nu - 1732 ) / 56 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 3\nu^{2} - 258\nu - 382 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + \beta _1 + 151 \)
|
| \(\nu^{3}\) | \(=\) |
\( 11\beta_{3} - 12\beta_{2} + 255\beta _1 - 71 \)
|
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 |
|
−20.6521 | 27.0000 | 298.510 | −179.861 | −557.607 | −319.187 | −3521.39 | 729.000 | 3714.51 | ||||||||||||||||||||||||||||||
| 1.2 | −8.19533 | 27.0000 | −60.8366 | 265.589 | −221.274 | −1237.51 | 1547.58 | 729.000 | −2176.59 | |||||||||||||||||||||||||||||||
| 1.3 | 1.29120 | 27.0000 | −126.333 | −86.1897 | 34.8624 | 1269.21 | −328.394 | 729.000 | −111.288 | |||||||||||||||||||||||||||||||
| 1.4 | 12.5562 | 27.0000 | 29.6594 | −309.538 | 339.019 | −748.513 | −1234.79 | 729.000 | −3886.64 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 51.8.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 153.8.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.8.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 153.8.a.f | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 15T_{2}^{3} - 214T_{2}^{2} - 1876T_{2} + 2744 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(51))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 15 T^{3} + \cdots + 2744 \)
|
| $3$ |
\( (T - 27)^{4} \)
|
| $5$ |
\( T^{4} + \cdots - 1274432000 \)
|
| $7$ |
\( T^{4} + \cdots - 375256425600 \)
|
| $11$ |
\( T^{4} + \cdots - 6774432270336 \)
|
| $13$ |
\( T^{4} + \cdots + 180456607428788 \)
|
| $17$ |
\( (T + 4913)^{4} \)
|
| $19$ |
\( T^{4} + \cdots - 54\!\cdots\!24 \)
|
| $23$ |
\( T^{4} + \cdots - 30\!\cdots\!24 \)
|
| $29$ |
\( T^{4} + \cdots - 68\!\cdots\!08 \)
|
| $31$ |
\( T^{4} + \cdots + 36\!\cdots\!76 \)
|
| $37$ |
\( T^{4} + \cdots - 76\!\cdots\!44 \)
|
| $41$ |
\( T^{4} + \cdots + 31\!\cdots\!68 \)
|
| $43$ |
\( T^{4} + \cdots - 13\!\cdots\!16 \)
|
| $47$ |
\( T^{4} + \cdots - 11\!\cdots\!48 \)
|
| $53$ |
\( T^{4} + \cdots + 90\!\cdots\!52 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!20 \)
|
| $61$ |
\( T^{4} + \cdots + 82\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 15\!\cdots\!60 \)
|
| $71$ |
\( T^{4} + \cdots - 16\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots - 44\!\cdots\!52 \)
|
| $79$ |
\( T^{4} + \cdots + 81\!\cdots\!64 \)
|
| $83$ |
\( T^{4} + \cdots + 15\!\cdots\!28 \)
|
| $89$ |
\( T^{4} + \cdots - 29\!\cdots\!72 \)
|
| $97$ |
\( T^{4} + \cdots - 14\!\cdots\!80 \)
|
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