Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [354,6,Mod(1,354)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(354, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("354.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 354.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: | \(56.7758722138\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.32832.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 9x^{2} + 10x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{4} \) |
| 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
\( x^{4} - 2x^{3} - 9x^{2} + 10x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu^{3} - 3\nu^{2} - 30\nu + 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{2} + 3\nu - 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( -3\nu^{3} + 6\nu^{2} + 24\nu - 21 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - \beta _1 + 4 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + \beta _1 + 50 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} + 4\beta_{2} - 2\beta _1 + 23 ) / 3 \)
|
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 |
|
4.00000 | 9.00000 | 16.0000 | −84.5867 | 36.0000 | 83.0564 | 64.0000 | 81.0000 | −338.347 | ||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | 9.00000 | 16.0000 | −38.5011 | 36.0000 | −5.25164 | 64.0000 | 81.0000 | −154.005 | |||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 9.00000 | 16.0000 | −28.1958 | 36.0000 | −61.0514 | 64.0000 | 81.0000 | −112.783 | |||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 9.00000 | 16.0000 | 47.2837 | 36.0000 | −178.753 | 64.0000 | 81.0000 | 189.135 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(59\) | \(1\) |
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 | 354.6.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1062.6.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.6.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1062.6.a.b | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 104T_{5}^{3} - 426T_{5}^{2} - 226264T_{5} - 4341815 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{4} \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} + 104 T^{3} + \cdots - 4341815 \)
|
| $7$ |
\( T^{4} + 162 T^{3} + \cdots - 4760127 \)
|
| $11$ |
\( T^{4} + \cdots - 6438510695 \)
|
| $13$ |
\( T^{4} + \cdots - 110744152479 \)
|
| $17$ |
\( T^{4} + \cdots + 129643070353 \)
|
| $19$ |
\( T^{4} + \cdots - 1515482739455 \)
|
| $23$ |
\( T^{4} + \cdots + 271268454601 \)
|
| $29$ |
\( T^{4} + \cdots + 15857050401385 \)
|
| $31$ |
\( T^{4} + \cdots - 334188288488855 \)
|
| $37$ |
\( T^{4} + \cdots - 34\!\cdots\!47 \)
|
| $41$ |
\( T^{4} + \cdots - 33\!\cdots\!35 \)
|
| $43$ |
\( T^{4} + \cdots + 260339333498865 \)
|
| $47$ |
\( T^{4} + \cdots - 10\!\cdots\!75 \)
|
| $53$ |
\( T^{4} + \cdots - 22\!\cdots\!19 \)
|
| $59$ |
\( (T + 3481)^{4} \)
|
| $61$ |
\( T^{4} + \cdots + 301384626340705 \)
|
| $67$ |
\( T^{4} + \cdots - 85\!\cdots\!23 \)
|
| $71$ |
\( T^{4} + \cdots + 68\!\cdots\!85 \)
|
| $73$ |
\( T^{4} + \cdots + 74\!\cdots\!45 \)
|
| $79$ |
\( T^{4} + \cdots + 21\!\cdots\!41 \)
|
| $83$ |
\( T^{4} + \cdots + 57\!\cdots\!81 \)
|
| $89$ |
\( T^{4} + \cdots - 19\!\cdots\!15 \)
|
| $97$ |
\( T^{4} + \cdots - 86\!\cdots\!11 \)
|
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