Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [342,10,Mod(1,342)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(342, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("342.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.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: | \(176.142255968\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - 2x^{3} - 8016x^{2} - 155839x + 2105804 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{29}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 38) |
| 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
\( x^{4} - 2x^{3} - 8016x^{2} - 155839x + 2105804 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 14\nu^{2} - 14774\nu - 194644 ) / 693 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 113\nu^{2} - 9131\nu + 199772 ) / 693 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{3} + 226\nu^{2} + 34894\nu - 407860 ) / 693 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 12 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 19\beta_{3} - 18\beta_{2} + 56\beta _1 + 32100 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3893\beta_{3} + 7198\beta_{2} + 4746\beta _1 + 1549236 ) / 12 \)
|
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 |
|
−16.0000 | 0 | 256.000 | −2263.93 | 0 | 5766.09 | −4096.00 | 0 | 36222.9 | ||||||||||||||||||||||||||||||
| 1.2 | −16.0000 | 0 | 256.000 | −745.613 | 0 | −3114.51 | −4096.00 | 0 | 11929.8 | |||||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 0 | 256.000 | 845.315 | 0 | −6607.98 | −4096.00 | 0 | −13525.0 | |||||||||||||||||||||||||||||||
| 1.4 | −16.0000 | 0 | 256.000 | 1298.23 | 0 | 6626.40 | −4096.00 | 0 | −20771.6 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(19\) | \(1\) |
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 | 342.10.a.i | 4 | |
| 3.b | odd | 2 | 1 | 38.10.a.e | ✓ | 4 | |
| 12.b | even | 2 | 1 | 304.10.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.10.a.e | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 304.10.a.d | 4 | 12.b | even | 2 | 1 | ||
| 342.10.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 866T_{5}^{3} - 3665655T_{5}^{2} - 315628300T_{5} + 1852446724000 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(342))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 1852446724000 \)
|
| $7$ |
\( T^{4} + \cdots + 786353549326443 \)
|
| $11$ |
\( T^{4} + \cdots + 30\!\cdots\!92 \)
|
| $13$ |
\( T^{4} + \cdots - 79\!\cdots\!24 \)
|
| $17$ |
\( T^{4} + \cdots + 16\!\cdots\!09 \)
|
| $19$ |
\( (T + 130321)^{4} \)
|
| $23$ |
\( T^{4} + \cdots - 60\!\cdots\!88 \)
|
| $29$ |
\( T^{4} + \cdots - 13\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 16\!\cdots\!44 \)
|
| $37$ |
\( T^{4} + \cdots + 88\!\cdots\!52 \)
|
| $41$ |
\( T^{4} + \cdots - 18\!\cdots\!92 \)
|
| $43$ |
\( T^{4} + \cdots - 70\!\cdots\!72 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots - 13\!\cdots\!08 \)
|
| $59$ |
\( T^{4} + \cdots - 41\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 32\!\cdots\!92 \)
|
| $67$ |
\( T^{4} + \cdots - 45\!\cdots\!48 \)
|
| $71$ |
\( T^{4} + \cdots - 13\!\cdots\!12 \)
|
| $73$ |
\( T^{4} + \cdots + 14\!\cdots\!57 \)
|
| $79$ |
\( T^{4} + \cdots + 32\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 94\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 24\!\cdots\!84 \)
|
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