Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [354,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("354.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(20.8866761420\) |
| 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: |
\( x^{4} - x^{3} - 44x^{2} + 19x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{4} - x^{3} - 44x^{2} + 19x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 45\nu - 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 43\nu + 14 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 22 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{2} + 45\beta _1 + 16 ) / 2 \)
|
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 |
|
2.00000 | −3.00000 | 4.00000 | −14.5171 | −6.00000 | −18.9849 | 8.00000 | 9.00000 | −29.0343 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.00000 | 4.00000 | 3.16153 | −6.00000 | 21.5427 | 8.00000 | 9.00000 | 6.32306 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.00000 | 4.00000 | 15.9851 | −6.00000 | −18.6951 | 8.00000 | 9.00000 | 31.9702 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −3.00000 | 4.00000 | 17.3705 | −6.00000 | 29.1373 | 8.00000 | 9.00000 | 34.7410 | |||||||||||||||||||||||||||||||
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 | 354.4.a.h | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1062.4.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.4.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1062.4.a.n | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 22T_{5}^{3} - 147T_{5}^{2} + 4684T_{5} - 12744 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(354))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} - 22 T^{3} + \cdots - 12744 \)
|
| $7$ |
\( T^{4} - 13 T^{3} + \cdots + 222784 \)
|
| $11$ |
\( T^{4} - 24 T^{3} + \cdots + 68272 \)
|
| $13$ |
\( T^{4} - 20 T^{3} + \cdots - 330872 \)
|
| $17$ |
\( T^{4} - 91 T^{3} + \cdots - 12374716 \)
|
| $19$ |
\( T^{4} - 141 T^{3} + \cdots + 4408272 \)
|
| $23$ |
\( T^{4} - 13 T^{3} + \cdots + 3954096 \)
|
| $29$ |
\( T^{4} - 295 T^{3} + \cdots + 2092376 \)
|
| $31$ |
\( T^{4} + \cdots - 1103280172 \)
|
| $37$ |
\( T^{4} - 609 T^{3} + \cdots + 441501528 \)
|
| $41$ |
\( T^{4} - 677 T^{3} + \cdots - 716976796 \)
|
| $43$ |
\( T^{4} - 170 T^{3} + \cdots + 61291392 \)
|
| $47$ |
\( T^{4} + \cdots + 1595260944 \)
|
| $53$ |
\( T^{4} - 166 T^{3} + \cdots - 890248168 \)
|
| $59$ |
\( (T - 59)^{4} \)
|
| $61$ |
\( T^{4} - 651 T^{3} + \cdots + 375354696 \)
|
| $67$ |
\( T^{4} + \cdots - 14066090032 \)
|
| $71$ |
\( T^{4} + \cdots - 53676221404 \)
|
| $73$ |
\( T^{4} + \cdots - 1837650372 \)
|
| $79$ |
\( T^{4} + \cdots + 12881904768 \)
|
| $83$ |
\( T^{4} + \cdots - 3649164368 \)
|
| $89$ |
\( T^{4} + \cdots + 72105812788 \)
|
| $97$ |
\( T^{4} + \cdots - 294308919036 \)
|
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