Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1521,4,Mod(1,1521)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1521, 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("1521.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1521 = 3^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1521.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: | \(89.7419051187\) |
| 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: |
\( x^{4} - 2x^{3} - 25x^{2} + 24x + 78 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 39) |
| 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
\( x^{4} - 2x^{3} - 25x^{2} + 24x + 78 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 19\nu + 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 21\beta _1 + 11 \)
|
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 |
|
−5.33039 | 0 | 20.4131 | 16.4131 | 0 | 9.67968 | −66.1667 | 0 | −87.4882 | ||||||||||||||||||||||||||||||
| 1.2 | −2.36176 | 0 | −2.42208 | −6.42208 | 0 | −29.4938 | 24.6145 | 0 | 15.1674 | |||||||||||||||||||||||||||||||
| 1.3 | 1.46610 | 0 | −5.85055 | −9.85055 | 0 | 29.9396 | −20.3063 | 0 | −14.4419 | |||||||||||||||||||||||||||||||
| 1.4 | 4.22605 | 0 | 9.85953 | 5.85953 | 0 | −24.1254 | 7.85849 | 0 | 24.7627 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(13\) | \(1\) |
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 | 1521.4.a.v | 4 | |
| 3.b | odd | 2 | 1 | 507.4.a.m | 4 | ||
| 13.b | even | 2 | 1 | 1521.4.a.bb | 4 | ||
| 13.c | even | 3 | 2 | 117.4.g.e | 8 | ||
| 39.d | odd | 2 | 1 | 507.4.a.i | 4 | ||
| 39.f | even | 4 | 2 | 507.4.b.h | 8 | ||
| 39.i | odd | 6 | 2 | 39.4.e.c | ✓ | 8 | |
| 156.p | even | 6 | 2 | 624.4.q.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.4.e.c | ✓ | 8 | 39.i | odd | 6 | 2 | |
| 117.4.g.e | 8 | 13.c | even | 3 | 2 | ||
| 507.4.a.i | 4 | 39.d | odd | 2 | 1 | ||
| 507.4.a.m | 4 | 3.b | odd | 2 | 1 | ||
| 507.4.b.h | 8 | 39.f | even | 4 | 2 | ||
| 624.4.q.i | 8 | 156.p | even | 6 | 2 | ||
| 1521.4.a.v | 4 | 1.a | even | 1 | 1 | trivial | |
| 1521.4.a.bb | 4 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):
|
\( T_{2}^{4} + 2T_{2}^{3} - 25T_{2}^{2} - 24T_{2} + 78 \)
|
|
\( T_{5}^{4} - 6T_{5}^{3} - 203T_{5}^{2} + 156T_{5} + 6084 \)
|
|
\( T_{7}^{4} + 14T_{7}^{3} - 1123T_{7}^{2} - 12652T_{7} + 206212 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 78 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 6 T^{3} + \cdots + 6084 \)
|
| $7$ |
\( T^{4} + 14 T^{3} + \cdots + 206212 \)
|
| $11$ |
\( T^{4} + 40 T^{3} + \cdots + 27408 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 98 T^{3} + \cdots + 22571952 \)
|
| $19$ |
\( T^{4} - 124 T^{3} + \cdots + 1720384 \)
|
| $23$ |
\( T^{4} + 104 T^{3} + \cdots - 2571504 \)
|
| $29$ |
\( T^{4} + 194 T^{3} + \cdots - 274591068 \)
|
| $31$ |
\( T^{4} - 26 T^{3} + \cdots + 328187792 \)
|
| $37$ |
\( T^{4} - 102 T^{3} + \cdots + 27176708 \)
|
| $41$ |
\( T^{4} + \cdots + 1021233552 \)
|
| $43$ |
\( T^{4} + \cdots - 2362804828 \)
|
| $47$ |
\( T^{4} + \cdots + 42871452048 \)
|
| $53$ |
\( T^{4} + 262 T^{3} + \cdots + 744728256 \)
|
| $59$ |
\( T^{4} + \cdots - 2116598016 \)
|
| $61$ |
\( T^{4} + \cdots - 5230543711 \)
|
| $67$ |
\( T^{4} + \cdots + 10235224388 \)
|
| $71$ |
\( T^{4} + \cdots - 99058755696 \)
|
| $73$ |
\( T^{4} + \cdots - 120133390247 \)
|
| $79$ |
\( T^{4} + 746 T^{3} + \cdots + 680937616 \)
|
| $83$ |
\( T^{4} + \cdots + 58964273856 \)
|
| $89$ |
\( T^{4} + \cdots - 268616132736 \)
|
| $97$ |
\( T^{4} - 2166 T^{3} + \cdots - 445091164 \)
|
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