Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1452,4,Mod(1,1452)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1452, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1452.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1452.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: | \(85.6707733283\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.885025.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 93x^{2} + 94x + 2189 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 132) |
| 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
\( x^{4} - 2x^{3} - 93x^{2} + 94x + 2189 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 49\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - \nu - 49 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 4\beta_{3} + \beta _1 + 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{3} + 4\beta_{2} + 50\beta _1 + 49 \)
|
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 |
|
0 | −3.00000 | 0 | −14.7487 | 0 | −36.1752 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −1.98361 | 0 | 12.2614 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | −1.66773 | 0 | 27.3211 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | 12.4000 | 0 | −14.4073 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 1452.4.a.o | 4 | |
| 11.b | odd | 2 | 1 | 1452.4.a.p | 4 | ||
| 11.c | even | 5 | 2 | 132.4.i.b | ✓ | 8 | |
| 33.h | odd | 10 | 2 | 396.4.j.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 132.4.i.b | ✓ | 8 | 11.c | even | 5 | 2 | |
| 396.4.j.b | 8 | 33.h | odd | 10 | 2 | ||
| 1452.4.a.o | 4 | 1.a | even | 1 | 1 | trivial | |
| 1452.4.a.p | 4 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1452))\):
|
\( T_{5}^{4} + 6T_{5}^{3} - 171T_{5}^{2} - 660T_{5} - 605 \)
|
|
\( T_{7}^{4} + 11T_{7}^{3} - 1146T_{7}^{2} - 3685T_{7} + 174595 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots - 605 \)
|
| $7$ |
\( T^{4} + 11 T^{3} + \cdots + 174595 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 158 T^{3} + \cdots + 314116 \)
|
| $17$ |
\( T^{4} + 117 T^{3} + \cdots - 13948880 \)
|
| $19$ |
\( T^{4} + 139 T^{3} + \cdots + 40780 \)
|
| $23$ |
\( T^{4} + 174 T^{3} + \cdots - 28151636 \)
|
| $29$ |
\( T^{4} + 177 T^{3} + \cdots - 270422636 \)
|
| $31$ |
\( T^{4} - 104 T^{3} + \cdots + 147373039 \)
|
| $37$ |
\( T^{4} + \cdots - 1403540784 \)
|
| $41$ |
\( T^{4} - 534 T^{3} + \cdots - 227885636 \)
|
| $43$ |
\( T^{4} - 342 T^{3} + \cdots + 819106020 \)
|
| $47$ |
\( T^{4} + \cdots + 3184072144 \)
|
| $53$ |
\( T^{4} + \cdots + 9427551109 \)
|
| $59$ |
\( T^{4} + \cdots + 17577710611 \)
|
| $61$ |
\( T^{4} + \cdots - 9607246916 \)
|
| $67$ |
\( T^{4} + \cdots + 14033195776 \)
|
| $71$ |
\( T^{4} + 273 T^{3} + \cdots - 65557580 \)
|
| $73$ |
\( T^{4} + \cdots + 21165828556 \)
|
| $79$ |
\( T^{4} - 63 T^{3} + \cdots - 487585251 \)
|
| $83$ |
\( T^{4} + \cdots - 199364597009 \)
|
| $89$ |
\( T^{4} + \cdots - 381112589036 \)
|
| $97$ |
\( T^{4} + \cdots - 351009166905 \)
|
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