Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2475,4,Mod(1,2475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, 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("2475.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2475.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: | \(146.029727264\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1772.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 12x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 495) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 12x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 8 \)
|
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 |
|
−3.32803 | 0 | 3.07579 | 0 | 0 | −12.1102 | 16.3879 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0.654334 | 0 | −7.57185 | 0 | 0 | −3.49576 | −10.1892 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 3.67370 | 0 | 5.49606 | 0 | 0 | 13.6060 | −9.19874 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
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 | 2475.4.a.x | 3 | |
| 3.b | odd | 2 | 1 | 2475.4.a.u | 3 | ||
| 5.b | even | 2 | 1 | 495.4.a.h | ✓ | 3 | |
| 15.d | odd | 2 | 1 | 495.4.a.j | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 495.4.a.h | ✓ | 3 | 5.b | even | 2 | 1 | |
| 495.4.a.j | yes | 3 | 15.d | odd | 2 | 1 | |
| 2475.4.a.u | 3 | 3.b | odd | 2 | 1 | ||
| 2475.4.a.x | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):
|
\( T_{2}^{3} - T_{2}^{2} - 12T_{2} + 8 \)
|
|
\( T_{7}^{3} + 2T_{7}^{2} - 170T_{7} - 576 \)
|
|
\( T_{29}^{3} + 34T_{29}^{2} - 23008T_{29} - 1455456 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - T^{2} - 12T + 8 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + 2 T^{2} + \cdots - 576 \)
|
| $11$ |
\( (T - 11)^{3} \)
|
| $13$ |
\( T^{3} - 66 T^{2} + \cdots + 42592 \)
|
| $17$ |
\( T^{3} + 100 T^{2} + \cdots - 53148 \)
|
| $19$ |
\( T^{3} + 70 T^{2} + \cdots - 6984 \)
|
| $23$ |
\( T^{3} + 10 T^{2} + \cdots - 609248 \)
|
| $29$ |
\( T^{3} + 34 T^{2} + \cdots - 1455456 \)
|
| $31$ |
\( T^{3} + 104 T^{2} + \cdots - 1002816 \)
|
| $37$ |
\( T^{3} - 270 T^{2} + \cdots + 266408 \)
|
| $41$ |
\( T^{3} + 82 T^{2} + \cdots + 156960 \)
|
| $43$ |
\( T^{3} - 428 T^{2} + \cdots - 2665692 \)
|
| $47$ |
\( T^{3} + 246 T^{2} + \cdots - 16904592 \)
|
| $53$ |
\( T^{3} + 298 T^{2} + \cdots + 230680 \)
|
| $59$ |
\( T^{3} + 204 T^{2} + \cdots - 13251872 \)
|
| $61$ |
\( T^{3} + 492 T^{2} + \cdots - 11332144 \)
|
| $67$ |
\( T^{3} - 812 T^{2} + \cdots - 18667072 \)
|
| $71$ |
\( T^{3} + 442 T^{2} + \cdots - 12388520 \)
|
| $73$ |
\( T^{3} - 1044 T^{2} + \cdots + 170236756 \)
|
| $79$ |
\( T^{3} + 1436 T^{2} + \cdots + 41550640 \)
|
| $83$ |
\( T^{3} + 734 T^{2} + \cdots + 18970200 \)
|
| $89$ |
\( T^{3} + 556 T^{2} + \cdots - 76817600 \)
|
| $97$ |
\( T^{3} - 1038 T^{2} + \cdots - 47060968 \)
|
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