Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(1,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, 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("448.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.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: | \(26.4328556826\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2981.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 11x + 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 224) |
| 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
\( x^{3} - x^{2} - 11x + 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} + 2\nu - 31 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 30 ) / 4 \)
|
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 | −8.18600 | 0 | −16.4526 | 0 | 7.00000 | 0 | 40.0106 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −4.64895 | 0 | 18.3341 | 0 | 7.00000 | 0 | −5.38731 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 4.83495 | 0 | −11.8815 | 0 | 7.00000 | 0 | −3.62330 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
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 | 448.4.a.u | 3 | |
| 4.b | odd | 2 | 1 | 448.4.a.x | 3 | ||
| 8.b | even | 2 | 1 | 224.4.a.h | yes | 3 | |
| 8.d | odd | 2 | 1 | 224.4.a.e | ✓ | 3 | |
| 24.f | even | 2 | 1 | 2016.4.a.s | 3 | ||
| 24.h | odd | 2 | 1 | 2016.4.a.t | 3 | ||
| 56.e | even | 2 | 1 | 1568.4.a.y | 3 | ||
| 56.h | odd | 2 | 1 | 1568.4.a.v | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.4.a.e | ✓ | 3 | 8.d | odd | 2 | 1 | |
| 224.4.a.h | yes | 3 | 8.b | even | 2 | 1 | |
| 448.4.a.u | 3 | 1.a | even | 1 | 1 | trivial | |
| 448.4.a.x | 3 | 4.b | odd | 2 | 1 | ||
| 1568.4.a.v | 3 | 56.h | odd | 2 | 1 | ||
| 1568.4.a.y | 3 | 56.e | even | 2 | 1 | ||
| 2016.4.a.s | 3 | 24.f | even | 2 | 1 | ||
| 2016.4.a.t | 3 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(448))\):
|
\( T_{3}^{3} + 8T_{3}^{2} - 24T_{3} - 184 \)
|
|
\( T_{5}^{3} + 10T_{5}^{2} - 324T_{5} - 3584 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 8 T^{2} + \cdots - 184 \)
|
| $5$ |
\( T^{3} + 10 T^{2} + \cdots - 3584 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} + 24 T^{2} + \cdots - 33856 \)
|
| $13$ |
\( T^{3} + 58 T^{2} + \cdots - 333392 \)
|
| $17$ |
\( T^{3} - 174 T^{2} + \cdots - 64104 \)
|
| $19$ |
\( T^{3} + 64 T^{2} + \cdots - 297704 \)
|
| $23$ |
\( T^{3} + 112 T^{2} + \cdots + 137856 \)
|
| $29$ |
\( T^{3} + 314 T^{2} + \cdots - 1238728 \)
|
| $31$ |
\( T^{3} + 280 T^{2} + \cdots - 3550016 \)
|
| $37$ |
\( T^{3} + 178 T^{2} + \cdots - 17098664 \)
|
| $41$ |
\( T^{3} - 318 T^{2} + \cdots + 22957208 \)
|
| $43$ |
\( T^{3} + 632 T^{2} + \cdots + 1740608 \)
|
| $47$ |
\( T^{3} + 664 T^{2} + \cdots + 8880832 \)
|
| $53$ |
\( T^{3} + 434 T^{2} + \cdots - 301352 \)
|
| $59$ |
\( T^{3} + 816 T^{2} + \cdots - 11275976 \)
|
| $61$ |
\( T^{3} + 450 T^{2} + \cdots - 377975904 \)
|
| $67$ |
\( T^{3} + 408 T^{2} + \cdots + 90099712 \)
|
| $71$ |
\( T^{3} - 184 T^{2} + \cdots + 8757248 \)
|
| $73$ |
\( T^{3} - 6 T^{2} + \cdots + 33883128 \)
|
| $79$ |
\( T^{3} + 552 T^{2} + \cdots - 5384192 \)
|
| $83$ |
\( T^{3} - 1736 T^{2} + \cdots + 114210264 \)
|
| $89$ |
\( T^{3} + 218 T^{2} + \cdots - 725061384 \)
|
| $97$ |
\( T^{3} + 1090 T^{2} + \cdots - 776495336 \)
|
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