Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [640,4,Mod(1,640)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(640, 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("640.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 640 = 2^{7} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 640.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: | \(37.7612224037\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1040528.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 25x^{2} + 14x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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
\( x^{4} - 2x^{3} - 25x^{2} + 14x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 44\nu - 36 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 12\nu^{2} + 26\nu - 117 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 3\beta _1 + 54 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{2} + 22\beta _1 + 58 ) / 2 \)
|
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 |
|
0 | −9.13763 | 0 | −5.00000 | 0 | 23.1776 | 0 | 56.4963 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.22671 | 0 | −5.00000 | 0 | −4.11169 | 0 | −22.0417 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 5.50114 | 0 | −5.00000 | 0 | −33.3372 | 0 | 3.26254 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.86321 | 0 | −5.00000 | 0 | 34.2713 | 0 | 70.2829 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 640.4.a.s | yes | 4 |
| 4.b | odd | 2 | 1 | 640.4.a.q | ✓ | 4 | |
| 8.b | even | 2 | 1 | 640.4.a.r | yes | 4 | |
| 8.d | odd | 2 | 1 | 640.4.a.t | yes | 4 | |
| 16.e | even | 4 | 2 | 1280.4.d.bb | 8 | ||
| 16.f | odd | 4 | 2 | 1280.4.d.bc | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 640.4.a.q | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 640.4.a.r | yes | 4 | 8.b | even | 2 | 1 | |
| 640.4.a.s | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 640.4.a.t | yes | 4 | 8.d | odd | 2 | 1 | |
| 1280.4.d.bb | 8 | 16.e | even | 4 | 2 | ||
| 1280.4.d.bc | 8 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(640))\):
|
\( T_{3}^{4} - 4T_{3}^{3} - 100T_{3}^{2} + 304T_{3} + 1104 \)
|
|
\( T_{7}^{4} - 20T_{7}^{3} - 1220T_{7}^{2} + 21872T_{7} + 108880 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 1104 \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} - 20 T^{3} + \cdots + 108880 \)
|
| $11$ |
\( T^{4} - 72 T^{3} + \cdots + 1613712 \)
|
| $13$ |
\( T^{4} + 32 T^{3} + \cdots + 258192 \)
|
| $17$ |
\( T^{4} - 96 T^{3} + \cdots - 3269232 \)
|
| $19$ |
\( T^{4} + 128 T^{3} + \cdots + 1164816 \)
|
| $23$ |
\( T^{4} - 44 T^{3} + \cdots - 13516080 \)
|
| $29$ |
\( T^{4} + 200 T^{3} + \cdots - 22368240 \)
|
| $31$ |
\( T^{4} - 24 T^{3} + \cdots + 6237440 \)
|
| $37$ |
\( T^{4} - 24 T^{3} + \cdots + 991516560 \)
|
| $41$ |
\( T^{4} + 32 T^{3} + \cdots + 607861648 \)
|
| $43$ |
\( T^{4} + \cdots - 2241857200 \)
|
| $47$ |
\( T^{4} + \cdots - 20842229424 \)
|
| $53$ |
\( T^{4} + \cdots + 20356308240 \)
|
| $59$ |
\( T^{4} + 176 T^{3} + \cdots + 628890768 \)
|
| $61$ |
\( T^{4} + \cdots - 34160636912 \)
|
| $67$ |
\( T^{4} + \cdots + 116617993040 \)
|
| $71$ |
\( T^{4} + \cdots + 2987836672 \)
|
| $73$ |
\( T^{4} + \cdots + 14073390480 \)
|
| $79$ |
\( T^{4} + \cdots - 490073812992 \)
|
| $83$ |
\( T^{4} + \cdots + 256975168080 \)
|
| $89$ |
\( T^{4} + \cdots - 415967432304 \)
|
| $97$ |
\( T^{4} + \cdots + 433240091664 \)
|
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