Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,6,Mod(1,320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.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: | \(51.3228223402\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.81998080.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 100x^{2} - 376x - 367 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11} \) |
| Twist minimal: | no (minimal twist has level 160) |
| 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
\( x^{4} - 100x^{2} - 376x - 367 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{3} - 4\nu^{2} - 190\nu - 364 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 26\nu^{3} - 68\nu^{2} - 2406\nu - 3932 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -32\nu^{3} + 64\nu^{2} + 3104\nu + 5824 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 16\beta_1 ) / 64 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} + 29\beta _1 + 800 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 103\beta_{3} - 24\beta_{2} + 1784\beta _1 + 18048 ) / 64 \)
|
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 | −27.0971 | 0 | −25.0000 | 0 | 250.932 | 0 | 491.252 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −10.6653 | 0 | −25.0000 | 0 | −176.978 | 0 | −129.252 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 10.6653 | 0 | −25.0000 | 0 | 176.978 | 0 | −129.252 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 27.0971 | 0 | −25.0000 | 0 | −250.932 | 0 | 491.252 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 320.6.a.z | 4 | |
| 4.b | odd | 2 | 1 | inner | 320.6.a.z | 4 | |
| 8.b | even | 2 | 1 | 160.6.a.h | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 160.6.a.h | ✓ | 4 | |
| 40.e | odd | 2 | 1 | 800.6.a.r | 4 | ||
| 40.f | even | 2 | 1 | 800.6.a.r | 4 | ||
| 40.i | odd | 4 | 2 | 800.6.c.l | 8 | ||
| 40.k | even | 4 | 2 | 800.6.c.l | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.6.a.h | ✓ | 4 | 8.b | even | 2 | 1 | |
| 160.6.a.h | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 320.6.a.z | 4 | 1.a | even | 1 | 1 | trivial | |
| 320.6.a.z | 4 | 4.b | odd | 2 | 1 | inner | |
| 800.6.a.r | 4 | 40.e | odd | 2 | 1 | ||
| 800.6.a.r | 4 | 40.f | even | 2 | 1 | ||
| 800.6.c.l | 8 | 40.i | odd | 4 | 2 | ||
| 800.6.c.l | 8 | 40.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 848T_{3}^{2} + 83520 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 848 T^{2} + 83520 \)
|
| $5$ |
\( (T + 25)^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 1972194880 \)
|
| $11$ |
\( T^{4} + \cdots + 16267617280 \)
|
| $13$ |
\( (T^{2} - 292 T - 844988)^{2} \)
|
| $17$ |
\( (T^{2} + 380 T - 830204)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 4329676800000 \)
|
| $23$ |
\( T^{4} + \cdots + 1972194880 \)
|
| $29$ |
\( (T^{2} - 84 T - 31185180)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 18\!\cdots\!20 \)
|
| $37$ |
\( (T^{2} + 9868 T + 20879140)^{2} \)
|
| $41$ |
\( (T^{2} - 23812 T + 99303940)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 863073883368000 \)
|
| $47$ |
\( T^{4} + \cdots + 90\!\cdots\!20 \)
|
| $53$ |
\( (T^{2} + 8748 T - 2525724)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!20 \)
|
| $61$ |
\( (T^{2} - 31012 T - 438746300)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 99\!\cdots\!20 \)
|
| $71$ |
\( T^{4} + \cdots + 13\!\cdots\!80 \)
|
| $73$ |
\( (T^{2} - 59700 T + 883225764)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 40\!\cdots\!80 \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} - 104660 T + 1958755300)^{2} \)
|
| $97$ |
\( (T^{2} + 29564 T - 2402062076)^{2} \)
|
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