Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,10,Mod(1,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.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: | \(197.773761087\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 10576x^{3} - 80944x^{2} + 15618603x - 183984404 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{29}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 10576x^{3} - 80944x^{2} + 15618603x - 183984404 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 334\nu^{4} + 5734\nu^{3} - 3266154\nu^{2} - 103642522\nu + 2768551592 ) / 808299 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -52\nu^{4} - 724\nu^{3} + 637404\nu^{2} + 12791116\nu - 984497054 ) / 79245 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -13012\nu^{4} + 1075916\nu^{3} + 85286844\nu^{2} - 6201600404\nu + 6531882256 ) / 4041495 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 151066\nu^{4} + 3066706\nu^{3} - 1529541774\nu^{2} - 40119007246\nu + 1450386559463 ) / 808299 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} + 27\beta_{2} - 425\beta _1 - 11 ) / 12288 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{4} - 26\beta_{3} + 2551\beta_{2} + 1587\beta _1 + 17326697 ) / 4096 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8365\beta_{4} + 19942\beta_{3} + 459903\beta_{2} - 2897717\beta _1 + 596598289 ) / 12288 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 104461\beta_{4} - 575242\beta_{3} + 25106927\beta_{2} - 1946133\beta _1 + 132068847825 ) / 4096 \)
|
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 | 81.0000 | 0 | −2198.43 | 0 | −5598.20 | 0 | 6561.00 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 81.0000 | 0 | −776.708 | 0 | 282.882 | 0 | 6561.00 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 81.0000 | 0 | −572.202 | 0 | 7556.27 | 0 | 6561.00 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 81.0000 | 0 | 1303.06 | 0 | −7990.94 | 0 | 6561.00 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 81.0000 | 0 | 2004.28 | 0 | 5788.00 | 0 | 6561.00 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
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 | 384.10.a.n | yes | 5 |
| 4.b | odd | 2 | 1 | 384.10.a.j | ✓ | 5 | |
| 8.b | even | 2 | 1 | 384.10.a.k | yes | 5 | |
| 8.d | odd | 2 | 1 | 384.10.a.o | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.10.a.j | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 384.10.a.k | yes | 5 | 8.b | even | 2 | 1 | |
| 384.10.a.n | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 384.10.a.o | yes | 5 | 8.d | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(384))\):
|
\( T_{5}^{5} + 240T_{5}^{4} - 5710656T_{5}^{3} - 1036113920T_{5}^{2} + 5674244014080T_{5} + 2551787185766400 \)
|
|
\( T_{7}^{5} - 38T_{7}^{4} - 92935832T_{7}^{3} + 23645721680T_{7}^{2} + 1957252651408208T_{7} - 553461243698564320 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T - 81)^{5} \)
|
| $5$ |
\( T^{5} + \cdots + 25\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots - 55\!\cdots\!20 \)
|
| $11$ |
\( T^{5} + \cdots - 10\!\cdots\!64 \)
|
| $13$ |
\( T^{5} + \cdots - 53\!\cdots\!40 \)
|
| $17$ |
\( T^{5} + \cdots + 72\!\cdots\!60 \)
|
| $19$ |
\( T^{5} + \cdots + 91\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 31\!\cdots\!76 \)
|
| $29$ |
\( T^{5} + \cdots - 78\!\cdots\!48 \)
|
| $31$ |
\( T^{5} + \cdots + 37\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 48\!\cdots\!40 \)
|
| $41$ |
\( T^{5} + \cdots - 52\!\cdots\!92 \)
|
| $43$ |
\( T^{5} + \cdots - 10\!\cdots\!08 \)
|
| $47$ |
\( T^{5} + \cdots + 13\!\cdots\!20 \)
|
| $53$ |
\( T^{5} + \cdots + 38\!\cdots\!80 \)
|
| $59$ |
\( T^{5} + \cdots + 34\!\cdots\!80 \)
|
| $61$ |
\( T^{5} + \cdots + 13\!\cdots\!12 \)
|
| $67$ |
\( T^{5} + \cdots - 16\!\cdots\!68 \)
|
| $71$ |
\( T^{5} + \cdots + 57\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots + 32\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots - 10\!\cdots\!80 \)
|
| $83$ |
\( T^{5} + \cdots - 36\!\cdots\!96 \)
|
| $89$ |
\( T^{5} + \cdots + 57\!\cdots\!40 \)
|
| $97$ |
\( T^{5} + \cdots + 23\!\cdots\!48 \)
|
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