Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(71.8519512762\) |
| 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: |
\( x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 56\nu^{4} - 64\nu^{3} - 8772\nu^{2} - 20218\nu - 3528 ) / 633 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 28\nu^{4} - 32\nu^{3} - 5652\nu^{2} - 8210\nu + 114075 ) / 633 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 36\nu^{4} + 200\nu^{3} - 8352\nu^{2} - 46908\nu + 206833 ) / 211 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{3} + \beta_{2} + 3\beta _1 + 366 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{4} - 45\beta_{3} + 9\beta_{2} + 630\beta _1 + 1298 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4\beta_{4} - 339\beta_{3} + 207\beta_{2} + 1552\beta _1 + 58325 ) / 4 \)
|
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 | −27.3187 | 0 | −52.2232 | 0 | 49.0000 | 0 | 503.312 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.95336 | 0 | −11.6830 | 0 | 49.0000 | 0 | −207.557 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.49210 | 0 | 99.5911 | 0 | 49.0000 | 0 | −236.789 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 19.9790 | 0 | −106.158 | 0 | 49.0000 | 0 | 156.159 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 25.7852 | 0 | 34.4729 | 0 | 49.0000 | 0 | 421.876 | 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 | 448.6.a.bf | 5 | |
| 4.b | odd | 2 | 1 | 448.6.a.be | 5 | ||
| 8.b | even | 2 | 1 | 224.6.a.i | ✓ | 5 | |
| 8.d | odd | 2 | 1 | 224.6.a.j | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.6.a.i | ✓ | 5 | 8.b | even | 2 | 1 | |
| 224.6.a.j | yes | 5 | 8.d | odd | 2 | 1 | |
| 448.6.a.be | 5 | 4.b | odd | 2 | 1 | ||
| 448.6.a.bf | 5 | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):
|
\( T_{3}^{5} - 10T_{3}^{4} - 876T_{3}^{3} + 7592T_{3}^{2} + 107952T_{3} + 208800 \)
|
|
\( T_{5}^{5} + 36T_{5}^{4} - 11972T_{5}^{3} - 342672T_{5}^{2} + 16702736T_{5} + 222365696 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 10 T^{4} + \cdots + 208800 \)
|
| $5$ |
\( T^{5} + 36 T^{4} + \cdots + 222365696 \)
|
| $7$ |
\( (T - 49)^{5} \)
|
| $11$ |
\( T^{5} + \cdots - 183286596608 \)
|
| $13$ |
\( T^{5} + \cdots - 247266196544 \)
|
| $17$ |
\( T^{5} + \cdots - 353511151373280 \)
|
| $19$ |
\( T^{5} + \cdots - 71\!\cdots\!08 \)
|
| $23$ |
\( T^{5} + \cdots + 11\!\cdots\!52 \)
|
| $29$ |
\( T^{5} + \cdots - 38\!\cdots\!20 \)
|
| $31$ |
\( T^{5} + \cdots + 34\!\cdots\!20 \)
|
| $37$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 13\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 77\!\cdots\!32 \)
|
| $47$ |
\( T^{5} + \cdots + 62\!\cdots\!40 \)
|
| $53$ |
\( T^{5} + \cdots + 57\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots + 14\!\cdots\!20 \)
|
| $61$ |
\( T^{5} + \cdots + 30\!\cdots\!16 \)
|
| $67$ |
\( T^{5} + \cdots - 16\!\cdots\!64 \)
|
| $71$ |
\( T^{5} + \cdots + 10\!\cdots\!36 \)
|
| $73$ |
\( T^{5} + \cdots + 81\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots - 26\!\cdots\!92 \)
|
| $83$ |
\( T^{5} + \cdots + 86\!\cdots\!00 \)
|
| $89$ |
\( T^{5} + \cdots + 64\!\cdots\!92 \)
|
| $97$ |
\( T^{5} + \cdots + 93\!\cdots\!00 \)
|
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