Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1078,4,Mod(1,1078)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1078, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1078.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1078 = 2 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1078.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: | \(63.6040589862\) |
| 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: |
\( x^{5} - x^{4} - 105x^{3} + 196x^{2} + 2156x - 3381 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 154) |
| 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
\( x^{5} - x^{4} - 105x^{3} + 196x^{2} + 2156x - 3381 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 6\nu^{2} - 49\nu - 189 ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{4} + 8\nu^{3} - 143\nu^{2} - 287\nu + 1302 ) / 63 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 7\nu^{3} - 85\nu^{2} - 322\nu + 1407 ) / 63 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{4} + \beta_{3} + 2\beta_{2} - \beta _1 + 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( 12\beta_{4} - 6\beta_{3} + 9\beta_{2} + 55\beta _1 - 63 \)
|
| \(\nu^{4}\) | \(=\) |
\( -191\beta_{4} + 127\beta_{3} + 107\beta_{2} - 148\beta _1 + 2604 \)
|
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 |
|
−2.00000 | −7.44076 | 4.00000 | −8.72246 | 14.8815 | 0 | −8.00000 | 28.3649 | 17.4449 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −6.37816 | 4.00000 | 17.2424 | 12.7563 | 0 | −8.00000 | 13.6809 | −34.4848 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −1.52836 | 4.00000 | −7.02863 | 3.05672 | 0 | −8.00000 | −24.6641 | 14.0573 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 4.97181 | 4.00000 | 11.0004 | −9.94361 | 0 | −8.00000 | −2.28115 | −22.0007 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 9.37547 | 4.00000 | −2.49169 | −18.7509 | 0 | −8.00000 | 60.8995 | 4.98338 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1078.4.a.u | 5 | |
| 7.b | odd | 2 | 1 | 1078.4.a.v | 5 | ||
| 7.d | odd | 6 | 2 | 154.4.e.d | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.e.d | ✓ | 10 | 7.d | odd | 6 | 2 | |
| 1078.4.a.u | 5 | 1.a | even | 1 | 1 | trivial | |
| 1078.4.a.v | 5 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + T_{3}^{4} - 105T_{3}^{3} - 196T_{3}^{2} + 2156T_{3} + 3381 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1078))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
|
| $3$ |
\( T^{5} + T^{4} + \cdots + 3381 \)
|
| $5$ |
\( T^{5} - 10 T^{4} + \cdots + 28974 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( (T - 11)^{5} \)
|
| $13$ |
\( T^{5} - 95 T^{4} + \cdots - 41909589 \)
|
| $17$ |
\( T^{5} - 91 T^{4} + \cdots + 260493492 \)
|
| $19$ |
\( T^{5} - 97 T^{4} + \cdots - 158912718 \)
|
| $23$ |
\( T^{5} - 70 T^{4} + \cdots - 176915034 \)
|
| $29$ |
\( T^{5} + \cdots - 1042605783 \)
|
| $31$ |
\( T^{5} + \cdots - 2550495216 \)
|
| $37$ |
\( T^{5} + \cdots - 81972268368 \)
|
| $41$ |
\( T^{5} + \cdots + 170842763574 \)
|
| $43$ |
\( T^{5} + \cdots - 6373649114432 \)
|
| $47$ |
\( T^{5} + \cdots + 69105344292 \)
|
| $53$ |
\( T^{5} + \cdots + 1132603137666 \)
|
| $59$ |
\( T^{5} + \cdots - 29584232790093 \)
|
| $61$ |
\( T^{5} + \cdots + 7276158445688 \)
|
| $67$ |
\( T^{5} + \cdots - 9806038534776 \)
|
| $71$ |
\( T^{5} + \cdots + 111018937184694 \)
|
| $73$ |
\( T^{5} + \cdots - 26663281726098 \)
|
| $79$ |
\( T^{5} + \cdots + 3002630981553 \)
|
| $83$ |
\( T^{5} + \cdots + 19379587900572 \)
|
| $89$ |
\( T^{5} + \cdots + 5211099594504 \)
|
| $97$ |
\( T^{5} + \cdots + 409729638076937 \)
|
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