Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1078,2,Mod(1077,1078)]
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([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1078.1077");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1078 = 2 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1078.c (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(8.60787333789\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{16})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{16}^{4} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{16}^{5} + \zeta_{16}^{3} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{16}^{6} + \zeta_{16}^{2} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{16}^{7} + \zeta_{16} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{16}^{7} + \zeta_{16} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{16}^{6} + \zeta_{16}^{2} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{16}^{5} + \zeta_{16}^{3} \)
|
| \(\zeta_{16}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} ) / 2 \)
|
| \(\zeta_{16}^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} ) / 2 \)
|
| \(\zeta_{16}^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{2} ) / 2 \)
|
| \(\zeta_{16}^{4}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{16}^{5}\) | \(=\) |
\( ( -\beta_{7} + \beta_{2} ) / 2 \)
|
| \(\zeta_{16}^{6}\) | \(=\) |
\( ( -\beta_{6} + \beta_{3} ) / 2 \)
|
| \(\zeta_{16}^{7}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(981\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1077.1 |
|
− | 1.00000i | − | 2.61313i | −1.00000 | 2.61313i | −2.61313 | 0 | 1.00000i | −3.82843 | 2.61313 | ||||||||||||||||||||||||||||||||||||||||
| 1077.2 | − | 1.00000i | − | 1.08239i | −1.00000 | 1.08239i | −1.08239 | 0 | 1.00000i | 1.82843 | 1.08239 | |||||||||||||||||||||||||||||||||||||||||
| 1077.3 | − | 1.00000i | 1.08239i | −1.00000 | − | 1.08239i | 1.08239 | 0 | 1.00000i | 1.82843 | −1.08239 | |||||||||||||||||||||||||||||||||||||||||
| 1077.4 | − | 1.00000i | 2.61313i | −1.00000 | − | 2.61313i | 2.61313 | 0 | 1.00000i | −3.82843 | −2.61313 | |||||||||||||||||||||||||||||||||||||||||
| 1077.5 | 1.00000i | − | 2.61313i | −1.00000 | 2.61313i | 2.61313 | 0 | − | 1.00000i | −3.82843 | −2.61313 | |||||||||||||||||||||||||||||||||||||||||
| 1077.6 | 1.00000i | − | 1.08239i | −1.00000 | 1.08239i | 1.08239 | 0 | − | 1.00000i | 1.82843 | −1.08239 | |||||||||||||||||||||||||||||||||||||||||
| 1077.7 | 1.00000i | 1.08239i | −1.00000 | − | 1.08239i | −1.08239 | 0 | − | 1.00000i | 1.82843 | 1.08239 | |||||||||||||||||||||||||||||||||||||||||
| 1077.8 | 1.00000i | 2.61313i | −1.00000 | − | 2.61313i | −2.61313 | 0 | − | 1.00000i | −3.82843 | 2.61313 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 77.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1078.2.c.a | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 1078.2.c.a | ✓ | 8 |
| 7.c | even | 3 | 2 | 1078.2.i.a | 16 | ||
| 7.d | odd | 6 | 2 | 1078.2.i.a | 16 | ||
| 11.b | odd | 2 | 1 | inner | 1078.2.c.a | ✓ | 8 |
| 77.b | even | 2 | 1 | inner | 1078.2.c.a | ✓ | 8 |
| 77.h | odd | 6 | 2 | 1078.2.i.a | 16 | ||
| 77.i | even | 6 | 2 | 1078.2.i.a | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1078.2.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1078.2.c.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 1078.2.c.a | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
| 1078.2.c.a | ✓ | 8 | 77.b | even | 2 | 1 | inner |
| 1078.2.i.a | 16 | 7.c | even | 3 | 2 | ||
| 1078.2.i.a | 16 | 7.d | odd | 6 | 2 | ||
| 1078.2.i.a | 16 | 77.h | odd | 6 | 2 | ||
| 1078.2.i.a | 16 | 77.i | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 8T_{3}^{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{4} + 8 T^{2} + 8)^{2} \)
|
| $5$ |
\( (T^{4} + 8 T^{2} + 8)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} + 14 T^{2} + 121)^{2} \)
|
| $13$ |
\( (T^{4} - 36 T^{2} + 162)^{2} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( (T^{4} - 36 T^{2} + 162)^{2} \)
|
| $23$ |
\( (T^{2} - 4 T - 14)^{4} \)
|
| $29$ |
\( (T^{4} + 108 T^{2} + 324)^{2} \)
|
| $31$ |
\( (T^{4} + 52 T^{2} + 98)^{2} \)
|
| $37$ |
\( (T - 6)^{8} \)
|
| $41$ |
\( (T^{4} - 144 T^{2} + 2592)^{2} \)
|
| $43$ |
\( (T^{2} + 18)^{4} \)
|
| $47$ |
\( (T^{4} + 68 T^{2} + 1058)^{2} \)
|
| $53$ |
\( (T^{2} + 8 T - 56)^{4} \)
|
| $59$ |
\( (T^{4} + 200 T^{2} + 7688)^{2} \)
|
| $61$ |
\( (T^{4} - 36 T^{2} + 162)^{2} \)
|
| $67$ |
\( (T^{2} - 12 T + 4)^{4} \)
|
| $71$ |
\( (T - 2)^{8} \)
|
| $73$ |
\( (T^{4} - 288 T^{2} + 10368)^{2} \)
|
| $79$ |
\( (T^{2} + 72)^{4} \)
|
| $83$ |
\( (T^{4} - 180 T^{2} + 162)^{2} \)
|
| $89$ |
\( (T^{4} + 164 T^{2} + 1922)^{2} \)
|
| $97$ |
\( (T^{4} + 100 T^{2} + 1922)^{2} \)
|
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