Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1456,2,Mod(337,1456)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1456, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1456.337");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1456 = 2^{4} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1456.k (of order \(2\), degree \(1\), not 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: | \(11.6262185343\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 17x^{6} + 72x^{4} + 36x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 728) |
| 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 in terms of a root \(\nu\) of
\( x^{8} + 17x^{6} + 72x^{4} + 36x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 15\nu^{4} + 52\nu^{2} + 12 ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 15\nu^{4} - 42\nu^{2} + 28 ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} - 35\nu^{5} - 159\nu^{3} - 114\nu ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{7} + 50\nu^{5} + 201\nu^{3} + 56\nu ) / 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 17\nu^{4} - 70\nu^{2} - 18 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -11\nu^{7} - 185\nu^{5} - 752\nu^{3} - 202\nu ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 3\beta_{5} - \beta_{4} - 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} - 9\beta_{3} - 14\beta_{2} + 33 \)
|
| \(\nu^{5}\) | \(=\) |
\( -15\beta_{7} - 49\beta_{5} + 9\beta_{4} + 74\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15\beta_{6} + 83\beta_{3} + 168\beta_{2} - 299 \)
|
| \(\nu^{7}\) | \(=\) |
\( 183\beta_{7} + 619\beta_{5} - 83\beta_{4} - 716\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).
| \(n\) | \(561\) | \(911\) | \(1093\) | \(1249\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
0 | −2.45513 | 0 | − | 1.45513i | 0 | 1.00000i | 0 | 3.02768 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 337.2 | 0 | −2.45513 | 0 | 1.45513i | 0 | − | 1.00000i | 0 | 3.02768 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 337.3 | 0 | −0.401251 | 0 | − | 0.598749i | 0 | − | 1.00000i | 0 | −2.83900 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 337.4 | 0 | −0.401251 | 0 | 0.598749i | 0 | 1.00000i | 0 | −2.83900 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 337.5 | 0 | 0.629066 | 0 | − | 1.62907i | 0 | − | 1.00000i | 0 | −2.60428 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 337.6 | 0 | 0.629066 | 0 | 1.62907i | 0 | 1.00000i | 0 | −2.60428 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 337.7 | 0 | 3.22732 | 0 | − | 4.22732i | 0 | − | 1.00000i | 0 | 7.41559 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 337.8 | 0 | 3.22732 | 0 | 4.22732i | 0 | 1.00000i | 0 | 7.41559 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1456.2.k.e | 8 | |
| 4.b | odd | 2 | 1 | 728.2.k.a | ✓ | 8 | |
| 13.b | even | 2 | 1 | inner | 1456.2.k.e | 8 | |
| 52.b | odd | 2 | 1 | 728.2.k.a | ✓ | 8 | |
| 52.f | even | 4 | 1 | 9464.2.a.x | 4 | ||
| 52.f | even | 4 | 1 | 9464.2.a.y | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 728.2.k.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 728.2.k.a | ✓ | 8 | 52.b | odd | 2 | 1 | |
| 1456.2.k.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 1456.2.k.e | 8 | 13.b | even | 2 | 1 | inner | |
| 9464.2.a.x | 4 | 52.f | even | 4 | 1 | ||
| 9464.2.a.y | 4 | 52.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - T_{3}^{3} - 8T_{3}^{2} + 2T_{3} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - T^{3} - 8 T^{2} + \cdots + 2)^{2} \)
|
| $5$ |
\( T^{8} + 23 T^{6} + \cdots + 36 \)
|
| $7$ |
\( (T^{2} + 1)^{4} \)
|
| $11$ |
\( T^{8} + 37 T^{6} + \cdots + 2500 \)
|
| $13$ |
\( T^{8} - 2 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( (T^{4} - 2 T^{3} + \cdots - 156)^{2} \)
|
| $19$ |
\( T^{8} + 47 T^{6} + \cdots + 400 \)
|
| $23$ |
\( (T^{4} + 10 T^{3} + \cdots - 97)^{2} \)
|
| $29$ |
\( (T^{4} + 3 T^{3} - 61 T^{2} + \cdots - 80)^{2} \)
|
| $31$ |
\( T^{8} + 92 T^{6} + \cdots + 961 \)
|
| $37$ |
\( T^{8} + 69 T^{6} + \cdots + 66564 \)
|
| $41$ |
\( T^{8} + 93 T^{6} + \cdots + 576 \)
|
| $43$ |
\( (T^{4} - 7 T^{3} - 33 T^{2} + \cdots - 52)^{2} \)
|
| $47$ |
\( T^{8} + 128 T^{6} + \cdots + 1521 \)
|
| $53$ |
\( (T^{4} + 13 T^{3} + \cdots + 186)^{2} \)
|
| $59$ |
\( T^{8} + 168 T^{6} + \cdots + 576 \)
|
| $61$ |
\( (T^{4} + 23 T^{3} + \cdots - 1480)^{2} \)
|
| $67$ |
\( T^{8} + 185 T^{6} + \cdots + 186624 \)
|
| $71$ |
\( T^{8} + 128 T^{6} + \cdots + 331776 \)
|
| $73$ |
\( T^{8} + 96 T^{6} + \cdots + 169 \)
|
| $79$ |
\( (T^{4} - 20 T^{3} + \cdots + 729)^{2} \)
|
| $83$ |
\( T^{8} + 415 T^{6} + \cdots + 3048516 \)
|
| $89$ |
\( T^{8} + 315 T^{6} + \cdots + 10562500 \)
|
| $97$ |
\( T^{8} + 632 T^{6} + \cdots + 226051225 \)
|
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