Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1120,2,Mod(561,1120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1120, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1120.561");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1120 = 2^{5} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1120.b (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: | \(8.94324502638\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.18939904.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 280) |
| 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
\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} - 7\nu^{2} + 6\nu - 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 19\nu^{2} - 12\nu + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 24\nu^{2} + 16\nu - 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \)
|
| \(\beta_{5}\) | \(=\) |
\( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 166\nu - 42 \)
|
| \(\beta_{6}\) | \(=\) |
\( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 168\nu - 43 \)
|
| \(\beta_{7}\) | \(=\) |
\( -28\nu^{7} + 98\nu^{6} - 342\nu^{5} + 610\nu^{4} - 890\nu^{3} + 774\nu^{2} - 440\nu + 109 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{5} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta _1 - 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} - 5\beta_{6} + 7\beta_{5} + 6\beta_{4} + 3\beta_{3} + 3\beta_{2} - 3\beta _1 - 10 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{7} - 6\beta_{6} + 8\beta_{5} + 6\beta_{4} - 4\beta_{3} - 4\beta_{2} + 2\beta _1 + 7 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5\beta_{7} + 11\beta_{6} - 13\beta_{5} - 20\beta_{4} - 25\beta_{3} - 25\beta_{2} + 15\beta _1 + 52 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 10\beta_{7} + 32\beta_{6} - 40\beta_{5} - 45\beta_{4} + 6\beta_{3} + 8\beta_{2} - \beta _1 - 6 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -3\beta_{7} + 11\beta_{6} - 19\beta_{5} + 133\beta_{3} + 147\beta_{2} - 63\beta _1 - 236 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).
| \(n\) | \(351\) | \(421\) | \(801\) | \(897\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 561.1 |
|
0 | − | 2.41421i | 0 | 1.00000i | 0 | 1.00000 | 0 | −2.82843 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 561.2 | 0 | − | 2.41421i | 0 | 1.00000i | 0 | 1.00000 | 0 | −2.82843 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 561.3 | 0 | − | 0.414214i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | 2.82843 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 561.4 | 0 | − | 0.414214i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | 2.82843 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 561.5 | 0 | 0.414214i | 0 | 1.00000i | 0 | 1.00000 | 0 | 2.82843 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 561.6 | 0 | 0.414214i | 0 | 1.00000i | 0 | 1.00000 | 0 | 2.82843 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 561.7 | 0 | 2.41421i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | −2.82843 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 561.8 | 0 | 2.41421i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | −2.82843 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1120.2.b.c | 8 | |
| 4.b | odd | 2 | 1 | 280.2.b.c | ✓ | 8 | |
| 8.b | even | 2 | 1 | inner | 1120.2.b.c | 8 | |
| 8.d | odd | 2 | 1 | 280.2.b.c | ✓ | 8 | |
| 16.e | even | 4 | 1 | 8960.2.a.bs | 4 | ||
| 16.e | even | 4 | 1 | 8960.2.a.bv | 4 | ||
| 16.f | odd | 4 | 1 | 8960.2.a.bt | 4 | ||
| 16.f | odd | 4 | 1 | 8960.2.a.bu | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.2.b.c | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 280.2.b.c | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 1120.2.b.c | 8 | 1.a | even | 1 | 1 | trivial | |
| 1120.2.b.c | 8 | 8.b | even | 2 | 1 | inner | |
| 8960.2.a.bs | 4 | 16.e | even | 4 | 1 | ||
| 8960.2.a.bt | 4 | 16.f | odd | 4 | 1 | ||
| 8960.2.a.bu | 4 | 16.f | odd | 4 | 1 | ||
| 8960.2.a.bv | 4 | 16.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\):
|
\( T_{3}^{4} + 6T_{3}^{2} + 1 \)
|
|
\( T_{13}^{8} + 52T_{13}^{6} + 166T_{13}^{4} + 52T_{13}^{2} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 6 T^{2} + 1)^{2} \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( T^{8} + 52 T^{6} + \cdots + 1681 \)
|
| $13$ |
\( T^{8} + 52 T^{6} + \cdots + 1 \)
|
| $17$ |
\( (T^{4} + 4 T^{3} - 22 T^{2} + \cdots + 41)^{2} \)
|
| $19$ |
\( T^{8} + 120 T^{6} + \cdots + 602176 \)
|
| $23$ |
\( (T^{4} - 4 T^{3} - 20 T^{2} + \cdots - 56)^{2} \)
|
| $29$ |
\( T^{8} + 156 T^{6} + \cdots + 160801 \)
|
| $31$ |
\( (T^{4} - 4 T^{3} - 44 T^{2} + \cdots - 56)^{2} \)
|
| $37$ |
\( T^{8} + 88 T^{6} + \cdots + 61504 \)
|
| $41$ |
\( (T^{4} - 12 T^{3} + \cdots + 392)^{2} \)
|
| $43$ |
\( T^{8} + 104 T^{6} + \cdots + 18496 \)
|
| $47$ |
\( (T^{4} - 74 T^{2} + \cdots + 721)^{2} \)
|
| $53$ |
\( T^{8} + 128 T^{6} + \cdots + 4096 \)
|
| $59$ |
\( T^{8} + 416 T^{6} + \cdots + 66064384 \)
|
| $61$ |
\( T^{8} + 424 T^{6} + \cdots + 47004736 \)
|
| $67$ |
\( T^{8} + 128 T^{6} + \cdots + 4096 \)
|
| $71$ |
\( (T^{4} + 8 T^{3} + \cdots + 3728)^{2} \)
|
| $73$ |
\( (T^{4} - 16 T^{3} + \cdots + 2048)^{2} \)
|
| $79$ |
\( (T^{4} + 24 T^{3} + \cdots - 1751)^{2} \)
|
| $83$ |
\( T^{8} + 176 T^{6} + \cdots + 12544 \)
|
| $89$ |
\( (T^{4} - 28 T^{3} + \cdots - 8696)^{2} \)
|
| $97$ |
\( (T^{4} - 12 T^{3} + \cdots - 10287)^{2} \)
|
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