Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [880,2,Mod(287,880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(880, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("880.287");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 880.bf (of order \(4\), degree \(2\), 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: | \(7.02683537787\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | 6.0.3534400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{5} - 70\nu^{4} + 183\nu^{3} + 120\nu^{2} - 966\nu + 240 ) / 445 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{5} - 5\nu^{4} - 184\nu^{3} + 390\nu^{2} + 643\nu - 1000 ) / 445 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 8\nu^{4} - 26\nu^{3} - \nu^{2} + 57\nu - 180 ) / 89 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -9\nu^{5} + 24\nu^{4} + 11\nu^{3} - 92\nu^{2} - 7\nu - 6 ) / 178 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{5} - 4\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{5} - 14\beta_{4} + 9\beta_{3} - 3\beta_{2} - 10\beta _1 - 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 32\beta_{4} + 6\beta_{3} - 17\beta_{2} - 25\beta _1 - 42 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).
| \(n\) | \(111\) | \(177\) | \(321\) | \(661\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{5}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 287.1 |
|
0 | −1.51702 | − | 1.51702i | 0 | 1.30135 | − | 1.81837i | 0 | −1.60270 | + | 1.60270i | 0 | 1.60270i | 0 | ||||||||||||||||||||||||||||||
| 287.2 | 0 | 0.743268 | + | 0.743268i | 0 | −0.447553 | + | 2.19082i | 0 | 1.89511 | − | 1.89511i | 0 | − | 1.89511i | 0 | ||||||||||||||||||||||||||||||
| 287.3 | 0 | 1.77375 | + | 1.77375i | 0 | 2.14620 | + | 0.627553i | 0 | −3.29240 | + | 3.29240i | 0 | 3.29240i | 0 | |||||||||||||||||||||||||||||||
| 463.1 | 0 | −1.51702 | + | 1.51702i | 0 | 1.30135 | + | 1.81837i | 0 | −1.60270 | − | 1.60270i | 0 | − | 1.60270i | 0 | ||||||||||||||||||||||||||||||
| 463.2 | 0 | 0.743268 | − | 0.743268i | 0 | −0.447553 | − | 2.19082i | 0 | 1.89511 | + | 1.89511i | 0 | 1.89511i | 0 | |||||||||||||||||||||||||||||||
| 463.3 | 0 | 1.77375 | − | 1.77375i | 0 | 2.14620 | − | 0.627553i | 0 | −3.29240 | − | 3.29240i | 0 | − | 3.29240i | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 20.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 880.2.bf.f | yes | 6 |
| 4.b | odd | 2 | 1 | 880.2.bf.e | ✓ | 6 | |
| 5.c | odd | 4 | 1 | 880.2.bf.e | ✓ | 6 | |
| 20.e | even | 4 | 1 | inner | 880.2.bf.f | yes | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 880.2.bf.e | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 880.2.bf.e | ✓ | 6 | 5.c | odd | 4 | 1 | |
| 880.2.bf.f | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 880.2.bf.f | yes | 6 | 20.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\):
|
\( T_{3}^{6} - 2T_{3}^{5} + 2T_{3}^{4} + 2T_{3}^{3} + 25T_{3}^{2} - 40T_{3} + 32 \)
|
|
\( T_{7}^{6} + 6T_{7}^{5} + 18T_{7}^{4} - 8T_{7}^{3} + 64T_{7}^{2} + 320T_{7} + 800 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 32 \)
|
| $5$ |
\( T^{6} - 6 T^{5} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 800 \)
|
| $11$ |
\( (T^{2} + 1)^{3} \)
|
| $13$ |
\( (T^{2} - 4 T + 8)^{3} \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots + 2048 \)
|
| $19$ |
\( (T^{3} - 6 T^{2} - 16 T + 16)^{2} \)
|
| $23$ |
\( T^{6} + 10 T^{5} + \cdots + 2312 \)
|
| $29$ |
\( T^{6} + 68 T^{4} + \cdots + 6400 \)
|
| $31$ |
\( T^{6} + 98 T^{4} + \cdots + 256 \)
|
| $37$ |
\( T^{6} + 8 T^{5} + \cdots + 2 \)
|
| $41$ |
\( (T^{3} + 18 T^{2} + \cdots + 80)^{2} \)
|
| $43$ |
\( T^{6} - 6 T^{5} + \cdots + 32 \)
|
| $47$ |
\( T^{6} + 12 T^{5} + \cdots + 32 \)
|
| $53$ |
\( T^{6} + 10 T^{5} + \cdots + 1260872 \)
|
| $59$ |
\( (T^{3} - 6 T^{2} + \cdots + 172)^{2} \)
|
| $61$ |
\( (T^{3} + 10 T^{2} + \cdots + 32)^{2} \)
|
| $67$ |
\( T^{6} - 38 T^{5} + \cdots + 336200 \)
|
| $71$ |
\( T^{6} + 102 T^{4} + \cdots + 35344 \)
|
| $73$ |
\( T^{6} - 8 T^{5} + \cdots + 131072 \)
|
| $79$ |
\( (T^{3} - 14 T^{2} + \cdots + 1448)^{2} \)
|
| $83$ |
\( T^{6} + 34 T^{5} + \cdots + 339488 \)
|
| $89$ |
\( T^{6} + 342 T^{4} + \cdots + 605284 \)
|
| $97$ |
\( T^{6} + 12 T^{5} + \cdots + 661250 \)
|
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