Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [32,9,Mod(31,32)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32.31");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 32.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: | \(13.0361155220\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{19})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{18} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 9x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -64\nu^{3} + 256\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -64\nu^{3} + 896\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{3} + 32\nu^{2} - 16\nu - 144 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 128 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 16\beta_{3} + 5\beta _1 + 2304 ) / 512 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{2} - 7\beta_1 ) / 64 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 89.7424i | 0 | 1102.91 | 0 | 3321.18i | 0 | −1492.70 | 0 | |||||||||||||||||||||||||||||
| 31.2 | 0 | − | 49.7424i | 0 | −570.909 | 0 | 2537.18i | 0 | 4086.70 | 0 | ||||||||||||||||||||||||||||||
| 31.3 | 0 | 49.7424i | 0 | −570.909 | 0 | − | 2537.18i | 0 | 4086.70 | 0 | ||||||||||||||||||||||||||||||
| 31.4 | 0 | 89.7424i | 0 | 1102.91 | 0 | − | 3321.18i | 0 | −1492.70 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 32.9.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 288.9.g.a | 4 | ||
| 4.b | odd | 2 | 1 | inner | 32.9.c.b | ✓ | 4 |
| 8.b | even | 2 | 1 | 64.9.c.e | 4 | ||
| 8.d | odd | 2 | 1 | 64.9.c.e | 4 | ||
| 12.b | even | 2 | 1 | 288.9.g.a | 4 | ||
| 16.e | even | 4 | 1 | 256.9.d.c | 4 | ||
| 16.e | even | 4 | 1 | 256.9.d.g | 4 | ||
| 16.f | odd | 4 | 1 | 256.9.d.c | 4 | ||
| 16.f | odd | 4 | 1 | 256.9.d.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.9.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 32.9.c.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 64.9.c.e | 4 | 8.b | even | 2 | 1 | ||
| 64.9.c.e | 4 | 8.d | odd | 2 | 1 | ||
| 256.9.d.c | 4 | 16.e | even | 4 | 1 | ||
| 256.9.d.c | 4 | 16.f | odd | 4 | 1 | ||
| 256.9.d.g | 4 | 16.e | even | 4 | 1 | ||
| 256.9.d.g | 4 | 16.f | odd | 4 | 1 | ||
| 288.9.g.a | 4 | 3.b | odd | 2 | 1 | ||
| 288.9.g.a | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 10528T_{3}^{2} + 19927296 \)
acting on \(S_{9}^{\mathrm{new}}(32, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 10528 T^{2} + 19927296 \)
|
| $5$ |
\( (T^{2} - 532 T - 629660)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 71004756250624 \)
|
| $11$ |
\( T^{4} + \cdots + 749474285334784 \)
|
| $13$ |
\( (T^{2} - 46292 T + 417367012)^{2} \)
|
| $17$ |
\( (T^{2} - 124260 T - 1320139836)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 20\!\cdots\!96 \)
|
| $23$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + 245356 T - 812539941020)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots - 2674936593180)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots + 2695753597060)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!76 \)
|
| $53$ |
\( (T^{2} + \cdots - 6045671785500)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 70\!\cdots\!24 \)
|
| $61$ |
\( (T^{2} + \cdots + 81632354350564)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 90\!\cdots\!04 \)
|
| $71$ |
\( T^{4} + \cdots + 16\!\cdots\!64 \)
|
| $73$ |
\( (T^{2} + \cdots - 84820874301756)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 19\!\cdots\!16 \)
|
| $83$ |
\( T^{4} + \cdots + 39\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} + \cdots - 748461593591100)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 85\!\cdots\!16)^{2} \)
|
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