Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(225,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.225");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.b (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: | \(26.4328556826\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\zeta_{12}^{3} + 4\zeta_{12} \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} - \beta_1 ) / 8 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( \beta_{3} + 3\beta _1 + 4 ) / 8 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( \beta_{3} - \beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(129\) | \(197\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 225.1 |
|
0 | − | 9.66025i | 0 | 11.8038i | 0 | −7.00000 | 0 | −66.3205 | 0 | |||||||||||||||||||||||||||||
| 225.2 | 0 | − | 7.66025i | 0 | − | 22.1962i | 0 | −7.00000 | 0 | −31.6795 | 0 | |||||||||||||||||||||||||||||
| 225.3 | 0 | 7.66025i | 0 | 22.1962i | 0 | −7.00000 | 0 | −31.6795 | 0 | |||||||||||||||||||||||||||||||
| 225.4 | 0 | 9.66025i | 0 | − | 11.8038i | 0 | −7.00000 | 0 | −66.3205 | 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 | 448.4.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 448.4.b.b | yes | 4 | |
| 8.b | even | 2 | 1 | inner | 448.4.b.a | ✓ | 4 |
| 8.d | odd | 2 | 1 | 448.4.b.b | yes | 4 | |
| 16.e | even | 4 | 1 | 1792.4.a.a | 2 | ||
| 16.e | even | 4 | 1 | 1792.4.a.d | 2 | ||
| 16.f | odd | 4 | 1 | 1792.4.a.b | 2 | ||
| 16.f | odd | 4 | 1 | 1792.4.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 448.4.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 448.4.b.a | ✓ | 4 | 8.b | even | 2 | 1 | inner |
| 448.4.b.b | yes | 4 | 4.b | odd | 2 | 1 | |
| 448.4.b.b | yes | 4 | 8.d | odd | 2 | 1 | |
| 1792.4.a.a | 2 | 16.e | even | 4 | 1 | ||
| 1792.4.a.b | 2 | 16.f | odd | 4 | 1 | ||
| 1792.4.a.c | 2 | 16.f | odd | 4 | 1 | ||
| 1792.4.a.d | 2 | 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 \(S_{4}^{\mathrm{new}}(448, [\chi])\):
|
\( T_{3}^{4} + 152T_{3}^{2} + 5476 \)
|
|
\( T_{23}^{2} - 288T_{23} + 20148 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 152T^{2} + 5476 \)
|
| $5$ |
\( T^{4} + 632 T^{2} + 68644 \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} + 2912 T^{2} + 2096704 \)
|
| $13$ |
\( T^{4} + 3752 T^{2} + 743044 \)
|
| $17$ |
\( (T^{2} + 140 T + 3172)^{2} \)
|
| $19$ |
\( T^{4} + 30312 T^{2} + 228070404 \)
|
| $23$ |
\( (T^{2} - 288 T + 20148)^{2} \)
|
| $29$ |
\( T^{4} + 9248 T^{2} + 5345344 \)
|
| $31$ |
\( (T^{2} - 280 T + 17872)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 10317683776 \)
|
| $41$ |
\( (T^{2} + 244 T - 17564)^{2} \)
|
| $43$ |
\( T^{4} + 66272 T^{2} + 239135296 \)
|
| $47$ |
\( (T^{2} - 392 T - 50336)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 1448868096 \)
|
| $59$ |
\( T^{4} + \cdots + 16809122500 \)
|
| $61$ |
\( T^{4} + 106136 T^{2} + 682985956 \)
|
| $67$ |
\( T^{4} + \cdots + 92922548224 \)
|
| $71$ |
\( (T^{2} - 248 T + 1504)^{2} \)
|
| $73$ |
\( (T^{2} - 204 T - 86796)^{2} \)
|
| $79$ |
\( (T^{2} - 440 T - 193568)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 22271980644 \)
|
| $89$ |
\( (T^{2} + 1156 T + 310852)^{2} \)
|
| $97$ |
\( (T^{2} + 428 T - 803276)^{2} \)
|
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