Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,2,Mod(31,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.bc (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: | \(4.98266508613\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu + 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 4\nu^{2} + 9\nu - 20 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta _1 - 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{3} - 5\beta_{2} + 13\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(209\) | \(469\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 1.00000i | 0 | −2.56155 | + | 2.56155i | 0 | 2.00000 | − | 2.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||
| 31.2 | 0 | − | 1.00000i | 0 | 1.56155 | − | 1.56155i | 0 | 2.00000 | − | 2.00000i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||
| 463.1 | 0 | 1.00000i | 0 | −2.56155 | − | 2.56155i | 0 | 2.00000 | + | 2.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||
| 463.2 | 0 | 1.00000i | 0 | 1.56155 | + | 1.56155i | 0 | 2.00000 | + | 2.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 52.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 624.2.bc.d | yes | 4 |
| 3.b | odd | 2 | 1 | 1872.2.bf.j | 4 | ||
| 4.b | odd | 2 | 1 | 624.2.bc.c | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1872.2.bf.i | 4 | ||
| 13.d | odd | 4 | 1 | 624.2.bc.c | ✓ | 4 | |
| 39.f | even | 4 | 1 | 1872.2.bf.i | 4 | ||
| 52.f | even | 4 | 1 | inner | 624.2.bc.d | yes | 4 |
| 156.l | odd | 4 | 1 | 1872.2.bf.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 624.2.bc.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 624.2.bc.c | ✓ | 4 | 13.d | odd | 4 | 1 | |
| 624.2.bc.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 624.2.bc.d | yes | 4 | 52.f | even | 4 | 1 | inner |
| 1872.2.bf.i | 4 | 12.b | even | 2 | 1 | ||
| 1872.2.bf.i | 4 | 39.f | even | 4 | 1 | ||
| 1872.2.bf.j | 4 | 3.b | odd | 2 | 1 | ||
| 1872.2.bf.j | 4 | 156.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):
|
\( T_{5}^{4} + 2T_{5}^{3} + 2T_{5}^{2} - 16T_{5} + 64 \)
|
|
\( T_{7}^{2} - 4T_{7} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 1)^{2} \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 64 \)
|
| $7$ |
\( (T^{2} - 4 T + 8)^{2} \)
|
| $11$ |
\( T^{4} - 2 T^{3} + \cdots + 64 \)
|
| $13$ |
\( T^{4} - 6 T^{3} + \cdots + 169 \)
|
| $17$ |
\( (T^{2} + 4)^{2} \)
|
| $19$ |
\( T^{4} - 4 T^{3} + \cdots + 1024 \)
|
| $23$ |
\( (T^{2} - 6 T - 8)^{2} \)
|
| $29$ |
\( (T^{2} - 2 T - 16)^{2} \)
|
| $31$ |
\( T^{4} - 4 T^{3} + \cdots + 1024 \)
|
| $37$ |
\( T^{4} + 1156 \)
|
| $41$ |
\( T^{4} + 10 T^{3} + \cdots + 16 \)
|
| $43$ |
\( (T^{2} + 4 T - 64)^{2} \)
|
| $47$ |
\( T^{4} - 10 T^{3} + \cdots + 4096 \)
|
| $53$ |
\( (T - 10)^{4} \)
|
| $59$ |
\( T^{4} - 18 T^{3} + \cdots + 1024 \)
|
| $61$ |
\( (T^{2} + 14 T + 32)^{2} \)
|
| $67$ |
\( (T^{2} - 4 T + 8)^{2} \)
|
| $71$ |
\( T^{4} + 34 T^{3} + \cdots + 18496 \)
|
| $73$ |
\( (T^{2} + 14 T + 98)^{2} \)
|
| $79$ |
\( T^{4} + 208T^{2} + 1024 \)
|
| $83$ |
\( T^{4} - 34 T^{3} + \cdots + 18496 \)
|
| $89$ |
\( T^{4} - 26 T^{3} + \cdots + 64 \)
|
| $97$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
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