Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [474,4,Mod(1,474)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(474, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("474.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 474 = 2 \cdot 3 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 474.a (trivial) |
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: | yes |
| Analytic conductor: | \(27.9669053427\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{21}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
2.00000 | 3.00000 | 4.00000 | −16.3739 | 6.00000 | 8.74773 | 8.00000 | 9.00000 | −32.7477 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.00000 | 4.00000 | −2.62614 | 6.00000 | −18.7477 | 8.00000 | 9.00000 | −5.25227 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(79\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 474.4.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 1422.4.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 474.4.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 1422.4.a.f | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 19T_{5} + 43 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(474))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( (T - 3)^{2} \)
|
| $5$ |
\( T^{2} + 19T + 43 \)
|
| $7$ |
\( T^{2} + 10T - 164 \)
|
| $11$ |
\( T^{2} + 87T + 1845 \)
|
| $13$ |
\( T^{2} + 23T - 125 \)
|
| $17$ |
\( T^{2} + 140T + 2800 \)
|
| $19$ |
\( T^{2} + 9T - 3261 \)
|
| $23$ |
\( T^{2} + 137T - 6905 \)
|
| $29$ |
\( T^{2} + 82T - 7580 \)
|
| $31$ |
\( T^{2} + 37T + 337 \)
|
| $37$ |
\( T^{2} + 304T + 12940 \)
|
| $41$ |
\( T^{2} - 162T - 39828 \)
|
| $43$ |
\( T^{2} - 374T + 27388 \)
|
| $47$ |
\( T^{2} + 444T - 3216 \)
|
| $53$ |
\( T^{2} + 344T + 24208 \)
|
| $59$ |
\( T^{2} + 528T - 11028 \)
|
| $61$ |
\( T^{2} + 18T - 371388 \)
|
| $67$ |
\( T^{2} - 61T - 818201 \)
|
| $71$ |
\( T^{2} + 2T - 657908 \)
|
| $73$ |
\( T^{2} - 345T + 12699 \)
|
| $79$ |
\( (T - 79)^{2} \)
|
| $83$ |
\( T^{2} - 1664 T + 409648 \)
|
| $89$ |
\( T^{2} + 301T - 646457 \)
|
| $97$ |
\( T^{2} - 1865 T + 673999 \)
|
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