Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [588,6,Mod(1,588)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("588.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 588.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: | \(94.3056860500\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{7081}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1770 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 84) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{7081})\). 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 |
|
0 | 9.00000 | 0 | −18.5743 | 0 | 0 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | 65.5743 | 0 | 0 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 588.6.a.l | 2 | |
| 7.b | odd | 2 | 1 | 588.6.a.h | 2 | ||
| 7.c | even | 3 | 2 | 84.6.i.b | ✓ | 4 | |
| 7.d | odd | 6 | 2 | 588.6.i.m | 4 | ||
| 21.h | odd | 6 | 2 | 252.6.k.e | 4 | ||
| 28.g | odd | 6 | 2 | 336.6.q.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.6.i.b | ✓ | 4 | 7.c | even | 3 | 2 | |
| 252.6.k.e | 4 | 21.h | odd | 6 | 2 | ||
| 336.6.q.g | 4 | 28.g | odd | 6 | 2 | ||
| 588.6.a.h | 2 | 7.b | odd | 2 | 1 | ||
| 588.6.a.l | 2 | 1.a | even | 1 | 1 | trivial | |
| 588.6.i.m | 4 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 47T_{5} - 1218 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( T^{2} - 47T - 1218 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 407T + 39642 \)
|
| $13$ |
\( T^{2} - 449T + 6144 \)
|
| $17$ |
\( T^{2} + 1868 T + 844032 \)
|
| $19$ |
\( T^{2} + 1463 T - 4069328 \)
|
| $23$ |
\( T^{2} + 44 T - 17702016 \)
|
| $29$ |
\( T^{2} - 767 T - 20885268 \)
|
| $31$ |
\( T^{2} + 11170 T + 20967261 \)
|
| $37$ |
\( T^{2} + 3113 T + 1642012 \)
|
| $41$ |
\( T^{2} + 7842 T + 15310512 \)
|
| $43$ |
\( T^{2} + 12629 T + 39092230 \)
|
| $47$ |
\( T^{2} - 9576 T + 20630700 \)
|
| $53$ |
\( T^{2} - 13395 T - 49605804 \)
|
| $59$ |
\( T^{2} + 47521 T - 204224580 \)
|
| $61$ |
\( T^{2} + 63652 T + 670173876 \)
|
| $67$ |
\( T^{2} - 44541 T + 421580414 \)
|
| $71$ |
\( T^{2} + \cdots + 3516363900 \)
|
| $73$ |
\( T^{2} - 6039 T - 989305390 \)
|
| $79$ |
\( T^{2} + \cdots - 4649155173 \)
|
| $83$ |
\( T^{2} + \cdots - 6214225254 \)
|
| $89$ |
\( T^{2} + \cdots + 1725081120 \)
|
| $97$ |
\( T^{2} + \cdots + 6255893750 \)
|
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