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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 699x^{2} - 686x + 59664 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2}\cdot 7 \) |
| 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 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
\( x^{4} - 2x^{3} - 699x^{2} - 686x + 59664 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{3} + 76\nu^{2} + 4535\nu - 33814 ) / 427 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 117\nu^{2} - 373\nu - 39257 ) / 427 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 5\nu^{2} + 495\nu - 435 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 13\beta_{2} + 16\beta _1 + 134 ) / 252 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{3} + 215\beta_{2} + 31\beta _1 + 22097 ) / 63 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -187\beta_{3} - 305\beta_{2} + 1220\beta _1 + 56950 ) / 36 \)
|
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 | −92.8255 | 0 | 0 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −9.00000 | 0 | −31.9616 | 0 | 0 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −9.00000 | 0 | 46.1154 | 0 | 0 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −9.00000 | 0 | 78.6718 | 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.n | 4 | |
| 7.b | odd | 2 | 1 | 588.6.a.p | 4 | ||
| 7.c | even | 3 | 2 | 84.6.i.c | ✓ | 8 | |
| 7.d | odd | 6 | 2 | 588.6.i.o | 8 | ||
| 21.h | odd | 6 | 2 | 252.6.k.f | 8 | ||
| 28.g | odd | 6 | 2 | 336.6.q.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.6.i.c | ✓ | 8 | 7.c | even | 3 | 2 | |
| 252.6.k.f | 8 | 21.h | odd | 6 | 2 | ||
| 336.6.q.i | 8 | 28.g | odd | 6 | 2 | ||
| 588.6.a.n | 4 | 1.a | even | 1 | 1 | trivial | |
| 588.6.a.p | 4 | 7.b | odd | 2 | 1 | ||
| 588.6.i.o | 8 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 8977T_{5}^{2} + 82500T_{5} + 10763676 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 9)^{4} \)
|
| $5$ |
\( T^{4} - 8977 T^{2} + \cdots + 10763676 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 16028038332 \)
|
| $13$ |
\( T^{4} + \cdots + 755795447424 \)
|
| $17$ |
\( T^{4} + \cdots - 15241024512 \)
|
| $19$ |
\( T^{4} + \cdots + 865904533024 \)
|
| $23$ |
\( T^{4} + \cdots - 13518157473792 \)
|
| $29$ |
\( T^{4} + \cdots - 333378202056336 \)
|
| $31$ |
\( T^{4} + \cdots + 100222024781907 \)
|
| $37$ |
\( T^{4} + \cdots + 40\!\cdots\!96 \)
|
| $41$ |
\( T^{4} + \cdots - 15\!\cdots\!88 \)
|
| $43$ |
\( T^{4} + \cdots - 359515701753932 \)
|
| $47$ |
\( T^{4} + \cdots - 76\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots - 29\!\cdots\!92 \)
|
| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!28 \)
|
| $61$ |
\( T^{4} + \cdots - 62\!\cdots\!84 \)
|
| $67$ |
\( T^{4} + \cdots - 56\!\cdots\!04 \)
|
| $71$ |
\( T^{4} + \cdots + 41\!\cdots\!92 \)
|
| $73$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 18\!\cdots\!69 \)
|
| $83$ |
\( T^{4} + \cdots + 53\!\cdots\!08 \)
|
| $89$ |
\( T^{4} + \cdots + 11\!\cdots\!28 \)
|
| $97$ |
\( T^{4} + \cdots - 37\!\cdots\!72 \)
|
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