Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [570,6,Mod(1,570)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(570, 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("570.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 570.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: | \(91.4187772934\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 17192x^{3} + 62959x^{2} + 59534416x + 102975568 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 17192x^{3} + 62959x^{2} + 59534416x + 102975568 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{4} + 104934\nu^{3} + 2999192\nu^{2} - 1126956099\nu - 17447507434 ) / 64212210 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -245\nu^{4} + 1842\nu^{3} + 6141346\nu^{2} - 31964757\nu - 24920789126 ) / 32106105 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1033\nu^{4} - 3882\nu^{3} + 11327504\nu^{2} - 52553103\nu - 5087759236 ) / 42808140 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -6\beta_{4} + 19\beta_{3} - \beta_{2} - 3\beta _1 + 13760 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 306\beta_{4} - 1034\beta_{3} + 2501\beta_{2} + 21620\beta _1 - 65041 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -149250\beta_{4} + 210290\beta_{3} - 15665\beta_{2} - 124395\beta _1 + 141108062 ) / 2 \)
|
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 |
|
4.00000 | 9.00000 | 16.0000 | −25.0000 | 36.0000 | −209.554 | 64.0000 | 81.0000 | −100.000 | |||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | 9.00000 | 16.0000 | −25.0000 | 36.0000 | −143.370 | 64.0000 | 81.0000 | −100.000 | ||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 9.00000 | 16.0000 | −25.0000 | 36.0000 | 9.46874 | 64.0000 | 81.0000 | −100.000 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 9.00000 | 16.0000 | −25.0000 | 36.0000 | 135.343 | 64.0000 | 81.0000 | −100.000 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 9.00000 | 16.0000 | −25.0000 | 36.0000 | 234.112 | 64.0000 | 81.0000 | −100.000 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(19\) | \(-1\) |
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 | 570.6.a.p | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 570.6.a.p | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{5} - 26T_{7}^{4} - 68504T_{7}^{3} + 732856T_{7}^{2} + 951170800T_{7} - 9013803008 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{5} \)
|
| $3$ |
\( (T - 9)^{5} \)
|
| $5$ |
\( (T + 25)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 9013803008 \)
|
| $11$ |
\( T^{5} + \cdots + 3659197956480 \)
|
| $13$ |
\( T^{5} + \cdots + 7426278576000 \)
|
| $17$ |
\( T^{5} + \cdots - 71830750247424 \)
|
| $19$ |
\( (T - 361)^{5} \)
|
| $23$ |
\( T^{5} + \cdots + 15\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots - 94\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 38\!\cdots\!36 \)
|
| $37$ |
\( T^{5} + \cdots + 22\!\cdots\!56 \)
|
| $41$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots - 12\!\cdots\!48 \)
|
| $47$ |
\( T^{5} + \cdots + 14\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots + 28\!\cdots\!12 \)
|
| $59$ |
\( T^{5} + \cdots - 25\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 24\!\cdots\!20 \)
|
| $67$ |
\( T^{5} + \cdots + 19\!\cdots\!40 \)
|
| $71$ |
\( T^{5} + \cdots - 16\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 65\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots - 65\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 70\!\cdots\!40 \)
|
| $89$ |
\( T^{5} + \cdots - 12\!\cdots\!40 \)
|
| $97$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
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