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: |
\( x^{5} - 6052x^{3} - 41130x^{2} + 7064712x + 22607640 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| 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
\( x^{5} - 6052x^{3} - 41130x^{2} + 7064712x + 22607640 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 462\nu^{3} - 14504\nu^{2} - 1359318\nu + 32631804 ) / 192888 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 462\nu^{3} + 14504\nu^{2} + 2516646\nu - 32631804 ) / 192888 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 139\nu^{4} + 78\nu^{3} - 491488\nu^{2} - 5743086\nu - 62748036 ) / 1157328 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -17\nu^{4} + 710\nu^{3} + 74912\nu^{2} - 1733558\nu - 45869700 ) / 42864 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + \beta_{2} - 6\beta _1 + 2417 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 117\beta_{4} + 405\beta_{3} + 1631\beta_{2} + 2063\beta _1 + 74079 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3514\beta_{4} + 18858\beta_{3} + 10117\beta_{2} - 14715\beta _1 + 8983802 \)
|
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 | −178.710 | 64.0000 | 81.0000 | 100.000 | |||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −9.00000 | 16.0000 | 25.0000 | −36.0000 | −61.5764 | 64.0000 | 81.0000 | 100.000 | ||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −9.00000 | 16.0000 | 25.0000 | −36.0000 | 46.3009 | 64.0000 | 81.0000 | 100.000 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | −9.00000 | 16.0000 | 25.0000 | −36.0000 | 73.3106 | 64.0000 | 81.0000 | 100.000 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | −9.00000 | 16.0000 | 25.0000 | −36.0000 | 208.674 | 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.o | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 570.6.a.o | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{5} - 88T_{7}^{4} - 39524T_{7}^{3} + 2492252T_{7}^{2} + 141820096T_{7} - 7794482032 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{5} \)
|
| $3$ |
\( (T + 9)^{5} \)
|
| $5$ |
\( (T - 25)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 7794482032 \)
|
| $11$ |
\( T^{5} + \cdots - 27318205411200 \)
|
| $13$ |
\( T^{5} + \cdots + 123295827765312 \)
|
| $17$ |
\( T^{5} + \cdots - 39814660703808 \)
|
| $19$ |
\( (T - 361)^{5} \)
|
| $23$ |
\( T^{5} + \cdots + 67\!\cdots\!88 \)
|
| $29$ |
\( T^{5} + \cdots + 24\!\cdots\!72 \)
|
| $31$ |
\( T^{5} + \cdots - 17\!\cdots\!44 \)
|
| $37$ |
\( T^{5} + \cdots + 92\!\cdots\!60 \)
|
| $41$ |
\( T^{5} + \cdots - 15\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 16\!\cdots\!20 \)
|
| $47$ |
\( T^{5} + \cdots - 54\!\cdots\!80 \)
|
| $53$ |
\( T^{5} + \cdots + 26\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots - 10\!\cdots\!56 \)
|
| $61$ |
\( T^{5} + \cdots + 17\!\cdots\!76 \)
|
| $67$ |
\( T^{5} + \cdots + 11\!\cdots\!40 \)
|
| $71$ |
\( T^{5} + \cdots - 15\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 48\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots - 77\!\cdots\!96 \)
|
| $83$ |
\( T^{5} + \cdots - 98\!\cdots\!80 \)
|
| $89$ |
\( T^{5} + \cdots + 30\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots + 16\!\cdots\!00 \)
|
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