Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [490,6,Mod(1,490)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(490, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("490.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 490.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: | \(78.5880717084\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{193})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 99x^{2} + 100x + 2114 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 7 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 99x^{2} + 100x + 2114 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} - 181\nu + 91 ) / 37 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -14\nu^{3} + 21\nu^{2} + 749\nu - 378 ) / 185 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 50 \)
|
| \(\nu\) | \(=\) |
\( ( -5\beta_{2} - 7\beta _1 + 7 ) / 14 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 14\beta_{3} - 5\beta_{2} - 7\beta _1 + 707 ) / 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 21\beta_{3} - 460\beta_{2} - 385\beta _1 + 1057 ) / 14 \)
|
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 | −28.1351 | 16.0000 | 25.0000 | −112.540 | 0 | 64.0000 | 548.583 | 100.000 | ||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −19.6498 | 16.0000 | 25.0000 | −78.5992 | 0 | 64.0000 | 143.115 | 100.000 | |||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −0.350197 | 16.0000 | 25.0000 | −1.40079 | 0 | 64.0000 | −242.877 | 100.000 | |||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 8.13508 | 16.0000 | 25.0000 | 32.5403 | 0 | 64.0000 | −176.820 | 100.000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 490.6.a.z | ✓ | 4 |
| 7.b | odd | 2 | 1 | 490.6.a.bc | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 490.6.a.z | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 490.6.a.bc | yes | 4 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 40T_{3}^{3} + 178T_{3}^{2} - 4440T_{3} - 1575 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(490))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{4} \)
|
| $3$ |
\( T^{4} + 40 T^{3} + \cdots - 1575 \)
|
| $5$ |
\( (T - 25)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 15203921449 \)
|
| $13$ |
\( T^{4} + \cdots - 1678775175 \)
|
| $17$ |
\( T^{4} + \cdots - 61894059439 \)
|
| $19$ |
\( T^{4} + \cdots + 148453264400 \)
|
| $23$ |
\( T^{4} + \cdots + 6328219032100 \)
|
| $29$ |
\( T^{4} + \cdots - 12721698396871 \)
|
| $31$ |
\( T^{4} + \cdots + 13952395382084 \)
|
| $37$ |
\( T^{4} + \cdots + 503875355864900 \)
|
| $41$ |
\( T^{4} + \cdots - 29\!\cdots\!44 \)
|
| $43$ |
\( T^{4} + \cdots + 462219528976 \)
|
| $47$ |
\( T^{4} + \cdots - 38\!\cdots\!75 \)
|
| $53$ |
\( T^{4} + \cdots - 12\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots + 21\!\cdots\!16 \)
|
| $61$ |
\( T^{4} + \cdots - 81\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 28\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots + 68\!\cdots\!56 \)
|
| $73$ |
\( T^{4} + \cdots - 12\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots + 18\!\cdots\!25 \)
|
| $83$ |
\( T^{4} + \cdots - 24\!\cdots\!56 \)
|
| $89$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 38\!\cdots\!71 \)
|
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