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: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{337}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 84 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = \sqrt{337}\). 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 |
|
−4.00000 | −8.35756 | 16.0000 | −25.0000 | 33.4302 | 0 | −64.0000 | −173.151 | 100.000 | ||||||||||||||||||||||||
| 1.2 | −4.00000 | 28.3576 | 16.0000 | −25.0000 | −113.430 | 0 | −64.0000 | 561.151 | 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.r | yes | 2 |
| 7.b | odd | 2 | 1 | 490.6.a.o | ✓ | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 490.6.a.o | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 490.6.a.r | yes | 2 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 20T_{3} - 237 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(490))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{2} \)
|
| $3$ |
\( T^{2} - 20T - 237 \)
|
| $5$ |
\( (T + 25)^{2} \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 860T + 176475 \)
|
| $13$ |
\( T^{2} + 1270 T + 240117 \)
|
| $17$ |
\( T^{2} - 250 T - 1930887 \)
|
| $19$ |
\( T^{2} - 460 T - 1505388 \)
|
| $23$ |
\( T^{2} + 2468 T + 1489056 \)
|
| $29$ |
\( T^{2} - 682T - 726219 \)
|
| $31$ |
\( T^{2} + 5740 T + 5131108 \)
|
| $37$ |
\( T^{2} - 2532 T - 123794944 \)
|
| $41$ |
\( T^{2} - 22560 T + 101937788 \)
|
| $43$ |
\( T^{2} - 6484 T - 216088236 \)
|
| $47$ |
\( T^{2} - 1840 T - 1704353 \)
|
| $53$ |
\( T^{2} - 26508 T - 342502684 \)
|
| $59$ |
\( T^{2} - 35520 T - 50484868 \)
|
| $61$ |
\( T^{2} + 68220 T + 891733952 \)
|
| $67$ |
\( T^{2} + 22640 T + 18651100 \)
|
| $71$ |
\( T^{2} + 33928 T - 928792704 \)
|
| $73$ |
\( T^{2} + 18140 T - 37442892 \)
|
| $79$ |
\( T^{2} + \cdots - 5678736425 \)
|
| $83$ |
\( T^{2} + \cdots - 1955662352 \)
|
| $89$ |
\( T^{2} + \cdots + 1207773600 \)
|
| $97$ |
\( T^{2} + \cdots + 4974289425 \)
|
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