Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [490,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("490.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(28.9109359028\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.115880.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 66x + 90 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 66x + 90 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 45 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 45 \)
|
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 |
|
2.00000 | −8.90730 | 4.00000 | −5.00000 | −17.8146 | 0 | 8.00000 | 52.3400 | −10.0000 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −2.37435 | 4.00000 | −5.00000 | −4.74870 | 0 | 8.00000 | −21.3625 | −10.0000 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 7.28165 | 4.00000 | −5.00000 | 14.5633 | 0 | 8.00000 | 26.0224 | −10.0000 | ||||||||||||||||||||||||||||
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 | 490.4.a.v | 3 | |
| 5.b | even | 2 | 1 | 2450.4.a.ce | 3 | ||
| 7.b | odd | 2 | 1 | 490.4.a.w | 3 | ||
| 7.c | even | 3 | 2 | 490.4.e.y | 6 | ||
| 7.d | odd | 6 | 2 | 70.4.e.e | ✓ | 6 | |
| 21.g | even | 6 | 2 | 630.4.k.r | 6 | ||
| 28.f | even | 6 | 2 | 560.4.q.m | 6 | ||
| 35.c | odd | 2 | 1 | 2450.4.a.cb | 3 | ||
| 35.i | odd | 6 | 2 | 350.4.e.k | 6 | ||
| 35.k | even | 12 | 4 | 350.4.j.i | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.4.e.e | ✓ | 6 | 7.d | odd | 6 | 2 | |
| 350.4.e.k | 6 | 35.i | odd | 6 | 2 | ||
| 350.4.j.i | 12 | 35.k | even | 12 | 4 | ||
| 490.4.a.v | 3 | 1.a | even | 1 | 1 | trivial | |
| 490.4.a.w | 3 | 7.b | odd | 2 | 1 | ||
| 490.4.e.y | 6 | 7.c | even | 3 | 2 | ||
| 560.4.q.m | 6 | 28.f | even | 6 | 2 | ||
| 630.4.k.r | 6 | 21.g | even | 6 | 2 | ||
| 2450.4.a.cb | 3 | 35.c | odd | 2 | 1 | ||
| 2450.4.a.ce | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(490))\):
|
\( T_{3}^{3} + 4T_{3}^{2} - 61T_{3} - 154 \)
|
|
\( T_{11}^{3} - 41T_{11}^{2} - 2496T_{11} + 102420 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} + 4 T^{2} + \cdots - 154 \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} - 41 T^{2} + \cdots + 102420 \)
|
| $13$ |
\( T^{3} - T^{2} + \cdots + 19180 \)
|
| $17$ |
\( T^{3} + 30 T^{2} + \cdots - 211896 \)
|
| $19$ |
\( T^{3} + 49 T^{2} + \cdots - 590688 \)
|
| $23$ |
\( T^{3} - 145 T^{2} + \cdots + 2380203 \)
|
| $29$ |
\( T^{3} - 268 T^{2} + \cdots + 562914 \)
|
| $31$ |
\( T^{3} - 28 T^{2} + \cdots + 1074880 \)
|
| $37$ |
\( T^{3} - 813 T^{2} + \cdots - 19088096 \)
|
| $41$ |
\( T^{3} + 313 T^{2} + \cdots - 15201837 \)
|
| $43$ |
\( T^{3} - 360 T^{2} + \cdots + 21473350 \)
|
| $47$ |
\( T^{3} - 977 T^{2} + \cdots + 38027916 \)
|
| $53$ |
\( T^{3} - 325 T^{2} + \cdots + 94403808 \)
|
| $59$ |
\( T^{3} - 272 T^{2} + \cdots + 49714560 \)
|
| $61$ |
\( T^{3} - 902 T^{2} + \cdots + 15889776 \)
|
| $67$ |
\( T^{3} + 170 T^{2} + \cdots + 14573256 \)
|
| $71$ |
\( T^{3} + 1080 T^{2} + \cdots - 45563904 \)
|
| $73$ |
\( T^{3} - 584 T^{2} + \cdots + 160200240 \)
|
| $79$ |
\( T^{3} + 310 T^{2} + \cdots + 26386400 \)
|
| $83$ |
\( T^{3} + 626 T^{2} + \cdots - 46561032 \)
|
| $89$ |
\( T^{3} - 400 T^{2} + \cdots + 478464102 \)
|
| $97$ |
\( T^{3} + 1626 T^{2} + \cdots - 35869960 \)
|
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