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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{177}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). 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 |
|
−2.00000 | −4.15207 | 4.00000 | 5.00000 | 8.30413 | 0 | −8.00000 | −9.76034 | −10.0000 | ||||||||||||||||||||||||
| 1.2 | −2.00000 | 9.15207 | 4.00000 | 5.00000 | −18.3041 | 0 | −8.00000 | 56.7603 | −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.u | yes | 2 |
| 5.b | even | 2 | 1 | 2450.4.a.bt | 2 | ||
| 7.b | odd | 2 | 1 | 490.4.a.p | ✓ | 2 | |
| 7.c | even | 3 | 2 | 490.4.e.t | 4 | ||
| 7.d | odd | 6 | 2 | 490.4.e.x | 4 | ||
| 35.c | odd | 2 | 1 | 2450.4.a.ca | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 490.4.a.p | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 490.4.a.u | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 490.4.e.t | 4 | 7.c | even | 3 | 2 | ||
| 490.4.e.x | 4 | 7.d | odd | 6 | 2 | ||
| 2450.4.a.bt | 2 | 5.b | even | 2 | 1 | ||
| 2450.4.a.ca | 2 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(490))\):
|
\( T_{3}^{2} - 5T_{3} - 38 \)
|
|
\( T_{11}^{2} + 5T_{11} - 1100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{2} \)
|
| $3$ |
\( T^{2} - 5T - 38 \)
|
| $5$ |
\( (T - 5)^{2} \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 5T - 1100 \)
|
| $13$ |
\( T^{2} - 95T + 1858 \)
|
| $17$ |
\( T^{2} + 25T + 112 \)
|
| $19$ |
\( T^{2} + 120T + 2892 \)
|
| $23$ |
\( T^{2} - 124T - 13856 \)
|
| $29$ |
\( T^{2} - 213T - 16314 \)
|
| $31$ |
\( T^{2} - 70T + 1048 \)
|
| $37$ |
\( T^{2} + 134T - 35336 \)
|
| $41$ |
\( T^{2} - 370T - 29672 \)
|
| $43$ |
\( T^{2} + 82T - 2744 \)
|
| $47$ |
\( T^{2} - 115T - 301532 \)
|
| $53$ |
\( T^{2} + 1024 T + 244444 \)
|
| $59$ |
\( T^{2} - 760T + 87052 \)
|
| $61$ |
\( T^{2} + 790T - 36728 \)
|
| $67$ |
\( T^{2} - 850T + 176200 \)
|
| $71$ |
\( T^{2} - 372T - 407904 \)
|
| $73$ |
\( T^{2} - 190T - 488168 \)
|
| $79$ |
\( T^{2} - 425T - 88700 \)
|
| $83$ |
\( T^{2} - 2110 T + 1111432 \)
|
| $89$ |
\( T^{2} - 560 T - 1054400 \)
|
| $97$ |
\( T^{2} - 675T - 205800 \)
|
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