Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1210,4,Mod(1,1210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, 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("1210.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.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: | \(71.3923111069\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.52525.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 24x^{2} + 25x + 125 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 110) |
| 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
\( x^{4} - 2x^{3} - 24x^{2} + 25x + 125 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 15 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 15\nu ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} + 5\beta_{2} + 16\beta _1 + 15 \)
|
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 | −8.01861 | 4.00000 | 5.00000 | 16.0372 | −0.227345 | −8.00000 | 37.2981 | −10.0000 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −1.93972 | 4.00000 | 5.00000 | 3.87944 | −2.87207 | −8.00000 | −23.2375 | −10.0000 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0.546474 | 4.00000 | 5.00000 | −1.09295 | 11.6093 | −8.00000 | −26.7014 | −10.0000 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 3.41185 | 4.00000 | 5.00000 | −6.82371 | 16.4901 | −8.00000 | −15.3593 | −10.0000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 1210.4.a.y | 4 | |
| 11.b | odd | 2 | 1 | 1210.4.a.ba | 4 | ||
| 11.c | even | 5 | 2 | 110.4.g.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.g.a | ✓ | 8 | 11.c | even | 5 | 2 | |
| 1210.4.a.y | 4 | 1.a | even | 1 | 1 | trivial | |
| 1210.4.a.ba | 4 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{4} + 6T_{3}^{3} - 22T_{3}^{2} - 43T_{3} + 29 \)
|
|
\( T_{7}^{4} - 25T_{7}^{3} + 105T_{7}^{2} + 575T_{7} + 125 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 29 \)
|
| $5$ |
\( (T - 5)^{4} \)
|
| $7$ |
\( T^{4} - 25 T^{3} + \cdots + 125 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 59 T^{3} + \cdots + 4543781 \)
|
| $17$ |
\( T^{4} - 23 T^{3} + \cdots + 2770889 \)
|
| $19$ |
\( T^{4} + 134 T^{3} + \cdots + 23589989 \)
|
| $23$ |
\( T^{4} + 259 T^{3} + \cdots - 3351821 \)
|
| $29$ |
\( T^{4} - 236 T^{3} + \cdots + 77783945 \)
|
| $31$ |
\( T^{4} + 405 T^{3} + \cdots + 3900181 \)
|
| $37$ |
\( T^{4} + 55 T^{3} + \cdots + 561149611 \)
|
| $41$ |
\( T^{4} + \cdots + 7390534509 \)
|
| $43$ |
\( T^{4} + 147 T^{3} + \cdots + 959787745 \)
|
| $47$ |
\( T^{4} + \cdots + 5588412559 \)
|
| $53$ |
\( T^{4} + \cdots + 7826405995 \)
|
| $59$ |
\( T^{4} + \cdots - 4300744001 \)
|
| $61$ |
\( T^{4} + \cdots - 39139679205 \)
|
| $67$ |
\( T^{4} + \cdots - 58505616781 \)
|
| $71$ |
\( T^{4} + \cdots - 101540482255 \)
|
| $73$ |
\( T^{4} + \cdots + 47681297019 \)
|
| $79$ |
\( T^{4} + \cdots - 54476878595 \)
|
| $83$ |
\( T^{4} - 1373 T^{3} + \cdots + 209588391 \)
|
| $89$ |
\( T^{4} + \cdots - 24048117271 \)
|
| $97$ |
\( T^{4} + \cdots - 1324233523521 \)
|
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