Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2070,4,Mod(1,2070)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2070, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2070.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2070.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: | \(122.133953712\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 60x^{2} - 45x + 108 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 230) |
| 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
\( x^{4} - 2x^{3} - 60x^{2} - 45x + 108 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 5\nu^{2} - 45\nu + 54 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} - \nu^{2} - 126\nu - 153 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 2\beta_{2} + 4\beta _1 + 29 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} - \beta_{2} + 65\beta _1 + 91 \)
|
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 | 0 | 4.00000 | −5.00000 | 0 | −24.6431 | −8.00000 | 0 | 10.0000 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 2.98517 | −8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 5.03071 | −8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 24.6272 | −8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(23\) | \(-1\) |
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 | 2070.4.a.bg | 4 | |
| 3.b | odd | 2 | 1 | 230.4.a.j | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1840.4.a.k | 4 | ||
| 15.d | odd | 2 | 1 | 1150.4.a.n | 4 | ||
| 15.e | even | 4 | 2 | 1150.4.b.o | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.4.a.j | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 1150.4.a.n | 4 | 15.d | odd | 2 | 1 | ||
| 1150.4.b.o | 8 | 15.e | even | 4 | 2 | ||
| 1840.4.a.k | 4 | 12.b | even | 2 | 1 | ||
| 2070.4.a.bg | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\):
|
\( T_{7}^{4} - 8T_{7}^{3} - 592T_{7}^{2} + 4865T_{7} - 9114 \)
|
|
\( T_{11}^{4} + 21T_{11}^{3} - 2605T_{11}^{2} + 43134T_{11} - 164232 \)
|
|
\( T_{17}^{4} + 56T_{17}^{3} - 8532T_{17}^{2} - 264771T_{17} + 7970400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} - 8 T^{3} + \cdots - 9114 \)
|
| $11$ |
\( T^{4} + 21 T^{3} + \cdots - 164232 \)
|
| $13$ |
\( T^{4} - 70 T^{3} + \cdots - 46062 \)
|
| $17$ |
\( T^{4} + 56 T^{3} + \cdots + 7970400 \)
|
| $19$ |
\( T^{4} - 173 T^{3} + \cdots + 9983784 \)
|
| $23$ |
\( (T - 23)^{4} \)
|
| $29$ |
\( T^{4} - 118 T^{3} + \cdots + 44611452 \)
|
| $31$ |
\( T^{4} + \cdots + 2678191911 \)
|
| $37$ |
\( T^{4} + \cdots - 2095186944 \)
|
| $41$ |
\( T^{4} + \cdots + 4789051317 \)
|
| $43$ |
\( T^{4} + \cdots + 5363873792 \)
|
| $47$ |
\( T^{4} + 367 T^{3} + \cdots + 668142000 \)
|
| $53$ |
\( T^{4} - 353 T^{3} + \cdots - 289523592 \)
|
| $59$ |
\( T^{4} + \cdots + 8963853984 \)
|
| $61$ |
\( T^{4} + \cdots - 1898667392 \)
|
| $67$ |
\( T^{4} + \cdots - 1366509568 \)
|
| $71$ |
\( T^{4} + \cdots + 106065123651 \)
|
| $73$ |
\( T^{4} + \cdots - 15845099784 \)
|
| $79$ |
\( T^{4} + \cdots + 60635801088 \)
|
| $83$ |
\( T^{4} - 1199 T^{3} + \cdots + 103460976 \)
|
| $89$ |
\( T^{4} + \cdots - 5919819264 \)
|
| $97$ |
\( T^{4} + \cdots - 22808262684 \)
|
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