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: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 273x^{2} + 1520x + 1508 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 690) |
| 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
\( x^{4} - x^{3} - 273x^{2} + 1520x + 1508 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} - 15\nu^{2} + 395\nu + 59 ) / 47 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{3} + \nu^{2} + 569\nu - 3084 ) / 94 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{3} + \nu^{2} + 945\nu - 3178 ) / 94 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 15\beta_{2} - 12\beta _1 + 541 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 95\beta_{3} - 155\beta_{2} - 2\beta _1 - 1871 ) / 2 \)
|
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 | −21.2752 | −8.00000 | 0 | 10.0000 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 5.52193 | −8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 12.1816 | −8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 29.5716 | −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.bh | 4 | |
| 3.b | odd | 2 | 1 | 690.4.a.t | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 690.4.a.t | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 2070.4.a.bh | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\):
|
\( T_{7}^{4} - 26T_{7}^{3} - 415T_{7}^{2} + 10580T_{7} - 42320 \)
|
|
\( T_{11}^{4} + 48T_{11}^{3} - 1186T_{11}^{2} - 50368T_{11} + 421376 \)
|
|
\( T_{17}^{4} + 62T_{17}^{3} - 957T_{17}^{2} - 95440T_{17} - 982972 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} - 26 T^{3} + \cdots - 42320 \)
|
| $11$ |
\( T^{4} + 48 T^{3} + \cdots + 421376 \)
|
| $13$ |
\( T^{4} - 52 T^{3} + \cdots + 75160 \)
|
| $17$ |
\( T^{4} + 62 T^{3} + \cdots - 982972 \)
|
| $19$ |
\( T^{4} - 68 T^{3} + \cdots + 10529280 \)
|
| $23$ |
\( (T + 23)^{4} \)
|
| $29$ |
\( T^{4} + 70 T^{3} + \cdots + 28757300 \)
|
| $31$ |
\( T^{4} - 118 T^{3} + \cdots + 91106496 \)
|
| $37$ |
\( T^{4} + \cdots - 2811798628 \)
|
| $41$ |
\( T^{4} + 146 T^{3} + \cdots + 9560044 \)
|
| $43$ |
\( T^{4} - 144 T^{3} + \cdots + 25697088 \)
|
| $47$ |
\( T^{4} + \cdots - 1231613568 \)
|
| $53$ |
\( T^{4} + \cdots + 32293128268 \)
|
| $59$ |
\( T^{4} + \cdots + 3069668800 \)
|
| $61$ |
\( T^{4} + \cdots + 106749336584 \)
|
| $67$ |
\( T^{4} - 214 T^{3} + \cdots - 950212624 \)
|
| $71$ |
\( T^{4} + \cdots - 51244534960 \)
|
| $73$ |
\( T^{4} + \cdots + 31148904040 \)
|
| $79$ |
\( T^{4} + \cdots + 106605708800 \)
|
| $83$ |
\( T^{4} + \cdots + 791196599024 \)
|
| $89$ |
\( T^{4} + \cdots - 2187251680 \)
|
| $97$ |
\( T^{4} + \cdots + 6495262688 \)
|
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