Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1710,4,Mod(1,1710)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1710, 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("1710.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1710.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: | \(100.893266110\) |
| 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: |
\( x^{4} - x^{3} - 410x^{2} + 4362x - 12540 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 570) |
| 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
\( x^{4} - x^{3} - 410x^{2} + 4362x - 12540 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 6\nu^{2} - 367\nu + 1824 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{3} - 11\nu^{2} + 749\nu - 3857 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + 4\beta_{2} - 15\beta _1 + 418 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -12\beta_{3} - 22\beta_{2} + 457\beta _1 - 6156 ) / 2 \)
|
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 | −15.8571 | −8.00000 | 0 | 10.0000 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | −1.42654 | −8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 17.1829 | −8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 36.1008 | −8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(19\) | \(-1\) |
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 | 1710.4.a.bc | 4 | |
| 3.b | odd | 2 | 1 | 570.4.a.s | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 570.4.a.s | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 1710.4.a.bc | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1710))\):
|
\( T_{7}^{4} - 36T_{7}^{3} - 278T_{7}^{2} + 9516T_{7} + 14032 \)
|
|
\( T_{11}^{4} + 54T_{11}^{3} - 570T_{11}^{2} - 36648T_{11} + 34560 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} - 36 T^{3} + \cdots + 14032 \)
|
| $11$ |
\( T^{4} + 54 T^{3} + \cdots + 34560 \)
|
| $13$ |
\( T^{4} - 46 T^{3} + \cdots - 3407016 \)
|
| $17$ |
\( T^{4} + 14 T^{3} + \cdots + 41635968 \)
|
| $19$ |
\( (T - 19)^{4} \)
|
| $23$ |
\( T^{4} + 104 T^{3} + \cdots + 1104000 \)
|
| $29$ |
\( T^{4} + 14 T^{3} + \cdots - 181111800 \)
|
| $31$ |
\( T^{4} - 30 T^{3} + \cdots + 16169536 \)
|
| $37$ |
\( T^{4} - 30 T^{3} + \cdots + 15035992 \)
|
| $41$ |
\( T^{4} + \cdots + 12058908048 \)
|
| $43$ |
\( T^{4} + \cdots + 4956833536 \)
|
| $47$ |
\( T^{4} + \cdots + 1082908800 \)
|
| $53$ |
\( T^{4} + \cdots + 17352011904 \)
|
| $59$ |
\( T^{4} + \cdots - 3721014720 \)
|
| $61$ |
\( T^{4} + \cdots - 5968735328 \)
|
| $67$ |
\( T^{4} + \cdots - 21765056256 \)
|
| $71$ |
\( T^{4} + 172 T^{3} + \cdots + 337920000 \)
|
| $73$ |
\( T^{4} + \cdots + 52421119600 \)
|
| $79$ |
\( T^{4} + \cdots + 26204800000 \)
|
| $83$ |
\( T^{4} + \cdots + 6066401280 \)
|
| $89$ |
\( T^{4} + \cdots + 2442413711520 \)
|
| $97$ |
\( T^{4} + \cdots + 994890105880 \)
|
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