Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [920,4,Mod(1,920)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(920, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("920.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 920 = 2^{3} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 920.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: | \(54.2817572053\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 81x^{4} + 161x^{3} + 1520x^{2} - 3915x + 588 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 81x^{4} + 161x^{3} + 1520x^{2} - 3915x + 588 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 82\nu^{3} + 79\nu^{2} + 1527\nu - 2172 ) / 72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{5} - 29\nu^{4} - 194\nu^{3} + 1139\nu^{2} - 1005\nu - 708 ) / 72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 13\nu^{4} + 10\nu^{3} - 559\nu^{2} + 1245\nu + 516 ) / 36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7\nu^{5} - 55\nu^{4} - 214\nu^{3} + 2185\nu^{2} - 3423\nu + 204 ) / 72 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 27 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{5} - 2\beta_{4} + 4\beta_{3} - 3\beta_{2} + 46\beta _1 - 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( -58\beta_{5} - 49\beta_{4} + 64\beta_{3} - 12\beta_{2} + 85\beta _1 + 1134 \)
|
| \(\nu^{5}\) | \(=\) |
\( -225\beta_{5} - 134\beta_{4} + 313\beta_{3} - 186\beta_{2} + 2251\beta _1 + 25 \)
|
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 |
|
0 | −6.42830 | 0 | −5.00000 | 0 | 8.32258 | 0 | 14.3231 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.38774 | 0 | −5.00000 | 0 | −5.78952 | 0 | 13.8032 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.160327 | 0 | −5.00000 | 0 | −2.26267 | 0 | −26.9743 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.66453 | 0 | −5.00000 | 0 | 34.7191 | 0 | −19.9003 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 4.43772 | 0 | −5.00000 | 0 | −11.9284 | 0 | −7.30664 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 7.55347 | 0 | −5.00000 | 0 | 4.93892 | 0 | 30.0549 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(23\) | \(1\) |
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 | 920.4.a.b | ✓ | 6 |
| 4.b | odd | 2 | 1 | 1840.4.a.s | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 920.4.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1840.4.a.s | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 2T_{3}^{5} - 81T_{3}^{4} + 161T_{3}^{3} + 1520T_{3}^{2} - 3915T_{3} + 588 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(920))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 588 \)
|
| $5$ |
\( (T + 5)^{6} \)
|
| $7$ |
\( T^{6} - 28 T^{5} + \cdots - 223000 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 1162512 \)
|
| $13$ |
\( T^{6} + 28 T^{5} + \cdots - 266086970 \)
|
| $17$ |
\( T^{6} + \cdots - 11777111616 \)
|
| $19$ |
\( T^{6} + \cdots - 21406390992 \)
|
| $23$ |
\( (T + 23)^{6} \)
|
| $29$ |
\( T^{6} + \cdots - 108803723556 \)
|
| $31$ |
\( T^{6} + \cdots - 96915684999 \)
|
| $37$ |
\( T^{6} + \cdots + 32607805056 \)
|
| $41$ |
\( T^{6} + \cdots - 488808379923 \)
|
| $43$ |
\( T^{6} + \cdots + 3224001804288 \)
|
| $47$ |
\( T^{6} + \cdots - 12\!\cdots\!36 \)
|
| $53$ |
\( T^{6} + \cdots - 59665103136 \)
|
| $59$ |
\( T^{6} + \cdots - 295068706077440 \)
|
| $61$ |
\( T^{6} + \cdots - 171409513711456 \)
|
| $67$ |
\( T^{6} + \cdots + 95\!\cdots\!44 \)
|
| $71$ |
\( T^{6} + \cdots - 36\!\cdots\!71 \)
|
| $73$ |
\( T^{6} + \cdots - 92144248176936 \)
|
| $79$ |
\( T^{6} + \cdots - 869715459643392 \)
|
| $83$ |
\( T^{6} + \cdots - 23\!\cdots\!40 \)
|
| $89$ |
\( T^{6} + \cdots + 22\!\cdots\!76 \)
|
| $97$ |
\( T^{6} + \cdots + 50\!\cdots\!96 \)
|
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