Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1856,2,Mod(1,1856)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1856, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1856.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1856 = 2^{6} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1856.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: | \(14.8202346151\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.68772992.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} + 17x^{2} - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 928) |
| 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
\( x^{6} - 9x^{4} + 17x^{2} - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 8\nu^{3} + 9\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{4} + 9\nu^{2} - 11 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{3} + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{5} + 9\beta_{3} + 32 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{4} + 16\beta_{2} + 39\beta_1 ) / 2 \)
|
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 |
|
0 | −3.15704 | 0 | −1.42586 | 0 | −1.70818 | 0 | 6.96693 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.95528 | 0 | 2.85952 | 0 | 2.57579 | 0 | 0.823126 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.458200 | 0 | −3.43366 | 0 | 5.14271 | 0 | −2.79005 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.458200 | 0 | −3.43366 | 0 | −5.14271 | 0 | −2.79005 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.95528 | 0 | 2.85952 | 0 | −2.57579 | 0 | 0.823126 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.15704 | 0 | −1.42586 | 0 | 1.70818 | 0 | 6.96693 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(29\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1856.2.a.bc | 6 | |
| 4.b | odd | 2 | 1 | inner | 1856.2.a.bc | 6 | |
| 8.b | even | 2 | 1 | 928.2.a.i | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 928.2.a.i | ✓ | 6 | |
| 24.f | even | 2 | 1 | 8352.2.a.bi | 6 | ||
| 24.h | odd | 2 | 1 | 8352.2.a.bi | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 928.2.a.i | ✓ | 6 | 8.b | even | 2 | 1 | |
| 928.2.a.i | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 1856.2.a.bc | 6 | 1.a | even | 1 | 1 | trivial | |
| 1856.2.a.bc | 6 | 4.b | odd | 2 | 1 | inner | |
| 8352.2.a.bi | 6 | 24.f | even | 2 | 1 | ||
| 8352.2.a.bi | 6 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1856))\):
|
\( T_{3}^{6} - 14T_{3}^{4} + 41T_{3}^{2} - 8 \)
|
|
\( T_{5}^{3} + 2T_{5}^{2} - 9T_{5} - 14 \)
|
|
\( T_{17}^{3} - 8T_{17}^{2} - 8T_{17} + 104 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 14 T^{4} + \cdots - 8 \)
|
| $5$ |
\( (T^{3} + 2 T^{2} - 9 T - 14)^{2} \)
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| $7$ |
\( T^{6} - 36 T^{4} + \cdots - 512 \)
|
| $11$ |
\( T^{6} - 14 T^{4} + \cdots - 8 \)
|
| $13$ |
\( (T^{3} + 2 T^{2} - 9 T - 14)^{2} \)
|
| $17$ |
\( (T^{3} - 8 T^{2} + \cdots + 104)^{2} \)
|
| $19$ |
\( T^{6} - 64 T^{4} + \cdots - 128 \)
|
| $23$ |
\( T^{6} - 196 T^{4} + \cdots - 270848 \)
|
| $29$ |
\( (T + 1)^{6} \)
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| $31$ |
\( T^{6} - 54 T^{4} + \cdots - 1352 \)
|
| $37$ |
\( (T^{3} + 2 T^{2} - 40 T + 32)^{2} \)
|
| $41$ |
\( (T^{3} - 12 T^{2} + \cdots + 184)^{2} \)
|
| $43$ |
\( T^{6} - 150 T^{4} + \cdots - 42632 \)
|
| $47$ |
\( T^{6} - 150 T^{4} + \cdots - 19208 \)
|
| $53$ |
\( (T^{3} + 6 T^{2} + \cdots + 178)^{2} \)
|
| $59$ |
\( T^{6} - 308 T^{4} + \cdots - 86528 \)
|
| $61$ |
\( (T^{3} - 8 T^{2} + \cdots + 104)^{2} \)
|
| $67$ |
\( T^{6} - 224 T^{4} + \cdots - 32768 \)
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| $71$ |
\( T^{6} - 248 T^{4} + \cdots - 204800 \)
|
| $73$ |
\( (T^{3} - 10 T^{2} + \cdots + 128)^{2} \)
|
| $79$ |
\( T^{6} - 46 T^{4} + \cdots - 200 \)
|
| $83$ |
\( T^{6} - 340 T^{4} + \cdots - 320000 \)
|
| $89$ |
\( (T^{3} - 4 T^{2} + \cdots + 1352)^{2} \)
|
| $97$ |
\( (T^{3} - 20 T^{2} + \cdots + 712)^{2} \)
|
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