Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [287,2,Mod(1,287)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("287.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 287 = 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 287.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: | \(2.29170653801\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.185257757.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 24x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 24x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + \nu^{2} + 5\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 10\nu^{3} + 2\nu^{2} + 24\nu - 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 10\nu^{3} - 8\nu^{2} - 24\nu + 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 6\beta_{3} + 6\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + 8\beta_{2} + 26\beta _1 - 6 \)
|
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 |
|
−2.47904 | 2.84004 | 4.14562 | −3.76023 | −7.04056 | −1.00000 | −5.31907 | 5.06582 | 9.32175 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.05073 | −1.62935 | 2.20548 | 4.18004 | 3.34135 | −1.00000 | −0.421375 | −0.345215 | −8.57212 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.644787 | −2.95586 | −1.58425 | −4.36552 | 1.90590 | −1.00000 | 2.31108 | 5.73713 | 2.81483 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.306800 | −1.50512 | −1.90587 | 0.333855 | 0.461772 | −1.00000 | 1.19832 | −0.734606 | −0.102427 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.01956 | 1.86075 | 2.07864 | −0.244521 | 3.75791 | −1.00000 | 0.158810 | 0.462403 | −0.493825 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.46179 | −2.61045 | 4.06039 | 2.85638 | −6.42638 | −1.00000 | 5.07224 | 3.81447 | 7.03179 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
| \(41\) | \(-1\) |
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 | 287.2.a.f | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2583.2.a.t | 6 | ||
| 4.b | odd | 2 | 1 | 4592.2.a.bg | 6 | ||
| 5.b | even | 2 | 1 | 7175.2.a.p | 6 | ||
| 7.b | odd | 2 | 1 | 2009.2.a.o | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 287.2.a.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2009.2.a.o | 6 | 7.b | odd | 2 | 1 | ||
| 2583.2.a.t | 6 | 3.b | odd | 2 | 1 | ||
| 4592.2.a.bg | 6 | 4.b | odd | 2 | 1 | ||
| 7175.2.a.p | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(287))\):
|
\( T_{2}^{6} + T_{2}^{5} - 10T_{2}^{4} - 10T_{2}^{3} + 23T_{2}^{2} + 24T_{2} + 5 \)
|
|
\( T_{3}^{6} + 4T_{3}^{5} - 8T_{3}^{4} - 46T_{3}^{3} - 13T_{3}^{2} + 111T_{3} + 100 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 10 T^{4} + \cdots + 5 \)
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| $3$ |
\( T^{6} + 4 T^{5} + \cdots + 100 \)
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| $5$ |
\( T^{6} + T^{5} + \cdots - 16 \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots + 2720 \)
|
| $13$ |
\( T^{6} - 7 T^{5} + \cdots - 1546 \)
|
| $17$ |
\( T^{6} - 7 T^{5} + \cdots + 2 \)
|
| $19$ |
\( T^{6} - 2 T^{5} + \cdots - 3212 \)
|
| $23$ |
\( T^{6} - 20 T^{5} + \cdots + 344 \)
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| $29$ |
\( T^{6} + 9 T^{5} + \cdots + 10448 \)
|
| $31$ |
\( T^{6} + 27 T^{5} + \cdots - 1280 \)
|
| $37$ |
\( T^{6} - 19 T^{5} + \cdots - 376 \)
|
| $41$ |
\( (T - 1)^{6} \)
|
| $43$ |
\( T^{6} - 19 T^{5} + \cdots + 29756 \)
|
| $47$ |
\( T^{6} + 19 T^{5} + \cdots - 512 \)
|
| $53$ |
\( T^{6} - 5 T^{5} + \cdots - 28432 \)
|
| $59$ |
\( T^{6} + 7 T^{5} + \cdots + 256 \)
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| $61$ |
\( T^{6} + 12 T^{5} + \cdots - 55952 \)
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| $67$ |
\( T^{6} - 27 T^{5} + \cdots - 26848 \)
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| $71$ |
\( T^{6} + 6 T^{5} + \cdots + 4672 \)
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| $73$ |
\( T^{6} - 52 T^{5} + \cdots + 656 \)
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| $79$ |
\( T^{6} - 152 T^{4} + \cdots + 2048 \)
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| $83$ |
\( T^{6} - 12 T^{5} + \cdots - 19744 \)
|
| $89$ |
\( T^{6} + 38 T^{5} + \cdots - 345082 \)
|
| $97$ |
\( T^{6} - 8 T^{5} + \cdots + 303494 \)
|
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