Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [85,4,Mod(1,85)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("85.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 85 = 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 85.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: | \(5.01516235049\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 17\nu^{2} - 13\nu + 42 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 23\nu^{2} - 13\nu + 96 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{4} + \nu^{3} + 97\nu^{2} - \nu - 300 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{4} + 4\beta_{3} + 6\beta_{2} + 11\beta _1 + 5 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{4} - 19\beta_{3} + 20\beta_{2} + \beta _1 + 115 \)
|
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 |
|
−5.14668 | −1.13153 | 18.4883 | 5.00000 | 5.82364 | −26.5778 | −53.9797 | −25.7196 | −25.7334 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.18655 | 6.08748 | −3.21900 | 5.00000 | −13.3106 | 10.8418 | 24.5309 | 10.0574 | −10.9327 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.290753 | −7.70584 | −7.91546 | 5.00000 | −2.24049 | 22.4661 | −4.62746 | 32.3800 | 1.45376 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.58234 | 2.57070 | 4.83317 | 5.00000 | 9.20914 | 25.2782 | −11.3447 | −20.3915 | 17.9117 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.46013 | −0.820806 | 21.8130 | 5.00000 | −4.48171 | −22.0083 | 75.4209 | −26.3263 | 27.3007 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 85.4.a.g | ✓ | 5 |
| 3.b | odd | 2 | 1 | 765.4.a.m | 5 | ||
| 4.b | odd | 2 | 1 | 1360.4.a.w | 5 | ||
| 5.b | even | 2 | 1 | 425.4.a.i | 5 | ||
| 5.c | odd | 4 | 2 | 425.4.b.i | 10 | ||
| 17.b | even | 2 | 1 | 1445.4.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.4.a.g | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 425.4.a.i | 5 | 5.b | even | 2 | 1 | ||
| 425.4.b.i | 10 | 5.c | odd | 4 | 2 | ||
| 765.4.a.m | 5 | 3.b | odd | 2 | 1 | ||
| 1360.4.a.w | 5 | 4.b | odd | 2 | 1 | ||
| 1445.4.a.l | 5 | 17.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(85))\):
|
\( T_{2}^{5} - 2T_{2}^{4} - 35T_{2}^{3} + 52T_{2}^{2} + 208T_{2} - 64 \)
|
|
\( T_{3}^{5} + T_{3}^{4} - 52T_{3}^{3} + 20T_{3}^{2} + 188T_{3} + 112 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 64 \)
|
| $3$ |
\( T^{5} + T^{4} + \cdots + 112 \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} - 10 T^{4} + \cdots - 3601472 \)
|
| $11$ |
\( T^{5} - 126 T^{4} + \cdots - 167488 \)
|
| $13$ |
\( T^{5} - 83 T^{4} + \cdots - 7643696 \)
|
| $17$ |
\( (T - 17)^{5} \)
|
| $19$ |
\( T^{5} + \cdots - 2274428800 \)
|
| $23$ |
\( T^{5} + \cdots + 2278533056 \)
|
| $29$ |
\( T^{5} + \cdots - 2844978640 \)
|
| $31$ |
\( T^{5} + \cdots - 73509549120 \)
|
| $37$ |
\( T^{5} + \cdots - 694556688448 \)
|
| $41$ |
\( T^{5} + \cdots - 478624610720 \)
|
| $43$ |
\( T^{5} + \cdots + 1497205751872 \)
|
| $47$ |
\( T^{5} + \cdots + 840515565952 \)
|
| $53$ |
\( T^{5} + \cdots - 474378006640 \)
|
| $59$ |
\( T^{5} + \cdots + 1194050514560 \)
|
| $61$ |
\( T^{5} + \cdots - 37656238184848 \)
|
| $67$ |
\( T^{5} + \cdots - 3947395137728 \)
|
| $71$ |
\( T^{5} + \cdots + 26612610106464 \)
|
| $73$ |
\( T^{5} + \cdots - 61959193488 \)
|
| $79$ |
\( T^{5} + \cdots + 10071329239040 \)
|
| $83$ |
\( T^{5} + \cdots - 968622815428096 \)
|
| $89$ |
\( T^{5} + \cdots - 30709087651760 \)
|
| $97$ |
\( T^{5} + \cdots + 17052960082000 \)
|
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