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: | \(3\) |
| Coefficient field: | 3.3.568.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 6x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 4 \)
|
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 |
|
−1.64002 | −5.37466 | −5.31032 | −5.00000 | 8.81456 | 2.20103 | 21.8292 | 1.88693 | 8.20012 | |||||||||||||||||||||||||||
| 1.2 | 0.135359 | 9.28467 | −7.98168 | −5.00000 | 1.25676 | 32.4157 | −2.16327 | 59.2051 | −0.676796 | ||||||||||||||||||||||||||||
| 1.3 | 4.50466 | 5.08998 | 12.2920 | −5.00000 | 22.9287 | −0.616696 | 19.3340 | −1.09206 | −22.5233 | ||||||||||||||||||||||||||||
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.f | ✓ | 3 |
| 3.b | odd | 2 | 1 | 765.4.a.k | 3 | ||
| 4.b | odd | 2 | 1 | 1360.4.a.p | 3 | ||
| 5.b | even | 2 | 1 | 425.4.a.f | 3 | ||
| 5.c | odd | 4 | 2 | 425.4.b.h | 6 | ||
| 17.b | even | 2 | 1 | 1445.4.a.k | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.4.a.f | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 425.4.a.f | 3 | 5.b | even | 2 | 1 | ||
| 425.4.b.h | 6 | 5.c | odd | 4 | 2 | ||
| 765.4.a.k | 3 | 3.b | odd | 2 | 1 | ||
| 1360.4.a.p | 3 | 4.b | odd | 2 | 1 | ||
| 1445.4.a.k | 3 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(85))\):
|
\( T_{2}^{3} - 3T_{2}^{2} - 7T_{2} + 1 \)
|
|
\( T_{3}^{3} - 9T_{3}^{2} - 30T_{3} + 254 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 3 T^{2} + \cdots + 1 \)
|
| $3$ |
\( T^{3} - 9 T^{2} + \cdots + 254 \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} - 34 T^{2} + \cdots + 44 \)
|
| $11$ |
\( T^{3} - 52 T^{2} + \cdots + 6784 \)
|
| $13$ |
\( T^{3} - 19 T^{2} + \cdots + 20408 \)
|
| $17$ |
\( (T + 17)^{3} \)
|
| $19$ |
\( T^{3} + 153 T^{2} + \cdots - 43112 \)
|
| $23$ |
\( T^{3} - 162 T^{2} + \cdots + 587260 \)
|
| $29$ |
\( T^{3} - 45 T^{2} + \cdots - 1563820 \)
|
| $31$ |
\( T^{3} + 67 T^{2} + \cdots + 634 \)
|
| $37$ |
\( T^{3} + 308 T^{2} + \cdots - 879328 \)
|
| $41$ |
\( T^{3} - 498 T^{2} + \cdots + 948856 \)
|
| $43$ |
\( T^{3} + 246 T^{2} + \cdots - 8063768 \)
|
| $47$ |
\( T^{3} - 101 T^{2} + \cdots - 37575724 \)
|
| $53$ |
\( T^{3} - 893 T^{2} + \cdots - 23102788 \)
|
| $59$ |
\( T^{3} - 355 T^{2} + \cdots + 23789032 \)
|
| $61$ |
\( T^{3} - 1019 T^{2} + \cdots - 32261500 \)
|
| $67$ |
\( T^{3} - 334 T^{2} + \cdots + 2242600 \)
|
| $71$ |
\( T^{3} - 313 T^{2} + \cdots - 104660798 \)
|
| $73$ |
\( T^{3} - 639 T^{2} + \cdots - 23564196 \)
|
| $79$ |
\( T^{3} + 92 T^{2} + \cdots + 310900832 \)
|
| $83$ |
\( T^{3} - 2736 T^{2} + \cdots - 751119952 \)
|
| $89$ |
\( T^{3} - 1623 T^{2} + \cdots + 25616512 \)
|
| $97$ |
\( T^{3} + 475 T^{2} + \cdots - 473709668 \)
|
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