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: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1304.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 11x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 11x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 7 \)
|
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 |
|
−5.40405 | −8.99158 | 21.2037 | −5.00000 | 48.5909 | 27.3417 | −71.3535 | 53.8486 | 27.0202 | |||||||||||||||||||||||||||
| 1.2 | −1.81763 | 6.14911 | −4.69622 | −5.00000 | −11.1768 | −34.0161 | 23.0770 | 10.8116 | 9.08815 | ||||||||||||||||||||||||||||
| 1.3 | 1.22168 | −1.15753 | −6.50750 | −5.00000 | −1.41413 | 14.6743 | −17.7235 | −25.6601 | −6.10839 | ||||||||||||||||||||||||||||
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.e | ✓ | 3 |
| 3.b | odd | 2 | 1 | 765.4.a.l | 3 | ||
| 4.b | odd | 2 | 1 | 1360.4.a.s | 3 | ||
| 5.b | even | 2 | 1 | 425.4.a.h | 3 | ||
| 5.c | odd | 4 | 2 | 425.4.b.g | 6 | ||
| 17.b | even | 2 | 1 | 1445.4.a.j | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.4.a.e | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 425.4.a.h | 3 | 5.b | even | 2 | 1 | ||
| 425.4.b.g | 6 | 5.c | odd | 4 | 2 | ||
| 765.4.a.l | 3 | 3.b | odd | 2 | 1 | ||
| 1360.4.a.s | 3 | 4.b | odd | 2 | 1 | ||
| 1445.4.a.j | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(85))\):
|
\( T_{2}^{3} + 6T_{2}^{2} + T_{2} - 12 \)
|
|
\( T_{3}^{3} + 4T_{3}^{2} - 52T_{3} - 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 6T^{2} + T - 12 \)
|
| $3$ |
\( T^{3} + 4 T^{2} + \cdots - 64 \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} - 8 T^{2} + \cdots + 13648 \)
|
| $11$ |
\( T^{3} + 118 T^{2} + \cdots + 31128 \)
|
| $13$ |
\( T^{3} + 10 T^{2} + \cdots - 4808 \)
|
| $17$ |
\( (T - 17)^{3} \)
|
| $19$ |
\( T^{3} + 160 T^{2} + \cdots - 1236992 \)
|
| $23$ |
\( T^{3} + 92 T^{2} + \cdots - 37824 \)
|
| $29$ |
\( T^{3} - 374 T^{2} + \cdots + 1059384 \)
|
| $31$ |
\( T^{3} - 70 T^{2} + \cdots - 869848 \)
|
| $37$ |
\( T^{3} + 246 T^{2} + \cdots - 107032 \)
|
| $41$ |
\( T^{3} + 354 T^{2} + \cdots + 268056 \)
|
| $43$ |
\( T^{3} + 286 T^{2} + \cdots - 3687392 \)
|
| $47$ |
\( T^{3} + 390 T^{2} + \cdots - 1611552 \)
|
| $53$ |
\( T^{3} + 1282 T^{2} + \cdots - 35263752 \)
|
| $59$ |
\( T^{3} + 304 T^{2} + \cdots - 121786368 \)
|
| $61$ |
\( T^{3} - 246 T^{2} + \cdots + 59018936 \)
|
| $67$ |
\( T^{3} + 718 T^{2} + \cdots - 59593088 \)
|
| $71$ |
\( T^{3} + 602 T^{2} + \cdots - 23858136 \)
|
| $73$ |
\( T^{3} - 282 T^{2} + \cdots + 187092392 \)
|
| $79$ |
\( T^{3} - 206 T^{2} + \cdots + 3736296 \)
|
| $83$ |
\( T^{3} + 2414 T^{2} + \cdots + 270648096 \)
|
| $89$ |
\( T^{3} + 870 T^{2} + \cdots - 7132536 \)
|
| $97$ |
\( T^{3} - 2230 T^{2} + \cdots + 724979000 \)
|
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