Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [705,2,Mod(1,705)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(705, 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("705.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 705 = 3 \cdot 5 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 705.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: | \(5.62945334250\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.414764096.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 29x^{2} - 42x - 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 29x^{2} - 42x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 8\nu^{2} + 9 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 10\nu^{3} + 21\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 8\beta_{2} + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + 10\beta_{3} + 39\beta_1 \)
|
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 |
|
−2.66262 | 1.00000 | 5.08952 | −1.00000 | −2.66262 | −0.272578 | −8.22622 | 1.00000 | 2.66262 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.77961 | 1.00000 | 1.16702 | −1.00000 | −1.77961 | 4.15308 | 1.48237 | 1.00000 | 1.77961 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.231521 | 1.00000 | −1.94640 | −1.00000 | −0.231521 | −3.28703 | 0.913673 | 1.00000 | 0.231521 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.52844 | 1.00000 | 0.336137 | −1.00000 | 1.52844 | 3.11578 | −2.54312 | 1.00000 | −1.52844 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.33118 | 1.00000 | 3.43441 | −1.00000 | 2.33118 | 3.47124 | 3.34386 | 1.00000 | −2.33118 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.81413 | 1.00000 | 5.91931 | −1.00000 | 2.81413 | −3.18049 | 11.0294 | 1.00000 | −2.81413 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(47\) | \(1\) |
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 | 705.2.a.m | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2115.2.a.s | 6 | ||
| 5.b | even | 2 | 1 | 3525.2.a.w | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 705.2.a.m | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2115.2.a.s | 6 | 3.b | odd | 2 | 1 | ||
| 3525.2.a.w | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(705))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 11T_{2}^{4} + 20T_{2}^{3} + 29T_{2}^{2} - 42T_{2} - 11 \)
|
|
\( T_{7}^{6} - 4T_{7}^{5} - 22T_{7}^{4} + 84T_{7}^{3} + 133T_{7}^{2} - 440T_{7} - 128 \)
|
|
\( T_{11}^{6} - 4T_{11}^{5} - 16T_{11}^{4} + 48T_{11}^{3} + 68T_{11}^{2} - 128T_{11} - 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots - 11 \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} - 4 T^{5} + \cdots - 128 \)
|
| $11$ |
\( T^{6} - 4 T^{5} + \cdots - 64 \)
|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots + 1348 \)
|
| $17$ |
\( T^{6} + 6 T^{5} + \cdots + 1348 \)
|
| $19$ |
\( T^{6} - 10 T^{5} + \cdots - 32 \)
|
| $23$ |
\( T^{6} + 4 T^{5} + \cdots - 128 \)
|
| $29$ |
\( T^{6} - 8 T^{5} + \cdots - 18268 \)
|
| $31$ |
\( T^{6} - 8 T^{5} + \cdots + 101888 \)
|
| $37$ |
\( T^{6} - 16 T^{5} + \cdots + 16 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots + 57860 \)
|
| $43$ |
\( T^{6} \)
|
| $47$ |
\( (T + 1)^{6} \)
|
| $53$ |
\( T^{6} + 10 T^{5} + \cdots + 7412 \)
|
| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 12032 \)
|
| $61$ |
\( T^{6} - 24 T^{5} + \cdots - 4700 \)
|
| $67$ |
\( T^{6} + 8 T^{5} + \cdots - 25600 \)
|
| $71$ |
\( T^{6} - 26 T^{5} + \cdots - 524672 \)
|
| $73$ |
\( T^{6} + 4 T^{5} + \cdots + 16720 \)
|
| $79$ |
\( T^{6} - 24 T^{5} + \cdots + 627968 \)
|
| $83$ |
\( T^{6} + 4 T^{5} + \cdots + 199936 \)
|
| $89$ |
\( T^{6} + 20 T^{5} + \cdots + 390704 \)
|
| $97$ |
\( T^{6} - 180 T^{4} + \cdots - 53824 \)
|
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