Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1352,2,Mod(1,1352)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1352, 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("1352.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1352 = 2^{3} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1352.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: | \(10.7957743533\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3728753.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 10x^{4} + 11x^{3} + 22x^{2} - 18x - 13 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} - x^{5} - 10x^{4} + 11x^{3} + 22x^{2} - 18x - 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 10\nu^{3} + \nu^{2} + 19\nu + 1 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 10\nu^{3} + 3\nu^{2} - 19\nu - 17 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 8\nu^{3} + 3\nu^{2} + 9\nu - 3 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 8\nu^{3} - 13\nu^{2} + 15\nu + 15 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 3\beta_{2} + 5\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 8\beta_{3} + 8\beta_{2} - 3\beta _1 + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{4} - 11\beta_{3} - 27\beta_{2} + 31\beta _1 - 25 \)
|
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 |
|
0 | −2.63830 | 0 | −1.20857 | 0 | −4.34343 | 0 | 3.96061 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.974624 | 0 | 0.130796 | 0 | −2.76527 | 0 | −2.05011 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.0716659 | 0 | −3.69476 | 0 | −1.67432 | 0 | −2.99486 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.589380 | 0 | 3.45555 | 0 | 4.70032 | 0 | −2.65263 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.76369 | 0 | 4.24972 | 0 | −2.37460 | 0 | 4.63797 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.33152 | 0 | −0.932734 | 0 | 3.45729 | 0 | 8.09903 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(13\) | \(-1\) |
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 | 1352.2.a.n | yes | 6 |
| 4.b | odd | 2 | 1 | 2704.2.a.bg | 6 | ||
| 13.b | even | 2 | 1 | 1352.2.a.m | ✓ | 6 | |
| 13.c | even | 3 | 2 | 1352.2.i.n | 12 | ||
| 13.d | odd | 4 | 2 | 1352.2.f.g | 12 | ||
| 13.e | even | 6 | 2 | 1352.2.i.m | 12 | ||
| 13.f | odd | 12 | 4 | 1352.2.o.h | 24 | ||
| 52.b | odd | 2 | 1 | 2704.2.a.bf | 6 | ||
| 52.f | even | 4 | 2 | 2704.2.f.r | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1352.2.a.m | ✓ | 6 | 13.b | even | 2 | 1 | |
| 1352.2.a.n | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 1352.2.f.g | 12 | 13.d | odd | 4 | 2 | ||
| 1352.2.i.m | 12 | 13.e | even | 6 | 2 | ||
| 1352.2.i.n | 12 | 13.c | even | 3 | 2 | ||
| 1352.2.o.h | 24 | 13.f | odd | 12 | 4 | ||
| 2704.2.a.bf | 6 | 52.b | odd | 2 | 1 | ||
| 2704.2.a.bg | 6 | 4.b | odd | 2 | 1 | ||
| 2704.2.f.r | 12 | 52.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1352))\):
|
\( T_{3}^{6} - 3T_{3}^{5} - 9T_{3}^{4} + 23T_{3}^{3} + 15T_{3}^{2} - 13T_{3} - 1 \)
|
|
\( T_{5}^{6} - 2T_{5}^{5} - 21T_{5}^{4} + 23T_{5}^{3} + 98T_{5}^{2} + 48T_{5} - 8 \)
|
|
\( T_{7}^{6} + 3T_{7}^{5} - 30T_{7}^{4} - 107T_{7}^{3} + 148T_{7}^{2} + 860T_{7} + 776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots - 1 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots - 8 \)
|
| $7$ |
\( T^{6} + 3 T^{5} + \cdots + 776 \)
|
| $11$ |
\( T^{6} + 13 T^{5} + \cdots - 127 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 11 T^{5} + \cdots + 1 \)
|
| $19$ |
\( T^{6} + 15 T^{5} + \cdots + 7069 \)
|
| $23$ |
\( T^{6} - 15 T^{5} + \cdots + 8 \)
|
| $29$ |
\( T^{6} - 9 T^{5} + \cdots - 24184 \)
|
| $31$ |
\( T^{6} - 3 T^{5} + \cdots - 664 \)
|
| $37$ |
\( T^{6} + 14 T^{5} + \cdots + 4264 \)
|
| $41$ |
\( T^{6} - 20 T^{5} + \cdots - 169 \)
|
| $43$ |
\( T^{6} - 10 T^{5} + \cdots + 2197 \)
|
| $47$ |
\( T^{6} - 10 T^{5} + \cdots + 2696 \)
|
| $53$ |
\( T^{6} - T^{5} + \cdots - 8632 \)
|
| $59$ |
\( T^{6} + 50 T^{5} + \cdots + 266923 \)
|
| $61$ |
\( T^{6} - 135 T^{4} + \cdots - 5312 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots - 67537 \)
|
| $71$ |
\( T^{6} + 9 T^{5} + \cdots + 112216 \)
|
| $73$ |
\( T^{6} - 6 T^{5} + \cdots - 282449 \)
|
| $79$ |
\( T^{6} - 13 T^{5} + \cdots - 246616 \)
|
| $83$ |
\( T^{6} + 36 T^{5} + \cdots + 70783 \)
|
| $89$ |
\( T^{6} - 18 T^{5} + \cdots + 1079 \)
|
| $97$ |
\( T^{6} - 14 T^{5} + \cdots - 124529 \)
|
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