Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6032,2,Mod(1,6032)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6032, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6032.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6032 = 2^{4} \cdot 13 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6032.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: | \(48.1657624992\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 15x^{3} - 15x^{2} - 8x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 377) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 15x^{3} - 15x^{2} - 8x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 8\nu^{2} + 5\nu - 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 9\nu^{3} + 10\nu^{2} - 7\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_{2} + 8\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 7\beta_{3} + 9\beta_{2} + 28\beta _1 + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} + 10\beta_{4} + 11\beta_{3} + 35\beta_{2} + 56\beta _1 + 32 \)
|
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 | −3.32938 | 0 | 1.40530 | 0 | −1.72070 | 0 | 8.08475 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.02334 | 0 | 1.17887 | 0 | −0.399284 | 0 | 1.09391 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.11702 | 0 | 2.27865 | 0 | 2.88565 | 0 | −1.75226 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.182751 | 0 | 1.30841 | 0 | −4.37827 | 0 | −2.96660 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.308654 | 0 | −3.95109 | 0 | −2.36054 | 0 | −2.90473 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.09954 | 0 | −3.72490 | 0 | 0.555254 | 0 | 1.40806 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.24430 | 0 | −0.495240 | 0 | −1.58211 | 0 | 2.03688 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(13\) | \(1\) |
| \(29\) | \(-1\) |
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 | 6032.2.a.u | 7 | |
| 4.b | odd | 2 | 1 | 377.2.a.e | ✓ | 7 | |
| 12.b | even | 2 | 1 | 3393.2.a.o | 7 | ||
| 20.d | odd | 2 | 1 | 9425.2.a.s | 7 | ||
| 52.b | odd | 2 | 1 | 4901.2.a.k | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 377.2.a.e | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 3393.2.a.o | 7 | 12.b | even | 2 | 1 | ||
| 4901.2.a.k | 7 | 52.b | odd | 2 | 1 | ||
| 6032.2.a.u | 7 | 1.a | even | 1 | 1 | trivial | |
| 9425.2.a.s | 7 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):
|
\( T_{3}^{7} + 2T_{3}^{6} - 11T_{3}^{5} - 16T_{3}^{4} + 30T_{3}^{3} + 33T_{3}^{2} - 6T_{3} - 2 \)
|
|
\( T_{5}^{7} + 2T_{5}^{6} - 18T_{5}^{5} - 7T_{5}^{4} + 106T_{5}^{3} - 111T_{5}^{2} - 8T_{5} + 36 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} + 2 T^{6} + \cdots - 2 \)
|
| $5$ |
\( T^{7} + 2 T^{6} + \cdots + 36 \)
|
| $7$ |
\( T^{7} + 7 T^{6} + \cdots + 18 \)
|
| $11$ |
\( T^{7} - 3 T^{6} + \cdots - 6508 \)
|
| $13$ |
\( (T + 1)^{7} \)
|
| $17$ |
\( T^{7} - 14 T^{6} + \cdots + 188 \)
|
| $19$ |
\( T^{7} - 12 T^{6} + \cdots + 3516 \)
|
| $23$ |
\( T^{7} + 5 T^{6} + \cdots - 8 \)
|
| $29$ |
\( (T - 1)^{7} \)
|
| $31$ |
\( T^{7} - 3 T^{6} + \cdots - 4 \)
|
| $37$ |
\( T^{7} - 9 T^{6} + \cdots + 52828 \)
|
| $41$ |
\( T^{7} + 9 T^{6} + \cdots - 56084 \)
|
| $43$ |
\( T^{7} + 24 T^{6} + \cdots - 8766 \)
|
| $47$ |
\( T^{7} - 9 T^{6} + \cdots + 15324 \)
|
| $53$ |
\( T^{7} + 13 T^{6} + \cdots - 568 \)
|
| $59$ |
\( T^{7} + 13 T^{6} + \cdots + 2766 \)
|
| $61$ |
\( T^{7} - 3 T^{6} + \cdots + 116 \)
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| $67$ |
\( T^{7} - 7 T^{6} + \cdots + 5498 \)
|
| $71$ |
\( T^{7} + 22 T^{6} + \cdots - 797206 \)
|
| $73$ |
\( T^{7} - 4 T^{6} + \cdots + 3020592 \)
|
| $79$ |
\( T^{7} + 48 T^{6} + \cdots - 565634 \)
|
| $83$ |
\( T^{7} - 30 T^{6} + \cdots + 4941174 \)
|
| $89$ |
\( T^{7} + 25 T^{6} + \cdots - 10810732 \)
|
| $97$ |
\( T^{7} - 4 T^{6} + \cdots - 824124 \)
|
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