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: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} - 10x^{8} + 21x^{7} + 28x^{6} - 67x^{5} - 20x^{4} + 76x^{3} - 8x^{2} - 26x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 3016) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} - 10x^{8} + 21x^{7} + 28x^{6} - 67x^{5} - 20x^{4} + 76x^{3} - 8x^{2} - 26x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{9} - 10\nu^{7} - \nu^{6} + 28\nu^{5} + 9\nu^{4} - 22\nu^{3} - 22\nu^{2} + 10 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{8} - 10\nu^{6} + 28\nu^{4} - 22\nu^{2} - 2\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{9} - \nu^{8} - 10\nu^{7} + 10\nu^{6} + 28\nu^{5} - 29\nu^{4} - 22\nu^{3} + 26\nu^{2} + 2\nu - 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} - 2\nu^{8} + 12\nu^{7} + 21\nu^{6} - 46\nu^{5} - 65\nu^{4} + 62\nu^{3} + 66\nu^{2} - 16\nu - 14 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{9} - \nu^{8} - 10\nu^{7} + 10\nu^{6} + 28\nu^{5} - 29\nu^{4} - 21\nu^{3} + 27\nu^{2} - 2\nu - 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{9} - 2\nu^{8} - 12\nu^{7} + 21\nu^{6} + 48\nu^{5} - 67\nu^{4} - 76\nu^{3} + 76\nu^{2} + 36\nu - 24 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{9} + 2\nu^{8} + 32\nu^{7} - 19\nu^{6} - 104\nu^{5} + 51\nu^{4} + 120\nu^{3} - 44\nu^{2} - 40\nu + 14 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{9} - 4\nu^{8} - 34\nu^{7} + 43\nu^{6} + 124\nu^{5} - 143\nu^{4} - 176\nu^{3} + 172\nu^{2} + 84\nu - 60 ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( 2\nu^{9} - 2\nu^{8} - 22\nu^{7} + 21\nu^{6} + 76\nu^{5} - 67\nu^{4} - 98\nu^{3} + 74\nu^{2} + 40\nu - 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} - \beta_{7} - 3\beta_{6} - \beta_{3} - \beta_{2} - 2\beta _1 + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{9} + 3\beta_{8} - \beta_{7} - 3\beta_{6} + 3\beta_{5} - 4\beta_{3} - \beta_{2} + \beta _1 + 13 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{9} - \beta_{8} - \beta_{7} - 3\beta_{6} + \beta_{5} - 2\beta_{3} - \beta_{2} - 3\beta _1 + 1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{9} + 9\beta_{8} - \beta_{7} - 9\beta_{6} + 9\beta_{5} - 13\beta_{3} - 4\beta_{2} + 7\beta _1 + 25 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 19 \beta_{9} - 12 \beta_{8} - 7 \beta_{7} - 18 \beta_{6} + 9 \beta_{5} + 3 \beta_{4} - 19 \beta_{3} + \cdots + 7 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{9} + 17\beta_{8} + \beta_{7} - 17\beta_{6} + 17\beta_{5} - 25\beta_{3} - 8\beta_{2} + 17\beta _1 + 40 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 116 \beta_{9} - 78 \beta_{8} - 38 \beta_{7} - 87 \beta_{6} + 51 \beta_{5} + 30 \beta_{4} - 116 \beta_{3} + \cdots + 38 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 121 \beta_{9} + 291 \beta_{8} + 46 \beta_{7} - 294 \beta_{6} + 291 \beta_{5} - 431 \beta_{3} + \cdots + 638 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( 231 \beta_{9} - 158 \beta_{8} - 72 \beta_{7} - 156 \beta_{6} + 98 \beta_{5} + 72 \beta_{4} - 232 \beta_{3} + \cdots + 75 \)
|
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 | −2.61420 | 0 | −3.25734 | 0 | −1.30647 | 0 | 3.83403 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.56651 | 0 | 0.728348 | 0 | 3.51001 | 0 | 3.58699 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.65416 | 0 | −0.127785 | 0 | −0.216876 | 0 | −0.263746 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.0834797 | 0 | 3.65561 | 0 | 2.67943 | 0 | −2.99303 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.381396 | 0 | 0.331991 | 0 | −2.55994 | 0 | −2.85454 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.619695 | 0 | −3.40954 | 0 | −0.0449896 | 0 | −2.61598 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.06017 | 0 | 1.60145 | 0 | −0.812633 | 0 | −1.87603 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.21819 | 0 | −3.58621 | 0 | −5.12377 | 0 | 1.92038 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.39322 | 0 | −2.03033 | 0 | 4.56146 | 0 | 2.72750 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 3.24568 | 0 | 1.09381 | 0 | −0.686226 | 0 | 7.53443 | 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.bb | 10 | |
| 4.b | odd | 2 | 1 | 3016.2.a.f | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3016.2.a.f | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 6032.2.a.bb | 10 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):
|
\( T_{3}^{10} - 3T_{3}^{9} - 15T_{3}^{8} + 47T_{3}^{7} + 60T_{3}^{6} - 229T_{3}^{5} + T_{3}^{4} + 320T_{3}^{3} - 212T_{3}^{2} + 28T_{3} + 4 \)
|
|
\( T_{5}^{10} + 5 T_{5}^{9} - 16 T_{5}^{8} - 97 T_{5}^{7} + 45 T_{5}^{6} + 455 T_{5}^{5} - 169 T_{5}^{4} + \cdots - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} - 3 T^{9} + \cdots + 4 \)
|
| $5$ |
\( T^{10} + 5 T^{9} + \cdots - 16 \)
|
| $7$ |
\( T^{10} - 38 T^{8} + \cdots - 4 \)
|
| $11$ |
\( T^{10} - 13 T^{9} + \cdots - 16256 \)
|
| $13$ |
\( (T + 1)^{10} \)
|
| $17$ |
\( T^{10} - 9 T^{9} + \cdots - 191952 \)
|
| $19$ |
\( T^{10} - 12 T^{9} + \cdots + 321344 \)
|
| $23$ |
\( T^{10} - 5 T^{9} + \cdots - 257152 \)
|
| $29$ |
\( (T - 1)^{10} \)
|
| $31$ |
\( T^{10} + 5 T^{9} + \cdots - 7232 \)
|
| $37$ |
\( T^{10} + 4 T^{9} + \cdots + 26800 \)
|
| $41$ |
\( T^{10} + 3 T^{9} + \cdots - 128224 \)
|
| $43$ |
\( T^{10} + \cdots - 144280076 \)
|
| $47$ |
\( T^{10} - 16 T^{9} + \cdots + 299664 \)
|
| $53$ |
\( T^{10} - 11 T^{9} + \cdots + 11916288 \)
|
| $59$ |
\( T^{10} - 27 T^{9} + \cdots - 83808 \)
|
| $61$ |
\( T^{10} + \cdots - 340376544 \)
|
| $67$ |
\( T^{10} + \cdots - 671267448 \)
|
| $71$ |
\( T^{10} - 21 T^{9} + \cdots + 135908 \)
|
| $73$ |
\( T^{10} - 288 T^{8} + \cdots - 1266688 \)
|
| $79$ |
\( T^{10} - 12 T^{9} + \cdots + 61732584 \)
|
| $83$ |
\( T^{10} - 24 T^{9} + \cdots + 11407576 \)
|
| $89$ |
\( T^{10} + \cdots - 16067537056 \)
|
| $97$ |
\( T^{10} + \cdots + 319280032 \)
|
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