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: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 13x^{7} + 13x^{6} + 51x^{5} - 50x^{4} - 59x^{3} + 45x^{2} + 20x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 13x^{7} + 13x^{6} + 51x^{5} - 50x^{4} - 59x^{3} + 45x^{2} + 20x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{8} + 13\nu^{6} - 49\nu^{4} - \nu^{3} + 44\nu^{2} + 11\nu + 3 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} - 13\nu^{6} + 51\nu^{4} - \nu^{3} - 58\nu^{2} - \nu + 9 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - 13\nu^{6} + 51\nu^{4} + \nu^{3} - 58\nu^{2} - 11\nu + 9 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{8} - 13\nu^{6} + 2\nu^{5} + 49\nu^{4} - 15\nu^{3} - 46\nu^{2} + 11\nu + 3 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} + 2\nu^{7} - 13\nu^{6} - 24\nu^{5} + 51\nu^{4} + 83\nu^{3} - 60\nu^{2} - 75\nu + 3 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{8} - 2\nu^{7} + 13\nu^{6} + 24\nu^{5} - 51\nu^{4} - 83\nu^{3} + 60\nu^{2} + 83\nu - 3 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -3\nu^{8} + 37\nu^{6} - 133\nu^{4} + \nu^{3} + 124\nu^{2} + 17\nu - 7 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{7} + 5\beta_{6} + 2\beta_{4} - 2\beta_{3} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 7\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 29\beta_{7} + 29\beta_{6} + 2\beta_{5} + 16\beta_{4} - 16\beta_{3} + 2\beta_{2} + 2\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -2\beta_{8} + \beta_{7} + \beta_{6} + 9\beta_{4} - 2\beta_{3} + 10\beta_{2} + 45\beta _1 + 85 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 175\beta_{7} + 179\beta_{6} + 24\beta_{5} + 108\beta_{4} - 110\beta_{3} + 24\beta_{2} + 26\beta _1 + 12 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( -26\beta_{8} + 16\beta_{7} + 16\beta_{6} + 67\beta_{4} - 25\beta_{3} + 79\beta_{2} + 286\beta _1 + 505 \)
|
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 | −2.63301 | 0 | 1.34149 | 0 | 0.562877 | 0 | 3.93275 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.21031 | 0 | 1.74097 | 0 | −4.95439 | 0 | 1.88546 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.75740 | 0 | −3.19666 | 0 | −3.77437 | 0 | 0.0884381 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.29118 | 0 | −2.09711 | 0 | 0.702804 | 0 | −1.33285 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.774078 | 0 | 4.01023 | 0 | −3.72578 | 0 | −2.40080 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.44677 | 0 | −0.302620 | 0 | 2.83956 | 0 | −0.906844 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.72113 | 0 | 2.94802 | 0 | −1.33933 | 0 | −0.0377024 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.17582 | 0 | 0.311009 | 0 | −2.97341 | 0 | 1.73418 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 3.32225 | 0 | −2.75533 | 0 | −4.33796 | 0 | 8.03736 | 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.y | 9 | |
| 4.b | odd | 2 | 1 | 377.2.a.f | ✓ | 9 | |
| 12.b | even | 2 | 1 | 3393.2.a.s | 9 | ||
| 20.d | odd | 2 | 1 | 9425.2.a.u | 9 | ||
| 52.b | odd | 2 | 1 | 4901.2.a.l | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 377.2.a.f | ✓ | 9 | 4.b | odd | 2 | 1 | |
| 3393.2.a.s | 9 | 12.b | even | 2 | 1 | ||
| 4901.2.a.l | 9 | 52.b | odd | 2 | 1 | ||
| 6032.2.a.y | 9 | 1.a | even | 1 | 1 | trivial | |
| 9425.2.a.u | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):
|
\( T_{3}^{9} - 19T_{3}^{7} - 6T_{3}^{6} + 120T_{3}^{5} + 59T_{3}^{4} - 304T_{3}^{3} - 184T_{3}^{2} + 264T_{3} + 184 \)
|
|
\( T_{5}^{9} - 2T_{5}^{8} - 24T_{5}^{7} + 39T_{5}^{6} + 178T_{5}^{5} - 247T_{5}^{4} - 400T_{5}^{3} + 536T_{5}^{2} + 32T_{5} - 48 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} - 19 T^{7} + \cdots + 184 \)
|
| $5$ |
\( T^{9} - 2 T^{8} + \cdots - 48 \)
|
| $7$ |
\( T^{9} + 17 T^{8} + \cdots - 1352 \)
|
| $11$ |
\( T^{9} + 3 T^{8} + \cdots + 48 \)
|
| $13$ |
\( (T - 1)^{9} \)
|
| $17$ |
\( T^{9} + 2 T^{8} + \cdots - 1392 \)
|
| $19$ |
\( T^{9} + 14 T^{8} + \cdots + 80 \)
|
| $23$ |
\( T^{9} + 13 T^{8} + \cdots - 44928 \)
|
| $29$ |
\( (T + 1)^{9} \)
|
| $31$ |
\( T^{9} + 13 T^{8} + \cdots - 2081264 \)
|
| $37$ |
\( T^{9} - 17 T^{8} + \cdots + 672064 \)
|
| $41$ |
\( T^{9} + 3 T^{8} + \cdots + 7488 \)
|
| $43$ |
\( T^{9} + 8 T^{8} + \cdots - 1687256 \)
|
| $47$ |
\( T^{9} - 7 T^{8} + \cdots + 11247504 \)
|
| $53$ |
\( T^{9} - 3 T^{8} + \cdots - 73872 \)
|
| $59$ |
\( T^{9} + 3 T^{8} + \cdots - 42360 \)
|
| $61$ |
\( T^{9} + 7 T^{8} + \cdots + 2618416 \)
|
| $67$ |
\( T^{9} + 39 T^{8} + \cdots + 450616 \)
|
| $71$ |
\( T^{9} + 4 T^{8} + \cdots - 3213192 \)
|
| $73$ |
\( T^{9} - 12 T^{8} + \cdots + 512 \)
|
| $79$ |
\( T^{9} + 16 T^{8} + \cdots + 595160 \)
|
| $83$ |
\( T^{9} + 10 T^{8} + \cdots - 2535672 \)
|
| $89$ |
\( T^{9} + 11 T^{8} + \cdots + 7965120 \)
|
| $97$ |
\( T^{9} + 4 T^{8} + \cdots + 9310144 \)
|
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