Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2008,2,Mod(1,2008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2008, 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("2008.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2008 = 2^{3} \cdot 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2008.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: | \(16.0339607259\) |
| 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: |
\( x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} - 4\nu^{7} - 3\nu^{6} + 28\nu^{5} - 16\nu^{4} - 47\nu^{3} + 36\nu^{2} + 14\nu + 3 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{8} + 4\nu^{7} + 7\nu^{6} - 32\nu^{5} - 16\nu^{4} + 67\nu^{3} + 24\nu^{2} - 30\nu - 11 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{8} - 8\nu^{7} - 21\nu^{6} + 60\nu^{5} + 32\nu^{4} - 117\nu^{3} + 50\nu - 7 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{8} + 4\nu^{7} + 29\nu^{6} - 28\nu^{5} - 88\nu^{4} + 49\nu^{3} + 92\nu^{2} - 14\nu - 13 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{8} + 12\nu^{7} + 39\nu^{6} - 88\nu^{5} - 84\nu^{4} + 159\nu^{3} + 56\nu^{2} - 46\nu - 3 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{8} - 8\nu^{7} - 21\nu^{6} + 58\nu^{5} + 34\nu^{4} - 101\nu^{3} - 10\nu^{2} + 22\nu - 1 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} - \beta_{3} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} - \beta_{6} - \beta_{4} + 6\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{8} + 10\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} - 8\beta_{3} + \beta_{2} + 19\beta _1 + 18 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{8} + 21\beta_{7} - 9\beta_{6} + 2\beta_{5} - 8\beta_{4} - 2\beta_{3} + 34\beta_{2} + 11\beta _1 + 77 \)
|
| \(\nu^{7}\) | \(=\) |
\( 55\beta_{8} + 77\beta_{7} - 13\beta_{6} + 19\beta_{5} - 10\beta_{4} - 51\beta_{3} + 15\beta_{2} + 101\beta _1 + 132 \)
|
| \(\nu^{8}\) | \(=\) |
\( 92\beta_{8} + 170\beta_{7} - 67\beta_{6} + 26\beta_{5} - 52\beta_{4} - 29\beta_{3} + 194\beta_{2} + 95\beta _1 + 462 \)
|
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 | −2.59002 | 0 | −1.92989 | 0 | −1.80522 | 0 | 3.70818 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.92428 | 0 | 0.603023 | 0 | 4.00229 | 0 | 0.702869 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.71581 | 0 | −0.169057 | 0 | −0.581118 | 0 | −0.0559961 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.540607 | 0 | −2.60958 | 0 | 0.631745 | 0 | −2.70774 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.0778757 | 0 | 1.70938 | 0 | −3.05552 | 0 | −2.99394 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.600027 | 0 | 2.23756 | 0 | 0.532333 | 0 | −2.63997 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.42772 | 0 | −3.45271 | 0 | 2.96003 | 0 | −0.961621 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.49215 | 0 | 0.0991275 | 0 | −1.31645 | 0 | −0.773492 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.17295 | 0 | −1.48785 | 0 | −1.36809 | 0 | 1.72170 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(251\) | \(-1\) |
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 | 2008.2.a.a | ✓ | 9 |
| 4.b | odd | 2 | 1 | 4016.2.a.h | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2008.2.a.a | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 4016.2.a.h | 9 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{9} + T_{3}^{8} - 11T_{3}^{7} - 7T_{3}^{6} + 40T_{3}^{5} + 11T_{3}^{4} - 53T_{3}^{3} + 2T_{3}^{2} + 13T_{3} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2008))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} + T^{8} - 11 T^{7} + \cdots - 1 \)
|
| $5$ |
\( T^{9} + 5 T^{8} + \cdots + 1 \)
|
| $7$ |
\( T^{9} - 21 T^{7} + \cdots + 23 \)
|
| $11$ |
\( T^{9} + 3 T^{8} + \cdots - 1240 \)
|
| $13$ |
\( T^{9} + 3 T^{8} + \cdots - 25 \)
|
| $17$ |
\( T^{9} + 11 T^{8} + \cdots + 5773 \)
|
| $19$ |
\( T^{9} - 4 T^{8} + \cdots - 64 \)
|
| $23$ |
\( T^{9} + 9 T^{8} + \cdots - 10211 \)
|
| $29$ |
\( T^{9} + 9 T^{8} + \cdots + 1000 \)
|
| $31$ |
\( T^{9} - 3 T^{8} + \cdots + 8948 \)
|
| $37$ |
\( T^{9} + 10 T^{8} + \cdots - 5464 \)
|
| $41$ |
\( T^{9} + 23 T^{8} + \cdots - 6803 \)
|
| $43$ |
\( T^{9} - 158 T^{7} + \cdots + 375784 \)
|
| $47$ |
\( T^{9} + 11 T^{8} + \cdots + 4600 \)
|
| $53$ |
\( T^{9} + 21 T^{8} + \cdots - 89504 \)
|
| $59$ |
\( T^{9} + 4 T^{8} + \cdots - 727040 \)
|
| $61$ |
\( T^{9} + 11 T^{8} + \cdots + 17439608 \)
|
| $67$ |
\( T^{9} + 4 T^{8} + \cdots - 16829 \)
|
| $71$ |
\( T^{9} + 19 T^{8} + \cdots - 42539456 \)
|
| $73$ |
\( T^{9} + 31 T^{8} + \cdots + 202816 \)
|
| $79$ |
\( T^{9} - 4 T^{8} + \cdots - 2471135 \)
|
| $83$ |
\( T^{9} + 22 T^{8} + \cdots + 1509952 \)
|
| $89$ |
\( T^{9} + 36 T^{8} + \cdots - 46507 \)
|
| $97$ |
\( T^{9} + \cdots + 1143623336 \)
|
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