Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2015,2,Mod(1,2015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, 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("2015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2015 = 5 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2015.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: | \(16.0898560073\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 7x^{6} + 12x^{5} + 15x^{4} - 19x^{3} - 13x^{2} + 8x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 7x^{6} + 12x^{5} + 15x^{4} - 19x^{3} - 13x^{2} + 8x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 7\nu^{4} + 4\nu^{3} + 12\nu^{2} - 2\nu - 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 10\nu^{3} + 9\nu^{2} - 9\nu - 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 6\nu^{5} - 10\nu^{4} - 9\nu^{3} + 10\nu^{2} + 4\nu - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 6\nu^{5} + 11\nu^{4} + 8\nu^{3} - 15\nu^{2} - \nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{3} + 6\beta_{2} + 6\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 7\beta_{3} + 9\beta_{2} + 20\beta _1 + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{7} + 8\beta_{6} - \beta_{5} + 2\beta_{4} + 10\beta_{3} + 35\beta_{2} + 36\beta _1 + 40 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12\beta_{7} + 11\beta_{6} - 8\beta_{5} + 10\beta_{4} + 43\beta_{3} + 65\beta_{2} + 110\beta _1 + 57 \)
|
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 |
|
−2.50052 | −0.363360 | 4.25261 | 1.00000 | 0.908590 | −0.636640 | −5.63271 | −2.86797 | −2.50052 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.77027 | −2.75040 | 1.13385 | 1.00000 | 4.86895 | 1.75040 | 1.53331 | 4.56471 | −1.77027 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.31231 | 1.69449 | −0.277833 | 1.00000 | −2.22371 | −2.69449 | 2.98923 | −0.128688 | −1.31231 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.836707 | 0.879795 | −1.29992 | 1.00000 | −0.736130 | −1.87979 | 2.76107 | −2.22596 | −0.836707 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.424732 | 2.53248 | −1.81960 | 1.00000 | 1.07563 | −3.53248 | −1.62231 | 3.41345 | 0.424732 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.805137 | −2.20564 | −1.35175 | 1.00000 | −1.77584 | 1.20564 | −2.69862 | 1.86484 | 0.805137 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.22431 | −0.580301 | −0.501065 | 1.00000 | −0.710468 | −0.419699 | −3.06208 | −2.66325 | 1.22431 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.96563 | −0.207067 | 1.86371 | 1.00000 | −0.407019 | −0.792933 | −0.267894 | −2.95712 | 1.96563 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(13\) | \(-1\) |
| \(31\) | \(1\) |
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 | 2015.2.a.g | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2015.2.a.g | ✓ | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2015))\):
|
\( T_{2}^{8} + 2T_{2}^{7} - 7T_{2}^{6} - 12T_{2}^{5} + 15T_{2}^{4} + 19T_{2}^{3} - 13T_{2}^{2} - 8T_{2} + 4 \)
|
|
\( T_{3}^{8} + T_{3}^{7} - 11T_{3}^{6} - 8T_{3}^{5} + 31T_{3}^{4} + 13T_{3}^{3} - 14T_{3}^{2} - 8T_{3} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 4 \)
|
| $3$ |
\( T^{8} + T^{7} - 11 T^{6} + \cdots - 1 \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} + 7 T^{7} + \cdots + 8 \)
|
| $11$ |
\( T^{8} + 3 T^{7} + \cdots - 428 \)
|
| $13$ |
\( (T - 1)^{8} \)
|
| $17$ |
\( T^{8} + 11 T^{7} + \cdots + 4463 \)
|
| $19$ |
\( T^{8} + 13 T^{7} + \cdots - 913 \)
|
| $23$ |
\( T^{8} - 3 T^{7} + \cdots + 82112 \)
|
| $29$ |
\( T^{8} + 12 T^{7} + \cdots - 11156 \)
|
| $31$ |
\( (T + 1)^{8} \)
|
| $37$ |
\( T^{8} + 12 T^{7} + \cdots - 19637 \)
|
| $41$ |
\( T^{8} + 20 T^{7} + \cdots + 7733 \)
|
| $43$ |
\( T^{8} + 13 T^{7} + \cdots - 555421 \)
|
| $47$ |
\( T^{8} + 3 T^{7} + \cdots + 113168 \)
|
| $53$ |
\( T^{8} + 17 T^{7} + \cdots + 42349 \)
|
| $59$ |
\( T^{8} + 9 T^{7} + \cdots + 15521 \)
|
| $61$ |
\( T^{8} + 15 T^{7} + \cdots - 158708 \)
|
| $67$ |
\( T^{8} + 5 T^{7} + \cdots + 527144 \)
|
| $71$ |
\( T^{8} + 9 T^{7} + \cdots + 227257 \)
|
| $73$ |
\( T^{8} + 27 T^{7} + \cdots - 85941 \)
|
| $79$ |
\( T^{8} + 8 T^{7} + \cdots - 2139964 \)
|
| $83$ |
\( T^{8} - 7 T^{7} + \cdots + 661657 \)
|
| $89$ |
\( T^{8} + 16 T^{7} + \cdots - 76184 \)
|
| $97$ |
\( T^{8} + 27 T^{7} + \cdots + 12989936 \)
|
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