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: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 9x^{5} + 13x^{4} + 22x^{3} - 15x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 9x^{5} + 13x^{4} + 22x^{3} - 15x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 9\nu^{4} + 12\nu^{3} + 23\nu^{2} - 10\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} - 3\nu^{5} - 19\nu^{4} + 16\nu^{3} + 50\nu^{2} - 5\nu - 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - 3\nu^{5} - 20\nu^{4} + 17\nu^{3} + 55\nu^{2} - 6\nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 10\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} + 2\beta_{5} - 2\beta_{4} + 9\beta_{3} + 10\beta_{2} + 39\beta _1 + 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( -11\beta_{6} + 13\beta_{5} - 3\beta_{4} + 15\beta_{3} + 39\beta_{2} + 83\beta _1 + 110 \)
|
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 |
|
−2.42914 | 2.12052 | 3.90072 | −1.00000 | −5.15104 | −3.12052 | −4.61712 | 1.49660 | 2.42914 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.18785 | −2.49278 | 2.78669 | −1.00000 | 5.45383 | 1.49278 | −1.72117 | 3.21394 | 2.18785 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.61440 | −0.497982 | 0.606299 | −1.00000 | 0.803944 | −0.502018 | 2.25000 | −2.75201 | 1.61440 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.610153 | −0.308845 | −1.62771 | −1.00000 | 0.188443 | −0.691155 | 2.21346 | −2.90461 | 0.610153 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.20110 | −2.80687 | −0.557366 | −1.00000 | −3.37132 | 1.80687 | −3.07164 | 4.87851 | −1.20110 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.45627 | 1.73310 | 0.120729 | −1.00000 | 2.52386 | −2.73310 | −2.73673 | 0.00363156 | −1.45627 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.18418 | 0.252856 | 2.77064 | −1.00000 | 0.552283 | −1.25286 | 1.68321 | −2.93606 | −2.18418 | ||||||||||||||||||||||||||||||||||||||||
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.f | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2015.2.a.f | ✓ | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2015))\):
|
\( T_{2}^{7} + 2T_{2}^{6} - 9T_{2}^{5} - 16T_{2}^{4} + 25T_{2}^{3} + 35T_{2}^{2} - 23T_{2} - 20 \)
|
|
\( T_{3}^{7} + 2T_{3}^{6} - 9T_{3}^{5} - 13T_{3}^{4} + 22T_{3}^{3} + 15T_{3}^{2} - T_{3} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots - 20 \)
|
| $3$ |
\( T^{7} + 2 T^{6} + \cdots - 1 \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( T^{7} + 5 T^{6} + \cdots + 10 \)
|
| $11$ |
\( T^{7} - 3 T^{6} + \cdots + 268 \)
|
| $13$ |
\( (T - 1)^{7} \)
|
| $17$ |
\( T^{7} + 2 T^{6} + \cdots + 5 \)
|
| $19$ |
\( T^{7} + 4 T^{6} + \cdots + 233 \)
|
| $23$ |
\( T^{7} + 13 T^{6} + \cdots - 40 \)
|
| $29$ |
\( T^{7} - 16 T^{6} + \cdots + 24500 \)
|
| $31$ |
\( (T - 1)^{7} \)
|
| $37$ |
\( T^{7} + 17 T^{6} + \cdots + 81911 \)
|
| $41$ |
\( T^{7} - 11 T^{6} + \cdots + 5 \)
|
| $43$ |
\( T^{7} + 14 T^{6} + \cdots - 359 \)
|
| $47$ |
\( T^{7} + T^{6} + \cdots + 1000 \)
|
| $53$ |
\( T^{7} + 12 T^{6} + \cdots - 1805 \)
|
| $59$ |
\( T^{7} - 30 T^{6} + \cdots + 8419 \)
|
| $61$ |
\( T^{7} + 13 T^{6} + \cdots - 956306 \)
|
| $67$ |
\( T^{7} + 13 T^{6} + \cdots - 14386 \)
|
| $71$ |
\( T^{7} - 2 T^{6} + \cdots - 316015 \)
|
| $73$ |
\( T^{7} + 38 T^{6} + \cdots - 100475 \)
|
| $79$ |
\( T^{7} + 2 T^{6} + \cdots - 186004 \)
|
| $83$ |
\( T^{7} + 14 T^{6} + \cdots + 128125 \)
|
| $89$ |
\( T^{7} + 34 T^{6} + \cdots + 65590 \)
|
| $97$ |
\( T^{7} + 39 T^{6} + \cdots - 644440 \)
|
| show more | |
| show less |