Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2014,2,Mod(1,2014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2014, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2014 = 2 \cdot 19 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2014.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: | \(16.0818709671\) |
| 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: |
\( x^{8} - x^{7} - 12x^{6} + 40x^{4} + 22x^{3} - 29x^{2} - 24x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 12x^{6} + 40x^{4} + 22x^{3} - 29x^{2} - 24x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{7} - 7\nu^{6} - 26\nu^{5} + 32\nu^{4} + 74\nu^{3} - 18\nu^{2} - 55\nu - 14 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{7} + 6\nu^{6} + 13\nu^{5} - 29\nu^{4} - 29\nu^{3} + 29\nu^{2} + 18\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{7} + 15\nu^{6} + 32\nu^{5} - 70\nu^{4} - 72\nu^{3} + 62\nu^{2} + 45\nu + 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{7} - 15\nu^{6} - 32\nu^{5} + 70\nu^{4} + 72\nu^{3} - 60\nu^{2} - 47\nu - 8 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{7} - 13\nu^{6} - 38\nu^{5} + 58\nu^{4} + 98\nu^{3} - 36\nu^{2} - 67\nu - 18 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -9\nu^{7} + 23\nu^{6} + 70\nu^{5} - 102\nu^{4} - 188\nu^{3} + 62\nu^{2} + 137\nu + 34 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + 2\beta_{5} + 3\beta_{4} - \beta_{3} - \beta_{2} + 5\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} - \beta_{6} + 9\beta_{5} + 13\beta_{4} - 4\beta_{3} - 3\beta_{2} + 12\beta _1 + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{7} - 6\beta_{6} + 26\beta_{5} + 43\beta_{4} - 16\beta_{3} - 16\beta_{2} + 44\beta _1 + 66 \)
|
| \(\nu^{6}\) | \(=\) |
\( -32\beta_{7} - 10\beta_{6} + 93\beta_{5} + 156\beta_{4} - 59\beta_{3} - 53\beta_{2} + 137\beta _1 + 240 \)
|
| \(\nu^{7}\) | \(=\) |
\( -124\beta_{7} - 40\beta_{6} + 303\beta_{5} + 530\beta_{4} - 209\beta_{3} - 205\beta_{2} + 474\beta _1 + 786 \)
|
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 |
|
1.00000 | −3.04978 | 1.00000 | 0.307563 | −3.04978 | −0.523411 | 1.00000 | 6.30114 | 0.307563 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.38245 | 1.00000 | 3.40594 | −2.38245 | −3.72146 | 1.00000 | 2.67605 | 3.40594 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.93337 | 1.00000 | −0.836498 | −1.93337 | −2.05843 | 1.00000 | 0.737907 | −0.836498 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.21087 | 1.00000 | −0.364343 | −1.21087 | 3.85179 | 1.00000 | −1.53380 | −0.364343 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.552690 | 1.00000 | −0.466887 | 0.552690 | −0.310719 | 1.00000 | −2.69453 | −0.466887 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.18505 | 1.00000 | −2.43542 | 1.18505 | −3.37563 | 1.00000 | −1.59566 | −2.43542 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 1.21968 | 1.00000 | −3.41661 | 1.21968 | 0.884407 | 1.00000 | −1.51237 | −3.41661 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 1.61903 | 1.00000 | 0.806256 | 1.61903 | −4.74655 | 1.00000 | −0.378732 | 0.806256 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(1\) |
| \(53\) | \(-1\) |
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 | 2014.2.a.h | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2014.2.a.h | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2014))\):
|
\( T_{3}^{8} + 4T_{3}^{7} - 5T_{3}^{6} - 27T_{3}^{5} + 12T_{3}^{4} + 61T_{3}^{3} - 25T_{3}^{2} - 44T_{3} + 22 \)
|
|
\( T_{7}^{8} + 10T_{7}^{7} + 16T_{7}^{6} - 124T_{7}^{5} - 479T_{7}^{4} - 384T_{7}^{3} + 313T_{7}^{2} + 340T_{7} + 68 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} + 4 T^{7} + \cdots + 22 \)
|
| $5$ |
\( T^{8} + 3 T^{7} + \cdots - 1 \)
|
| $7$ |
\( T^{8} + 10 T^{7} + \cdots + 68 \)
|
| $11$ |
\( T^{8} + 12 T^{7} + \cdots + 617 \)
|
| $13$ |
\( T^{8} + 3 T^{7} + \cdots + 458 \)
|
| $17$ |
\( T^{8} + 9 T^{7} + \cdots - 1973 \)
|
| $19$ |
\( (T + 1)^{8} \)
|
| $23$ |
\( T^{8} + 9 T^{7} + \cdots - 2699 \)
|
| $29$ |
\( T^{8} + 23 T^{7} + \cdots + 202082 \)
|
| $31$ |
\( T^{8} + 10 T^{7} + \cdots + 309861 \)
|
| $37$ |
\( T^{8} + T^{7} + \cdots + 92044 \)
|
| $41$ |
\( T^{8} + 11 T^{7} + \cdots - 34042 \)
|
| $43$ |
\( T^{8} + 19 T^{7} + \cdots - 79101 \)
|
| $47$ |
\( T^{8} - T^{7} + \cdots - 296242 \)
|
| $53$ |
\( (T - 1)^{8} \)
|
| $59$ |
\( T^{8} + 27 T^{7} + \cdots + 52129 \)
|
| $61$ |
\( T^{8} + 14 T^{7} + \cdots - 180716 \)
|
| $67$ |
\( T^{8} + 49 T^{7} + \cdots - 7331576 \)
|
| $71$ |
\( T^{8} + 19 T^{7} + \cdots + 20094232 \)
|
| $73$ |
\( T^{8} + 13 T^{7} + \cdots - 4218624 \)
|
| $79$ |
\( T^{8} + 11 T^{7} + \cdots + 8319809 \)
|
| $83$ |
\( T^{8} - 17 T^{7} + \cdots - 192033242 \)
|
| $89$ |
\( T^{8} + 13 T^{7} + \cdots - 267047 \)
|
| $97$ |
\( T^{8} - 32 T^{7} + \cdots - 19123577 \)
|
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