Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2031,2,Mod(1,2031)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2031, 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("2031.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2031 = 3 \cdot 677 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2031.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: | \(16.2176166505\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.32354821.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 5x^{5} + 9x^{4} + 6x^{3} - 8x^{2} - x + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 5x^{5} + 9x^{4} + 6x^{3} - 8x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 4\nu^{4} + 8\nu^{3} + 2\nu^{2} - 5\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 6\nu^{4} + 4\nu^{3} + 9\nu^{2} - 2\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 5\nu^{4} + 8\nu^{3} + 6\nu^{2} - 4\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 5\nu^{4} + 8\nu^{3} + 7\nu^{2} - 5\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 5\nu^{4} + 9\nu^{3} + 6\nu^{2} - 8\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{4} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{5} - 5\beta_{4} + \beta_{2} + 5\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{6} + \beta_{5} - 7\beta_{4} + \beta_{3} + \beta_{2} + 16\beta _1 + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( 16\beta_{5} - 24\beta_{4} + 2\beta_{3} + 7\beta_{2} + 23\beta _1 + 29 \)
|
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 |
|
−2.20690 | −1.00000 | 2.87039 | −1.61210 | 2.20690 | 0.450247 | −1.92086 | 1.00000 | 3.55774 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.84468 | −1.00000 | 1.40285 | 1.75647 | 1.84468 | 1.52307 | 1.10155 | 1.00000 | −3.24012 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.47087 | −1.00000 | 0.163459 | −0.168975 | 1.47087 | 3.77659 | 2.70131 | 1.00000 | 0.248541 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.146874 | −1.00000 | −1.97843 | −2.25454 | 0.146874 | −1.49119 | 0.584326 | 1.00000 | 0.331132 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.455154 | −1.00000 | −1.79283 | −3.54297 | −0.455154 | 0.0807299 | −1.72633 | 1.00000 | −1.61260 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.31629 | −1.00000 | −0.267381 | −1.10800 | −1.31629 | 1.45284 | −2.98453 | 1.00000 | −1.45845 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.89788 | −1.00000 | 1.60194 | −3.06987 | −1.89788 | 2.20772 | −0.755467 | 1.00000 | −5.82624 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(677\) | \( +1 \) |
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 | 2031.2.a.e | ✓ | 7 |
| 3.b | odd | 2 | 1 | 6093.2.a.i | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2031.2.a.e | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 6093.2.a.i | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2031))\):
|
\( T_{2}^{7} + 2T_{2}^{6} - 6T_{2}^{5} - 11T_{2}^{4} + 10T_{2}^{3} + 14T_{2}^{2} - 5T_{2} - 1 \)
|
|
\( T_{7}^{7} - 8T_{7}^{6} + 19T_{7}^{5} - 4T_{7}^{4} - 40T_{7}^{3} + 48T_{7}^{2} - 16T_{7} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots - 1 \)
|
| $3$ |
\( (T + 1)^{7} \)
|
| $5$ |
\( T^{7} + 10 T^{6} + \cdots - 13 \)
|
| $7$ |
\( T^{7} - 8 T^{6} + \cdots + 1 \)
|
| $11$ |
\( T^{7} - 10 T^{6} + \cdots - 52 \)
|
| $13$ |
\( T^{7} + 3 T^{6} + \cdots + 53 \)
|
| $17$ |
\( T^{7} - 8 T^{6} + \cdots + 491 \)
|
| $19$ |
\( T^{7} + 5 T^{6} + \cdots + 3361 \)
|
| $23$ |
\( T^{7} + T^{6} + \cdots + 64 \)
|
| $29$ |
\( T^{7} - 5 T^{6} + \cdots + 10211 \)
|
| $31$ |
\( T^{7} + 24 T^{6} + \cdots + 5123 \)
|
| $37$ |
\( T^{7} + 26 T^{6} + \cdots - 109 \)
|
| $41$ |
\( T^{7} + 12 T^{6} + \cdots - 16384 \)
|
| $43$ |
\( T^{7} + 26 T^{6} + \cdots + 64831 \)
|
| $47$ |
\( T^{7} - 2 T^{6} + \cdots - 681236 \)
|
| $53$ |
\( T^{7} - 16 T^{6} + \cdots - 23923 \)
|
| $59$ |
\( T^{7} + 17 T^{6} + \cdots + 83 \)
|
| $61$ |
\( T^{7} + 10 T^{6} + \cdots + 38876 \)
|
| $67$ |
\( T^{7} - 20 T^{6} + \cdots + 242153 \)
|
| $71$ |
\( T^{7} + 5 T^{6} + \cdots - 11239 \)
|
| $73$ |
\( T^{7} + 13 T^{6} + \cdots - 1792972 \)
|
| $79$ |
\( T^{7} + 32 T^{6} + \cdots - 163148 \)
|
| $83$ |
\( T^{7} - 11 T^{6} + \cdots + 5492 \)
|
| $89$ |
\( T^{7} + 8 T^{6} + \cdots - 4 \)
|
| $97$ |
\( T^{7} + 41 T^{6} + \cdots + 1612139 \)
|
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