Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2873,2,Mod(1,2873)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2873, 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("2873.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2873 = 13^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2873.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: | \(22.9410205007\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.434581.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 4x^{4} + 5x^{3} + 4x^{2} - 2x - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 4x^{4} + 5x^{3} + 4x^{2} - 2x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 3\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 3\nu^{3} + 3\nu^{2} + \nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 3\nu^{3} + 4\nu^{2} - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 2\nu^{3} + 8\nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} + 7\beta_{4} - 5\beta_{3} - \beta_{2} + 8\beta _1 + 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} + 17\beta_{4} - 9\beta_{3} - 5\beta_{2} + 24\beta _1 + 1 \)
|
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 |
|
−2.37002 | 0.513325 | 3.61700 | 3.07557 | −1.21659 | −1.60388 | −3.83233 | −2.73650 | −7.28916 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.84349 | 0.768618 | 1.39845 | −1.15479 | −1.41694 | 4.49396 | 1.10896 | −2.40923 | 2.12883 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.16937 | 3.18252 | −0.632563 | 4.00814 | −3.72156 | 1.10992 | 3.07845 | 7.12843 | −4.68702 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.169375 | 0.174377 | −1.97131 | −2.00814 | 0.0295350 | 1.10992 | −0.672639 | −2.96959 | −0.340128 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0.843487 | 2.92340 | −1.28853 | 3.15479 | 2.46585 | 4.49396 | −2.77383 | 5.54629 | 2.66102 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.37002 | −1.56224 | −0.123042 | −1.07557 | −2.14031 | −1.60388 | −2.90861 | −0.559399 | −1.47355 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(-1\) |
| \(17\) | \(1\) |
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 | 2873.2.a.m | ✓ | 6 |
| 13.b | even | 2 | 1 | 2873.2.a.o | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2873.2.a.m | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2873.2.a.o | yes | 6 | 13.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2873))\):
|
\( T_{2}^{6} + 3T_{2}^{5} - 2T_{2}^{4} - 9T_{2}^{3} + T_{2}^{2} + 6T_{2} - 1 \)
|
|
\( T_{3}^{6} - 6T_{3}^{5} + 7T_{3}^{4} + 12T_{3}^{3} - 21T_{3}^{2} + 9T_{3} - 1 \)
|
|
\( T_{5}^{6} - 6T_{5}^{5} - 3T_{5}^{4} + 52T_{5}^{3} + 8T_{5}^{2} - 136T_{5} - 97 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{5} + \cdots - 1 \)
|
| $3$ |
\( T^{6} - 6 T^{5} + \cdots - 1 \)
|
| $5$ |
\( T^{6} - 6 T^{5} + \cdots - 97 \)
|
| $7$ |
\( (T^{3} - 4 T^{2} - 4 T + 8)^{2} \)
|
| $11$ |
\( T^{6} + 16 T^{5} + \cdots - 5419 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T + 1)^{6} \)
|
| $19$ |
\( T^{6} - 14 T^{5} + \cdots - 3821 \)
|
| $23$ |
\( T^{6} + T^{5} + \cdots - 181 \)
|
| $29$ |
\( T^{6} + 23 T^{5} + \cdots - 2731 \)
|
| $31$ |
\( T^{6} - 10 T^{5} + \cdots - 853 \)
|
| $37$ |
\( T^{6} - 13 T^{5} + \cdots + 1 \)
|
| $41$ |
\( T^{6} - 3 T^{5} + \cdots + 14951 \)
|
| $43$ |
\( T^{6} + 20 T^{5} + \cdots - 461 \)
|
| $47$ |
\( T^{6} - 18 T^{5} + \cdots + 1723 \)
|
| $53$ |
\( T^{6} - 18 T^{5} + \cdots + 13483 \)
|
| $59$ |
\( T^{6} - 39 T^{5} + \cdots - 21797 \)
|
| $61$ |
\( T^{6} - 33 T^{5} + \cdots - 116117 \)
|
| $67$ |
\( T^{6} - 15 T^{5} + \cdots - 923 \)
|
| $71$ |
\( T^{6} + 25 T^{5} + \cdots + 26797 \)
|
| $73$ |
\( T^{6} - 17 T^{5} + \cdots + 349579 \)
|
| $79$ |
\( T^{6} + 28 T^{5} + \cdots - 20439 \)
|
| $83$ |
\( T^{6} + 3 T^{5} + \cdots + 85637 \)
|
| $89$ |
\( T^{6} - 28 T^{5} + \cdots + 879103 \)
|
| $97$ |
\( T^{6} - 36 T^{5} + \cdots - 1669669 \)
|
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