Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1881,2,Mod(1,1881)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1881, 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("1881.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1881 = 3^{2} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1881.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: | \(15.0198606202\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - 2x^{6} - 8x^{5} + 14x^{4} + 17x^{3} - 24x^{2} - 7x + 2 \)
|
| 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
\( x^{7} - 2x^{6} - 8x^{5} + 14x^{4} + 17x^{3} - 24x^{2} - 7x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 7\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 3\nu^{2} + 10\nu + 1 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 8\nu^{4} + 5\nu^{3} + 15\nu^{2} - 6\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 9\beta_{3} + 11\beta_{2} + 19\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 9\beta_{4} + 20\beta_{3} + 47\beta_{2} + 13\beta _1 + 75 \)
|
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 |
|
−2.10677 | 0 | 2.43847 | 1.80464 | 0 | −0.302128 | −0.923745 | 0 | −3.80195 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.65416 | 0 | 0.736234 | −0.687408 | 0 | −2.34156 | 2.09047 | 0 | 1.13708 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.418452 | 0 | −1.82490 | 2.54380 | 0 | 2.12534 | 1.60054 | 0 | −1.06446 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.185421 | 0 | −1.96562 | −2.57457 | 0 | −2.38915 | −0.735311 | 0 | −0.477381 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.57695 | 0 | 0.486784 | 1.60147 | 0 | 3.17843 | −2.38627 | 0 | 2.52545 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.77601 | 0 | 1.15422 | −3.21402 | 0 | −1.43801 | −1.50211 | 0 | −5.70814 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.64099 | 0 | 4.97481 | 0.526092 | 0 | 3.16708 | 7.85644 | 0 | 1.38940 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(11\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 1881.2.a.q | yes | 7 |
| 3.b | odd | 2 | 1 | 1881.2.a.o | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1881.2.a.o | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 1881.2.a.q | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1881))\):
|
\( T_{2}^{7} - 2T_{2}^{6} - 8T_{2}^{5} + 14T_{2}^{4} + 17T_{2}^{3} - 24T_{2}^{2} - 7T_{2} + 2 \)
|
|
\( T_{5}^{7} - 15T_{5}^{5} + 8T_{5}^{4} + 60T_{5}^{3} - 56T_{5}^{2} - 29T_{5} + 22 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots + 2 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - 15 T^{5} + \cdots + 22 \)
|
| $7$ |
\( T^{7} - 2 T^{6} + \cdots - 52 \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} - 3 T^{6} + \cdots + 197 \)
|
| $17$ |
\( T^{7} - 7 T^{6} + \cdots - 2021 \)
|
| $19$ |
\( (T + 1)^{7} \)
|
| $23$ |
\( T^{7} - 18 T^{6} + \cdots + 144 \)
|
| $29$ |
\( T^{7} - 14 T^{6} + \cdots + 5902 \)
|
| $31$ |
\( T^{7} + 8 T^{6} + \cdots + 33808 \)
|
| $37$ |
\( T^{7} + 2 T^{6} + \cdots - 12808 \)
|
| $41$ |
\( T^{7} - 18 T^{6} + \cdots - 11464 \)
|
| $43$ |
\( T^{7} + 4 T^{6} + \cdots - 5888 \)
|
| $47$ |
\( T^{7} - 10 T^{6} + \cdots - 692866 \)
|
| $53$ |
\( T^{7} - 23 T^{6} + \cdots - 437171 \)
|
| $59$ |
\( T^{7} - 15 T^{6} + \cdots - 32573 \)
|
| $61$ |
\( T^{7} - 12 T^{6} + \cdots + 4414 \)
|
| $67$ |
\( T^{7} + 6 T^{6} + \cdots - 288404 \)
|
| $71$ |
\( T^{7} - 39 T^{6} + \cdots - 31927 \)
|
| $73$ |
\( T^{7} + 4 T^{6} + \cdots - 33884 \)
|
| $79$ |
\( T^{7} - 21 T^{6} + \cdots + 57821 \)
|
| $83$ |
\( T^{7} - 15 T^{6} + \cdots + 100073 \)
|
| $89$ |
\( T^{7} - 13 T^{6} + \cdots + 244951 \)
|
| $97$ |
\( T^{7} + 4 T^{6} + \cdots - 3978512 \)
|
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