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: | \(1\) |
| 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: |
\( x^{7} - x^{6} - 9x^{5} + 7x^{4} + 22x^{3} - 12x^{2} - 9x - 1 \)
|
| 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
\( x^{7} - x^{6} - 9x^{5} + 7x^{4} + 22x^{3} - 12x^{2} - 9x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 8\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{6} + 2\nu^{5} + 7\nu^{4} - 12\nu^{3} - 13\nu^{2} + 16\nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 9\nu^{4} - 7\nu^{3} - 21\nu^{2} + 12\nu + 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 9\nu^{4} + 8\nu^{3} + 21\nu^{2} - 17\nu - 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{6} + 6\beta_{5} + 5\beta_{3} - 4\beta_{2} + 2\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{6} + 8\beta_{5} + \beta_{4} + 2\beta_{3} + 25\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 35\beta_{6} + 33\beta_{5} + \beta_{4} + 26\beta_{3} - 15\beta_{2} + 20\beta _1 + 36 \)
|
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 |
|
−2.44300 | 0 | 3.96824 | −0.800323 | 0 | 0.588123 | −4.80840 | 0 | 1.95519 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.31577 | 0 | 3.36278 | −4.44606 | 0 | 0.922010 | −3.15587 | 0 | 10.2960 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.877508 | 0 | −1.22998 | 2.15163 | 0 | 1.62983 | 2.83433 | 0 | −1.88807 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.659361 | 0 | −1.56524 | −1.36783 | 0 | −4.12178 | 2.35078 | 0 | 0.901896 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.518355 | 0 | −1.73131 | −0.575947 | 0 | 0.279802 | −1.93414 | 0 | −0.298545 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.71314 | 0 | 0.934849 | −3.26605 | 0 | 3.36585 | −1.82475 | 0 | −5.59520 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.06414 | 0 | 2.26067 | 0.304582 | 0 | −4.66384 | 0.538053 | 0 | 0.628700 | ||||||||||||||||||||||||||||||||||||||||
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.n | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1881.2.a.r | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1881.2.a.n | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1881.2.a.r | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1881))\):
|
\( T_{2}^{7} + 2T_{2}^{6} - 8T_{2}^{5} - 14T_{2}^{4} + 17T_{2}^{3} + 24T_{2}^{2} - 3T_{2} - 6 \)
|
|
\( T_{5}^{7} + 8T_{5}^{6} + 13T_{5}^{5} - 28T_{5}^{4} - 80T_{5}^{3} - 48T_{5}^{2} + 3T_{5} + 6 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots - 6 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + 8 T^{6} + \cdots + 6 \)
|
| $7$ |
\( T^{7} + 2 T^{6} + \cdots - 16 \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} + 7 T^{6} + \cdots - 97 \)
|
| $17$ |
\( T^{7} + 5 T^{6} + \cdots + 7827 \)
|
| $19$ |
\( (T - 1)^{7} \)
|
| $23$ |
\( T^{7} + 22 T^{6} + \cdots - 4332 \)
|
| $29$ |
\( T^{7} + 18 T^{6} + \cdots + 906 \)
|
| $31$ |
\( T^{7} - 74 T^{5} + \cdots + 2092 \)
|
| $37$ |
\( T^{7} - 6 T^{6} + \cdots + 78824 \)
|
| $41$ |
\( T^{7} + 6 T^{6} + \cdots - 67044 \)
|
| $43$ |
\( T^{7} - 8 T^{6} + \cdots - 196 \)
|
| $47$ |
\( T^{7} + 14 T^{6} + \cdots + 3558 \)
|
| $53$ |
\( T^{7} - T^{6} + \cdots - 60177 \)
|
| $59$ |
\( T^{7} + 23 T^{6} + \cdots + 1160193 \)
|
| $61$ |
\( T^{7} + 28 T^{6} + \cdots + 5255906 \)
|
| $67$ |
\( T^{7} - 14 T^{6} + \cdots + 463208 \)
|
| $71$ |
\( T^{7} + 31 T^{6} + \cdots + 1585779 \)
|
| $73$ |
\( T^{7} + 12 T^{6} + \cdots + 259184 \)
|
| $79$ |
\( T^{7} + 17 T^{6} + \cdots - 50681 \)
|
| $83$ |
\( T^{7} + 5 T^{6} + \cdots - 15399 \)
|
| $89$ |
\( T^{7} - 3 T^{6} + \cdots - 1646619 \)
|
| $97$ |
\( T^{7} - 12 T^{6} + \cdots + 511388 \)
|
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