Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2805,2,Mod(1,2805)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2805, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2805.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2805 = 3 \cdot 5 \cdot 11 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2805.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: | \(22.3980377670\) |
| 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: |
\( x^{7} - 2x^{6} - 9x^{5} + 16x^{4} + 22x^{3} - 33x^{2} - 8x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{7} - 2x^{6} - 9x^{5} + 16x^{4} + 22x^{3} - 33x^{2} - 8x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 8\nu^{4} + 4\nu^{3} + 16\nu^{2} + \nu - 5 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 6\nu^{4} - 18\nu^{3} - 8\nu^{2} + 21\nu + 1 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 8\nu^{3} + 26\nu^{2} - 15\nu - 9 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{4} + 2\beta_{3} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 2\beta_{4} + 9\beta_{3} + 10\beta_{2} + 17\beta _1 + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( -9\beta_{6} + \beta_{5} + 12\beta_{4} + 21\beta_{3} + 46\beta_{2} + 85 \)
|
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 |
|
−2.15525 | 1.00000 | 2.64510 | −1.00000 | −2.15525 | −2.51408 | −1.39036 | 1.00000 | 2.15525 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.85657 | 1.00000 | 1.44685 | −1.00000 | −1.85657 | 0.736534 | 1.02696 | 1.00000 | 1.85657 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.595118 | 1.00000 | −1.64583 | −1.00000 | −0.595118 | 3.15683 | 2.16970 | 1.00000 | 0.595118 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.510037 | 1.00000 | −1.73986 | −1.00000 | 0.510037 | −4.75958 | −1.90747 | 1.00000 | −0.510037 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.29651 | 1.00000 | −0.319072 | −1.00000 | 1.29651 | 2.76110 | −3.00669 | 1.00000 | −1.29651 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.18700 | 1.00000 | 2.78296 | −1.00000 | 2.18700 | −1.57271 | 1.71233 | 1.00000 | −2.18700 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.61340 | 1.00000 | 4.82985 | −1.00000 | 2.61340 | 1.19190 | 7.39554 | 1.00000 | −2.61340 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 2805.2.a.q | ✓ | 7 |
| 3.b | odd | 2 | 1 | 8415.2.a.bm | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2805.2.a.q | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 8415.2.a.bm | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2805))\):
|
\( T_{2}^{7} - 2T_{2}^{6} - 9T_{2}^{5} + 16T_{2}^{4} + 22T_{2}^{3} - 33T_{2}^{2} - 8T_{2} + 9 \)
|
|
\( T_{7}^{7} + T_{7}^{6} - 25T_{7}^{5} - T_{7}^{4} + 157T_{7}^{3} - 52T_{7}^{2} - 235T_{7} + 144 \)
|
|
\( T_{19}^{7} - 9T_{19}^{6} - 59T_{19}^{5} + 595T_{19}^{4} + 265T_{19}^{3} - 6640T_{19}^{2} - 1645T_{19} + 15600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots + 9 \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( T^{7} + T^{6} + \cdots + 144 \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} + 2 T^{6} + \cdots - 30 \)
|
| $17$ |
\( (T - 1)^{7} \)
|
| $19$ |
\( T^{7} - 9 T^{6} + \cdots + 15600 \)
|
| $23$ |
\( T^{7} - 8 T^{6} + \cdots + 200 \)
|
| $29$ |
\( T^{7} - 12 T^{6} + \cdots + 216 \)
|
| $31$ |
\( T^{7} + 3 T^{6} + \cdots + 520 \)
|
| $37$ |
\( T^{7} - T^{6} + \cdots - 53594 \)
|
| $41$ |
\( T^{7} - 20 T^{6} + \cdots + 19600 \)
|
| $43$ |
\( T^{7} - 12 T^{6} + \cdots - 16384 \)
|
| $47$ |
\( T^{7} - T^{6} + \cdots + 32640 \)
|
| $53$ |
\( T^{7} - 6 T^{6} + \cdots - 240 \)
|
| $59$ |
\( T^{7} - 30 T^{6} + \cdots - 837440 \)
|
| $61$ |
\( T^{7} + 15 T^{6} + \cdots + 94 \)
|
| $67$ |
\( T^{7} - 4 T^{6} + \cdots - 1500 \)
|
| $71$ |
\( T^{7} - 9 T^{6} + \cdots + 32960 \)
|
| $73$ |
\( T^{7} - 10 T^{6} + \cdots - 615256 \)
|
| $79$ |
\( T^{7} - 2 T^{6} + \cdots + 69120 \)
|
| $83$ |
\( T^{7} - 13 T^{6} + \cdots + 82620 \)
|
| $89$ |
\( T^{7} - 20 T^{6} + \cdots - 38312 \)
|
| $97$ |
\( T^{7} + 13 T^{6} + \cdots - 50298 \)
|
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