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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 9x^{6} + 29x^{5} + 18x^{4} - 73x^{3} + 5x^{2} + 27x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 9x^{6} + 29x^{5} + 18x^{4} - 73x^{3} + 5x^{2} + 27x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 10\nu^{5} + 6\nu^{4} + 24\nu^{3} - 5\nu^{2} - \nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 6\nu^{3} + 24\nu^{2} - 5\nu - 3 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{7} + \nu^{6} + 11\nu^{5} - 7\nu^{4} - 33\nu^{3} + 10\nu^{2} + 19\nu - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 26\nu^{4} - 56\nu^{3} + 56\nu^{2} + 15\nu - 13 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{3} + 8\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} + 11\beta_{3} + 12\beta_{2} + 28\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -11\beta_{7} + 12\beta_{6} + 13\beta_{5} + 2\beta_{4} + 25\beta_{3} + 62\beta_{2} + 13\beta _1 + 83 \)
|
| \(\nu^{7}\) | \(=\) |
\( -15\beta_{7} + 26\beta_{6} + 17\beta_{5} + 24\beta_{4} + 99\beta_{3} + 115\beta_{2} + 168\beta _1 + 28 \)
|
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 |
|
−2.38053 | −1.00000 | 3.66691 | 1.00000 | 2.38053 | 4.88343 | −3.96812 | 1.00000 | −2.38053 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.88191 | −1.00000 | 1.54158 | 1.00000 | 1.88191 | −2.74904 | 0.862712 | 1.00000 | −1.88191 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.634036 | −1.00000 | −1.59800 | 1.00000 | 0.634036 | 3.80254 | 2.28126 | 1.00000 | −0.634036 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.112491 | −1.00000 | −1.98735 | 1.00000 | −0.112491 | 1.07041 | −0.448541 | 1.00000 | 0.112491 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.747869 | −1.00000 | −1.44069 | 1.00000 | −0.747869 | −1.99392 | −2.57319 | 1.00000 | 0.747869 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.07244 | −1.00000 | 2.29503 | 1.00000 | −2.07244 | −0.713527 | 0.611429 | 1.00000 | 2.07244 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.16282 | −1.00000 | 2.67780 | 1.00000 | −2.16282 | 3.84329 | 1.46595 | 1.00000 | 2.16282 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.80084 | −1.00000 | 5.84473 | 1.00000 | −2.80084 | 0.856821 | 10.7685 | 1.00000 | 2.80084 | |||||||||||||||||||||||||||||||||||||||||||
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.u | ✓ | 8 |
| 3.b | odd | 2 | 1 | 8415.2.a.bp | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2805.2.a.u | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 8415.2.a.bp | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2805))\):
|
\( T_{2}^{8} - 3T_{2}^{7} - 9T_{2}^{6} + 29T_{2}^{5} + 18T_{2}^{4} - 73T_{2}^{3} + 5T_{2}^{2} + 27T_{2} - 3 \)
|
|
\( T_{7}^{8} - 9T_{7}^{7} + 7T_{7}^{6} + 113T_{7}^{5} - 187T_{7}^{4} - 376T_{7}^{3} + 569T_{7}^{2} + 144T_{7} - 256 \)
|
|
\( T_{19}^{8} - 7T_{19}^{7} - 63T_{19}^{6} + 509T_{19}^{5} + 789T_{19}^{4} - 10120T_{19}^{3} + 3763T_{19}^{2} + 60032T_{19} - 70768 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 3 T^{7} + \cdots - 3 \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} - 9 T^{7} + \cdots - 256 \)
|
| $11$ |
\( (T - 1)^{8} \)
|
| $13$ |
\( T^{8} - 10 T^{7} + \cdots - 4156 \)
|
| $17$ |
\( (T - 1)^{8} \)
|
| $19$ |
\( T^{8} - 7 T^{7} + \cdots - 70768 \)
|
| $23$ |
\( T^{8} - 6 T^{7} + \cdots + 89088 \)
|
| $29$ |
\( T^{8} - 6 T^{7} + \cdots + 2224 \)
|
| $31$ |
\( T^{8} - 9 T^{7} + \cdots - 256 \)
|
| $37$ |
\( T^{8} - 3 T^{7} + \cdots - 15924 \)
|
| $41$ |
\( T^{8} + 16 T^{7} + \cdots - 787936 \)
|
| $43$ |
\( T^{8} - 30 T^{7} + \cdots + 1408896 \)
|
| $47$ |
\( T^{8} - 9 T^{7} + \cdots - 1882112 \)
|
| $53$ |
\( T^{8} + 2 T^{7} + \cdots + 6286368 \)
|
| $59$ |
\( T^{8} + 6 T^{7} + \cdots + 5567168 \)
|
| $61$ |
\( T^{8} - T^{7} + \cdots + 2683428 \)
|
| $67$ |
\( T^{8} - 34 T^{7} + \cdots - 4837072 \)
|
| $71$ |
\( T^{8} - 19 T^{7} + \cdots - 81152 \)
|
| $73$ |
\( T^{8} - 20 T^{7} + \cdots - 20016 \)
|
| $79$ |
\( T^{8} + 6 T^{7} + \cdots + 121718784 \)
|
| $83$ |
\( T^{8} - 23 T^{7} + \cdots + 275984 \)
|
| $89$ |
\( T^{8} + 16 T^{7} + \cdots - 31951344 \)
|
| $97$ |
\( T^{8} - 9 T^{7} + \cdots + 1160708 \)
|
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