Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1005,2,Mod(1,1005)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1005, 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("1005.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1005 = 3 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1005.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: | \(8.02496540314\) |
| 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: |
\( x^{8} - 3x^{7} - 9x^{6} + 30x^{5} + 14x^{4} - 68x^{3} - 2x^{2} + 37x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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
\( x^{8} - 3x^{7} - 9x^{6} + 30x^{5} + 14x^{4} - 68x^{3} - 2x^{2} + 37x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 19\nu^{4} + 13\nu^{3} - 35\nu^{2} + 7\nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} + 11\nu^{5} - 17\nu^{4} - 31\nu^{3} + 19\nu^{2} + 23\nu + 12 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - 11\nu^{5} + 19\nu^{4} + 31\nu^{3} - 35\nu^{2} - 23\nu + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 10\nu^{5} - 18\nu^{4} - 23\nu^{3} + 28\nu^{2} + 14\nu \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 12\nu^{5} + 8\nu^{4} + 41\nu^{3} - 5\nu^{2} - 42\nu - 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 8\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{6} - \beta_{5} + 9\beta_{4} - 8\beta_{3} + 9\beta_{2} + 39\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 8\beta_{5} + 10\beta_{4} + 2\beta_{3} + 57\beta_{2} - 2\beta _1 + 103 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{7} - 66\beta_{6} - 12\beta_{5} + 69\beta_{4} - 53\beta_{3} + 65\beta_{2} + 262\beta _1 - 8 \)
|
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 |
|
−2.66043 | −1.00000 | 5.07789 | −1.00000 | 2.66043 | 1.09595 | −8.18850 | 1.00000 | 2.66043 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.48832 | −1.00000 | 4.19171 | −1.00000 | 2.48832 | −2.54202 | −5.45368 | 1.00000 | 2.48832 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.49428 | −1.00000 | 0.232860 | −1.00000 | 1.49428 | 4.86123 | 2.64059 | 1.00000 | 1.49428 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.35736 | −1.00000 | −0.157569 | −1.00000 | 1.35736 | −3.58687 | 2.92860 | 1.00000 | 1.35736 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.238687 | −1.00000 | −1.94303 | −1.00000 | −0.238687 | −4.78250 | −0.941150 | 1.00000 | −0.238687 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.640143 | −1.00000 | −1.59022 | −1.00000 | −0.640143 | 2.60243 | −2.29825 | 1.00000 | −0.640143 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.47139 | −1.00000 | 0.164991 | −1.00000 | −1.47139 | 1.54470 | −2.70002 | 1.00000 | −1.47139 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.65016 | −1.00000 | 5.02336 | −1.00000 | −2.65016 | −2.19291 | 8.01240 | 1.00000 | −2.65016 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(67\) | \(-1\) |
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 | 1005.2.a.j | ✓ | 8 |
| 3.b | odd | 2 | 1 | 3015.2.a.n | 8 | ||
| 5.b | even | 2 | 1 | 5025.2.a.be | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1005.2.a.j | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 3015.2.a.n | 8 | 3.b | odd | 2 | 1 | ||
| 5025.2.a.be | 8 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 3T_{2}^{7} - 9T_{2}^{6} - 30T_{2}^{5} + 14T_{2}^{4} + 68T_{2}^{3} - 2T_{2}^{2} - 37T_{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 3 T^{7} + \cdots + 8 \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} + 3 T^{7} + \cdots + 2048 \)
|
| $11$ |
\( T^{8} + T^{7} + \cdots - 640 \)
|
| $13$ |
\( T^{8} - T^{7} + \cdots + 7760 \)
|
| $17$ |
\( T^{8} + 7 T^{7} + \cdots - 2200 \)
|
| $19$ |
\( T^{8} - 28 T^{7} + \cdots - 36032 \)
|
| $23$ |
\( T^{8} + 11 T^{7} + \cdots + 2368 \)
|
| $29$ |
\( T^{8} - 2 T^{7} + \cdots - 177064 \)
|
| $31$ |
\( T^{8} - 15 T^{7} + \cdots + 1728 \)
|
| $37$ |
\( T^{8} + 3 T^{7} + \cdots + 5857160 \)
|
| $41$ |
\( T^{8} - 5 T^{7} + \cdots - 338224 \)
|
| $43$ |
\( T^{8} - 15 T^{7} + \cdots - 21507968 \)
|
| $47$ |
\( T^{8} + 6 T^{7} + \cdots - 937024 \)
|
| $53$ |
\( T^{8} + 19 T^{7} + \cdots - 960352 \)
|
| $59$ |
\( T^{8} - 18 T^{7} + \cdots - 1444720 \)
|
| $61$ |
\( T^{8} - 13 T^{7} + \cdots + 164480 \)
|
| $67$ |
\( (T - 1)^{8} \)
|
| $71$ |
\( T^{8} - 4 T^{7} + \cdots - 30976 \)
|
| $73$ |
\( T^{8} - 11 T^{7} + \cdots + 1257064 \)
|
| $79$ |
\( T^{8} - 33 T^{7} + \cdots - 3520 \)
|
| $83$ |
\( T^{8} - 44 T^{7} + \cdots - 20268032 \)
|
| $89$ |
\( T^{8} - 5 T^{7} + \cdots - 2382808 \)
|
| $97$ |
\( T^{8} - 7 T^{7} + \cdots + 15706064 \)
|
| show more | |
| show less |