Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4842,2,Mod(1,4842)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4842, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4842.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4842 = 2 \cdot 3^{2} \cdot 269 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4842.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: | \(38.6635646587\) |
| 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: |
\( x^{8} - x^{7} - 21x^{6} + 3x^{5} + 135x^{4} + 76x^{3} - 180x^{2} - 110x + 41 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1614) |
| 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
\( x^{8} - x^{7} - 21x^{6} + 3x^{5} + 135x^{4} + 76x^{3} - 180x^{2} - 110x + 41 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} + 17\nu^{5} - 22\nu^{4} - 87\nu^{3} + 49\nu^{2} + 91\nu - 31 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - 19\nu^{5} + 24\nu^{4} + 109\nu^{3} - 55\nu^{2} - 121\nu + 43 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 20\nu^{5} - 36\nu^{4} - 129\nu^{3} + 64\nu^{2} + 147\nu - 34 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 6\nu^{6} + 53\nu^{5} - 66\nu^{4} - 287\nu^{3} + 135\nu^{2} + 321\nu - 87 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} + 8\nu^{6} + 55\nu^{5} - 94\nu^{4} - 323\nu^{3} + 197\nu^{2} + 365\nu - 125 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 18\nu^{5} - 22\nu^{4} - 99\nu^{3} + 42\nu^{2} + 107\nu - 28 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} + 9\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 13\beta_{7} + 11\beta_{6} - 13\beta_{5} - 11\beta_{4} + \beta_{3} - 11\beta_{2} + 18\beta _1 + 46 \)
|
| \(\nu^{5}\) | \(=\) |
\( 32\beta_{7} + 19\beta_{6} - 43\beta_{5} - 19\beta_{4} - 9\beta_{2} + 99\beta _1 + 92 \)
|
| \(\nu^{6}\) | \(=\) |
\( 155\beta_{7} + 124\beta_{6} - 164\beta_{5} - 122\beta_{4} + 14\beta_{3} - 114\beta_{2} + 262\beta _1 + 506 \)
|
| \(\nu^{7}\) | \(=\) |
\( 443\beta_{7} + 291\beta_{6} - 561\beta_{5} - 287\beta_{4} + 6\beta_{3} - 190\beta_{2} + 1168\beta _1 + 1343 \)
|
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 |
|
−1.00000 | 0 | 1.00000 | −3.42609 | 0 | −1.66749 | −1.00000 | 0 | 3.42609 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −2.70217 | 0 | −0.273844 | −1.00000 | 0 | 2.70217 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | −2.47011 | 0 | 2.18495 | −1.00000 | 0 | 2.47011 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | −2.07331 | 0 | 1.27215 | −1.00000 | 0 | 2.07331 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0 | 1.00000 | −0.489622 | 0 | 4.55820 | −1.00000 | 0 | 0.489622 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0 | 1.00000 | 0.956644 | 0 | −1.43206 | −1.00000 | 0 | −0.956644 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 0 | 1.00000 | 2.82271 | 0 | 4.56941 | −1.00000 | 0 | −2.82271 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 0 | 1.00000 | 3.38195 | 0 | −0.211316 | −1.00000 | 0 | −3.38195 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(269\) | \( +1 \) |
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 | 4842.2.a.p | 8 | |
| 3.b | odd | 2 | 1 | 1614.2.a.j | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1614.2.a.j | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 4842.2.a.p | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4842))\):
|
\( T_{5}^{8} + 4T_{5}^{7} - 17T_{5}^{6} - 82T_{5}^{5} + 42T_{5}^{4} + 450T_{5}^{3} + 265T_{5}^{2} - 402T_{5} - 212 \)
|
|
\( T_{7}^{8} - 9T_{7}^{7} + 14T_{7}^{6} + 52T_{7}^{5} - 90T_{7}^{4} - 105T_{7}^{3} + 106T_{7}^{2} + 64T_{7} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 4 T^{7} + \cdots - 212 \)
|
| $7$ |
\( T^{8} - 9 T^{7} + \cdots + 8 \)
|
| $11$ |
\( T^{8} + 2 T^{7} + \cdots + 4423 \)
|
| $13$ |
\( T^{8} - 9 T^{7} + \cdots + 32 \)
|
| $17$ |
\( T^{8} - 2 T^{7} + \cdots + 19628 \)
|
| $19$ |
\( T^{8} - 12 T^{7} + \cdots - 644 \)
|
| $23$ |
\( T^{8} + 6 T^{7} + \cdots - 11296 \)
|
| $29$ |
\( T^{8} + T^{7} + \cdots + 424 \)
|
| $31$ |
\( T^{8} - 8 T^{7} + \cdots + 1688 \)
|
| $37$ |
\( T^{8} - 10 T^{7} + \cdots - 44512 \)
|
| $41$ |
\( T^{8} - T^{7} + \cdots + 3832 \)
|
| $43$ |
\( T^{8} - 8 T^{7} + \cdots + 1288 \)
|
| $47$ |
\( T^{8} + 2 T^{7} + \cdots - 165728 \)
|
| $53$ |
\( T^{8} - 9 T^{7} + \cdots + 3392944 \)
|
| $59$ |
\( T^{8} - 3 T^{7} + \cdots - 570752 \)
|
| $61$ |
\( T^{8} - 397 T^{6} + \cdots - 2592512 \)
|
| $67$ |
\( T^{8} - 6 T^{7} + \cdots + 24929728 \)
|
| $71$ |
\( T^{8} - 17 T^{7} + \cdots - 61627 \)
|
| $73$ |
\( T^{8} + T^{7} + \cdots + 4667021 \)
|
| $79$ |
\( T^{8} - T^{7} + \cdots + 102452 \)
|
| $83$ |
\( T^{8} - 17 T^{7} + \cdots + 3780448 \)
|
| $89$ |
\( T^{8} - 4 T^{7} + \cdots + 119672 \)
|
| $97$ |
\( T^{8} - 13 T^{7} + \cdots - 20041 \)
|
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