Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2583,2,Mod(1,2583)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2583, 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("2583.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2583 = 3^{2} \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2583.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: | \(20.6253588421\) |
| 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: |
\( x^{8} - 4x^{7} - 6x^{6} + 28x^{5} + 16x^{4} - 58x^{3} - 28x^{2} + 26x + 10 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 6x^{6} + 28x^{5} + 16x^{4} - 58x^{3} - 28x^{2} + 26x + 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 8\nu^{3} + 13\nu^{2} + 5\nu - 7 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} + \nu^{5} + 21\nu^{4} - 18\nu^{3} - 15\nu^{2} + 19\nu - 1 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{6} + 6\nu^{5} + 13\nu^{4} - 28\nu^{3} - 29\nu^{2} + 20\nu + 11 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} - 5\nu^{5} + 25\nu^{4} + 8\nu^{3} - 41\nu^{2} - 6\nu + 10 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} - 2\nu^{5} + 33\nu^{4} - 9\nu^{3} - 66\nu^{2} + 13\nu + 29 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 3\beta_{2} + 5\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{7} - 4\beta_{6} - 3\beta_{5} - 2\beta_{3} + 13\beta_{2} + 11\beta _1 + 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( 18\beta_{7} - 19\beta_{6} - 15\beta_{5} + \beta_{4} - 11\beta_{3} + 44\beta_{2} + 40\beta _1 + 51 \)
|
| \(\nu^{6}\) | \(=\) |
\( 66\beta_{7} - 69\beta_{6} - 52\beta_{5} + 3\beta_{4} - 32\beta_{3} + 160\beta_{2} + 117\beta _1 + 189 \)
|
| \(\nu^{7}\) | \(=\) |
\( 246\beta_{7} - 260\beta_{6} - 200\beta_{5} + 17\beta_{4} - 125\beta_{3} + 552\beta_{2} + 400\beta _1 + 592 \)
|
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 |
|
−2.43957 | 0 | 3.95150 | 3.04347 | 0 | 1.00000 | −4.76083 | 0 | −7.42475 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.28120 | 0 | −0.358525 | −0.278215 | 0 | 1.00000 | 3.02174 | 0 | 0.356450 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.04374 | 0 | −0.910616 | −2.36717 | 0 | 1.00000 | 3.03791 | 0 | 2.47070 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.259377 | 0 | −1.93272 | 3.60445 | 0 | 1.00000 | −1.02006 | 0 | 0.934912 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.35175 | 0 | −0.172766 | −1.36361 | 0 | 1.00000 | −2.93704 | 0 | −1.84326 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.94000 | 0 | 1.76360 | 3.48399 | 0 | 1.00000 | −0.458617 | 0 | 6.75895 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.41978 | 0 | 3.85532 | 1.32904 | 0 | 1.00000 | 4.48946 | 0 | 3.21599 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.79360 | 0 | 5.80420 | 0.548043 | 0 | 1.00000 | 10.6274 | 0 | 1.53101 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(41\) | \(1\) |
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 | 2583.2.a.w | yes | 8 |
| 3.b | odd | 2 | 1 | 2583.2.a.v | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2583.2.a.v | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 2583.2.a.w | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2583))\):
|
\( T_{2}^{8} - 4T_{2}^{7} - 6T_{2}^{6} + 36T_{2}^{5} - 4T_{2}^{4} - 82T_{2}^{3} + 28T_{2}^{2} + 56T_{2} - 15 \)
|
|
\( T_{5}^{8} - 8T_{5}^{7} + 10T_{5}^{6} + 56T_{5}^{5} - 124T_{5}^{4} - 60T_{5}^{3} + 202T_{5}^{2} - 32T_{5} - 25 \)
|
|
\( T_{11}^{8} - 2T_{11}^{7} - 44T_{11}^{6} + 80T_{11}^{5} + 560T_{11}^{4} - 896T_{11}^{3} - 1920T_{11}^{2} + 2880T_{11} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 4 T^{7} + \cdots - 15 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 8 T^{7} + \cdots - 25 \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( T^{8} - 2 T^{7} + \cdots + 16 \)
|
| $13$ |
\( T^{8} + 2 T^{7} + \cdots - 19683 \)
|
| $17$ |
\( T^{8} - 26 T^{7} + \cdots + 36 \)
|
| $19$ |
\( T^{8} - 72 T^{6} + \cdots + 5508 \)
|
| $23$ |
\( T^{8} - 18 T^{7} + \cdots - 5431 \)
|
| $29$ |
\( T^{8} - 2 T^{7} + \cdots + 23391 \)
|
| $31$ |
\( T^{8} - 6 T^{7} + \cdots + 33444 \)
|
| $37$ |
\( T^{8} + 12 T^{7} + \cdots - 3875 \)
|
| $41$ |
\( (T + 1)^{8} \)
|
| $43$ |
\( T^{8} - 4 T^{7} + \cdots - 64756 \)
|
| $47$ |
\( T^{8} - 40 T^{7} + \cdots - 1741983 \)
|
| $53$ |
\( T^{8} - 12 T^{7} + \cdots - 1211601 \)
|
| $59$ |
\( T^{8} - 6 T^{7} + \cdots + 625052 \)
|
| $61$ |
\( T^{8} - 22 T^{7} + \cdots + 8468 \)
|
| $67$ |
\( T^{8} + 16 T^{7} + \cdots - 127213 \)
|
| $71$ |
\( T^{8} + 4 T^{7} + \cdots + 1722300 \)
|
| $73$ |
\( T^{8} + 28 T^{7} + \cdots - 976572 \)
|
| $79$ |
\( T^{8} - 2 T^{7} + \cdots - 4110025 \)
|
| $83$ |
\( T^{8} - 28 T^{7} + \cdots + 192832 \)
|
| $89$ |
\( T^{8} + 2 T^{7} + \cdots + 89136 \)
|
| $97$ |
\( T^{8} - 4 T^{7} + \cdots - 201023 \)
|
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