Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5054,2,Mod(1,5054)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5054, 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("5054.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5054 = 2 \cdot 7 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5054.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: | \(40.3563931816\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.19520000000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \)
|
| 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
\( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 8\nu + 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 4\nu^{3} + 4\nu^{2} - 18\nu - 4 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 10\nu^{2} + 16\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 8\nu^{2} + 18\nu + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - 12\nu^{5} + 17\nu^{4} + 48\nu^{3} - 32\nu^{2} - 64\nu - 16 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{7} + 5\nu^{6} + 22\nu^{5} - 43\nu^{4} - 84\nu^{3} + 86\nu^{2} + 116\nu + 24 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{5} + 8\beta_{4} + 2\beta_{3} + 4\beta_{2} + 10\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -14\beta_{5} + 16\beta_{4} + 12\beta_{3} + 16\beta_{2} + 42\beta _1 + 20 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + 4\beta_{6} - 66\beta_{5} + 70\beta_{4} + 30\beta_{3} + 56\beta_{2} + 92\beta _1 + 102 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4\beta_{7} + 10\beta_{6} - 148\beta_{5} + 180\beta_{4} + 122\beta_{3} + 188\beta_{2} + 326\beta _1 + 188 \)
|
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 | −2.13415 | 1.00000 | −1.04552 | −2.13415 | −1.00000 | 1.00000 | 1.55458 | −1.04552 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.04815 | 1.00000 | 1.23629 | −2.04815 | −1.00000 | 1.00000 | 1.19490 | 1.23629 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.578669 | 1.00000 | 1.36891 | −0.578669 | −1.00000 | 1.00000 | −2.66514 | 1.36891 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.0295377 | 1.00000 | −2.02989 | −0.0295377 | −1.00000 | 1.00000 | −2.99913 | −2.02989 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.717767 | 1.00000 | 2.91631 | 0.717767 | −1.00000 | 1.00000 | −2.48481 | 2.91631 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.30521 | 1.00000 | 3.15124 | 2.30521 | −1.00000 | 1.00000 | 2.31400 | 3.15124 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.40760 | 1.00000 | 1.76144 | 2.40760 | −1.00000 | 1.00000 | 2.79656 | 1.76144 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 3.35992 | 1.00000 | −1.35877 | 3.35992 | −1.00000 | 1.00000 | 8.28904 | −1.35877 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(19\) | \(1\) |
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 | 5054.2.a.bi | yes | 8 |
| 19.b | odd | 2 | 1 | 5054.2.a.bf | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5054.2.a.bf | ✓ | 8 | 19.b | odd | 2 | 1 | |
| 5054.2.a.bi | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):
|
\( T_{3}^{8} - 4T_{3}^{7} - 8T_{3}^{6} + 38T_{3}^{5} + 15T_{3}^{4} - 98T_{3}^{3} + 2T_{3}^{2} + 34T_{3} + 1 \)
|
|
\( T_{5}^{8} - 6T_{5}^{7} + 2T_{5}^{6} + 42T_{5}^{5} - 50T_{5}^{4} - 82T_{5}^{3} + 122T_{5}^{2} + 46T_{5} - 79 \)
|
|
\( T_{13}^{8} - 6T_{13}^{7} - 38T_{13}^{6} + 182T_{13}^{5} + 355T_{13}^{4} - 422T_{13}^{3} - 888T_{13}^{2} - 144T_{13} + 181 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 1 \)
|
| $5$ |
\( T^{8} - 6 T^{7} + \cdots - 79 \)
|
| $7$ |
\( (T + 1)^{8} \)
|
| $11$ |
\( T^{8} - 4 T^{7} + \cdots - 419 \)
|
| $13$ |
\( T^{8} - 6 T^{7} + \cdots + 181 \)
|
| $17$ |
\( T^{8} - 2 T^{7} + \cdots - 779 \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} - 20 T^{7} + \cdots + 1525 \)
|
| $29$ |
\( T^{8} - 16 T^{7} + \cdots - 7619 \)
|
| $31$ |
\( T^{8} - 22 T^{7} + \cdots - 919 \)
|
| $37$ |
\( T^{8} + 12 T^{7} + \cdots - 410699 \)
|
| $41$ |
\( T^{8} - 36 T^{7} + \cdots + 515341 \)
|
| $43$ |
\( T^{8} - 190 T^{6} + \cdots + 3305 \)
|
| $47$ |
\( T^{8} - 6 T^{7} + \cdots + 121 \)
|
| $53$ |
\( T^{8} - 16 T^{7} + \cdots - 24979 \)
|
| $59$ |
\( T^{8} - 14 T^{7} + \cdots - 3744859 \)
|
| $61$ |
\( T^{8} - 10 T^{7} + \cdots - 411095 \)
|
| $67$ |
\( T^{8} + 20 T^{7} + \cdots - 4692995 \)
|
| $71$ |
\( T^{8} - 240 T^{6} + \cdots + 14480 \)
|
| $73$ |
\( T^{8} + 18 T^{7} + \cdots - 58519 \)
|
| $79$ |
\( T^{8} - 4 T^{7} + \cdots + 1204961 \)
|
| $83$ |
\( T^{8} - 36 T^{7} + \cdots + 24520336 \)
|
| $89$ |
\( T^{8} - 18 T^{7} + \cdots - 122399 \)
|
| $97$ |
\( T^{8} - 26 T^{7} + \cdots - 7416859 \)
|
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