Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2151,2,Mod(1,2151)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2151, 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("2151.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2151 = 3^{2} \cdot 239 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2151.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: | \(17.1758214748\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.2585660609.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 4x^{6} + 15x^{5} + x^{4} - 19x^{3} + 6x^{2} + 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 717) |
| 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
\( x^{8} - 3x^{7} - 4x^{6} + 15x^{5} + x^{4} - 19x^{3} + 6x^{2} + 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{7} + 3\nu^{6} + 4\nu^{5} - 15\nu^{4} - \nu^{3} + 18\nu^{2} - 5\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 6\nu^{5} + 9\nu^{4} + 10\nu^{3} - 9\nu^{2} - 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{7} + 3\nu^{6} + 5\nu^{5} - 16\nu^{4} - 8\nu^{3} + 22\nu^{2} + 5\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{7} - 5\nu^{6} - 10\nu^{5} + 24\nu^{4} + 12\nu^{3} - 29\nu^{2} - 2\nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{7} - 5\nu^{6} - 11\nu^{5} + 26\nu^{4} + 16\nu^{3} - 35\nu^{2} - 4\nu + 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( -3\nu^{7} + 8\nu^{6} + 14\nu^{5} - 38\nu^{4} - 15\nu^{3} + 45\nu^{2} - 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( -3\nu^{7} + 7\nu^{6} + 16\nu^{5} - 33\nu^{4} - 22\nu^{3} + 38\nu^{2} + 4\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{4} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta _1 + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{7} + 2\beta_{6} - 2\beta_{5} - \beta_{4} - 2\beta_{3} - 5\beta_{2} + 4 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -9\beta_{7} + 8\beta_{6} - 6\beta_{5} + 3\beta_{4} - 6\beta_{3} - 7\beta_{2} - 4\beta _1 + 22 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( -15\beta_{7} + 9\beta_{6} - 8\beta_{5} + 3\beta_{4} - 7\beta_{3} - 16\beta_{2} - \beta _1 + 18 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -69\beta_{7} + 57\beta_{6} - 41\beta_{5} + 30\beta_{4} - 37\beta_{3} - 68\beta_{2} - 17\beta _1 + 125 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -200\beta_{7} + 139\beta_{6} - 113\beta_{5} + 75\beta_{4} - 93\beta_{3} - 217\beta_{2} - 19\beta _1 + 273 ) / 2 \)
|
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 |
|
−1.77982 | 0 | 1.16777 | 3.42938 | 0 | −4.73145 | 1.48122 | 0 | −6.10369 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.04938 | 0 | −0.898808 | 3.87926 | 0 | 2.44573 | 3.04194 | 0 | −4.07080 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.777975 | 0 | −1.39476 | −0.966236 | 0 | −3.17897 | 2.64103 | 0 | 0.751707 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.711649 | 0 | −1.49356 | −0.309852 | 0 | −4.19834 | −2.48618 | 0 | −0.220506 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.846183 | 0 | −1.28397 | 1.31908 | 0 | 0.998785 | −2.77884 | 0 | 1.11619 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.92536 | 0 | 1.70701 | 3.65027 | 0 | 3.22894 | −0.564105 | 0 | 7.02808 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.37665 | 0 | 3.64845 | −0.0257644 | 0 | 0.278803 | 3.91777 | 0 | −0.0612329 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.74734 | 0 | 5.54786 | 2.02387 | 0 | −1.84350 | 9.74717 | 0 | 5.56025 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(239\) | \(1\) |
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 | 2151.2.a.g | 8 | |
| 3.b | odd | 2 | 1 | 717.2.a.f | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 717.2.a.f | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 2151.2.a.g | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):
|
\( T_{2}^{8} - 5T_{2}^{7} + T_{2}^{6} + 25T_{2}^{5} - 22T_{2}^{4} - 33T_{2}^{3} + 31T_{2}^{2} + 12T_{2} - 11 \)
|
|
\( T_{5}^{8} - 13T_{5}^{7} + 61T_{5}^{6} - 113T_{5}^{5} + 20T_{5}^{4} + 151T_{5}^{3} - 81T_{5}^{2} - 41T_{5} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 5 T^{7} + \cdots - 11 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 13 T^{7} + \cdots - 1 \)
|
| $7$ |
\( T^{8} + 7 T^{7} + \cdots + 256 \)
|
| $11$ |
\( T^{8} - 19 T^{7} + \cdots + 125 \)
|
| $13$ |
\( T^{8} + 3 T^{7} + \cdots + 8912 \)
|
| $17$ |
\( T^{8} - 13 T^{7} + \cdots - 76013 \)
|
| $19$ |
\( T^{8} + 6 T^{7} + \cdots - 69776 \)
|
| $23$ |
\( T^{8} - 18 T^{7} + \cdots - 32528 \)
|
| $29$ |
\( T^{8} - 10 T^{7} + \cdots - 85537 \)
|
| $31$ |
\( T^{8} + 2 T^{7} + \cdots - 88763 \)
|
| $37$ |
\( T^{8} + 8 T^{7} + \cdots - 16 \)
|
| $41$ |
\( T^{8} - 22 T^{7} + \cdots + 258608 \)
|
| $43$ |
\( T^{8} + 24 T^{7} + \cdots + 63184 \)
|
| $47$ |
\( T^{8} - 17 T^{7} + \cdots - 1377328 \)
|
| $53$ |
\( T^{8} - 382 T^{6} + \cdots - 23116976 \)
|
| $59$ |
\( T^{8} - 24 T^{7} + \cdots + 1041776 \)
|
| $61$ |
\( T^{8} - 10 T^{7} + \cdots + 945611 \)
|
| $67$ |
\( T^{8} + 48 T^{7} + \cdots + 1011184 \)
|
| $71$ |
\( T^{8} - 17 T^{7} + \cdots - 891136 \)
|
| $73$ |
\( T^{8} - 2 T^{7} + \cdots - 495952 \)
|
| $79$ |
\( T^{8} - 17 T^{7} + \cdots - 2003312 \)
|
| $83$ |
\( T^{8} - 37 T^{7} + \cdots - 20237 \)
|
| $89$ |
\( T^{8} - 41 T^{7} + \cdots - 2989312 \)
|
| $97$ |
\( T^{8} + 20 T^{7} + \cdots + 340592 \)
|
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