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: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 22x^{3} - 5x^{2} - 7x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 22x^{3} - 5x^{2} - 7x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 7\nu^{2} - 5\nu + 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 10\nu^{4} + 7\nu^{3} + 21\nu^{2} + 2\nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 10\nu^{4} - 7\nu^{3} - 21\nu^{2} + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 10\nu^{4} + 7\nu^{3} + 22\nu^{2} + \nu - 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 10\nu^{4} + 8\nu^{3} + 22\nu^{2} - 4\nu - 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - \nu^{5} - 21\nu^{4} + 6\nu^{3} + 51\nu^{2} + 11\nu - 12 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{4} + \beta_{3} - \beta_{2} + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{5} - 2\beta_{4} + 5\beta_{3} + 5\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{5} + 16\beta_{4} + 7\beta_{3} - 7\beta_{2} + 2\beta _1 + 36 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{6} + 14\beta_{5} - 18\beta_{4} + 31\beta_{3} + 31\beta_{2} + 2\beta _1 - 6 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2\beta_{6} - 20\beta_{5} + 114\beta_{4} + 43\beta_{3} - 53\beta_{2} + 22\beta _1 + 234 ) / 2 \)
|
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 |
|
−2.42067 | 0 | 3.85962 | −2.85962 | 0 | −2.49298 | −4.50152 | 0 | 6.92219 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.03375 | 0 | 2.13614 | −1.13614 | 0 | 4.00620 | −0.276881 | 0 | 2.31063 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.42661 | 0 | 0.0352187 | 0.964781 | 0 | 2.45740 | 2.80298 | 0 | −1.37637 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.0204073 | 0 | −1.99958 | 2.99958 | 0 | −1.53276 | −0.0816207 | 0 | 0.0612134 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.51425 | 0 | 0.292958 | 0.707042 | 0 | 3.59927 | −2.58489 | 0 | 1.07064 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.83393 | 0 | 1.36331 | −0.363312 | 0 | −2.69764 | −1.16764 | 0 | −0.666290 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.51244 | 0 | 4.31233 | −3.31233 | 0 | −0.339486 | 5.80958 | 0 | −8.32202 | ||||||||||||||||||||||||||||||||||||||||
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.f | 7 | |
| 3.b | odd | 2 | 1 | 717.2.a.e | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 717.2.a.e | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 2151.2.a.f | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2151))\):
|
\( T_{2}^{7} - 12T_{2}^{5} + 44T_{2}^{3} - T_{2}^{2} - 49T_{2} + 1 \)
|
|
\( T_{5}^{7} + 3T_{5}^{6} - 11T_{5}^{5} - 31T_{5}^{4} + 19T_{5}^{3} + 38T_{5}^{2} - 12T_{5} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 12 T^{5} + \cdots + 1 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + 3 T^{6} + \cdots - 8 \)
|
| $7$ |
\( T^{7} - 3 T^{6} + \cdots - 124 \)
|
| $11$ |
\( T^{7} + 11 T^{6} + \cdots - 592 \)
|
| $13$ |
\( T^{7} + 5 T^{6} + \cdots + 152 \)
|
| $17$ |
\( T^{7} + 11 T^{6} + \cdots + 6376 \)
|
| $19$ |
\( T^{7} + 8 T^{6} + \cdots - 620 \)
|
| $23$ |
\( T^{7} + 26 T^{6} + \cdots - 512 \)
|
| $29$ |
\( T^{7} + 6 T^{6} + \cdots - 73880 \)
|
| $31$ |
\( T^{7} + 2 T^{6} + \cdots - 128 \)
|
| $37$ |
\( T^{7} - 12 T^{6} + \cdots + 11608 \)
|
| $41$ |
\( T^{7} + 26 T^{6} + \cdots - 9208 \)
|
| $43$ |
\( T^{7} - 10 T^{6} + \cdots + 2564 \)
|
| $47$ |
\( T^{7} + 7 T^{6} + \cdots - 464896 \)
|
| $53$ |
\( T^{7} + 2 T^{6} + \cdots - 928 \)
|
| $59$ |
\( T^{7} + 6 T^{6} + \cdots + 24320 \)
|
| $61$ |
\( T^{7} + 8 T^{6} + \cdots - 47728 \)
|
| $67$ |
\( T^{7} - 24 T^{6} + \cdots - 3733504 \)
|
| $71$ |
\( T^{7} - 25 T^{6} + \cdots + 37696 \)
|
| $73$ |
\( T^{7} + 16 T^{6} + \cdots + 3496 \)
|
| $79$ |
\( T^{7} - 19 T^{6} + \cdots + 115900 \)
|
| $83$ |
\( T^{7} + 37 T^{6} + \cdots + 69584 \)
|
| $89$ |
\( T^{7} + 29 T^{6} + \cdots + 3800 \)
|
| $97$ |
\( T^{7} + 12 T^{6} + \cdots - 335144 \)
|
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