Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2672,2,Mod(1,2672)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2672, 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("2672.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2672 = 2^{4} \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2672.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: | \(21.3360274201\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - x^{6} - 9x^{5} + 5x^{4} + 25x^{3} - 4x^{2} - 17x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1336) |
| 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
\( x^{7} - x^{6} - 9x^{5} + 5x^{4} + 25x^{3} - 4x^{2} - 17x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 3\nu^{2} + 4\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 9\nu^{3} + 11\nu^{2} - 7\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 7\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 6\beta_{3} + 7\beta_{2} + 25\beta _1 + 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} + 8\beta_{4} + 9\beta_{3} + 24\beta_{2} + 43\beta _1 + 63 \)
|
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 |
|
0 | −1.41051 | 0 | 0.329231 | 0 | 2.82300 | 0 | −1.01045 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.19297 | 0 | −2.67228 | 0 | −2.81671 | 0 | −1.57681 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.0395540 | 0 | 0.949625 | 0 | −1.23977 | 0 | −2.99844 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.941682 | 0 | −2.86037 | 0 | −1.50273 | 0 | −2.11324 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.07427 | 0 | 3.79451 | 0 | 2.17413 | 0 | 1.30261 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.71785 | 0 | 2.12216 | 0 | −1.38168 | 0 | 4.38672 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.83013 | 0 | −1.66287 | 0 | 2.94376 | 0 | 5.00961 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(167\) | \(-1\) |
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 | 2672.2.a.l | 7 | |
| 4.b | odd | 2 | 1 | 1336.2.a.b | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1336.2.a.b | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 2672.2.a.l | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2672))\):
|
\( T_{3}^{7} - 6T_{3}^{6} + 6T_{3}^{5} + 20T_{3}^{4} - 30T_{3}^{3} - 17T_{3}^{2} + 26T_{3} - 1 \)
|
|
\( T_{7}^{7} - T_{7}^{6} - 17T_{7}^{5} + 8T_{7}^{4} + 95T_{7}^{3} + 16T_{7}^{2} - 183T_{7} - 131 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 6 T^{6} + \cdots - 1 \)
|
| $5$ |
\( T^{7} - 19 T^{5} + \cdots + 32 \)
|
| $7$ |
\( T^{7} - T^{6} + \cdots - 131 \)
|
| $11$ |
\( T^{7} - 6 T^{6} + \cdots - 41 \)
|
| $13$ |
\( T^{7} + 2 T^{6} + \cdots - 800 \)
|
| $17$ |
\( T^{7} + 9 T^{6} + \cdots - 32 \)
|
| $19$ |
\( T^{7} - 3 T^{6} + \cdots + 17659 \)
|
| $23$ |
\( T^{7} - 10 T^{6} + \cdots - 4768 \)
|
| $29$ |
\( T^{7} + 5 T^{6} + \cdots - 25 \)
|
| $31$ |
\( T^{7} - 21 T^{6} + \cdots - 1447 \)
|
| $37$ |
\( T^{7} - 19 T^{6} + \cdots - 1024 \)
|
| $41$ |
\( T^{7} + 22 T^{6} + \cdots + 1855520 \)
|
| $43$ |
\( T^{7} - 19 T^{6} + \cdots + 20000 \)
|
| $47$ |
\( T^{7} - 13 T^{6} + \cdots - 383908 \)
|
| $53$ |
\( T^{7} - 5 T^{6} + \cdots - 2656 \)
|
| $59$ |
\( T^{7} - 18 T^{6} + \cdots - 8288 \)
|
| $61$ |
\( T^{7} - 26 T^{6} + \cdots + 453613 \)
|
| $67$ |
\( T^{7} - 27 T^{6} + \cdots - 459232 \)
|
| $71$ |
\( T^{7} - 46 T^{6} + \cdots - 137728 \)
|
| $73$ |
\( T^{7} + 25 T^{6} + \cdots - 124640 \)
|
| $79$ |
\( T^{7} - 22 T^{6} + \cdots + 404960 \)
|
| $83$ |
\( T^{7} + T^{6} + \cdots - 68768 \)
|
| $89$ |
\( T^{7} + 3 T^{6} + \cdots + 7516100 \)
|
| $97$ |
\( T^{7} - 11 T^{6} + \cdots + 59188 \)
|
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