Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [538,2,Mod(1,538)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(538, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("538.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 538 = 2 \cdot 269 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 538.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: | \(4.29595162874\) |
| 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: |
\( x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + \nu^{5} + 8\nu^{4} - 5\nu^{3} - 11\nu^{2} + 5\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 12\nu^{4} - 9\nu^{3} + 37\nu^{2} + 11\nu - 30 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - \nu^{5} + 12\nu^{4} + 11\nu^{3} - 37\nu^{2} - 23\nu + 28 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 12\nu^{4} - 11\nu^{3} + 39\nu^{2} + 21\nu - 32 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{6} + 3\nu^{5} + 48\nu^{4} - 11\nu^{3} - 113\nu^{2} - 3\nu + 58 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 8\beta_{5} + 8\beta_{4} + \beta_{3} - 2\beta_{2} + 11\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{6} + 3\beta_{5} + 10\beta_{4} + 18\beta_{3} - 3\beta_{2} + 43\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{6} + 56\beta_{5} + 58\beta_{4} + 16\beta_{3} - 21\beta_{2} + 95\beta _1 + 38 \)
|
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 | −3.38066 | 1.00000 | −2.51475 | 3.38066 | 2.26176 | −1.00000 | 8.42889 | 2.51475 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.79016 | 1.00000 | 2.36234 | 2.79016 | −3.88429 | −1.00000 | 4.78499 | −2.36234 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.20937 | 1.00000 | −2.40321 | 1.20937 | 1.52017 | −1.00000 | −1.53742 | 2.40321 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −0.370477 | 1.00000 | 2.98038 | 0.370477 | −2.56529 | −1.00000 | −2.86275 | −2.98038 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.292058 | 1.00000 | −2.36976 | −0.292058 | 3.96531 | −1.00000 | −2.91470 | 2.36976 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0.980985 | 1.00000 | −0.685853 | −0.980985 | −1.46090 | −1.00000 | −2.03767 | 0.685853 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.47763 | 1.00000 | −3.36915 | −2.47763 | −2.83676 | −1.00000 | 3.13865 | 3.36915 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(269\) | \( +1 \) |
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 | 538.2.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 4842.2.a.o | 7 | ||
| 4.b | odd | 2 | 1 | 4304.2.a.i | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 538.2.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 4304.2.a.i | 7 | 4.b | odd | 2 | 1 | ||
| 4842.2.a.o | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + 4T_{3}^{6} - 6T_{3}^{5} - 30T_{3}^{4} + 31T_{3}^{2} + 2T_{3} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(538))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{7} \)
|
| $3$ |
\( T^{7} + 4 T^{6} + \cdots - 3 \)
|
| $5$ |
\( T^{7} + 6 T^{6} + \cdots + 233 \)
|
| $7$ |
\( T^{7} + 3 T^{6} + \cdots - 563 \)
|
| $11$ |
\( T^{7} + 12 T^{6} + \cdots - 480 \)
|
| $13$ |
\( T^{7} - 3 T^{6} + \cdots - 89 \)
|
| $17$ |
\( T^{7} + 8 T^{6} + \cdots - 32 \)
|
| $19$ |
\( T^{7} + 7 T^{6} + \cdots + 6788 \)
|
| $23$ |
\( T^{7} + 22 T^{6} + \cdots + 47584 \)
|
| $29$ |
\( T^{7} + 7 T^{6} + \cdots + 35296 \)
|
| $31$ |
\( T^{7} + 3 T^{6} + \cdots + 44411 \)
|
| $37$ |
\( T^{7} + 13 T^{6} + \cdots - 73351 \)
|
| $41$ |
\( T^{7} + 11 T^{6} + \cdots + 32644 \)
|
| $43$ |
\( T^{7} + 14 T^{6} + \cdots + 55392 \)
|
| $47$ |
\( T^{7} + 23 T^{6} + \cdots - 149280 \)
|
| $53$ |
\( T^{7} + 21 T^{6} + \cdots + 1074432 \)
|
| $59$ |
\( T^{7} + 20 T^{6} + \cdots + 149669 \)
|
| $61$ |
\( T^{7} - 13 T^{6} + \cdots + 495031 \)
|
| $67$ |
\( T^{7} + 30 T^{6} + \cdots + 1051936 \)
|
| $71$ |
\( T^{7} + 23 T^{6} + \cdots - 67860 \)
|
| $73$ |
\( T^{7} - 9 T^{6} + \cdots + 378085 \)
|
| $79$ |
\( T^{7} - 11 T^{6} + \cdots - 76640 \)
|
| $83$ |
\( T^{7} + 3 T^{6} + \cdots - 3711332 \)
|
| $89$ |
\( T^{7} - 16 T^{6} + \cdots - 818177 \)
|
| $97$ |
\( T^{7} - 21 T^{6} + \cdots + 5604 \)
|
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