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: | \(0\) |
| 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: |
\( x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 15\nu^{4} - 12\nu^{3} + 51\nu^{2} + 21\nu - 48 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{6} - \nu^{5} + 45\nu^{4} + 10\nu^{3} - 151\nu^{2} - 17\nu + 138 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{6} + \nu^{5} - 73\nu^{4} - 8\nu^{3} + 225\nu^{2} + 13\nu - 186 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17\nu^{6} + 3\nu^{5} - 251\nu^{4} - 24\nu^{3} + 801\nu^{2} + 47\nu - 696 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -21\nu^{6} - 3\nu^{5} + 311\nu^{4} + 20\nu^{3} - 997\nu^{2} - 39\nu + 856 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} + 2\beta_{4} - 3\beta_{3} - 2\beta_{2} + 6\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 13\beta_{6} + 13\beta_{5} + \beta_{4} - 11\beta_{3} - 12\beta_{2} + 44 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} - 27\beta_{5} + 26\beta_{4} - 37\beta_{3} - 22\beta_{2} + 55\beta _1 - 41 \)
|
| \(\nu^{6}\) | \(=\) |
\( 145\beta_{6} + 147\beta_{5} + 13\beta_{4} - 113\beta_{3} - 129\beta_{2} - 4\beta _1 + 458 \)
|
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 | −3.13760 | 1.00000 | 3.00565 | −3.13760 | −0.324393 | 1.00000 | 6.84453 | 3.00565 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.60058 | 1.00000 | −3.80694 | −2.60058 | 0.487379 | 1.00000 | 3.76302 | −3.80694 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.383827 | 1.00000 | 4.15424 | −0.383827 | 4.87818 | 1.00000 | −2.85268 | 4.15424 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.366193 | 1.00000 | −0.953965 | 0.366193 | 3.52894 | 1.00000 | −2.86590 | −0.953965 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.65255 | 1.00000 | 0.725043 | 1.65255 | 1.29978 | 1.00000 | −0.269065 | 0.725043 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.12827 | 1.00000 | 3.84433 | 2.12827 | −4.95403 | 1.00000 | 1.52953 | 3.84433 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.97499 | 1.00000 | 0.0316474 | 2.97499 | 1.08414 | 1.00000 | 5.85056 | 0.0316474 | ||||||||||||||||||||||||||||||||||||||||
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.e | ✓ | 7 |
| 3.b | odd | 2 | 1 | 4842.2.a.n | 7 | ||
| 4.b | odd | 2 | 1 | 4304.2.a.h | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 538.2.a.e | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 4304.2.a.h | 7 | 4.b | odd | 2 | 1 | ||
| 4842.2.a.n | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - T_{3}^{6} - 16T_{3}^{5} + 19T_{3}^{4} + 63T_{3}^{3} - 87T_{3}^{2} - 10T_{3} + 12 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(538))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( T^{7} - T^{6} + \cdots + 12 \)
|
| $5$ |
\( T^{7} - 7 T^{6} + \cdots - 4 \)
|
| $7$ |
\( T^{7} - 6 T^{6} + \cdots - 19 \)
|
| $11$ |
\( T^{7} + 3 T^{6} + \cdots + 288 \)
|
| $13$ |
\( T^{7} + 9 T^{6} + \cdots + 4 \)
|
| $17$ |
\( T^{7} - 8 T^{6} + \cdots - 22016 \)
|
| $19$ |
\( T^{7} + 11 T^{6} + \cdots - 20 \)
|
| $23$ |
\( T^{7} - 12 T^{6} + \cdots + 2432 \)
|
| $29$ |
\( T^{7} + 5 T^{6} + \cdots - 27040 \)
|
| $31$ |
\( T^{7} - 14 T^{6} + \cdots + 250907 \)
|
| $37$ |
\( T^{7} - 13 T^{6} + \cdots + 2236 \)
|
| $41$ |
\( T^{7} - 12 T^{6} + \cdots - 288917 \)
|
| $43$ |
\( T^{7} + 11 T^{6} + \cdots - 227616 \)
|
| $47$ |
\( T^{7} - 2 T^{6} + \cdots + 64896 \)
|
| $53$ |
\( T^{7} - 19 T^{6} + \cdots - 26988 \)
|
| $59$ |
\( T^{7} + 9 T^{6} + \cdots - 146660 \)
|
| $61$ |
\( T^{7} + 3 T^{6} + \cdots + 210068 \)
|
| $67$ |
\( T^{7} + 33 T^{6} + \cdots - 251936 \)
|
| $71$ |
\( T^{7} - 28 T^{6} + \cdots + 28293 \)
|
| $73$ |
\( T^{7} + 14 T^{6} + \cdots - 382603 \)
|
| $79$ |
\( T^{7} - 2 T^{6} + \cdots + 14720 \)
|
| $83$ |
\( T^{7} + 7 T^{6} + \cdots - 441152 \)
|
| $89$ |
\( T^{7} - 18 T^{6} + \cdots + 1143155 \)
|
| $97$ |
\( T^{7} + 4 T^{6} + \cdots + 15927 \)
|
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