Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5525,2,Mod(1,5525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5525, 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("5525.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5525 = 5^{2} \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5525.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: | \(44.1173471168\) |
| 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: |
\( x^{7} - 3x^{6} - 5x^{5} + 14x^{4} + 9x^{3} - 15x^{2} - 4x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1105) |
| 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
\( x^{7} - 3x^{6} - 5x^{5} + 14x^{4} + 9x^{3} - 15x^{2} - 4x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 11\nu^{2} + 5\nu - 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 11\nu^{3} + 6\nu^{2} - 6\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} + 5\nu^{4} - 13\nu^{3} - 10\nu^{2} + 9\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 2\beta_{3} + 8\beta_{2} + 11\beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{6} + 3\beta_{5} + \beta_{4} + 10\beta_{3} + 21\beta_{2} + 37\beta _1 + 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( 13\beta_{6} + 14\beta_{5} + 3\beta_{4} + 27\beta_{3} + 67\beta_{2} + 100\beta _1 + 71 \)
|
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 |
|
−1.93904 | 0.371029 | 1.75990 | 0 | −0.719442 | 3.31415 | 0.465574 | −2.86234 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.16721 | 2.90813 | −0.637621 | 0 | −3.39439 | −0.0566511 | 3.07866 | 5.45719 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0.0916017 | −1.72881 | −1.99161 | 0 | −0.158361 | 3.11320 | −0.365638 | −0.0112325 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.492383 | 2.41044 | −1.75756 | 0 | 1.18686 | −1.56612 | −1.85016 | 2.81022 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.71486 | −0.985811 | 0.940735 | 0 | −1.69052 | −2.23606 | −1.81649 | −2.02818 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.14822 | 2.52588 | 2.61483 | 0 | 5.42613 | 4.07283 | 1.32079 | 3.38005 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.65920 | −2.50086 | 5.07133 | 0 | −6.65027 | 3.35866 | 8.16726 | 3.25428 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 5525.2.a.bc | 7 | |
| 5.b | even | 2 | 1 | 1105.2.a.h | ✓ | 7 | |
| 15.d | odd | 2 | 1 | 9945.2.a.bd | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1105.2.a.h | ✓ | 7 | 5.b | even | 2 | 1 | |
| 5525.2.a.bc | 7 | 1.a | even | 1 | 1 | trivial | |
| 9945.2.a.bd | 7 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5525))\):
|
\( T_{2}^{7} - 4T_{2}^{6} - 2T_{2}^{5} + 21T_{2}^{4} - 10T_{2}^{3} - 22T_{2}^{2} + 13T_{2} - 1 \)
|
|
\( T_{3}^{7} - 3T_{3}^{6} - 11T_{3}^{5} + 31T_{3}^{4} + 38T_{3}^{3} - 81T_{3}^{2} - 52T_{3} + 28 \)
|
|
\( T_{11}^{7} + 14T_{11}^{6} + 37T_{11}^{5} - 240T_{11}^{4} - 1100T_{11}^{3} + 401T_{11}^{2} + 4060T_{11} - 2284 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 4 T^{6} + \cdots - 1 \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 28 \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} - 10 T^{6} + \cdots + 28 \)
|
| $11$ |
\( T^{7} + 14 T^{6} + \cdots - 2284 \)
|
| $13$ |
\( (T - 1)^{7} \)
|
| $17$ |
\( (T + 1)^{7} \)
|
| $19$ |
\( T^{7} + 8 T^{6} + \cdots + 304 \)
|
| $23$ |
\( T^{7} - 18 T^{6} + \cdots + 76 \)
|
| $29$ |
\( T^{7} - T^{6} + \cdots - 3764 \)
|
| $31$ |
\( T^{7} + 4 T^{6} + \cdots - 117268 \)
|
| $37$ |
\( T^{7} - 12 T^{6} + \cdots + 6836 \)
|
| $41$ |
\( T^{7} - 10 T^{6} + \cdots - 60116 \)
|
| $43$ |
\( T^{7} - 14 T^{6} + \cdots + 83872 \)
|
| $47$ |
\( T^{7} - 28 T^{6} + \cdots - 42544 \)
|
| $53$ |
\( T^{7} - 6 T^{6} + \cdots + 20444 \)
|
| $59$ |
\( T^{7} + 17 T^{6} + \cdots - 9392 \)
|
| $61$ |
\( T^{7} - 19 T^{6} + \cdots + 66532 \)
|
| $67$ |
\( T^{7} - 27 T^{6} + \cdots - 4544032 \)
|
| $71$ |
\( T^{7} + 19 T^{6} + \cdots - 1824292 \)
|
| $73$ |
\( T^{7} - 2 T^{6} + \cdots - 68 \)
|
| $79$ |
\( T^{7} + 6 T^{6} + \cdots + 98116 \)
|
| $83$ |
\( T^{7} - 42 T^{6} + \cdots - 12516112 \)
|
| $89$ |
\( T^{7} - 2 T^{6} + \cdots + 422164 \)
|
| $97$ |
\( T^{7} + 4 T^{6} + \cdots + 1894844 \)
|
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