Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5577,2,Mod(1,5577)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5577, 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("5577.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5577 = 3 \cdot 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5577.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: | \(44.5325692073\) |
| 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: |
\( x^{7} - 3x^{6} - 7x^{5} + 21x^{4} + 13x^{3} - 33x^{2} - 7x + 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 429) |
| 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
\( x^{7} - 3x^{6} - 7x^{5} + 21x^{4} + 13x^{3} - 33x^{2} - 7x + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 4\nu^{2} + 9\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 12\nu^{3} + 11\nu^{2} - 14\nu - 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 7\beta_{2} + 9\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 8\beta_{3} + 10\beta_{2} + 38\beta _1 + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} + 9\beta_{4} + 11\beta_{3} + 46\beta_{2} + 70\beta _1 + 87 \)
|
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 |
|
−2.73878 | 1.00000 | 5.50093 | 2.84154 | −2.73878 | −3.93129 | −9.58828 | 1.00000 | −7.78236 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.53441 | 1.00000 | 4.42325 | −3.70100 | −2.53441 | 0.957295 | −6.14151 | 1.00000 | 9.37985 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.42819 | 1.00000 | 0.0397381 | −0.0606573 | −1.42819 | 1.70646 | 2.79963 | 1.00000 | 0.0866304 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.409068 | 1.00000 | −1.83266 | −4.13953 | −0.409068 | −5.18273 | 1.56782 | 1.00000 | 1.69335 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.584778 | 1.00000 | −1.65803 | 1.95350 | 0.584778 | −1.51078 | −2.13914 | 1.00000 | 1.14237 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.36814 | 1.00000 | −0.128197 | −2.18365 | 1.36814 | 4.27070 | −2.91167 | 1.00000 | −2.98753 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.15754 | 1.00000 | 2.65498 | −0.710210 | 2.15754 | −2.30964 | 1.41315 | 1.00000 | −1.53231 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
| \(13\) | \(-1\) |
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 | 5577.2.a.x | 7 | |
| 13.b | even | 2 | 1 | 5577.2.a.y | 7 | ||
| 13.d | odd | 4 | 2 | 429.2.b.b | ✓ | 14 | |
| 39.f | even | 4 | 2 | 1287.2.b.c | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 429.2.b.b | ✓ | 14 | 13.d | odd | 4 | 2 | |
| 1287.2.b.c | 14 | 39.f | even | 4 | 2 | ||
| 5577.2.a.x | 7 | 1.a | even | 1 | 1 | trivial | |
| 5577.2.a.y | 7 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):
|
\( T_{2}^{7} + 3T_{2}^{6} - 7T_{2}^{5} - 21T_{2}^{4} + 13T_{2}^{3} + 33T_{2}^{2} - 7T_{2} - 7 \)
|
|
\( T_{5}^{7} + 6T_{5}^{6} - 6T_{5}^{5} - 74T_{5}^{4} - 32T_{5}^{3} + 198T_{5}^{2} + 144T_{5} + 8 \)
|
|
\( T_{7}^{7} + 6T_{7}^{6} - 18T_{7}^{5} - 136T_{7}^{4} - 16T_{7}^{3} + 524T_{7}^{2} + 160T_{7} - 496 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 3 T^{6} + \cdots - 7 \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( T^{7} + 6 T^{6} + \cdots + 8 \)
|
| $7$ |
\( T^{7} + 6 T^{6} + \cdots - 496 \)
|
| $11$ |
\( (T + 1)^{7} \)
|
| $13$ |
\( T^{7} \)
|
| $17$ |
\( T^{7} + 2 T^{6} + \cdots - 1256 \)
|
| $19$ |
\( T^{7} + 8 T^{6} + \cdots + 80 \)
|
| $23$ |
\( T^{7} - 4 T^{6} + \cdots - 1024 \)
|
| $29$ |
\( T^{7} + 12 T^{6} + \cdots + 104 \)
|
| $31$ |
\( T^{7} - 10 T^{6} + \cdots - 392 \)
|
| $37$ |
\( T^{7} + 6 T^{6} + \cdots - 125824 \)
|
| $41$ |
\( T^{7} + 2 T^{6} + \cdots - 39760 \)
|
| $43$ |
\( T^{7} + 16 T^{6} + \cdots - 692704 \)
|
| $47$ |
\( T^{7} + 18 T^{6} + \cdots + 122368 \)
|
| $53$ |
\( T^{7} - 10 T^{6} + \cdots + 11456 \)
|
| $59$ |
\( T^{7} + 2 T^{6} + \cdots - 145408 \)
|
| $61$ |
\( T^{7} + 10 T^{6} + \cdots + 32 \)
|
| $67$ |
\( T^{7} + 8 T^{6} + \cdots - 500296 \)
|
| $71$ |
\( T^{7} + 36 T^{6} + \cdots + 593536 \)
|
| $73$ |
\( T^{7} + 20 T^{6} + \cdots + 555776 \)
|
| $79$ |
\( T^{7} - 6 T^{6} + \cdots + 55424 \)
|
| $83$ |
\( T^{7} + 30 T^{6} + \cdots - 35840 \)
|
| $89$ |
\( T^{7} + 34 T^{6} + \cdots + 87880 \)
|
| $97$ |
\( T^{7} + 16 T^{6} + \cdots + 216704 \)
|
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