Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5733,2,Mod(1,5733)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5733, 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("5733.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5733.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: | \(45.7782354788\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.4507648.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 637) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 4\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 6\beta_{3} + 8\beta_{2} + 18\beta _1 + 8 \)
|
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 |
|
−2.18322 | 0 | 2.76645 | −2.11065 | 0 | 0 | −1.67333 | 0 | 4.60802 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.83237 | 0 | 1.35758 | −2.62555 | 0 | 0 | 1.17715 | 0 | 4.81098 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.264627 | 0 | −1.92997 | 1.43515 | 0 | 0 | 1.03998 | 0 | −0.379780 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.656184 | 0 | −1.56942 | 1.35996 | 0 | 0 | −2.34220 | 0 | 0.892385 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.17619 | 0 | −0.616586 | −3.14862 | 0 | 0 | −3.07759 | 0 | −3.70337 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.44785 | 0 | 3.99195 | −0.910286 | 0 | 0 | 4.87599 | 0 | −2.22824 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(-1\) |
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 | 5733.2.a.br | 6 | |
| 3.b | odd | 2 | 1 | 637.2.a.n | yes | 6 | |
| 7.b | odd | 2 | 1 | 5733.2.a.bu | 6 | ||
| 21.c | even | 2 | 1 | 637.2.a.m | ✓ | 6 | |
| 21.g | even | 6 | 2 | 637.2.e.o | 12 | ||
| 21.h | odd | 6 | 2 | 637.2.e.n | 12 | ||
| 39.d | odd | 2 | 1 | 8281.2.a.cd | 6 | ||
| 273.g | even | 2 | 1 | 8281.2.a.cc | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 637.2.a.m | ✓ | 6 | 21.c | even | 2 | 1 | |
| 637.2.a.n | yes | 6 | 3.b | odd | 2 | 1 | |
| 637.2.e.n | 12 | 21.h | odd | 6 | 2 | ||
| 637.2.e.o | 12 | 21.g | even | 6 | 2 | ||
| 5733.2.a.br | 6 | 1.a | even | 1 | 1 | trivial | |
| 5733.2.a.bu | 6 | 7.b | odd | 2 | 1 | ||
| 8281.2.a.cc | 6 | 273.g | even | 2 | 1 | ||
| 8281.2.a.cd | 6 | 39.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5733))\):
|
\( T_{2}^{6} - 8T_{2}^{4} + 14T_{2}^{2} - 4T_{2} - 2 \)
|
|
\( T_{5}^{6} + 6T_{5}^{5} + 5T_{5}^{4} - 24T_{5}^{3} - 31T_{5}^{2} + 26T_{5} + 31 \)
|
|
\( T_{11}^{6} + 4T_{11}^{5} - 38T_{11}^{4} - 156T_{11}^{3} + 186T_{11}^{2} + 692T_{11} - 562 \)
|
|
\( T_{17}^{6} + 16T_{17}^{5} + 56T_{17}^{4} - 324T_{17}^{3} - 2792T_{17}^{2} - 6792T_{17} - 5294 \)
|
|
\( T_{19}^{6} - 2T_{19}^{5} - 17T_{19}^{4} + 16T_{19}^{3} + 83T_{19}^{2} - 6T_{19} - 73 \)
|
|
\( T_{31}^{6} - 6T_{31}^{5} - 115T_{31}^{4} + 508T_{31}^{3} + 4181T_{31}^{2} - 10046T_{31} - 44249 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 8 T^{4} + \cdots - 2 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 6 T^{5} + \cdots + 31 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + 4 T^{5} + \cdots - 562 \)
|
| $13$ |
\( (T - 1)^{6} \)
|
| $17$ |
\( T^{6} + 16 T^{5} + \cdots - 5294 \)
|
| $19$ |
\( T^{6} - 2 T^{5} + \cdots - 73 \)
|
| $23$ |
\( T^{6} - 6 T^{5} + \cdots + 529 \)
|
| $29$ |
\( T^{6} - 6 T^{5} + \cdots + 529 \)
|
| $31$ |
\( T^{6} - 6 T^{5} + \cdots - 44249 \)
|
| $37$ |
\( T^{6} - 82 T^{4} + \cdots + 254 \)
|
| $41$ |
\( T^{6} - 8 T^{5} + \cdots + 28784 \)
|
| $43$ |
\( T^{6} - 2 T^{5} + \cdots + 35153 \)
|
| $47$ |
\( T^{6} + 30 T^{5} + \cdots - 135617 \)
|
| $53$ |
\( T^{6} - 14 T^{5} + \cdots - 1319 \)
|
| $59$ |
\( T^{6} + 24 T^{5} + \cdots + 1532 \)
|
| $61$ |
\( T^{6} - 246 T^{4} + \cdots - 216584 \)
|
| $67$ |
\( T^{6} - 16 T^{5} + \cdots + 6112 \)
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| $71$ |
\( T^{6} + 8 T^{5} + \cdots - 1206162 \)
|
| $73$ |
\( T^{6} + 6 T^{5} + \cdots - 142657 \)
|
| $79$ |
\( T^{6} + 22 T^{5} + \cdots + 7913 \)
|
| $83$ |
\( T^{6} + 50 T^{5} + \cdots - 167041 \)
|
| $89$ |
\( T^{6} + 26 T^{5} + \cdots + 9959 \)
|
| $97$ |
\( T^{6} + 14 T^{5} + \cdots + 217287 \)
|
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