Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [531,3,Mod(235,531)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(531, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("531.235");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 531 = 3^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 531.c (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(14.4687020375\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.7196038400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 27x^{4} + 215x^{2} + 509 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 59) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 27x^{4} + 215x^{2} + 509 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 18\nu^{3} - 57\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 18\nu^{2} + 61 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 22\nu^{2} + 97 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 22\nu^{3} + 105\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{2} - 12\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -18\beta_{4} + 22\beta_{3} + 101 \)
|
| \(\nu^{5}\) | \(=\) |
\( -18\beta_{5} - 22\beta_{2} + 159\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).
| \(n\) | \(119\) | \(415\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 235.1 |
|
− | 3.84580i | 0 | −10.7902 | 3.40857 | 0 | 6.79021 | 26.1138i | 0 | − | 13.1087i | ||||||||||||||||||||||||||||||||||
| 235.2 | − | 2.79188i | 0 | −3.79459 | 4.43134 | 0 | −0.205410 | − | 0.573480i | 0 | − | 12.3718i | ||||||||||||||||||||||||||||||||||
| 235.3 | − | 2.10124i | 0 | −0.415197 | −3.83991 | 0 | −3.58480 | − | 7.53252i | 0 | 8.06856i | |||||||||||||||||||||||||||||||||||
| 235.4 | 2.10124i | 0 | −0.415197 | −3.83991 | 0 | −3.58480 | 7.53252i | 0 | − | 8.06856i | ||||||||||||||||||||||||||||||||||||
| 235.5 | 2.79188i | 0 | −3.79459 | 4.43134 | 0 | −0.205410 | 0.573480i | 0 | 12.3718i | |||||||||||||||||||||||||||||||||||||
| 235.6 | 3.84580i | 0 | −10.7902 | 3.40857 | 0 | 6.79021 | − | 26.1138i | 0 | 13.1087i | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 59.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 531.3.c.b | 6 | |
| 3.b | odd | 2 | 1 | 59.3.b.b | ✓ | 6 | |
| 12.b | even | 2 | 1 | 944.3.h.d | 6 | ||
| 59.b | odd | 2 | 1 | inner | 531.3.c.b | 6 | |
| 177.d | even | 2 | 1 | 59.3.b.b | ✓ | 6 | |
| 708.b | odd | 2 | 1 | 944.3.h.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 59.3.b.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 59.3.b.b | ✓ | 6 | 177.d | even | 2 | 1 | |
| 531.3.c.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 531.3.c.b | 6 | 59.b | odd | 2 | 1 | inner | |
| 944.3.h.d | 6 | 12.b | even | 2 | 1 | ||
| 944.3.h.d | 6 | 708.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 27T_{2}^{4} + 215T_{2}^{2} + 509 \)
acting on \(S_{3}^{\mathrm{new}}(531, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 27 T^{4} + \cdots + 509 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} - 4 T^{2} - 15 T + 58)^{2} \)
|
| $7$ |
\( (T^{3} - 3 T^{2} - 25 T - 5)^{2} \)
|
| $11$ |
\( T^{6} + 281 T^{4} + \cdots + 814400 \)
|
| $13$ |
\( T^{6} + 505 T^{4} + \cdots + 521216 \)
|
| $17$ |
\( (T^{3} + 19 T^{2} + \cdots - 3140)^{2} \)
|
| $19$ |
\( (T^{3} - 20 T^{2} + \cdots + 1250)^{2} \)
|
| $23$ |
\( T^{6} + 2596 T^{4} + \cdots + 81440000 \)
|
| $29$ |
\( (T^{3} + 14 T^{2} + \cdots - 940)^{2} \)
|
| $31$ |
\( T^{6} + 3460 T^{4} + \cdots + 81440000 \)
|
| $37$ |
\( T^{6} + \cdots + 1290042176 \)
|
| $41$ |
\( (T^{3} - 65 T^{2} + \cdots + 20225)^{2} \)
|
| $43$ |
\( T^{6} + 1457 T^{4} + \cdots + 91505984 \)
|
| $47$ |
\( T^{6} + 3392 T^{4} + \cdots + 33357824 \)
|
| $53$ |
\( (T^{3} + 106 T^{2} + \cdots - 9400)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 42180533641 \)
|
| $61$ |
\( T^{6} + \cdots + 1303040000 \)
|
| $67$ |
\( T^{6} + 17860 T^{4} + \cdots + 68930816 \)
|
| $71$ |
\( (T^{3} - 111 T^{2} + \cdots + 31860)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 7257411584 \)
|
| $79$ |
\( (T^{3} - 185 T^{2} + \cdots - 126851)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 51389812736 \)
|
| $89$ |
\( T^{6} + \cdots + 1707555737600 \)
|
| $97$ |
\( T^{6} + 17412 T^{4} + \cdots + 453588224 \)
|
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