Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [459,2,Mod(154,459)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(459, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("459.154");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 459 = 3^{3} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 459.e (of order \(3\), degree \(2\), not 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: | \(3.66513345278\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.152695449.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 3x^{5} - 5x^{4} + 6x^{3} - 16x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 153) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 3x^{5} - 5x^{4} + 6x^{3} - 16x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} + 11\nu^{4} - 3\nu^{3} - 4\nu^{2} - 16\nu - 24 ) / 24 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + \nu^{4} + 6\nu^{3} - 11\nu^{2} + 4\nu + 12 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 2\nu^{6} + 8\nu^{5} - 9\nu^{4} - \nu^{3} + 6\nu^{2} - 16\nu + 32 ) / 24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{7} - 4\nu^{6} - 4\nu^{5} + 7\nu^{4} - 7\nu^{3} + 16\nu^{2} + 12\nu - 64 ) / 24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{7} - 4\nu^{5} - 7\nu^{4} + 11\nu^{3} - 4\nu^{2} + 4\nu + 32 ) / 24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} + 4\nu^{6} + 12\nu^{5} - 5\nu^{4} + 21\nu^{3} - 8\nu^{2} - 44\nu + 72 ) / 24 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -9\nu^{7} + 10\nu^{6} + 8\nu^{5} - 27\nu^{4} + 29\nu^{3} - 30\nu^{2} - 40\nu + 128 ) / 24 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + \beta_{3} + 2\beta_{2} - 2\beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} - 2\beta _1 + 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 2\beta_{6} + \beta_{5} + 7\beta_{4} + 3\beta_{2} - \beta _1 + 2 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2\beta_{7} + 2\beta_{6} - 5\beta_{5} + \beta_{4} + 6\beta_{3} + 6\beta_{2} - 4\beta _1 - 4 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -2\beta_{7} + 4\beta_{6} - 4\beta_{5} - 10\beta_{4} - 11\beta_{3} - 7\beta_{2} - 7\beta _1 - 8 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 8\beta_{7} - 3\beta_{6} - 9\beta_{5} + 18\beta_{4} + \beta_{3} + 5\beta_{2} + \beta _1 + 21 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/459\mathbb{Z}\right)^\times\).
| \(n\) | \(137\) | \(190\) |
| \(\chi(n)\) | \(-1 - \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 154.1 |
|
−1.17642 | + | 2.03762i | 0 | −1.76793 | − | 3.06215i | 1.17642 | + | 2.03762i | 0 | 1.76793 | − | 3.06215i | 3.61366 | 0 | −5.53587 | ||||||||||||||||||||||||||||||||||
| 154.2 | −0.580136 | + | 1.00483i | 0 | 0.326884 | + | 0.566179i | 0.580136 | + | 1.00483i | 0 | −0.326884 | + | 0.566179i | −3.07909 | 0 | −1.34623 | |||||||||||||||||||||||||||||||||||
| 154.3 | 0.281864 | − | 0.488204i | 0 | 0.841105 | + | 1.45684i | −0.281864 | − | 0.488204i | 0 | −0.841105 | + | 1.45684i | 2.07577 | 0 | −0.317790 | |||||||||||||||||||||||||||||||||||
| 154.4 | 0.974693 | − | 1.68822i | 0 | −0.900054 | − | 1.55894i | −0.974693 | − | 1.68822i | 0 | 0.900054 | − | 1.55894i | 0.389667 | 0 | −3.80011 | |||||||||||||||||||||||||||||||||||
| 307.1 | −1.17642 | − | 2.03762i | 0 | −1.76793 | + | 3.06215i | 1.17642 | − | 2.03762i | 0 | 1.76793 | + | 3.06215i | 3.61366 | 0 | −5.53587 | |||||||||||||||||||||||||||||||||||
| 307.2 | −0.580136 | − | 1.00483i | 0 | 0.326884 | − | 0.566179i | 0.580136 | − | 1.00483i | 0 | −0.326884 | − | 0.566179i | −3.07909 | 0 | −1.34623 | |||||||||||||||||||||||||||||||||||
| 307.3 | 0.281864 | + | 0.488204i | 0 | 0.841105 | − | 1.45684i | −0.281864 | + | 0.488204i | 0 | −0.841105 | − | 1.45684i | 2.07577 | 0 | −0.317790 | |||||||||||||||||||||||||||||||||||
| 307.4 | 0.974693 | + | 1.68822i | 0 | −0.900054 | + | 1.55894i | −0.974693 | + | 1.68822i | 0 | 0.900054 | + | 1.55894i | 0.389667 | 0 | −3.80011 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 459.2.e.b | 8 | |
| 3.b | odd | 2 | 1 | 153.2.e.b | ✓ | 8 | |
| 9.c | even | 3 | 1 | inner | 459.2.e.b | 8 | |
| 9.c | even | 3 | 1 | 1377.2.a.f | 4 | ||
| 9.d | odd | 6 | 1 | 153.2.e.b | ✓ | 8 | |
| 9.d | odd | 6 | 1 | 1377.2.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 153.2.e.b | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 153.2.e.b | ✓ | 8 | 9.d | odd | 6 | 1 | |
| 459.2.e.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 459.2.e.b | 8 | 9.c | even | 3 | 1 | inner | |
| 1377.2.a.e | 4 | 9.d | odd | 6 | 1 | ||
| 1377.2.a.f | 4 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + T_{2}^{7} + 6T_{2}^{6} + T_{2}^{5} + 25T_{2}^{4} + 9T_{2}^{3} + 24T_{2}^{2} - 9T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(459, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{7} + 6 T^{6} + \cdots + 9 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - T^{7} + 6 T^{6} + \cdots + 9 \)
|
| $7$ |
\( T^{8} - 3 T^{7} + \cdots + 49 \)
|
| $11$ |
\( T^{8} - 3 T^{7} + \cdots + 81 \)
|
| $13$ |
\( T^{8} - 9 T^{7} + \cdots + 529 \)
|
| $17$ |
\( (T - 1)^{8} \)
|
| $19$ |
\( (T^{4} + 7 T^{3} + \cdots + 451)^{2} \)
|
| $23$ |
\( T^{8} - 10 T^{7} + \cdots + 25281 \)
|
| $29$ |
\( T^{8} + 15 T^{7} + \cdots + 729 \)
|
| $31$ |
\( T^{8} - 15 T^{7} + \cdots + 9409 \)
|
| $37$ |
\( (T^{4} + 12 T^{3} + \cdots - 1223)^{2} \)
|
| $41$ |
\( T^{8} + 6 T^{7} + \cdots + 23409 \)
|
| $43$ |
\( T^{8} - 18 T^{7} + \cdots + 2611456 \)
|
| $47$ |
\( T^{8} - 12 T^{7} + \cdots + 29241 \)
|
| $53$ |
\( (T^{4} + 12 T^{3} + \cdots - 81)^{2} \)
|
| $59$ |
\( T^{8} + 114 T^{6} + \cdots + 263169 \)
|
| $61$ |
\( T^{8} + 5 T^{7} + \cdots + 33674809 \)
|
| $67$ |
\( T^{8} - 6 T^{7} + \cdots + 790321 \)
|
| $71$ |
\( (T^{4} - 25 T^{3} + \cdots - 1107)^{2} \)
|
| $73$ |
\( (T^{4} - 4 T^{3} - 95 T^{2} + \cdots - 27)^{2} \)
|
| $79$ |
\( T^{8} - 8 T^{7} + \cdots + 720801 \)
|
| $83$ |
\( T^{8} + 7 T^{7} + \cdots + 53361 \)
|
| $89$ |
\( (T^{4} - 14 T^{3} + \cdots - 4131)^{2} \)
|
| $97$ |
\( T^{8} - 16 T^{7} + \cdots + 20241001 \)
|
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