Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [693,2,Mod(232,693)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(693, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 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("693.232");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 693.j (of order \(3\), degree \(2\), 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: | \(5.53363286007\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.4406832.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 6\nu^{4} - 36\nu^{3} + 24\nu^{2} + 5\nu - 30 ) / 149 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -6\nu^{5} + 36\nu^{4} - 67\nu^{3} + 144\nu^{2} + 30\nu + 416 ) / 149 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 30\nu^{5} - 31\nu^{4} + 186\nu^{3} + 174\nu^{2} + 744\nu + 155 ) / 149 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 89\nu^{5} - 87\nu^{4} + 522\nu^{3} + 695\nu^{2} + 2088\nu + 435 ) / 149 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 6\beta_{2} - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{5} + 19\beta_{4} + 6\beta_{3} - 12\beta _1 - 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{5} + 42\beta_{4} + 43\beta_{2} - 43\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).
| \(n\) | \(155\) | \(199\) | \(442\) |
| \(\chi(n)\) | \(-\beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 232.1 |
|
−1.37241 | + | 2.37709i | 1.50000 | − | 0.866025i | −2.76704 | − | 4.79265i | −1.97779 | − | 3.42563i | 4.75418i | −0.500000 | + | 0.866025i | 9.70041 | 1.50000 | − | 2.59808i | 10.8574 | ||||||||||||||||||||||||
| 232.2 | 0.697966 | − | 1.20891i | 1.50000 | − | 0.866025i | 0.0256871 | + | 0.0444914i | −0.629755 | − | 1.09077i | − | 2.41782i | −0.500000 | + | 0.866025i | 2.86358 | 1.50000 | − | 2.59808i | −1.75819 | ||||||||||||||||||||||||
| 232.3 | 1.17445 | − | 2.03420i | 1.50000 | − | 0.866025i | −1.75865 | − | 3.04607i | 2.10755 | + | 3.65038i | − | 4.06840i | −0.500000 | + | 0.866025i | −3.56399 | 1.50000 | − | 2.59808i | 9.90081 | ||||||||||||||||||||||||
| 463.1 | −1.37241 | − | 2.37709i | 1.50000 | + | 0.866025i | −2.76704 | + | 4.79265i | −1.97779 | + | 3.42563i | − | 4.75418i | −0.500000 | − | 0.866025i | 9.70041 | 1.50000 | + | 2.59808i | 10.8574 | ||||||||||||||||||||||||
| 463.2 | 0.697966 | + | 1.20891i | 1.50000 | + | 0.866025i | 0.0256871 | − | 0.0444914i | −0.629755 | + | 1.09077i | 2.41782i | −0.500000 | − | 0.866025i | 2.86358 | 1.50000 | + | 2.59808i | −1.75819 | |||||||||||||||||||||||||
| 463.3 | 1.17445 | + | 2.03420i | 1.50000 | + | 0.866025i | −1.75865 | + | 3.04607i | 2.10755 | − | 3.65038i | 4.06840i | −0.500000 | − | 0.866025i | −3.56399 | 1.50000 | + | 2.59808i | 9.90081 | |||||||||||||||||||||||||
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 | 693.2.j.d | ✓ | 6 |
| 9.c | even | 3 | 1 | inner | 693.2.j.d | ✓ | 6 |
| 9.c | even | 3 | 1 | 6237.2.a.o | 3 | ||
| 9.d | odd | 6 | 1 | 6237.2.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 693.2.j.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 693.2.j.d | ✓ | 6 | 9.c | even | 3 | 1 | inner |
| 6237.2.a.o | 3 | 9.c | even | 3 | 1 | ||
| 6237.2.a.p | 3 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - T_{2}^{5} + 8T_{2}^{4} - 11T_{2}^{3} + 58T_{2}^{2} - 63T_{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} + \cdots + 81 \)
|
| $3$ |
\( (T^{2} - 3 T + 3)^{3} \)
|
| $5$ |
\( T^{6} + T^{5} + \cdots + 441 \)
|
| $7$ |
\( (T^{2} + T + 1)^{3} \)
|
| $11$ |
\( (T^{2} - T + 1)^{3} \)
|
| $13$ |
\( T^{6} + 8 T^{4} + \cdots + 16 \)
|
| $17$ |
\( (T^{3} + 6 T^{2} + 4 T - 12)^{2} \)
|
| $19$ |
\( (T^{3} + 4 T^{2} - 12 T - 36)^{2} \)
|
| $23$ |
\( T^{6} + 8 T^{5} + \cdots + 186624 \)
|
| $29$ |
\( T^{6} + 8 T^{5} + \cdots + 144 \)
|
| $31$ |
\( T^{6} - T^{5} + \cdots + 1849 \)
|
| $37$ |
\( (T + 1)^{6} \)
|
| $41$ |
\( T^{6} - 4 T^{5} + \cdots + 17424 \)
|
| $43$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $47$ |
\( T^{6} + 13 T^{5} + \cdots + 1089 \)
|
| $53$ |
\( (T^{3} + 7 T^{2} - 13 T - 3)^{2} \)
|
| $59$ |
\( T^{6} - 15 T^{5} + \cdots + 9 \)
|
| $61$ |
\( T^{6} + 6 T^{5} + \cdots + 784 \)
|
| $67$ |
\( T^{6} + 5 T^{5} + \cdots + 1369 \)
|
| $71$ |
\( (T^{3} + 13 T^{2} - 13 T + 3)^{2} \)
|
| $73$ |
\( (T^{3} + 4 T^{2} - 80 T + 64)^{2} \)
|
| $79$ |
\( T^{6} + 12 T^{5} + \cdots + 1024 \)
|
| $83$ |
\( T^{6} + 24 T^{5} + \cdots + 82944 \)
|
| $89$ |
\( (T^{3} - 30 T^{2} + \cdots - 648)^{2} \)
|
| $97$ |
\( T^{6} - 19 T^{5} + \cdots + 281961 \)
|
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