Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [861,2,Mod(247,861)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(861, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("861.247");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 861 = 3 \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 861.i (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: | \(6.87511961403\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.16638075.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 7x^{4} - 4x^{3} + 41x^{2} - 30x + 25 \)
|
| 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
\( x^{6} - x^{5} + 7x^{4} - 4x^{3} + 41x^{2} - 30x + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{4} + 49\nu^{3} - 41\nu^{2} + 30\nu - 210 ) / 257 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6\nu^{5} - 42\nu^{4} + 37\nu^{3} - 246\nu^{2} + 180\nu - 1003 ) / 257 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -42\nu^{5} + 37\nu^{4} - 259\nu^{3} - 77\nu^{2} - 1517\nu - 175 ) / 1285 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -138\nu^{5} - 62\nu^{4} - 851\nu^{3} - 253\nu^{2} - 5168\nu - 575 ) / 1285 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 4\beta_{4} - \beta_{3} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 6\beta_{2} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{5} + 23\beta_{4} - \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} - 3\beta_{4} + 8\beta_{3} - 37\beta_{2} - 37\beta _1 - 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/861\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(493\) | \(575\) |
| \(\chi(n)\) | \(1\) | \(\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 247.1 |
|
−1.28789 | + | 2.23069i | 0.500000 | + | 0.866025i | −2.31730 | − | 4.01369i | 0.500000 | − | 0.866025i | −2.57577 | −2.00000 | − | 1.73205i | 6.78616 | −0.500000 | + | 0.866025i | 1.28789 | + | 2.23069i | ||||||||||||||||||||||
| 247.2 | −0.406358 | + | 0.703833i | 0.500000 | + | 0.866025i | 0.669746 | + | 1.16003i | 0.500000 | − | 0.866025i | −0.812716 | −2.00000 | − | 1.73205i | −2.71406 | −0.500000 | + | 0.866025i | 0.406358 | + | 0.703833i | |||||||||||||||||||||||
| 247.3 | 1.19424 | − | 2.06849i | 0.500000 | + | 0.866025i | −1.85244 | − | 3.20852i | 0.500000 | − | 0.866025i | 2.38849 | −2.00000 | − | 1.73205i | −4.07210 | −0.500000 | + | 0.866025i | −1.19424 | − | 2.06849i | |||||||||||||||||||||||
| 739.1 | −1.28789 | − | 2.23069i | 0.500000 | − | 0.866025i | −2.31730 | + | 4.01369i | 0.500000 | + | 0.866025i | −2.57577 | −2.00000 | + | 1.73205i | 6.78616 | −0.500000 | − | 0.866025i | 1.28789 | − | 2.23069i | |||||||||||||||||||||||
| 739.2 | −0.406358 | − | 0.703833i | 0.500000 | − | 0.866025i | 0.669746 | − | 1.16003i | 0.500000 | + | 0.866025i | −0.812716 | −2.00000 | + | 1.73205i | −2.71406 | −0.500000 | − | 0.866025i | 0.406358 | − | 0.703833i | |||||||||||||||||||||||
| 739.3 | 1.19424 | + | 2.06849i | 0.500000 | − | 0.866025i | −1.85244 | + | 3.20852i | 0.500000 | + | 0.866025i | 2.38849 | −2.00000 | + | 1.73205i | −4.07210 | −0.500000 | − | 0.866025i | −1.19424 | + | 2.06849i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 861.2.i.b | ✓ | 6 |
| 7.c | even | 3 | 1 | inner | 861.2.i.b | ✓ | 6 |
| 7.c | even | 3 | 1 | 6027.2.a.r | 3 | ||
| 7.d | odd | 6 | 1 | 6027.2.a.t | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 861.2.i.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 861.2.i.b | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 6027.2.a.r | 3 | 7.c | even | 3 | 1 | ||
| 6027.2.a.t | 3 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} + 7T_{2}^{4} + 4T_{2}^{3} + 41T_{2}^{2} + 30T_{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots + 25 \)
|
| $3$ |
\( (T^{2} - T + 1)^{3} \)
|
| $5$ |
\( (T^{2} - T + 1)^{3} \)
|
| $7$ |
\( (T^{2} + 4 T + 7)^{3} \)
|
| $11$ |
\( T^{6} - 3 T^{5} + \cdots + 225 \)
|
| $13$ |
\( (T^{3} + 8 T^{2} + 15 T + 5)^{2} \)
|
| $17$ |
\( T^{6} - 3 T^{5} + \cdots + 2209 \)
|
| $19$ |
\( T^{6} + 5 T^{5} + \cdots + 1 \)
|
| $23$ |
\( T^{6} - T^{5} + \cdots + 13689 \)
|
| $29$ |
\( (T^{3} - 2 T^{2} - 35 T - 45)^{2} \)
|
| $31$ |
\( T^{6} + T^{5} + \cdots + 37249 \)
|
| $37$ |
\( T^{6} + 16 T^{5} + \cdots + 12321 \)
|
| $41$ |
\( (T + 1)^{6} \)
|
| $43$ |
\( (T^{3} - 8 T^{2} + \cdots + 312)^{2} \)
|
| $47$ |
\( T^{6} - 2 T^{5} + \cdots + 4761 \)
|
| $53$ |
\( T^{6} + 13 T^{5} + \cdots + 81 \)
|
| $59$ |
\( T^{6} - 11 T^{5} + \cdots + 225 \)
|
| $61$ |
\( T^{6} - 8 T^{5} + \cdots + 87025 \)
|
| $67$ |
\( T^{6} - 17 T^{5} + \cdots + 11881 \)
|
| $71$ |
\( (T^{3} - 29 T^{2} + \cdots - 115)^{2} \)
|
| $73$ |
\( T^{6} - 22 T^{5} + \cdots + 94249 \)
|
| $79$ |
\( T^{6} - 24 T^{5} + \cdots + 142129 \)
|
| $83$ |
\( (T^{3} + 9 T^{2} + \cdots - 2963)^{2} \)
|
| $89$ |
\( T^{6} + 4 T^{5} + \cdots + 16129 \)
|
| $97$ |
\( (T^{3} + 15 T^{2} + \cdots - 1647)^{2} \)
|
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