Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [966,2,Mod(277,966)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("966.277");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 966.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: | \(7.71354883526\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.1768034304.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - x^{6} - 6x^{5} + 14x^{4} + 18x^{3} - 31x^{2} - 14x + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - x^{6} - 6x^{5} + 14x^{4} + 18x^{3} - 31x^{2} - 14x + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{7} - 67\nu^{6} + 173\nu^{5} - 208\nu^{4} - 56\nu^{3} - 965\nu^{2} + 1303\nu + 3997 ) / 1659 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 34\nu^{7} - 33\nu^{6} + 176\nu^{5} - 218\nu^{4} + 154\nu^{3} - 1474\nu^{2} - 522\nu - 623 ) / 3318 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 128\nu^{7} - 417\nu^{6} - 1094\nu^{5} - 40\nu^{4} + 5264\nu^{3} + 7918\nu^{2} + 1548\nu - 22351 ) / 9954 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 20\nu^{7} - 117\nu^{6} - 8\nu^{5} - 184\nu^{4} + 704\nu^{3} + 304\nu^{2} - 1506\nu + 1655 ) / 1422 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -332\nu^{7} + 615\nu^{6} + 38\nu^{5} + 1348\nu^{4} - 6188\nu^{3} - 9028\nu^{2} + 21492\nu + 16135 ) / 9954 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 338\nu^{7} + 843\nu^{6} - 1178\nu^{5} - 1972\nu^{4} - 7252\nu^{3} + 11110\nu^{2} + 666\nu - 22393 ) / 9954 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -254\nu^{7} - 339\nu^{6} + 149\nu^{5} + 3190\nu^{4} + 3241\nu^{3} - 1870\nu^{2} - 8982\nu + 4459 ) / 4977 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} - \beta_{4} - 2\beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} - 4\beta_{6} - 2\beta_{5} - 3\beta_{4} + \beta_{3} - 3\beta _1 + 9 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{7} + 3\beta_{6} - 3\beta_{4} + 3\beta_{3} + 5\beta_{2} + 3\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -4\beta_{6} - 18\beta_{5} - 23\beta_{4} - 3\beta_{3} - 12\beta_{2} + 17\beta _1 - 3 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{7} - 12\beta_{6} - 8\beta_{5} - 15\beta_{4} - 3\beta_{3} - 15\beta _1 + 40 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 32\beta_{7} + 54\beta_{6} - 37\beta_{4} + 59\beta_{3} + 148\beta_{2} + 37\beta _1 + 207 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).
| \(n\) | \(323\) | \(829\) | \(925\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 277.1 |
|
−0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | −0.914214 | − | 1.58346i | 1.00000 | −1.75255 | − | 1.98206i | 1.00000 | −0.500000 | − | 0.866025i | −0.914214 | + | 1.58346i | ||||||||||||||||||||||||||||
| 277.2 | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | −0.914214 | − | 1.58346i | 1.00000 | 2.45965 | − | 0.974732i | 1.00000 | −0.500000 | − | 0.866025i | −0.914214 | + | 1.58346i | |||||||||||||||||||||||||||||
| 277.3 | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 1.91421 | + | 3.31552i | 1.00000 | −1.87485 | + | 1.86680i | 1.00000 | −0.500000 | − | 0.866025i | 1.91421 | − | 3.31552i | |||||||||||||||||||||||||||||
| 277.4 | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 1.91421 | + | 3.31552i | 1.00000 | 1.16774 | − | 2.37411i | 1.00000 | −0.500000 | − | 0.866025i | 1.91421 | − | 3.31552i | |||||||||||||||||||||||||||||
| 415.1 | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | −0.914214 | + | 1.58346i | 1.00000 | −1.75255 | + | 1.98206i | 1.00000 | −0.500000 | + | 0.866025i | −0.914214 | − | 1.58346i | |||||||||||||||||||||||||||||
| 415.2 | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | −0.914214 | + | 1.58346i | 1.00000 | 2.45965 | + | 0.974732i | 1.00000 | −0.500000 | + | 0.866025i | −0.914214 | − | 1.58346i | |||||||||||||||||||||||||||||
| 415.3 | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 1.91421 | − | 3.31552i | 1.00000 | −1.87485 | − | 1.86680i | 1.00000 | −0.500000 | + | 0.866025i | 1.91421 | + | 3.31552i | |||||||||||||||||||||||||||||
| 415.4 | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 1.91421 | − | 3.31552i | 1.00000 | 1.16774 | + | 2.37411i | 1.00000 | −0.500000 | + | 0.866025i | 1.91421 | + | 3.31552i | |||||||||||||||||||||||||||||
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 | 966.2.i.l | ✓ | 8 |
| 7.c | even | 3 | 1 | inner | 966.2.i.l | ✓ | 8 |
| 7.c | even | 3 | 1 | 6762.2.a.cp | 4 | ||
| 7.d | odd | 6 | 1 | 6762.2.a.cm | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 966.2.i.l | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 966.2.i.l | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 6762.2.a.cm | 4 | 7.d | odd | 6 | 1 | ||
| 6762.2.a.cp | 4 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\):
|
\( T_{5}^{4} - 2T_{5}^{3} + 11T_{5}^{2} + 14T_{5} + 49 \)
|
|
\( T_{11}^{8} + 6T_{11}^{7} + 55T_{11}^{6} + 198T_{11}^{5} + 1485T_{11}^{4} + 5220T_{11}^{3} + 20764T_{11}^{2} + 29328T_{11} + 35344 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{4} \)
|
| $3$ |
\( (T^{2} + T + 1)^{4} \)
|
| $5$ |
\( (T^{4} - 2 T^{3} + 11 T^{2} + \cdots + 49)^{2} \)
|
| $7$ |
\( T^{8} - 12 T^{5} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} + 6 T^{7} + \cdots + 35344 \)
|
| $13$ |
\( (T^{4} - 6 T^{3} - T^{2} + \cdots - 2)^{2} \)
|
| $17$ |
\( T^{8} - 10 T^{7} + \cdots + 21316 \)
|
| $19$ |
\( T^{8} - 4 T^{7} + \cdots + 4096 \)
|
| $23$ |
\( (T^{2} + T + 1)^{4} \)
|
| $29$ |
\( (T^{4} - 6 T^{3} - T^{2} + \cdots + 16)^{2} \)
|
| $31$ |
\( T^{8} + 2 T^{7} + \cdots + 35344 \)
|
| $37$ |
\( T^{8} + 58 T^{6} + \cdots + 18496 \)
|
| $41$ |
\( (T^{4} - 4 T^{3} - 30 T^{2} + \cdots + 64)^{2} \)
|
| $43$ |
\( (T^{4} - 20 T^{3} + \cdots - 5696)^{2} \)
|
| $47$ |
\( T^{8} + 10 T^{7} + \cdots + 1817104 \)
|
| $53$ |
\( T^{8} + 58 T^{6} + \cdots + 305809 \)
|
| $59$ |
\( T^{8} + 6 T^{7} + \cdots + 12544 \)
|
| $61$ |
\( T^{8} + 232 T^{6} + \cdots + 78287104 \)
|
| $67$ |
\( T^{8} + 18 T^{7} + \cdots + 125350416 \)
|
| $71$ |
\( (T^{4} - 42 T^{3} + \cdots + 10654)^{2} \)
|
| $73$ |
\( T^{8} + 6 T^{7} + \cdots + 1024 \)
|
| $79$ |
\( T^{8} - 6 T^{7} + \cdots + 256 \)
|
| $83$ |
\( (T^{4} + 10 T^{3} + \cdots + 1252)^{2} \)
|
| $89$ |
\( T^{8} + 106 T^{6} + \cdots + 399424 \)
|
| $97$ |
\( (T^{4} + 10 T^{3} + \cdots + 64)^{2} \)
|
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