Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [946,2,Mod(221,946)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(946, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("946.221");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 946 = 2 \cdot 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 946.e (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: | \(7.55384803121\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.14648745024.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 7x^{6} + 47x^{4} + 14x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{8} + 7x^{6} + 47x^{4} + 14x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{6} + 47\nu^{4} + 329\nu^{2} + 4 ) / 94 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 13\nu^{6} + 94\nu^{4} + 611\nu^{2} + 268\nu + 182 ) / 94 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 13\nu^{6} + 94\nu^{4} + 611\nu^{2} - 268\nu + 182 ) / 94 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{7} - 47\nu^{5} - 329\nu^{3} - 4\nu ) / 47 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -21\nu^{7} - \nu^{6} - 141\nu^{5} - 940\nu^{3} - 12\nu + 174 ) / 94 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 20\nu^{7} - 14\nu^{6} + 141\nu^{5} - 94\nu^{4} + 940\nu^{3} - 611\nu^{2} + 280\nu - 8 ) / 94 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{3} + 4\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{4} + 7\beta_{3} - 26\beta_{2} - 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{7} - 7\beta_{6} + 20\beta_{5} + 7\beta_{3} - 20\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -47\beta_{7} - 47\beta_{6} - 47\beta_{4} + 174 \)
|
| \(\nu^{7}\) | \(=\) |
\( 47\beta_{4} - 47\beta_{3} + 134\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/946\mathbb{Z}\right)^\times\).
| \(n\) | \(89\) | \(431\) |
| \(\chi(n)\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 221.1 |
|
1.00000 | −1.70407 | + | 2.95154i | 1.00000 | 1.05737 | − | 1.83141i | −1.70407 | + | 2.95154i | 1.50000 | + | 2.59808i | 1.00000 | −4.30774 | − | 7.46122i | 1.05737 | − | 1.83141i | ||||||||||||||||||||||||||||||
| 221.2 | 1.00000 | −0.146707 | + | 0.254104i | 1.00000 | −2.05737 | + | 3.56346i | −0.146707 | + | 0.254104i | 1.50000 | + | 2.59808i | 1.00000 | 1.45695 | + | 2.52352i | −2.05737 | + | 3.56346i | |||||||||||||||||||||||||||||||
| 221.3 | 1.00000 | 0.221350 | − | 0.383390i | 1.00000 | 0.408080 | − | 0.706815i | 0.221350 | − | 0.383390i | 1.50000 | + | 2.59808i | 1.00000 | 1.40201 | + | 2.42835i | 0.408080 | − | 0.706815i | |||||||||||||||||||||||||||||||
| 221.4 | 1.00000 | 1.12943 | − | 1.95623i | 1.00000 | −1.40808 | + | 2.43887i | 1.12943 | − | 1.95623i | 1.50000 | + | 2.59808i | 1.00000 | −1.05123 | − | 1.82078i | −1.40808 | + | 2.43887i | |||||||||||||||||||||||||||||||
| 595.1 | 1.00000 | −1.70407 | − | 2.95154i | 1.00000 | 1.05737 | + | 1.83141i | −1.70407 | − | 2.95154i | 1.50000 | − | 2.59808i | 1.00000 | −4.30774 | + | 7.46122i | 1.05737 | + | 1.83141i | |||||||||||||||||||||||||||||||
| 595.2 | 1.00000 | −0.146707 | − | 0.254104i | 1.00000 | −2.05737 | − | 3.56346i | −0.146707 | − | 0.254104i | 1.50000 | − | 2.59808i | 1.00000 | 1.45695 | − | 2.52352i | −2.05737 | − | 3.56346i | |||||||||||||||||||||||||||||||
| 595.3 | 1.00000 | 0.221350 | + | 0.383390i | 1.00000 | 0.408080 | + | 0.706815i | 0.221350 | + | 0.383390i | 1.50000 | − | 2.59808i | 1.00000 | 1.40201 | − | 2.42835i | 0.408080 | + | 0.706815i | |||||||||||||||||||||||||||||||
| 595.4 | 1.00000 | 1.12943 | + | 1.95623i | 1.00000 | −1.40808 | − | 2.43887i | 1.12943 | + | 1.95623i | 1.50000 | − | 2.59808i | 1.00000 | −1.05123 | + | 1.82078i | −1.40808 | − | 2.43887i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 946.2.e.k | ✓ | 8 |
| 43.c | even | 3 | 1 | inner | 946.2.e.k | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 946.2.e.k | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 946.2.e.k | ✓ | 8 | 43.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\):
|
\( T_{3}^{8} + T_{3}^{7} + 9T_{3}^{6} - 10T_{3}^{5} + 62T_{3}^{4} - 10T_{3}^{3} + 9T_{3}^{2} + T_{3} + 1 \)
|
|
\( T_{5}^{8} + 4T_{5}^{7} + 23T_{5}^{6} + 16T_{5}^{5} + 117T_{5}^{4} - 6T_{5}^{3} + 624T_{5}^{2} - 440T_{5} + 400 \)
|
|
\( T_{7}^{2} - 3T_{7} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} + T^{7} + 9 T^{6} + \cdots + 1 \)
|
| $5$ |
\( T^{8} + 4 T^{7} + \cdots + 400 \)
|
| $7$ |
\( (T^{2} - 3 T + 9)^{4} \)
|
| $11$ |
\( (T + 1)^{8} \)
|
| $13$ |
\( T^{8} - 2 T^{7} + \cdots + 625 \)
|
| $17$ |
\( T^{8} - T^{7} + \cdots + 10609 \)
|
| $19$ |
\( T^{8} + 17 T^{7} + \cdots + 93025 \)
|
| $23$ |
\( T^{8} + 6 T^{7} + \cdots + 250000 \)
|
| $29$ |
\( T^{8} + 7 T^{7} + \cdots + 323761 \)
|
| $31$ |
\( T^{8} - 17 T^{7} + \cdots + 38809 \)
|
| $37$ |
\( T^{8} + T^{7} + \cdots + 12769 \)
|
| $41$ |
\( (T^{4} + 8 T^{3} + \cdots + 1660)^{2} \)
|
| $43$ |
\( (T^{4} + 7 T^{3} + \cdots + 1849)^{2} \)
|
| $47$ |
\( (T^{4} + 2 T^{3} + \cdots + 7232)^{2} \)
|
| $53$ |
\( T^{8} - 8 T^{7} + \cdots + 425104 \)
|
| $59$ |
\( (T^{4} - 16 T^{3} + \cdots + 80)^{2} \)
|
| $61$ |
\( T^{8} - 19 T^{7} + \cdots + 5329 \)
|
| $67$ |
\( T^{8} + T^{7} + \cdots + 218089 \)
|
| $71$ |
\( T^{8} + T^{7} + \cdots + 143641 \)
|
| $73$ |
\( T^{8} + 2 T^{7} + \cdots + 112954384 \)
|
| $79$ |
\( T^{8} - 25 T^{7} + \cdots + 34225 \)
|
| $83$ |
\( T^{8} + 4 T^{7} + \cdots + 9517225 \)
|
| $89$ |
\( T^{8} - 42 T^{7} + \cdots + 6370576 \)
|
| $97$ |
\( (T^{4} + 14 T^{3} + \cdots + 784)^{2} \)
|
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