Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [322,2,Mod(93,322)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("322.93");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 322 = 2 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 322.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: | \(2.57118294509\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.310217769.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -49\nu^{7} + 64\nu^{6} - 256\nu^{5} + 338\nu^{4} - 1088\nu^{3} + 1216\nu^{2} - 1385\nu + 256 ) / 289 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 60\nu^{7} + 16\nu^{6} + 225\nu^{5} - 60\nu^{4} + 884\nu^{3} + 15\nu^{2} + 304\nu + 64 ) / 289 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 63\nu^{7} - 41\nu^{6} + 164\nu^{5} - 352\nu^{4} + 697\nu^{3} - 779\nu^{2} - 490\nu - 164 ) / 289 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -64\nu^{7} + 60\nu^{6} - 240\nu^{5} + 353\nu^{4} - 1020\nu^{3} + 1140\nu^{2} - 305\nu + 240 ) / 289 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -164\nu^{7} - 63\nu^{6} - 615\nu^{5} + 164\nu^{4} - 2108\nu^{3} - 41\nu^{2} - 41\nu + 37 ) / 289 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 180\nu^{7} + 48\nu^{6} + 675\nu^{5} - 180\nu^{4} + 2363\nu^{3} + 45\nu^{2} + 45\nu + 481 ) / 289 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{2} - \beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + 3\beta_{3} - 3\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{5} - 4\beta_{4} + 4\beta_{2} + 5\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{7} + \beta_{6} + 6\beta_{5} + \beta_{4} - 11\beta_{3} - 5\beta_{2} - 5\beta _1 - 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( -16\beta_{7} - 15\beta_{6} + 7\beta_{3} - 7\beta _1 + 27 \)
|
| \(\nu^{7}\) | \(=\) |
\( -30\beta_{5} - 8\beta_{4} + 23\beta_{2} + 65\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).
| \(n\) | \(185\) | \(281\) |
| \(\chi(n)\) | \(-1 + \beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 93.1 |
|
−0.500000 | + | 0.866025i | −1.28821 | − | 2.23124i | −0.500000 | − | 0.866025i | −0.663319 | + | 1.14890i | 2.57641 | −1.46157 | + | 2.20541i | 1.00000 | −1.81896 | + | 3.15053i | −0.663319 | − | 1.14890i | ||||||||||||||||||||||||||||
| 93.2 | −0.500000 | + | 0.866025i | −0.873734 | − | 1.51335i | −0.500000 | − | 0.866025i | −2.13304 | + | 3.69453i | 1.74747 | 1.89234 | − | 1.84906i | 1.00000 | −0.0268230 | + | 0.0464588i | −2.13304 | − | 3.69453i | |||||||||||||||||||||||||||||
| 93.3 | −0.500000 | + | 0.866025i | 0.0985631 | + | 0.170716i | −0.500000 | − | 0.866025i | 0.154437 | − | 0.267494i | −0.197126 | −1.71031 | − | 2.01862i | 1.00000 | 1.48057 | − | 2.56442i | 0.154437 | + | 0.267494i | |||||||||||||||||||||||||||||
| 93.4 | −0.500000 | + | 0.866025i | 0.563379 | + | 0.975800i | −0.500000 | − | 0.866025i | −0.858079 | + | 1.48624i | −1.12676 | 0.779537 | + | 2.52830i | 1.00000 | 0.865209 | − | 1.49859i | −0.858079 | − | 1.48624i | |||||||||||||||||||||||||||||
| 277.1 | −0.500000 | − | 0.866025i | −1.28821 | + | 2.23124i | −0.500000 | + | 0.866025i | −0.663319 | − | 1.14890i | 2.57641 | −1.46157 | − | 2.20541i | 1.00000 | −1.81896 | − | 3.15053i | −0.663319 | + | 1.14890i | |||||||||||||||||||||||||||||
| 277.2 | −0.500000 | − | 0.866025i | −0.873734 | + | 1.51335i | −0.500000 | + | 0.866025i | −2.13304 | − | 3.69453i | 1.74747 | 1.89234 | + | 1.84906i | 1.00000 | −0.0268230 | − | 0.0464588i | −2.13304 | + | 3.69453i | |||||||||||||||||||||||||||||
| 277.3 | −0.500000 | − | 0.866025i | 0.0985631 | − | 0.170716i | −0.500000 | + | 0.866025i | 0.154437 | + | 0.267494i | −0.197126 | −1.71031 | + | 2.01862i | 1.00000 | 1.48057 | + | 2.56442i | 0.154437 | − | 0.267494i | |||||||||||||||||||||||||||||
| 277.4 | −0.500000 | − | 0.866025i | 0.563379 | − | 0.975800i | −0.500000 | + | 0.866025i | −0.858079 | − | 1.48624i | −1.12676 | 0.779537 | − | 2.52830i | 1.00000 | 0.865209 | + | 1.49859i | −0.858079 | + | 1.48624i | |||||||||||||||||||||||||||||
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 | 322.2.e.a | ✓ | 8 |
| 7.c | even | 3 | 1 | inner | 322.2.e.a | ✓ | 8 |
| 7.c | even | 3 | 1 | 2254.2.a.z | 4 | ||
| 7.d | odd | 6 | 1 | 2254.2.a.x | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.2.e.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 322.2.e.a | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 2254.2.a.x | 4 | 7.d | odd | 6 | 1 | ||
| 2254.2.a.z | 4 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 3T_{3}^{7} + 10T_{3}^{6} + 7T_{3}^{5} + 15T_{3}^{4} - T_{3}^{3} + 26T_{3}^{2} - 5T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{4} \)
|
| $3$ |
\( T^{8} + 3 T^{7} + \cdots + 1 \)
|
| $5$ |
\( T^{8} + 7 T^{7} + \cdots + 9 \)
|
| $7$ |
\( T^{8} + T^{7} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} + 2 T^{7} + \cdots + 81 \)
|
| $13$ |
\( (T^{4} - T^{3} - 42 T^{2} + 343)^{2} \)
|
| $17$ |
\( T^{8} + 5 T^{7} + \cdots + 178929 \)
|
| $19$ |
\( T^{8} + 11 T^{7} + \cdots + 10201 \)
|
| $23$ |
\( (T^{2} - T + 1)^{4} \)
|
| $29$ |
\( (T^{4} - 2 T^{3} - 11 T^{2} + \cdots - 21)^{2} \)
|
| $31$ |
\( T^{8} + 6 T^{7} + \cdots + 499849 \)
|
| $37$ |
\( T^{8} - 8 T^{7} + \cdots + 1868689 \)
|
| $41$ |
\( (T^{4} - 9 T^{3} + \cdots + 417)^{2} \)
|
| $43$ |
\( (T^{4} - 4 T^{3} + \cdots + 1021)^{2} \)
|
| $47$ |
\( T^{8} + 11 T^{7} + \cdots + 471969 \)
|
| $53$ |
\( T^{8} + T^{7} + \cdots + 870489 \)
|
| $59$ |
\( T^{8} + 12 T^{7} + \cdots + 6561 \)
|
| $61$ |
\( T^{8} + 21 T^{7} + \cdots + 155575729 \)
|
| $67$ |
\( T^{8} - 3 T^{7} + \cdots + 33953929 \)
|
| $71$ |
\( (T^{4} - 11 T^{3} + \cdots - 189)^{2} \)
|
| $73$ |
\( T^{8} - 16 T^{7} + \cdots + 2229049 \)
|
| $79$ |
\( T^{8} - 21 T^{7} + \cdots + 2809 \)
|
| $83$ |
\( (T^{4} - 4 T^{3} + \cdots + 5913)^{2} \)
|
| $89$ |
\( T^{8} + 27 T^{7} + \cdots + 15992001 \)
|
| $97$ |
\( (T^{4} - 6 T^{3} - 33 T^{2} + \cdots - 23)^{2} \)
|
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