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: | 8.0.6498455769.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 25x^{4} - 3x^{3} + 6x^{2} + x + 1 \)
|
| 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
\( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 25x^{4} - 3x^{3} + 6x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -14\nu^{7} + 36\nu^{6} - 126\nu^{5} + 105\nu^{4} - 396\nu^{3} + 504\nu^{2} - 672\nu + 90 ) / 119 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -15\nu^{7} + 7\nu^{6} - 84\nu^{5} - 66\nu^{4} - 434\nu^{3} - 21\nu^{2} - 6\nu + 315 ) / 119 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 20\nu^{7} - 15\nu^{6} + 112\nu^{5} + 88\nu^{4} + 522\nu^{3} + 28\nu^{2} + 8\nu + 22 ) / 119 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -22\nu^{7} + 42\nu^{6} - 147\nu^{5} + 46\nu^{4} - 462\nu^{3} + 588\nu^{2} - 104\nu + 105 ) / 119 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 24\nu^{7} - 69\nu^{6} + 182\nu^{5} - 180\nu^{4} + 402\nu^{3} - 1085\nu^{2} + 81\nu + 6 ) / 119 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 105\nu^{7} - 83\nu^{6} + 588\nu^{5} + 462\nu^{4} + 2579\nu^{3} + 147\nu^{2} + 42\nu + 209 ) / 119 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} + \beta_{4} + \beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + 6\beta_{4} + \beta_{3} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{6} - 10\beta_{5} + 6\beta_{3} - \beta_{2} - 10\beta _1 - 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{7} - 10\beta_{6} - 13\beta_{5} - 38\beta_{4} - 6\beta_{2} - 38\beta _1 + 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{7} - 81\beta_{4} - 38\beta_{3} + 98 \)
|
| \(\nu^{7}\) | \(=\) |
\( 81\beta_{6} + 114\beta_{5} - 81\beta_{3} + 38\beta_{2} + 255\beta _1 + 81 \)
|
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 | −1.33821 | − | 2.31784i | −0.500000 | − | 0.866025i | 1.93020 | − | 3.34320i | −2.67641 | −2.21184 | − | 1.45181i | −1.00000 | −2.08159 | + | 3.60541i | −1.93020 | − | 3.34320i | ||||||||||||||||||||||||||||
| 93.2 | 0.500000 | − | 0.866025i | −0.271028 | − | 0.469434i | −0.500000 | − | 0.866025i | 0.298300 | − | 0.516670i | −0.542055 | −2.61586 | + | 0.396592i | −1.00000 | 1.35309 | − | 2.34362i | −0.298300 | − | 0.516670i | |||||||||||||||||||||||||||||
| 93.3 | 0.500000 | − | 0.866025i | 0.186817 | + | 0.323577i | −0.500000 | − | 0.866025i | −1.58159 | + | 2.73939i | 0.373635 | 2.36323 | + | 1.18960i | −1.00000 | 1.43020 | − | 2.47718i | 1.58159 | + | 2.73939i | |||||||||||||||||||||||||||||
| 93.4 | 0.500000 | − | 0.866025i | 0.922415 | + | 1.59767i | −0.500000 | − | 0.866025i | 1.85309 | − | 3.20964i | 1.84483 | 0.964471 | + | 2.46370i | −1.00000 | −0.201700 | + | 0.349355i | −1.85309 | − | 3.20964i | |||||||||||||||||||||||||||||
| 277.1 | 0.500000 | + | 0.866025i | −1.33821 | + | 2.31784i | −0.500000 | + | 0.866025i | 1.93020 | + | 3.34320i | −2.67641 | −2.21184 | + | 1.45181i | −1.00000 | −2.08159 | − | 3.60541i | −1.93020 | + | 3.34320i | |||||||||||||||||||||||||||||
| 277.2 | 0.500000 | + | 0.866025i | −0.271028 | + | 0.469434i | −0.500000 | + | 0.866025i | 0.298300 | + | 0.516670i | −0.542055 | −2.61586 | − | 0.396592i | −1.00000 | 1.35309 | + | 2.34362i | −0.298300 | + | 0.516670i | |||||||||||||||||||||||||||||
| 277.3 | 0.500000 | + | 0.866025i | 0.186817 | − | 0.323577i | −0.500000 | + | 0.866025i | −1.58159 | − | 2.73939i | 0.373635 | 2.36323 | − | 1.18960i | −1.00000 | 1.43020 | + | 2.47718i | 1.58159 | − | 2.73939i | |||||||||||||||||||||||||||||
| 277.4 | 0.500000 | + | 0.866025i | 0.922415 | − | 1.59767i | −0.500000 | + | 0.866025i | 1.85309 | + | 3.20964i | 1.84483 | 0.964471 | − | 2.46370i | −1.00000 | −0.201700 | − | 0.349355i | −1.85309 | + | 3.20964i | |||||||||||||||||||||||||||||
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.d | ✓ | 8 |
| 7.c | even | 3 | 1 | inner | 322.2.e.d | ✓ | 8 |
| 7.c | even | 3 | 1 | 2254.2.a.u | 4 | ||
| 7.d | odd | 6 | 1 | 2254.2.a.r | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.2.e.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 322.2.e.d | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 2254.2.a.r | 4 | 7.d | odd | 6 | 1 | ||
| 2254.2.a.u | 4 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + T_{3}^{7} + 6T_{3}^{6} - 3T_{3}^{5} + 25T_{3}^{4} + 3T_{3}^{3} + 6T_{3}^{2} - T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{4} \)
|
| $3$ |
\( T^{8} + T^{7} + 6 T^{6} + \cdots + 1 \)
|
| $5$ |
\( T^{8} - 5 T^{7} + \cdots + 729 \)
|
| $7$ |
\( T^{8} + 3 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} - 2 T^{7} + \cdots + 729 \)
|
| $13$ |
\( (T^{4} - 7 T^{3} + 38 T - 9)^{2} \)
|
| $17$ |
\( T^{8} - 7 T^{7} + \cdots + 81 \)
|
| $19$ |
\( T^{8} - T^{7} + 18 T^{6} + \cdots + 9 \)
|
| $23$ |
\( (T^{2} - T + 1)^{4} \)
|
| $29$ |
\( (T^{4} + 6 T^{3} + \cdots + 147)^{2} \)
|
| $31$ |
\( T^{8} - 4 T^{7} + \cdots + 36481 \)
|
| $37$ |
\( T^{8} + 70 T^{6} + \cdots + 301401 \)
|
| $41$ |
\( (T^{4} + 15 T^{3} + \cdots - 7647)^{2} \)
|
| $43$ |
\( (T^{4} + 12 T^{3} + \cdots - 547)^{2} \)
|
| $47$ |
\( T^{8} - 15 T^{7} + \cdots + 2424249 \)
|
| $53$ |
\( T^{8} + 21 T^{7} + \cdots + 531441 \)
|
| $59$ |
\( T^{8} - 32 T^{7} + \cdots + 1565001 \)
|
| $61$ |
\( T^{8} - 3 T^{7} + \cdots + 729 \)
|
| $67$ |
\( T^{8} + 13 T^{7} + \cdots + 9 \)
|
| $71$ |
\( (T^{4} + 7 T^{3} + \cdots + 807)^{2} \)
|
| $73$ |
\( T^{8} - 16 T^{7} + \cdots + 1750329 \)
|
| $79$ |
\( T^{8} + 3 T^{7} + \cdots + 1495729 \)
|
| $83$ |
\( (T^{4} - 8 T^{3} + 9 T^{2} + \cdots + 9)^{2} \)
|
| $89$ |
\( T^{8} - 33 T^{7} + \cdots + 13461561 \)
|
| $97$ |
\( (T^{4} + 12 T^{3} + \cdots - 199)^{2} \)
|
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