Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,8,Mod(68,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.68");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 69.c (of order \(2\), degree \(1\), 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: | \(21.5545667584\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.8869743.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{3} + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{6} - 3x^{3} + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{2} + 4\nu ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{3} - \nu^{2} + 4\nu - 4 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} - 4\nu^{3} - \nu^{2} + 4\nu + 4 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} - 2\beta_{3} + 4\beta_1 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 2\beta_{3} + 12\beta_{2} + 2\beta_1 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -7\beta_{5} - 7\beta_{4} + 10\beta_{3} + 4\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(47\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 68.1 |
|
− | 21.8836i | 5.84430 | − | 46.3988i | −350.893 | 0 | −1015.37 | − | 127.894i | 0 | 4877.70i | −2118.69 | − | 542.337i | 0 | |||||||||||||||||||||||||||||
| 68.2 | − | 15.9248i | 37.2603 | − | 28.2607i | −125.600 | 0 | −450.047 | − | 593.365i | 0 | − | 38.2170i | 589.667 | − | 2106.01i | 0 | |||||||||||||||||||||||||||||
| 68.3 | − | 5.95879i | −43.1046 | − | 18.1381i | 92.4928 | 0 | −108.081 | + | 256.851i | 0 | − | 1313.87i | 1529.02 | + | 1563.67i | 0 | |||||||||||||||||||||||||||||
| 68.4 | 5.95879i | −43.1046 | + | 18.1381i | 92.4928 | 0 | −108.081 | − | 256.851i | 0 | 1313.87i | 1529.02 | − | 1563.67i | 0 | |||||||||||||||||||||||||||||||
| 68.5 | 15.9248i | 37.2603 | + | 28.2607i | −125.600 | 0 | −450.047 | + | 593.365i | 0 | 38.2170i | 589.667 | + | 2106.01i | 0 | |||||||||||||||||||||||||||||||
| 68.6 | 21.8836i | 5.84430 | + | 46.3988i | −350.893 | 0 | −1015.37 | + | 127.894i | 0 | − | 4877.70i | −2118.69 | + | 542.337i | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
| 3.b | odd | 2 | 1 | inner |
| 69.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 69.8.c.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 69.8.c.a | ✓ | 6 |
| 23.b | odd | 2 | 1 | CM | 69.8.c.a | ✓ | 6 |
| 69.c | even | 2 | 1 | inner | 69.8.c.a | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.8.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 69.8.c.a | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 69.8.c.a | ✓ | 6 | 23.b | odd | 2 | 1 | CM |
| 69.8.c.a | ✓ | 6 | 69.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 768T_{2}^{4} + 147456T_{2}^{2} + 4312247 \)
acting on \(S_{8}^{\mathrm{new}}(69, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 768 T^{4} + \cdots + 4312247 \)
|
| $3$ |
\( T^{6} + \cdots + 10460353203 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( (T^{3} - 188245551 T + 84809369662)^{2} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( (T^{2} + 3404825447)^{3} \)
|
| $29$ |
\( T^{6} + \cdots + 19\!\cdots\!92 \)
|
| $31$ |
\( (T^{3} + \cdots - 76\!\cdots\!64)^{2} \)
|
| $37$ |
\( T^{6} \)
|
| $41$ |
\( T^{6} + \cdots + 26\!\cdots\!88 \)
|
| $43$ |
\( T^{6} \)
|
| $47$ |
\( T^{6} + \cdots + 39\!\cdots\!92 \)
|
| $53$ |
\( T^{6} \)
|
| $59$ |
\( (T^{2} + 9953995454012)^{3} \)
|
| $61$ |
\( T^{6} \)
|
| $67$ |
\( T^{6} \)
|
| $71$ |
\( T^{6} + \cdots + 27\!\cdots\!08 \)
|
| $73$ |
\( (T^{3} + \cdots - 19\!\cdots\!62)^{2} \)
|
| $79$ |
\( T^{6} \)
|
| $83$ |
\( T^{6} \)
|
| $89$ |
\( T^{6} \)
|
| $97$ |
\( T^{6} \)
|
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