Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [88,3,Mod(3,88)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(88, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 5, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("88.3");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 88 = 2^{3} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 88.l (of order \(10\), degree \(4\), 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: | \(2.39782632637\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | 8.0.64000000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{10}]$ |
$q$-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
\( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 4\beta_{7} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).
| \(n\) | \(23\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
1.61803 | − | 1.17557i | −0.527486 | − | 1.62344i | 1.23607 | − | 3.80423i | 0 | −2.76195 | − | 2.00668i | 0 | −2.47214 | − | 7.60845i | 4.92385 | − | 3.57739i | 0 | ||||||||||||||||||||||||||||||
| 3.2 | 1.61803 | − | 1.17557i | 1.52749 | + | 4.70112i | 1.23607 | − | 3.80423i | 0 | 7.99802 | + | 5.81090i | 0 | −2.47214 | − | 7.60845i | −12.4862 | + | 9.07172i | 0 | |||||||||||||||||||||||||||||||
| 27.1 | −0.618034 | − | 1.90211i | −3.85250 | − | 2.79901i | −3.23607 | + | 2.35114i | 0 | −2.94305 | + | 9.05777i | 0 | 6.47214 | + | 4.70228i | 4.22618 | + | 13.0068i | 0 | |||||||||||||||||||||||||||||||
| 27.2 | −0.618034 | − | 1.90211i | 4.85250 | + | 3.52555i | −3.23607 | + | 2.35114i | 0 | 3.70698 | − | 11.4089i | 0 | 6.47214 | + | 4.70228i | 8.33613 | + | 25.6560i | 0 | |||||||||||||||||||||||||||||||
| 59.1 | 1.61803 | + | 1.17557i | −0.527486 | + | 1.62344i | 1.23607 | + | 3.80423i | 0 | −2.76195 | + | 2.00668i | 0 | −2.47214 | + | 7.60845i | 4.92385 | + | 3.57739i | 0 | |||||||||||||||||||||||||||||||
| 59.2 | 1.61803 | + | 1.17557i | 1.52749 | − | 4.70112i | 1.23607 | + | 3.80423i | 0 | 7.99802 | − | 5.81090i | 0 | −2.47214 | + | 7.60845i | −12.4862 | − | 9.07172i | 0 | |||||||||||||||||||||||||||||||
| 75.1 | −0.618034 | + | 1.90211i | −3.85250 | + | 2.79901i | −3.23607 | − | 2.35114i | 0 | −2.94305 | − | 9.05777i | 0 | 6.47214 | − | 4.70228i | 4.22618 | − | 13.0068i | 0 | |||||||||||||||||||||||||||||||
| 75.2 | −0.618034 | + | 1.90211i | 4.85250 | − | 3.52555i | −3.23607 | − | 2.35114i | 0 | 3.70698 | + | 11.4089i | 0 | 6.47214 | − | 4.70228i | 8.33613 | − | 25.6560i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 11.c | even | 5 | 1 | inner |
| 88.l | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 88.3.l.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 352.3.t.a | 8 | ||
| 8.b | even | 2 | 1 | 352.3.t.a | 8 | ||
| 8.d | odd | 2 | 1 | CM | 88.3.l.a | ✓ | 8 |
| 11.c | even | 5 | 1 | inner | 88.3.l.a | ✓ | 8 |
| 44.h | odd | 10 | 1 | 352.3.t.a | 8 | ||
| 88.l | odd | 10 | 1 | inner | 88.3.l.a | ✓ | 8 |
| 88.o | even | 10 | 1 | 352.3.t.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.3.l.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 88.3.l.a | ✓ | 8 | 8.d | odd | 2 | 1 | CM |
| 88.3.l.a | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 88.3.l.a | ✓ | 8 | 88.l | odd | 10 | 1 | inner |
| 352.3.t.a | 8 | 4.b | odd | 2 | 1 | ||
| 352.3.t.a | 8 | 8.b | even | 2 | 1 | ||
| 352.3.t.a | 8 | 44.h | odd | 10 | 1 | ||
| 352.3.t.a | 8 | 88.o | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 4T_{3}^{7} + 12T_{3}^{6} + 58T_{3}^{5} + 350T_{3}^{4} - 668T_{3}^{3} + 19497T_{3}^{2} + 17834T_{3} + 58081 \)
acting on \(S_{3}^{\mathrm{new}}(88, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 58081 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + 14 T^{7} + \cdots + 214358881 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} + \cdots + 169613009281 \)
|
| $19$ |
\( T^{8} + \cdots + 9729652321 \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( T^{8} + \cdots + 674502480961 \)
|
| $43$ |
\( (T^{4} + 14 T^{3} + \cdots + 15320401)^{2} \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( T^{8} \)
|
| $59$ |
\( T^{8} + \cdots + 126163192953121 \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( (T^{4} + 62 T^{3} + \cdots + 29253361)^{2} \)
|
| $71$ |
\( T^{8} \)
|
| $73$ |
\( T^{8} + \cdots + 127878123839041 \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( T^{8} + \cdots + 369567942241 \)
|
| $89$ |
\( (T^{4} + 146 T^{3} + \cdots - 76137119)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!61 \)
|
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