Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [168,2,Mod(85,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("168.85");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 168.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: | \(1.34148675396\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.386672896.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + \nu^{4} + 2\nu^{3} - 2\nu^{2} + 4\nu + 4 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} + 3\nu^{5} - 4\nu^{4} + 2\nu^{3} - 4\nu - 24 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} - \nu^{5} + 2\nu^{3} - 4\nu - 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} + 2\nu^{4} - 2\nu^{3} + 4\nu^{2} + 4\nu ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 2\nu^{6} + \nu^{5} - 2\nu^{3} + 20\nu + 8 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - 6\nu^{6} - 3\nu^{5} - 2\nu^{3} + 8\nu^{2} + 20\nu + 40 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} - 2\nu^{6} - \nu^{5} + 2\nu^{3} + 8\nu^{2} + 12\nu + 8 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} - \beta_{5} + \beta_{3} + 2\beta_{2} + 2\beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{7} - \beta_{5} + 4\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 - 2 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2\beta_{7} - \beta_{5} + 4\beta_{4} - 3\beta_{3} + 6\beta_{2} + 2\beta _1 + 6 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2\beta_{7} - 4\beta_{6} + 3\beta_{5} + 5\beta_{3} - 2\beta_{2} - 2\beta _1 + 10 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -10\beta_{7} + 8\beta_{6} - \beta_{5} + 4\beta_{4} + 5\beta_{3} + 6\beta_{2} + 2\beta _1 - 2 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).
| \(n\) | \(73\) | \(85\) | \(113\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 85.1 |
|
−1.40961 | − | 0.114062i | 1.00000i | 1.97398 | + | 0.321565i | 1.12875i | 0.114062 | − | 1.40961i | 1.00000 | −2.74586 | − | 0.678435i | −1.00000 | 0.128747 | − | 1.59109i | ||||||||||||||||||||||||||||||||
| 85.2 | −1.40961 | + | 0.114062i | − | 1.00000i | 1.97398 | − | 0.321565i | − | 1.12875i | 0.114062 | + | 1.40961i | 1.00000 | −2.74586 | + | 0.678435i | −1.00000 | 0.128747 | + | 1.59109i | |||||||||||||||||||||||||||||||
| 85.3 | −0.621372 | − | 1.27039i | 1.00000i | −1.22779 | + | 1.57877i | − | 3.69833i | 1.27039 | − | 0.621372i | 1.00000 | 2.76858 | + | 0.578773i | −1.00000 | −4.69833 | + | 2.29804i | ||||||||||||||||||||||||||||||||
| 85.4 | −0.621372 | + | 1.27039i | − | 1.00000i | −1.22779 | − | 1.57877i | 3.69833i | 1.27039 | + | 0.621372i | 1.00000 | 2.76858 | − | 0.578773i | −1.00000 | −4.69833 | − | 2.29804i | ||||||||||||||||||||||||||||||||
| 85.5 | 0.835949 | − | 1.14070i | − | 1.00000i | −0.602380 | − | 1.90713i | − | 0.467138i | −1.14070 | − | 0.835949i | 1.00000 | −2.67901 | − | 0.907128i | −1.00000 | −0.532862 | − | 0.390503i | |||||||||||||||||||||||||||||||
| 85.6 | 0.835949 | + | 1.14070i | 1.00000i | −0.602380 | + | 1.90713i | 0.467138i | −1.14070 | + | 0.835949i | 1.00000 | −2.67901 | + | 0.907128i | −1.00000 | −0.532862 | + | 0.390503i | |||||||||||||||||||||||||||||||||
| 85.7 | 1.19503 | − | 0.756243i | 1.00000i | 0.856193 | − | 1.80747i | 4.10245i | 0.756243 | + | 1.19503i | 1.00000 | −0.343707 | − | 2.80747i | −1.00000 | 3.10245 | + | 4.90255i | |||||||||||||||||||||||||||||||||
| 85.8 | 1.19503 | + | 0.756243i | − | 1.00000i | 0.856193 | + | 1.80747i | − | 4.10245i | 0.756243 | − | 1.19503i | 1.00000 | −0.343707 | + | 2.80747i | −1.00000 | 3.10245 | − | 4.90255i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 168.2.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 504.2.c.f | 8 | ||
| 4.b | odd | 2 | 1 | 672.2.c.b | 8 | ||
| 7.b | odd | 2 | 1 | 1176.2.c.c | 8 | ||
| 8.b | even | 2 | 1 | inner | 168.2.c.b | ✓ | 8 |
| 8.d | odd | 2 | 1 | 672.2.c.b | 8 | ||
| 12.b | even | 2 | 1 | 2016.2.c.e | 8 | ||
| 16.e | even | 4 | 1 | 5376.2.a.bm | 4 | ||
| 16.e | even | 4 | 1 | 5376.2.a.bp | 4 | ||
| 16.f | odd | 4 | 1 | 5376.2.a.bl | 4 | ||
| 16.f | odd | 4 | 1 | 5376.2.a.bq | 4 | ||
| 24.f | even | 2 | 1 | 2016.2.c.e | 8 | ||
| 24.h | odd | 2 | 1 | 504.2.c.f | 8 | ||
| 28.d | even | 2 | 1 | 4704.2.c.c | 8 | ||
| 56.e | even | 2 | 1 | 4704.2.c.c | 8 | ||
| 56.h | odd | 2 | 1 | 1176.2.c.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 168.2.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 168.2.c.b | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 504.2.c.f | 8 | 3.b | odd | 2 | 1 | ||
| 504.2.c.f | 8 | 24.h | odd | 2 | 1 | ||
| 672.2.c.b | 8 | 4.b | odd | 2 | 1 | ||
| 672.2.c.b | 8 | 8.d | odd | 2 | 1 | ||
| 1176.2.c.c | 8 | 7.b | odd | 2 | 1 | ||
| 1176.2.c.c | 8 | 56.h | odd | 2 | 1 | ||
| 2016.2.c.e | 8 | 12.b | even | 2 | 1 | ||
| 2016.2.c.e | 8 | 24.f | even | 2 | 1 | ||
| 4704.2.c.c | 8 | 28.d | even | 2 | 1 | ||
| 4704.2.c.c | 8 | 56.e | even | 2 | 1 | ||
| 5376.2.a.bl | 4 | 16.f | odd | 4 | 1 | ||
| 5376.2.a.bm | 4 | 16.e | even | 4 | 1 | ||
| 5376.2.a.bp | 4 | 16.e | even | 4 | 1 | ||
| 5376.2.a.bq | 4 | 16.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 32T_{5}^{6} + 276T_{5}^{4} + 352T_{5}^{2} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{6} + \cdots + 16 \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{8} + 32 T^{6} + \cdots + 64 \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( T^{8} + 64 T^{6} + \cdots + 40000 \)
|
| $13$ |
\( T^{8} + 56 T^{6} + \cdots + 1024 \)
|
| $17$ |
\( (T^{4} - 2 T^{3} - 30 T^{2} + \cdots - 8)^{2} \)
|
| $19$ |
\( T^{8} + 88 T^{6} + \cdots + 16384 \)
|
| $23$ |
\( (T^{4} - 6 T^{3} - 18 T^{2} + \cdots - 64)^{2} \)
|
| $29$ |
\( T^{8} + 216 T^{6} + \cdots + 6885376 \)
|
| $31$ |
\( (T^{4} - 4 T^{3} + \cdots + 256)^{2} \)
|
| $37$ |
\( T^{8} + 104 T^{6} + \cdots + 65536 \)
|
| $41$ |
\( (T^{4} + 2 T^{3} - 30 T^{2} + \cdots - 8)^{2} \)
|
| $43$ |
\( T^{8} + 176 T^{6} + \cdots + 565504 \)
|
| $47$ |
\( (T^{4} - 144 T^{2} + \cdots + 3584)^{2} \)
|
| $53$ |
\( T^{8} + 88 T^{6} + \cdots + 16384 \)
|
| $59$ |
\( (T^{2} + 16)^{4} \)
|
| $61$ |
\( T^{8} + 216 T^{6} + \cdots + 409600 \)
|
| $67$ |
\( T^{8} + 192 T^{6} + \cdots + 12544 \)
|
| $71$ |
\( (T^{4} + 14 T^{3} + \cdots - 3136)^{2} \)
|
| $73$ |
\( (T^{4} + 4 T^{3} + \cdots - 2608)^{2} \)
|
| $79$ |
\( (T^{4} + 20 T^{3} + \cdots - 2560)^{2} \)
|
| $83$ |
\( T^{8} + 416 T^{6} + \cdots + 55115776 \)
|
| $89$ |
\( (T^{4} - 10 T^{3} + \cdots - 6280)^{2} \)
|
| $97$ |
\( (T^{4} - 20 T^{3} + \cdots - 13808)^{2} \)
|
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