Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [208,2,Mod(77,208)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(208, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("208.77");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.p (of order \(4\), 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: | \(1.66088836204\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.959512576.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 7x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 7x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + \nu ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 16\nu^{2} ) / 45 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{4} - \nu^{2} + 3 ) / 15 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} + 13\nu^{3} ) / 135 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 16\nu^{3} + 45\nu ) / 45 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 3\nu^{4} - \nu^{2} - 3 ) / 15 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - 16\nu^{3} + 45\nu ) / 45 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{5} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} + 6\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{5} + 3\beta_{4} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{6} + 5\beta_{3} - 2 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{7} - \beta_{5} + 30\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{6} - 8\beta_{3} - 3\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -13\beta_{7} + 13\beta_{5} - 96\beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 77.1 |
|
− | 1.41421i | −1.15831 | − | 1.15831i | −2.00000 | 0.707107 | + | 0.707107i | −1.63810 | + | 1.63810i | −4.46653 | 2.82843i | − | 0.316625i | 1.00000 | − | 1.00000i | ||||||||||||||||||||||||||||||||
| 77.2 | − | 1.41421i | 2.15831 | + | 2.15831i | −2.00000 | 0.707107 | + | 0.707107i | 3.05231 | − | 3.05231i | 0.223888 | 2.82843i | 6.31662i | 1.00000 | − | 1.00000i | ||||||||||||||||||||||||||||||||||
| 77.3 | 1.41421i | −1.15831 | − | 1.15831i | −2.00000 | −0.707107 | − | 0.707107i | 1.63810 | − | 1.63810i | 4.46653 | − | 2.82843i | − | 0.316625i | 1.00000 | − | 1.00000i | |||||||||||||||||||||||||||||||||
| 77.4 | 1.41421i | 2.15831 | + | 2.15831i | −2.00000 | −0.707107 | − | 0.707107i | −3.05231 | + | 3.05231i | −0.223888 | − | 2.82843i | 6.31662i | 1.00000 | − | 1.00000i | ||||||||||||||||||||||||||||||||||
| 181.1 | − | 1.41421i | −1.15831 | + | 1.15831i | −2.00000 | −0.707107 | + | 0.707107i | 1.63810 | + | 1.63810i | 4.46653 | 2.82843i | 0.316625i | 1.00000 | + | 1.00000i | ||||||||||||||||||||||||||||||||||
| 181.2 | − | 1.41421i | 2.15831 | − | 2.15831i | −2.00000 | −0.707107 | + | 0.707107i | −3.05231 | − | 3.05231i | −0.223888 | 2.82843i | − | 6.31662i | 1.00000 | + | 1.00000i | |||||||||||||||||||||||||||||||||
| 181.3 | 1.41421i | −1.15831 | + | 1.15831i | −2.00000 | 0.707107 | − | 0.707107i | −1.63810 | − | 1.63810i | −4.46653 | − | 2.82843i | 0.316625i | 1.00000 | + | 1.00000i | ||||||||||||||||||||||||||||||||||
| 181.4 | 1.41421i | 2.15831 | − | 2.15831i | −2.00000 | 0.707107 | − | 0.707107i | 3.05231 | + | 3.05231i | 0.223888 | − | 2.82843i | − | 6.31662i | 1.00000 | + | 1.00000i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
| 16.e | even | 4 | 1 | inner |
| 208.p | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 208.2.p.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 832.2.p.a | 8 | ||
| 13.b | even | 2 | 1 | inner | 208.2.p.a | ✓ | 8 |
| 16.e | even | 4 | 1 | inner | 208.2.p.a | ✓ | 8 |
| 16.f | odd | 4 | 1 | 832.2.p.a | 8 | ||
| 52.b | odd | 2 | 1 | 832.2.p.a | 8 | ||
| 208.o | odd | 4 | 1 | 832.2.p.a | 8 | ||
| 208.p | even | 4 | 1 | inner | 208.2.p.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 208.2.p.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 208.2.p.a | ✓ | 8 | 13.b | even | 2 | 1 | inner |
| 208.2.p.a | ✓ | 8 | 16.e | even | 4 | 1 | inner |
| 208.2.p.a | ✓ | 8 | 208.p | even | 4 | 1 | inner |
| 832.2.p.a | 8 | 4.b | odd | 2 | 1 | ||
| 832.2.p.a | 8 | 16.f | odd | 4 | 1 | ||
| 832.2.p.a | 8 | 52.b | odd | 2 | 1 | ||
| 832.2.p.a | 8 | 208.o | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 10T_{3} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(208, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{4} \)
|
| $3$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 25)^{2} \)
|
| $5$ |
\( (T^{4} + 1)^{2} \)
|
| $7$ |
\( (T^{4} - 20 T^{2} + 1)^{2} \)
|
| $11$ |
\( (T^{4} + 16)^{2} \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots + 169)^{2} \)
|
| $17$ |
\( (T - 3)^{8} \)
|
| $19$ |
\( T^{8} + 1592T^{4} + 16 \)
|
| $23$ |
\( (T^{4} + 40 T^{2} + 4)^{2} \)
|
| $29$ |
\( (T^{4} + 12 T^{3} + \cdots + 16)^{2} \)
|
| $31$ |
\( (T^{2} + 18)^{4} \)
|
| $37$ |
\( T^{8} + 7586 T^{4} + 390625 \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( (T^{4} + 2 T^{3} + \cdots + 2401)^{2} \)
|
| $47$ |
\( (T^{4} + 124 T^{2} + 1369)^{2} \)
|
| $53$ |
\( (T^{4} + 12 T^{3} + \cdots + 4900)^{2} \)
|
| $59$ |
\( T^{8} + 20792 T^{4} + 92236816 \)
|
| $61$ |
\( (T^{2} - 8 T + 32)^{4} \)
|
| $67$ |
\( (T^{4} + 1936)^{2} \)
|
| $71$ |
\( (T^{4} - 180 T^{2} + 81)^{2} \)
|
| $73$ |
\( (T^{4} - 188 T^{2} + 2500)^{2} \)
|
| $79$ |
\( (T^{2} + 2 T - 98)^{4} \)
|
| $83$ |
\( T^{8} + 82616 T^{4} + 6250000 \)
|
| $89$ |
\( (T^{2} - 198)^{4} \)
|
| $97$ |
\( (T^{2} + 72)^{4} \)
|
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