Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [255,2,Mod(98,255)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(255, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("255.98");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 255 = 3 \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 255.k (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: | \(2.03618525154\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.1871773696.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 31x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{8} + 31x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 19\nu ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 40\nu^{2} ) / 63 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 12 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{6} - 97\nu^{2} ) / 63 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 40\nu^{3} ) / 63 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{7} + 97\nu^{3} ) / 189 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 4\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{7} + 4\beta_{6} \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{4} - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 21\beta_{2} - 19\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -40\beta_{5} - 97\beta_{3} \)
|
| \(\nu^{7}\) | \(=\) |
\( 120\beta_{7} - 97\beta_{6} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/255\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(86\) | \(241\) |
| \(\chi(n)\) | \(\beta_{3}\) | \(-1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 98.1 |
|
−1.62831 | + | 1.62831i | 1.41421 | − | 1.00000i | − | 3.30278i | 0.707107 | + | 2.12132i | −0.674467 | + | 3.93108i | 1.00000i | 2.12132 | + | 2.12132i | 1.00000 | − | 2.82843i | −4.60555 | − | 2.30278i | |||||||||||||||||||||||||||
| 98.2 | −0.921201 | + | 0.921201i | −1.41421 | − | 1.00000i | 0.302776i | −0.707107 | − | 2.12132i | 2.22398 | − | 0.381574i | 1.00000i | −2.12132 | − | 2.12132i | 1.00000 | + | 2.82843i | 2.60555 | + | 1.30278i | |||||||||||||||||||||||||||||
| 98.3 | 0.921201 | − | 0.921201i | 1.41421 | − | 1.00000i | 0.302776i | 0.707107 | + | 2.12132i | 0.381574 | − | 2.22398i | 1.00000i | 2.12132 | + | 2.12132i | 1.00000 | − | 2.82843i | 2.60555 | + | 1.30278i | |||||||||||||||||||||||||||||
| 98.4 | 1.62831 | − | 1.62831i | −1.41421 | − | 1.00000i | − | 3.30278i | −0.707107 | − | 2.12132i | −3.93108 | + | 0.674467i | 1.00000i | −2.12132 | − | 2.12132i | 1.00000 | + | 2.82843i | −4.60555 | − | 2.30278i | ||||||||||||||||||||||||||||
| 242.1 | −1.62831 | − | 1.62831i | 1.41421 | + | 1.00000i | 3.30278i | 0.707107 | − | 2.12132i | −0.674467 | − | 3.93108i | − | 1.00000i | 2.12132 | − | 2.12132i | 1.00000 | + | 2.82843i | −4.60555 | + | 2.30278i | ||||||||||||||||||||||||||||
| 242.2 | −0.921201 | − | 0.921201i | −1.41421 | + | 1.00000i | − | 0.302776i | −0.707107 | + | 2.12132i | 2.22398 | + | 0.381574i | − | 1.00000i | −2.12132 | + | 2.12132i | 1.00000 | − | 2.82843i | 2.60555 | − | 1.30278i | |||||||||||||||||||||||||||
| 242.3 | 0.921201 | + | 0.921201i | 1.41421 | + | 1.00000i | − | 0.302776i | 0.707107 | − | 2.12132i | 0.381574 | + | 2.22398i | − | 1.00000i | 2.12132 | − | 2.12132i | 1.00000 | + | 2.82843i | 2.60555 | − | 1.30278i | |||||||||||||||||||||||||||
| 242.4 | 1.62831 | + | 1.62831i | −1.41421 | + | 1.00000i | 3.30278i | −0.707107 | + | 2.12132i | −3.93108 | − | 0.674467i | − | 1.00000i | −2.12132 | + | 2.12132i | 1.00000 | − | 2.82843i | −4.60555 | + | 2.30278i | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 85.f | odd | 4 | 1 | inner |
| 255.k | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 255.2.k.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 255.2.k.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | 255.2.r.a | yes | 8 | |
| 15.e | even | 4 | 1 | 255.2.r.a | yes | 8 | |
| 17.c | even | 4 | 1 | 255.2.r.a | yes | 8 | |
| 51.f | odd | 4 | 1 | 255.2.r.a | yes | 8 | |
| 85.f | odd | 4 | 1 | inner | 255.2.k.a | ✓ | 8 |
| 255.k | even | 4 | 1 | inner | 255.2.k.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 255.2.k.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 255.2.k.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 255.2.k.a | ✓ | 8 | 85.f | odd | 4 | 1 | inner |
| 255.2.k.a | ✓ | 8 | 255.k | even | 4 | 1 | inner |
| 255.2.r.a | yes | 8 | 5.c | odd | 4 | 1 | |
| 255.2.r.a | yes | 8 | 15.e | even | 4 | 1 | |
| 255.2.r.a | yes | 8 | 17.c | even | 4 | 1 | |
| 255.2.r.a | yes | 8 | 51.f | odd | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 31T_{2}^{4} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(255, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 31T^{4} + 81 \)
|
| $3$ |
\( (T^{4} - 2 T^{2} + 9)^{2} \)
|
| $5$ |
\( (T^{4} + 8 T^{2} + 25)^{2} \)
|
| $7$ |
\( (T^{2} + 1)^{4} \)
|
| $11$ |
\( T^{8} + 994T^{4} + 6561 \)
|
| $13$ |
\( (T^{2} - 2 T + 2)^{4} \)
|
| $17$ |
\( T^{8} - 254 T^{4} + 83521 \)
|
| $19$ |
\( (T^{2} - 13)^{4} \)
|
| $23$ |
\( (T^{4} + 56 T^{2} + 576)^{2} \)
|
| $29$ |
\( (T^{4} + 625)^{2} \)
|
| $31$ |
\( (T^{4} - 8 T^{3} + \cdots + 324)^{2} \)
|
| $37$ |
\( (T^{2} - 4 T - 9)^{4} \)
|
| $41$ |
\( T^{8} + 6034 T^{4} + 6765201 \)
|
| $43$ |
\( (T^{4} + 676)^{2} \)
|
| $47$ |
\( T^{8} + 11186 T^{4} + 3418801 \)
|
| $53$ |
\( T^{8} + 33026 T^{4} + 163047361 \)
|
| $59$ |
\( (T^{4} + 244 T^{2} + 7396)^{2} \)
|
| $61$ |
\( (T^{4} + 4 T^{3} + \cdots + 576)^{2} \)
|
| $67$ |
\( (T^{4} + 12 T^{3} + \cdots + 64)^{2} \)
|
| $71$ |
\( T^{8} + 53536 T^{4} + 20736 \)
|
| $73$ |
\( (T^{4} + 266 T^{2} + 10201)^{2} \)
|
| $79$ |
\( (T^{2} - 4 T + 8)^{4} \)
|
| $83$ |
\( T^{8} + 15904 T^{4} + 1679616 \)
|
| $89$ |
\( (T^{4} - 68 T^{2} + 324)^{2} \)
|
| $97$ |
\( (T + 12)^{8} \)
|
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